2.1 Band theory of solids

Band theory of solids explains how electrons behave in crystalline materials and why some materials conduct electricity while others don't. This theory is the foundation for understanding semiconductor devices and modern electronics.

The core idea: when atoms come together to form a crystal, their individual energy levels merge into broad energy bands. The structure of these bands dictates a material's electrical, optical, and thermal properties.

Energy bands in solids

In an isolated atom, electrons occupy discrete energy levels. But in a solid, billions of atoms sit close together, and their electron wave functions overlap. This overlap causes each discrete level to split into a huge number of closely spaced states, forming a near-continuous energy band.

Between these bands are band gaps, energy ranges where no electron states exist. An electron can't have an energy within the gap unless it receives enough external energy (thermal, optical, etc.) to jump across it.

Origin of energy bands

The periodic arrangement of atoms in a crystal creates a periodic potential that electrons must navigate. As you bring atoms closer together:

1. Their outer electron orbitals begin to overlap.
2. Each atomic energy level splits into as many sub-levels as there are atoms (for $N$ atoms, you get $N$ closely spaced levels).
3. These sub-levels are so close in energy that they effectively form a continuous band.

The width of each band depends on how strongly the orbitals overlap. Outer orbitals overlap more, so they form wider bands. Inner (core) orbitals overlap less and remain narrow.

Allowed vs. forbidden energy states

Allowed states are the energy values where electrons can exist inside a band. They correspond to valid solutions of the Schrödinger equation for the crystal's periodic potential.

Forbidden states (band gaps) are energy ranges with no valid solutions. An electron sitting at the top of one band cannot move to the next band without gaining at least the band gap energy $E_g$. This is what makes the gap so important for determining conductivity.

Band structure

The band structure of a solid maps out how electron energy $E$ varies with wave vector $k$ (which is related to the electron's crystal momentum). This $E(k)$ relationship gives you a complete picture of which energies are allowed, where the gaps are, and how electrons move through the material.

Valence vs. conduction bands

• The valence band is the highest energy band that is fully occupied at absolute zero (0 K). These electrons participate in chemical bonding and are relatively localized.
• The conduction band is the next band up, separated from the valence band by the band gap. At 0 K it's empty, but electrons that reach it are free to move and carry current.

The key distinction: electrons in the valence band are "stuck" in bonds, while electrons in the conduction band are mobile. Electrical conduction requires carriers in the conduction band (or empty states, i.e., holes, in the valence band).

Energy band gap

The band gap $E_g$ is the energy difference between the top of the valence band and the bottom of the conduction band. It's the minimum energy needed to promote an electron from bonding states into conducting states, simultaneously creating a free electron and a hole (the empty state left behind).

Band gap values determine material behavior:

• Si: $E_g \approx 1.12 \text{ eV}$ (semiconductor)
• GaAs: $E_g \approx 1.42 \text{ eV}$ (semiconductor)
• Diamond: $E_g \approx 5.5 \text{ eV}$ (insulator)

Larger gaps mean fewer thermally excited carriers and lower conductivity at room temperature.

Direct vs. indirect band gaps

This distinction matters enormously for optical devices.

• Direct band gap (e.g., GaAs, InP): The conduction band minimum and valence band maximum occur at the same $k$-value. An electron can transition between bands by simply absorbing or emitting a photon, with no change in momentum needed. This makes direct-gap materials efficient light emitters and absorbers.
• Indirect band gap (e.g., Si, Ge): The conduction band minimum and valence band maximum occur at different $k$-values. Transitions require both a photon and a phonon (lattice vibration) to conserve momentum. This two-particle process is much less probable, which is why silicon is a poor light emitter compared to GaAs.

Brillouin zones

Brillouin zones organize the allowed wave vectors ($k$-values) of electrons in a crystal. They're defined in reciprocal space (momentum space), which is the natural framework for describing wave-like behavior in periodic structures.

