Band Theory of Solids

Band Theory of Solids

• Band theory explains band structures in solids.
• Band theory classifies solids based on their band structure - insulators, semiconductors, metals, and semimetals.
• The theory introduces concepts like effective mass and electron velocity.

Failures of Free Electron Theory

• Free electron theory fails to explain the existence of band structures.
• It fails to explain the concept of effective mass of electrons.
• It cannot explain the positive Hall coefficient observed in some metals.
• It fails to classify solids as insulators, conductors and semiconductors.
• Failures are attributed to oversimplified assumption of neglecting the presence/effect of ion cores; also considering potential from ion cores as constant/zero.
• Band theory considers the impact of ion cores on electron movement, addressing limitations of free electron theory.

Periodic Potential, Bloch's Theorem, and Bloch Functions

• Lattice sites are periodically arranged, resulting in a periodic potential.
• In 1D, with lattice constant 'a', the potential repeats at x + a.
• Bloch's Theorem states that wave function in periodic potential is a modulated plane wave.
• Bloch functions are wave functions which are modulated plane waves.
• The modulating factor (Ukx) has the same periodicity as the potential.
• In 3D, V(r) = V(r + T), where T is the translation vector in the crystal lattice.
• Bloch functions in 3D are represented as modulated plane waves: ψ(r) = Ukr * e^(ik.r).
• The period of Ukr is the same as that of the potential.

Kronig-Penney Model

• Considers a 1D case with a series of potential wells and barriers.
• A potential well of width 'a' and potential barrier of width 'b' with height V0 is considered.
• The potential has a period of a + b.
• Bloch's theorem suggests wave functions will be Bloch functions.
• Schrödinger's equation is solved for both regions and Bloch wave functions are put in those equations
• After applying boundary conditions the equation is: (α² + β²) / 2αβ * sinh(βb) * sin(αa) + cosh(βb) * cos(αa) = cos(k(a+b)).
• In a special case where V0 approaches infinity and b approaches zero and the barrier becomes a Dirac delta function, the equation simplifies to P * sin(αa) / αa + cos(αa) = cos(ka).
• P represents the binding strength of electrons to ion cores; higher values indicate stronger binding.
• When P is infinite, α = nπ/a, resulting in energy levels corresponding to an infinite potential well.
• Strong electron binding leads to discrete energy levels, similar to isolated atoms.
• When P = 0, free electron case: energy becomes E = ħ²k²/2m.
• The energy is continuous and all energy values are allowed and bands become continuous.
• For values between the extreme cases, allowed and forbidden energy bands appear.
• Certain values of alpha are allowed and Forbidden, leading to allowed and forbidden bands.
• The model verifies/proves the existence of band structures.

Velocity and Effective Mass of Electrons

• Group velocity of electrons is given by vg = dω/dk which is the same as (1/ħ)(dE/dk).
• Effective mass formula m* = ħ² / (d²E/dk²).
• Effective mass can be positive, negative, zero, or infinite.
• Effective mass describes how electrons move considering the influence of ion cores and external fields.
• Slower movement mean heavier mass than free electron theory.
• Faster movement means lighter mass than in free electron theory.
• Zero effective mass means the electron moves at an ultra-relativistic speed.
• Negative mass characterizes the behavior as holes which accounts for positive Hall coefficient in some materials.

Number of Wave Functions (Energy States) in a Band

• The total number of wave functions, energy states, or allowed k-values in a band within the first Brillouin zone is equal to the number of primitive unit cells (N).
• For a 1D crystal of length L with N primitive unit cells, the lattice constant a = L/N.
• Due to periodicity, ψ(x+L) = ψ(x) meaning that k = 2πn/L from which N = L/a is derived.
• Integrating dN = (L / 2π)dk over limits -π/a to π/a which is finding the number of energy states in the first Brillouin zone leads to: N=L/a

Classification of Solids

• Insulators at T=0 have fully filled bands with energy gap of >5eV.
• Semiconductors have band gap of 0.2eV to 2.5eV.
• Topmost filled band is the valence band, first empty band is the conduction band.
• Metals have partially-filled valence bands or zero bandgap between conduction and valence bands.
• Semimetals feature a valence band that is partly empty and a conduction band that is partly filled.