Kronig-Penney Model: Energy Bands in Periodic Potentials

Effective mass (m*) relates to E-K diagram curvature: m* = ℏ²/(d²E/dk²). Steeper curvature (narrower parabola) means lower m*. E-K diagram for electrons is parabolic; for holes it is inverted. Effective mass affects mobility and transport properties. The relationship between effective mass and mobility is: μ = qτ/m*, where τ is average collision time.

The effective mass of an electron in a semiconductor is a concept that describes how electrons behave as if they have a different mass than their actual rest mass, due to interactions with the crystal lattice; this effective mass is derived from the curvature of the energy band (E-k relationship) and is given by the formula m* = ℏ²/(d²E/dk²), where m* is the effective mass, ℏ is the reduced Planck's constant, and E is the energy as a function of the wave vector k.

The effective mass m* is defined as m* = ℏ² / (d²E/dk²). The effective mass is inversely proportional to the radius of curvature of the E(k) curve. A larger radius of curvature corresponds to a smaller effective mass. For the given dispersion relations, the effective mass order is m*_A > m*_B > m*_C because the radius of curvature is smallest for A and largest for C.

Substituting dk/dt = F/ħ into the acceleration equation yields a = (1/ħ²) × (d²E/dk²) × F. Comparing with F = M*a gives 1/M* = (1/ħ²) × (d²E/dk²), so M* = ħ² / (d²E/dk²). This fundamental formula shows effective mass is inversely proportional to the curvature of the energy-wave vector relationship. When plotting M* versus k, effective mass is minimum at k = 0 and increases toward maximum wave vectors. This behavior is characteristic of electrons in periodic potentials, where band curvature determines effective mass.

The effective mass concept emerges from treating electrons as matter waves in periodic potentials. Starting with E = ħω and v_g = dω/dk, we derive that acceleration a = -(eF/ħ²)(d²E/dk²). Comparing to F = ma gives m* = ħ²/(d²E/dk²). This shows effective mass depends on band curvature. Unlike rest mass, effective mass can be negative or infinite, explaining unusual electron behavior in solids.

Materials are classified by their band gap size: conductors have overlapping bands (zero band gap), insulators have large gaps (~5-10 eV), and semiconductors have intermediate gaps (~1 eV). Conductors have high conductivity due to overlapping bands allowing free electron movement. Insulators have negligible conductivity because electrons cannot overcome the large band gap. Semiconductors offer controllable conductivity through doping, temperature, light, and electric fields, making them essential for electronic devices.

Materials are classified into conductors, semiconductors, and insulators based on their band gap. Conductors have zero band gap, semiconductors have about 1 eV (silicon: 1.1 eV, germanium: 0.7 eV), and insulators have 6 eV or more (diamond: 6 eV). The valence band contains bound electrons attracted to the nucleus, while the conduction band contains free electrons enabling conductivity. Insulators have completely filled valence band and empty conduction band. Conductors have filled valence band and filled/partially filled conduction band. Semiconductors have partially filled bands at room temperature but behave like insulators at absolute zero.

All materials are classified into three categories based on their band gap size: (1) Conductors have very small or zero band gaps (<0.2 eV), allowing electrons to easily move between bands; (2) Semiconductors have moderate band gaps (0.2-3 eV), where some electrons can gain enough energy to jump to the conduction band; (3) Insulators have large band gaps (>3 eV), preventing electrons from crossing to the conduction band. This classification determines whether a material conducts electricity efficiently, partially, or not at all.

Materials are classified based on their band gap size: (1) Conductors (metals) have no band gap or overlapping valence and conduction bands, allowing easy electron movement; (2) Semiconductors have a very small energy gap; (3) Insulators have a large energy gap, making it difficult for electrons to jump from valence to conduction band.

Materials are classified into three categories based on their band structure: (1) Insulators have completely filled valence bands and completely empty conduction bands separated by large forbidden gaps (>6 eV, e.g., diamond with ~7 eV gap); no electrons can cross this gap under normal conditions. (2) Conductors/metals have overlapping valence and conduction bands (zero band gap), providing continuous electron flow. (3) Semiconductors have partly filled valence bands and empty conduction bands with moderate gaps (~1 eV for Ge/Si); at absolute zero they behave like insulators, but at room temperature thermal excitation moves electrons to the conduction band, enabling conductivity. The band gap determines material classification and electrical behavior.

In 2D systems, the Fermi surface is deformed by the periodic potential such that it intersects Bragg planes perpendicularly, because the gradient of energy is parallel to Bragg planes. This deformation changes the shape from spherical (free electron case) to a more complex shape that determines electronic properties. When a crystal has a basis (multiple atoms per primitive cell), the Fourier coefficients of the periodic potential can become zero for certain reciprocal lattice vectors when the structure factor equals zero. These 'extinction rules' mean certain Bragg reflections are absent, and correspondingly, certain band gaps do not open. The nearly free electron model works well for alkali metals (column 1) and alkaline earth metals (column 2) where valence electrons are loosely bound, but fails for transition metals where d-orbitals create additional bands that are neither completely free nor completely bound.

Weak periodic potentials produce minimal Fermi surface distortion. Intermediate potentials cause rounding of sharp corners as electrons occupy lower-energy states near zone boundaries. Strong potentials produce dramatic distortions, with Fermi surfaces acquiring complex shapes. The distortion always occurs perpendicularly to zone boundaries due to zero group velocity at these points. States near zone boundaries experience larger energy changes, causing more pronounced deformation. In FCC metals, neck diameters increase with stronger lattice potentials (Cu < Ag < Au).

This section explores how weak periodic potentials modify the free electron Fermi surface across different dimensions. In 1D, parabolic bands remain approximately preserved. In 2D square lattices, electrons redistribute to lower energy states near Brillouin zone boundaries, causing Fermi surface distortion. Along zone diagonals, energy becomes lower than free electron predictions, with electrons transferring from high-energy regions to lower-energy states. In extreme cases, the Fermi surface becomes open rather than closed. In 3D simple cubic metals, the Fermi sphere distorts so that states near zone diagonals move toward zone faces, creating open-ended Fermi surfaces intersecting Brillouin zone planes. This dimensional analysis reveals how periodic potentials fundamentally alter Fermi surface geometry in real materials.

When electrons move through periodic potentials from ions, energy bands form with band gaps between them. Without periodic potentials, electrons form a Fermi surface (free electron gas). With periodic potentials, band gaps open at Brillouin zone boundaries, with gap size proportional to potential strength. This determines whether materials are metals, semiconductors, or insulators. For free electrons, the Fermi surface is a sphere with radius k_F = (3π²n)^(1/3). At finite temperature, electrons near the Fermi surface create particle-hole excitations. For tightly bound electrons (tight-binding), Fermi surfaces have different shapes (diamond-like) depending on lattice symmetry.

The Fermi surface is a surface of constant energy in k-space that separates filled from unfilled electron states at absolute zero; for free electrons, it is spherical with radius k_f = (3π²n)^(1/3), but when periodic lattice potential is introduced, the Fermi surface becomes distorted and extends into higher Brillouin zones, requiring zone-folding techniques (reduced zone scheme) to map all portions back to the first Brillouin zone for proper visualization and analysis.