5.3 Nearly free electron model

The nearly free electron model extends the free electron model by incorporating weak periodic potentials from crystal lattices. It explains energy bands, gaps, and Brillouin zones, providing a more accurate description of electronic properties in solids.

This model introduces concepts like Bloch functions, effective mass, and Fermi surfaces. While it has limitations for strongly correlated systems, it remains useful for understanding semiconductors, insulators, and simple metals, forming a basis for more advanced theories.

Origin of nearly free electron model

• Developed as an extension of the free electron model to account for the periodic potential of the crystal lattice
• Incorporates the effect of the weak periodic potential on the electron wave functions and energy levels
• Provides a more accurate description of the electronic properties of solids compared to the free electron model

Assumptions in nearly free electron model

Weak periodic potential

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• The periodic potential due to the crystal lattice is assumed to be weak compared to the kinetic energy of the electrons
• The weak periodic potential causes small perturbations to the free electron wave functions and energy levels
• The perturbations lead to the formation of energy bands and gaps in the electronic structure

Electron wave functions

• The electron wave functions are assumed to be similar to those of free electrons, with slight modifications due to the periodic potential
• The wave functions are described by Bloch functions, which are plane waves modulated by a periodic function with the same periodicity as the lattice
• The Bloch functions satisfy the Schrödinger equation in the presence of the weak periodic potential

Consequences of weak periodic potential

Bragg planes

• The weak periodic potential leads to the formation of Bragg planes in reciprocal space
• Bragg planes are defined by the condition $2\mathbf{k} \cdot \mathbf{G} = G^2$, where $\mathbf{k}$ is the electron wave vector and $\mathbf{G}$ is a reciprocal lattice vector
• Electron waves undergo Bragg reflection at the Bragg planes, leading to the formation of standing waves and energy gaps

Energy gaps

• The Bragg reflection at the Bragg planes results in the opening of energy gaps at specific wave vectors
• The energy gaps occur at the boundaries of the Brillouin zones, which are defined by the Bragg planes
• The size of the energy gaps depends on the strength of the periodic potential and the magnitude of the reciprocal lattice vectors

Brillouin zones

• The Brillouin zones are the primitive cells of the reciprocal lattice, constructed using the Bragg planes as boundaries
• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice, containing all unique wave vectors
• Higher-order Brillouin zones are obtained by translating the first Brillouin zone by reciprocal lattice vectors

E-k diagram

Reduced and periodic zone schemes

• The E-k diagram represents the relationship between the electron energy and wave vector in the nearly free electron model
• The reduced zone scheme shows the energy bands and gaps within the first Brillouin zone
  • The energy bands are folded back into the first Brillouin zone at the zone boundaries
  • The reduced zone scheme emphasizes the periodicity of the energy bands in reciprocal space
• The periodic zone scheme shows the energy bands and gaps in extended k-space
  • The energy bands are repeated periodically in reciprocal space
  • The periodic zone scheme provides a more intuitive representation of the electronic structure

Energy bands and gaps

• The E-k diagram consists of energy bands separated by energy gaps
• The energy bands represent the allowed energy states for electrons in the solid
• The energy gaps represent the forbidden energy states, where no electron states exist
• The width of the energy bands and the size of the energy gaps depend on the strength of the periodic potential

Effective mass

Definition of effective mass

• The effective mass is a concept used to describe the response of electrons to external forces in a solid
• It is defined as the inverse of the second derivative of the energy with respect to the wave vector: $m^* = \hbar^2 \left(\frac{\partial^2 E}{\partial k^2}\right)^{-1}$
• The effective mass can be positive (electron-like) or negative (hole-like), depending on the curvature of the energy bands

Calculation from E-k diagram

• The effective mass can be calculated from the E-k diagram by fitting a parabola to the energy bands near the band extrema
• For a parabolic band, the effective mass is constant and given by $m^* = \hbar^2 \left(\frac{\partial^2 E}{\partial k^2}\right)^{-1}$
• For non-parabolic bands, the effective mass varies with the wave vector and can be obtained by numerical differentiation of the E-k diagram

Density of states

Effect of energy bands on density of states

• The density of states (DOS) represents the number of electronic states per unit energy interval
• The shape of the energy bands in the E-k diagram determines the density of states
• The DOS is high near the band edges, where the energy bands are relatively flat
• The DOS is low in the middle of the energy bands, where the bands are more dispersive

Van Hove singularities

• Van Hove singularities are sharp peaks or discontinuities in the density of states
• They occur at critical points in the Brillouin zone, where the gradient of the energy band vanishes $(\nabla_k E = 0)$
• The type of Van Hove singularity depends on the local topology of the energy band (saddle points, maxima, or minima)
• Van Hove singularities can have significant effects on the electronic and optical properties of solids

Fermi surfaces

Construction of Fermi surfaces

• The Fermi surface is the surface in reciprocal space that separates the occupied and unoccupied electronic states at zero temperature
• It is constructed by plotting the wave vectors of the highest occupied electronic states in the Brillouin zone
• The shape of the Fermi surface depends on the electronic structure and the filling of the energy bands
• Fermi surfaces can have various topologies (closed surfaces, open surfaces, or a combination of both)

Electron and hole pockets

• Electron pockets are regions of the Fermi surface where the energy bands are electron-like (positive curvature)
• Hole pockets are regions of the Fermi surface where the energy bands are hole-like (negative curvature)
• The presence of electron and hole pockets indicates the existence of multiple types of charge carriers in the solid
• The size and shape of the electron and hole pockets can be determined from the Fermi surface topology

Limitations of nearly free electron model

Strongly correlated systems

• The nearly free electron model assumes weak electron-electron interactions and a weak periodic potential
• In strongly correlated systems (transition metal oxides, heavy fermion materials), the electron-electron interactions are strong and cannot be neglected
• The nearly free electron model fails to capture the complex electronic properties of strongly correlated systems, such as metal-insulator transitions and unconventional superconductivity

Failures in describing band gaps

• The nearly free electron model often underestimates the size of the band gaps in semiconductors and insulators
• This is because the model assumes a weak periodic potential, which may not be sufficient to describe the strong ionic potentials in these materials
• More advanced models (tight-binding model, density functional theory) are needed to accurately predict the band gaps and electronic structure of semiconductors and insulators

Applications of nearly free electron model

Semiconductors and insulators

• The nearly free electron model can provide a qualitative understanding of the electronic structure of semiconductors and insulators
• It explains the formation of energy bands and gaps, which are crucial for the electronic properties of these materials
• The model can be used to estimate the effective masses of electrons and holes, which determine the transport properties
• However, quantitative predictions of band gaps and other properties often require more sophisticated models

Metals and alloys

• The nearly free electron model is particularly useful for describing the electronic structure of simple metals (alkali metals) and alloys
• It can explain the formation of energy bands, the shape of the Fermi surface, and the presence of electron and hole pockets
• The model can be used to calculate the density of states, which is important for understanding the electronic and thermal properties of metals
• The nearly free electron model provides a foundation for more advanced models (pseudopotential theory) used to study the electronic structure of complex metals and alloys