Band theory of solids explains how electrons behave in crystalline materials. It's crucial for understanding why some materials conduct electricity while others don't. This theory forms the basis for modern electronics and semiconductor technology.

Energy bands in solids arise from the interaction of electron wave functions with the periodic potential of the crystal lattice. These bands determine a material's electrical, optical, and thermal properties, influencing its behavior in various applications.

Energy bands in solids

• Energy bands are a fundamental concept in solid state physics and semiconductor device physics, describing the range of energies that electrons can have in a solid material
• The formation of energy bands is a direct consequence of the periodic potential experienced by electrons in a crystalline solid, leading to a range of allowed energy states separated by forbidden regions
• Understanding energy bands is crucial for explaining the electrical, optical, and thermal properties of semiconductors and their applications in electronic and optoelectronic devices

Origin of energy bands

• Energy bands arise from the interaction between the wave functions of electrons in a periodic potential created by the atomic lattice of a solid
• As atoms are brought together to form a solid, their discrete atomic energy levels split and broaden into continuous energy bands due to the overlap of electronic wave functions
• The width of the energy bands and the separation between them depend on the strength of the atomic potential and the degree of overlap between the atomic orbitals

Allowed vs forbidden energy states

• Within an energy band, electrons can occupy a continuum of allowed energy states, which are separated by forbidden energy regions called band gaps
• Allowed energy states correspond to the solutions of the Schrödinger equation for an electron in a periodic potential, satisfying the boundary conditions imposed by the crystal lattice
• Forbidden energy states, or band gaps, represent energy ranges where no electronic states exist, and electrons cannot occupy these regions without external excitation or thermal energy

Band structure

• The band structure of a solid describes the relationship between the energy of electrons and their wave vector (momentum) in the crystalline lattice
• It provides a comprehensive picture of the allowed energy states, band gaps, and the movement of electrons within the solid
• The band structure is a critical determinant of the electrical, optical, and thermal properties of a material, and it varies depending on the crystal structure and chemical composition of the solid

Valence vs conduction bands

• The valence band is the highest occupied energy band in a solid at absolute zero temperature, containing the electrons involved in chemical bonding
• The conduction band is the lowest unoccupied energy band, separated from the valence band by an energy band gap
• Electrons in the conduction band are free to move throughout the solid, contributing to electrical conductivity, while electrons in the valence band are more localized and do not participate in conduction

Energy band gap

• The energy band gap ($E_g$) is the energy difference between the top of the valence band and the bottom of the conduction band
• It represents the minimum energy required to excite an electron from the valence band to the conduction band, creating a free electron and a hole (absence of an electron)
• The size of the band gap determines the electrical conductivity and optical properties of a material, with larger band gaps corresponding to insulators and smaller band gaps to semiconductors

Direct vs indirect band gaps

• In a direct band gap semiconductor (GaAs, InP), the minimum of the conduction band and the maximum of the valence band occur at the same wave vector (momentum) in the Brillouin zone
• In an indirect band gap semiconductor (Si, Ge), the minimum of the conduction band and the maximum of the valence band occur at different wave vectors
• Direct band gap semiconductors exhibit more efficient optical transitions, as electrons can be directly excited from the valence band to the conduction band by absorbing a photon with energy greater than the band gap

Brillouin zones

• Brillouin zones are a fundamental concept in the study of the electronic structure of crystalline solids, representing the primitive cell in the reciprocal lattice
• They provide a convenient way to describe the wave vector (momentum) space of electrons in a periodic potential and to analyze the band structure of a solid
• The shape and size of the Brillouin zones depend on the crystal structure of the material, and they play a crucial role in determining the electrical, optical, and thermal properties of the solid

Reciprocal lattice

• The reciprocal lattice is a mathematical construct that represents the Fourier transform of the real-space lattice of a crystalline solid
• It is defined as the set of all wave vectors (G) that satisfy the condition $e^{iG \cdot R} = 1$, where R is any vector in the real-space lattice
• The reciprocal lattice vectors are perpendicular to the planes of the real-space lattice and have magnitudes inversely proportional to the interplanar distances

First Brillouin zone

• The first Brillouin zone is the primitive cell of the reciprocal lattice, containing all the unique wave vectors that characterize the electronic states in a crystalline solid
• It is defined as the Wigner-Seitz cell of the reciprocal lattice, constructed by drawing perpendicular bisector planes to the reciprocal lattice vectors from the origin
• The first Brillouin zone is of particular importance because it contains all the information necessary to describe the electronic properties of the solid, including the band structure and the Fermi surface

Higher order Brillouin zones

• Higher order Brillouin zones are the neighboring cells of the reciprocal lattice, obtained by translating the first Brillouin zone by reciprocal lattice vectors
• They contain the same information as the first Brillouin zone but at higher wave vectors (momenta)
• Higher order Brillouin zones are often used to describe the extended band structure of a solid and to analyze the effects of external perturbations, such as electric and magnetic fields, on the electronic properties of the material

