⚛️Solid State Physics Unit 5 – Electronic band structure

Key Concepts and Definitions

• Electronic band structure describes the allowed energy states and forbidden energy gaps for electrons in a solid material
• Energy bands form due to the interaction between the atomic orbitals of the constituent atoms in the solid
• Valence band represents the highest occupied energy band at absolute zero temperature
• Conduction band represents the lowest unoccupied energy band at absolute zero temperature
• Band gap is the energy difference between the top of the valence band and the bottom of the conduction band
  • Determines the electrical properties of the material (metal, semiconductor, or insulator)
• Fermi level is the energy level with a 50% probability of being occupied by an electron at thermodynamic equilibrium
• Density of states (DOS) describes the number of electronic states per unit energy interval

Origins of Band Structure

• Band structure arises from the quantum mechanical behavior of electrons in a periodic potential created by the atomic lattice
• Schrödinger equation describes the wave-like behavior of electrons in a solid
  • Solutions to the Schrödinger equation for a periodic potential yield the allowed energy states and wavefunctions of electrons
• Tight-binding approximation considers the overlap of atomic orbitals between neighboring atoms to form energy bands
  • Assumes electrons are tightly bound to their respective atoms
• Nearly-free electron approximation treats electrons as almost free particles perturbed by a weak periodic potential
  • Assumes electrons are weakly affected by the atomic lattice
• Interaction between atomic orbitals leads to the formation of bonding and antibonding orbitals
  • Bonding orbitals have lower energy and contribute to the valence band
  • Antibonding orbitals have higher energy and contribute to the conduction band

Free Electron Model

• Simplest model for describing the behavior of electrons in a solid
• Assumes electrons are completely free to move throughout the solid without any influence from the atomic lattice
• Electrons are treated as a gas of non-interacting particles obeying the Pauli exclusion principle
• Energy of an electron in the free electron model is given by $E = \frac{\hbar^2k^2}{2m}$, where $\hbar$ is the reduced Planck's constant, $k$ is the wave vector, and $m$ is the electron mass
• Density of states in the free electron model is proportional to the square root of energy, $g(E) \propto \sqrt{E}$
• Fermi energy $E_F$ is the energy of the highest occupied state at absolute zero temperature
  • Depends on the electron density $n$ as $E_F = \frac{\hbar^2}{2m}(3\pi^2n)^{2/3}$
• Free electron model provides a good approximation for the behavior of electrons in metals

Bloch's Theorem and Periodic Potentials

• Bloch's theorem states that the wavefunction of an electron in a periodic potential can be expressed as the product of a plane wave and a periodic function
  • $\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}}u_{n\mathbf{k}}(\mathbf{r})$, where $u_{n\mathbf{k}}(\mathbf{r})$ has the same periodicity as the lattice
• Periodic potential $V(\mathbf{r})$ represents the influence of the atomic lattice on the electrons
  • $V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R})$, where $\mathbf{R}$ is any lattice translation vector
• Bloch wavefunctions are labeled by the band index $n$ and the wave vector $\mathbf{k}$
  • Band index distinguishes between different energy bands
  • Wave vector determines the phase of the plane wave part of the wavefunction
• Bloch's theorem simplifies the problem of solving the Schrödinger equation for a periodic potential
  • Allows the use of periodic boundary conditions and the concept of reciprocal space

Energy Bands and Band Gaps

• Energy bands are formed by the splitting and broadening of atomic energy levels due to the interaction between atoms in a solid
• Valence band is the highest occupied energy band at absolute zero temperature
  • Electrons in the valence band are responsible for bonding and electrical properties
• Conduction band is the lowest unoccupied energy band at absolute zero temperature
  • Electrons in the conduction band are free to move and contribute to electrical conduction
• Band gap is the energy difference between the top of the valence band and the bottom of the conduction band
  • Determines the electrical properties of the material (metal, semiconductor, or insulator)
• Direct band gap occurs when the minimum of the conduction band and the maximum of the valence band occur at the same wave vector $\mathbf{k}$
  • Allows for efficient optical transitions (GaAs, InP)
• Indirect band gap occurs when the minimum of the conduction band and the maximum of the valence band occur at different wave vectors
  • Optical transitions require the assistance of phonons to conserve momentum (Si, Ge)

Brillouin Zones and k-space

• Brillouin zones are the primitive cells of the reciprocal lattice, which is the Fourier transform of the real-space lattice
• First Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice
  • Contains all the unique values of the wave vector $\mathbf{k}$ that characterize the electronic states
• Higher-order Brillouin zones are obtained by translating the first Brillouin zone by reciprocal lattice vectors
• Brillouin zone boundaries occur at the planes bisecting the reciprocal lattice vectors
  • Represent the points where Bragg reflection occurs, leading to the opening of band gaps
• High-symmetry points in the Brillouin zone (Γ, X, L, etc.) are used to label the electronic states and band structure
• Reduced zone scheme folds the band structure from higher Brillouin zones into the first Brillouin zone
  • Allows for a more compact representation of the band structure

Metals, Semiconductors, and Insulators

• Metals have overlapping valence and conduction bands, or a partially filled conduction band
  • Electrons can easily move to unoccupied states, leading to high electrical conductivity (Cu, Al, Au)
• Semiconductors have a small band gap (typically less than 4 eV) between the valence and conduction bands
  • Electrical conductivity can be controlled by temperature, doping, or applied electric fields (Si, GaAs, GaN)
  • Intrinsic semiconductors have equal numbers of electrons and holes at thermal equilibrium
  • Extrinsic semiconductors are doped with impurities to create excess electrons (n-type) or holes (p-type)
• Insulators have a large band gap (typically greater than 4 eV) between the valence and conduction bands
  • Negligible electrical conductivity at room temperature due to the lack of charge carriers (diamond, SiO2, Al2O3)

Applications and Real-World Examples

• Semiconductor devices (transistors, diodes, solar cells) rely on the control of band structure and charge carrier properties
  • Band engineering allows for the design of materials with specific electronic and optical properties (quantum wells, superlattices)
• Light-emitting diodes (LEDs) exploit the direct band gap of certain semiconductors to generate light through electroluminescence (GaAs, GaN, InGaN)
• Photovoltaic cells (solar cells) convert light into electricity using the band gap of semiconductors to generate electron-hole pairs (Si, GaAs, CdTe, perovskites)
• Thermoelectric materials utilize the band structure to generate electricity from temperature gradients or vice versa (Bi2Te3, PbTe, SiGe)
• Topological insulators have an insulating bulk but conductive surface states protected by topology (Bi2Se3, Bi2Te3, Sb2Te3)
  • Potential applications in spintronics and quantum computing
• Superconductors have a gap in the electronic excitation spectrum, leading to zero electrical resistance below a critical temperature (Nb, MgB2, cuprates, iron-based superconductors)