Electrons in energy bands

• The behavior of electrons in a crystalline solid is governed by the energy band structure, which determines the allowed energy states and the movement of electrons in the solid
• Electrons occupy the available energy states in the bands according to the Pauli exclusion principle, with the most energetically favorable states being filled first
• The motion of electrons in the energy bands is described by their wave functions, which are solutions to the Schrödinger equation for a periodic potential

Electron wave functions

• Electron wave functions in a crystalline solid are described by Bloch functions, which are the product of a plane wave and a periodic function that has the same periodicity as the lattice
• Bloch functions $\psi_{n,k}(r) = e^{ik \cdot r} u_{n,k}(r)$ are characterized by a band index $n$ and a wave vector $k$, where $u_{n,k}(r)$ is a periodic function
• The band index $n$ denotes the energy band, while the wave vector $k$ represents the electron's momentum and determines its direction of motion in the solid

Bloch theorem

• The Bloch theorem states that the eigenfunctions of the Schrödinger equation for a periodic potential are Bloch functions
• It follows from the translational symmetry of the crystal lattice and the periodicity of the potential experienced by the electrons
• The Bloch theorem allows the electronic structure of a crystalline solid to be described in terms of the wave vector $k$ within the first Brillouin zone, simplifying the analysis of the band structure and the properties of the material

Effective mass of electrons

• The effective mass ($m^$) is a concept used to describe the motion of electrons in a crystalline solid, accounting for the influence of the periodic potential on the electron's dynamics
• It is defined as the inverse of the second derivative of the energy band with respect to the wave vector: $\frac{1}{m^} = \frac{1}{\hbar^2} \frac{d^2E}{dk^2}$
• The effective mass can be different from the free electron mass and can even be negative in some cases, depending on the curvature of the energy band
• The concept of effective mass is essential for understanding the transport properties of electrons in semiconductors, such as mobility and conductivity

Density of states

• The density of states (DOS) is a fundamental concept in solid state physics that describes the number of electronic states available per unit energy interval in a solid
• It is a critical factor in determining the electrical, optical, and thermal properties of a material, as it governs the distribution of electrons among the available energy states
• The DOS varies with the dimensionality of the system (3D, 2D, 1D, or 0D) and is influenced by the band structure and the presence of defects or impurities in the solid

Definition of density of states

• The density of states $g(E)$ is defined as the number of electronic states per unit volume per unit energy interval at energy $E$
• Mathematically, it is expressed as $g(E) = \frac{dN}{dE}$, where $N$ is the total number of states with energies less than or equal to $E$
• The DOS is often given in units of states per unit volume per unit energy (e.g., $cm^{-3}eV^{-1}$) and can be calculated from the band structure of the solid using various computational methods

Density of states in bands

• In a crystalline solid, the DOS varies within each energy band and depends on the dispersion relation $E(k)$ of the band
• For a parabolic band in three dimensions, the DOS is proportional to the square root of the energy measured from the band edge: $g(E) \propto \sqrt{E - E_0}$, where $E_0$ is the energy at the band edge
• The DOS is generally higher near the band edges and decreases towards the middle of the band, reflecting the changing curvature of the energy band with wave vector

Fermi-Dirac distribution

• The Fermi-Dirac distribution $f(E)$ describes the probability of an electronic state with energy $E$ being occupied at a given temperature $T$
• It is given by the equation $f(E) = \frac{1}{e^{(E - \mu)/k_BT} + 1}$, where $\mu$ is the chemical potential (or Fermi level) and $k_B$ is the Boltzmann constant
• At absolute zero temperature, the Fermi-Dirac distribution is a step function, with all states below the Fermi level being occupied and all states above it being empty
• As the temperature increases, the distribution becomes smoother, with some states above the Fermi level being occupied and some states below it being empty

Types of solids

• Solids can be classified into three main categories based on their electrical conductivity and band structure: metals, semiconductors, and insulators
• The distinction between these types of solids arises from the relative positions of the valence and conduction bands, the size of the band gap, and the location of the Fermi level
• Understanding the differences between these solid types is crucial for designing and optimizing electronic and optoelectronic devices, as well as for predicting the behavior of materials under various conditions

Metals vs semiconductors vs insulators

• Metals are characterized by a partially filled conduction band or overlapping valence and conduction bands, resulting in high electrical conductivity
• Semiconductors have a small band gap (typically less than 3 eV) between the valence and conduction bands, allowing for moderate electrical conductivity that can be controlled by temperature, doping, or external fields
• Insulators have a large band gap (greater than 3 eV) between the valence and conduction bands, resulting in very low electrical conductivity under normal conditions

Conductivity and band structure

• The electrical conductivity of a solid is directly related to its band structure and the availability of charge carriers (electrons or holes) in the conduction band
• In metals, the high conductivity is due to the presence of a large number of free electrons in the partially filled conduction band or overlapping bands
• In semiconductors, the conductivity can be controlled by the addition of impurities (doping) or by external factors such as temperature or electric fields, which can excite electrons from the valence band to the conduction band
• In insulators, the large band gap prevents the excitation of electrons to the conduction band under normal conditions, resulting in very low conductivity

Fermi level in different solids

• The Fermi level is the energy level at which the probability of an electronic state being occupied is 50% at thermodynamic equilibrium
• In metals, the Fermi level lies within the conduction band, indicating a partially filled band and a large number of free electrons available for conduction
• In intrinsic semiconductors, the Fermi level lies approximately halfway between the valence and conduction bands, with an equal number of electrons and holes at thermal equilibrium
• In insulators, the Fermi level lies within the band gap, far from both the valence and conduction bands, indicating a very low concentration of charge carriers

Semiconductor band structure

• The band structure of semiconductors is of particular interest due to their widespread use in electronic and optoelectronic devices
• Semiconductors are characterized by a small band gap between the valence and conduction bands, which allows for the control of electrical conductivity through doping, temperature, or external fields
• The specific features of the valence and conduction bands in semiconductors, such as the effective mass and density of states, determine the material's electrical and optical properties

Valence band characteristics

• The valence band in semiconductors is the highest occupied energy band at absolute zero temperature, containing the electrons involved in chemical bonding
• The top of the valence band is characterized by a maximum in the energy dispersion relation $E(k)$, which determines the hole effective mass and mobility
• In many semiconductors (Si, Ge, GaAs), the valence band is split into multiple subbands (heavy hole, light hole, and split-off bands) due to spin-orbit coupling, resulting in different hole effective masses and optical transitions

Conduction band characteristics

• The conduction band in semiconductors is the lowest unoccupied energy band, separated from the valence band by the band gap
• The bottom of the conduction band is characterized by a minimum in the energy dispersion relation $E(k)$, which determines the electron effective mass and mobility
• In some semiconductors (GaAs, InP), the conduction band minimum occurs at the center of the Brillouin zone (Γ point), resulting in a direct band gap and efficient optical transitions

Intrinsic vs extrinsic semiconductors

• Intrinsic semiconductors are pure materials with no intentional doping, where the electrical conductivity is determined by the thermal excitation of electrons from the valence band to the conduction band
• In intrinsic semiconductors, the concentration of electrons in the conduction band is equal to the concentration of holes in the valence band, and the Fermi level lies approximately halfway between the bands
• Extrinsic semiconductors are materials that have been doped with impurities to control the concentration and type of charge carriers (electrons or holes)
• N-type semiconductors are doped with donor impurities that provide extra electrons to the conduction band, while p-type semiconductors are doped with acceptor impurities that create holes in the valence band

Band structure engineering

• Band structure engineering involves the modification of the electronic structure of semiconductors to optimize their electrical, optical, or thermal properties for specific applications
• This can be achieved through various methods, such as alloying, strain application, or quantum confinement, which alter the band gap, effective mass, or density of states of the material
• Band structure engineering is a powerful tool for designing novel semiconductor devices, such as high-efficiency solar cells, light-emitting diodes, or high-speed transistors

Alloying effects on band structure

• Alloying is the process of mixing two or more semiconductors to form a new material with properties intermediate between those of the constituent compounds
• The band gap of the alloy can be tuned by varying the composition, allowing for the creation of materials with specific band gaps for various applications (e.g., AlGaAs for red LEDs, InGaAsP for optical communication)
• Alloying can also affect the effective mass, mobility, and density of states of the material, influencing its electrical and optical properties

Strain effects on band structure

• Strain is the deformation of a material due to applied stress, which can be induced by lattice mismatch in epitaxial growth or by external mechanical forces
• Strain can modify the band structure of semiconductors by shifting the positions of the valence and conduction bands, changing the band gap, or splitting degenerate bands
• Tensile strain can reduce the band gap and increase the electron mobility, while compressive strain can increase the band gap and hole mobility, allowing for the optimization of device performance

Quantum confinement effects

• Quantum confinement occurs when the size of a semiconductor structure becomes comparable to the de Broglie wavelength of the charge carriers, leading to the discretization of energy levels and the modification of the band structure
• Quantum confinement can be achieved in various low-dimensional structures, such as quantum wells (2D), quantum wires (1D), or quantum dots (0D), which are created by sandwiching a narrow-gap semiconductor between wide-gap barriers
• Quantum confinement effects can increase the band gap, modify the effective mass and density of states, and enhance the optical properties of the material, enabling the development of novel optoelectronic devices, such as quantum well lasers or quantum dot solar cells