Reciprocal lattice

The reciprocal lattice is the Fourier transform of the real-space crystal lattice. Its lattice vectors $\mathbf{G}$ satisfy:

$e^{i\mathbf{G} \cdot \mathbf{R}} = 1$

where $\mathbf{R}$ is any real-space lattice vector. Reciprocal lattice vectors point perpendicular to planes in the real lattice, and their magnitudes are inversely proportional to the spacing between those planes. Think of it this way: closely spaced planes in real space correspond to large reciprocal lattice vectors.

First Brillouin zone

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. You construct it by:

1. Drawing lines from the origin to all neighboring reciprocal lattice points.
2. Constructing perpendicular bisector planes on each of those lines.
3. The smallest enclosed volume around the origin is the first Brillouin zone.

This zone contains all the unique $k$-values you need. Because of the crystal's periodicity, any $k$ outside this zone is equivalent to one inside it (they differ by a reciprocal lattice vector). So the entire band structure can be represented within the first Brillouin zone.

Higher order Brillouin zones

Higher order zones are the next shells outward in reciprocal space, obtained by translating the first zone by reciprocal lattice vectors. They contain the same physical information as the first zone but at higher $k$-values. These are useful for visualizing the extended zone scheme of the band structure and for analyzing how external fields affect electron dynamics.

Electrons in energy bands

Electron wave functions

In a crystal, electron wave functions aren't simple plane waves. They're described by Bloch functions:

$\psi_{n,\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} \, u_{n,\mathbf{k}}(\mathbf{r})$

Here, $e^{i\mathbf{k} \cdot \mathbf{r}}$ is a plane wave that carries the electron's crystal momentum, and $u_{n,\mathbf{k}}(\mathbf{r})$ is a periodic function with the same periodicity as the lattice. The index $n$ labels the band, and $\mathbf{k}$ specifies the momentum within that band.

The physical picture: the electron travels through the crystal like a modulated wave. The plane wave part gives it a definite momentum direction, while the periodic part reflects the influence of the atomic cores.

Bloch theorem

Bloch's theorem states that the solutions to the Schrödinger equation in a periodic potential must take the Bloch function form shown above. This follows directly from the translational symmetry of the lattice.

Why this matters: it means you only need to solve for the wave function within one unit cell and one Brillouin zone. The rest of the crystal is just periodic repetition. This dramatically simplifies the problem of finding the electronic structure.

Effective mass of electrons

An electron in a crystal doesn't respond to external forces the same way a free electron does, because the periodic potential is constantly influencing it. The effective mass $m^*$ captures this modified response:

$\frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2E}{dk^2}$

The effective mass depends on the curvature of the $E(k)$ band:

• High curvature (wide, gently curving band) → small $m^*$ → electrons accelerate easily → high mobility.
• Low curvature (flat band) → large $m^*$ → electrons are sluggish.
• Negative curvature (near the top of a band) → negative $m^*$. This is where the hole concept becomes useful: a missing electron near the top of the valence band behaves like a positive particle with positive effective mass.

For example, in GaAs the electron effective mass is about $0.067 \, m_0$ (where $m_0$ is the free electron mass), which is why GaAs has such high electron mobility.

Density of states

The density of states (DOS) tells you how many electronic states are available at each energy. Even if a band exists at some energy, conduction depends on how many states are packed into that energy range.

Definition of density of states

The DOS $g(E)$ is defined as the number of electronic states per unit volume per unit energy:

$g(E) = \frac{dN}{dE}$

It's typically given in units of $\text{cm}^{-3}\text{eV}^{-1}$. A high DOS at a particular energy means lots of states are available there for electrons to occupy.

Density of states in bands

For a parabolic band (a good approximation near band edges) in three dimensions, the DOS follows a square-root dependence:

$g(E) \propto \sqrt{E - E_0}$

where $E_0$ is the band edge energy. This means the DOS starts at zero right at the band edge and increases as you move deeper into the band. Near the edges is where most of the action happens in semiconductors, since that's where carriers are thermally excited.

Fermi-Dirac distribution

The DOS tells you how many states exist; the Fermi-Dirac distribution tells you which ones are actually occupied:

$f(E) = \frac{1}{e^{(E - \mu)/k_BT} + 1}$

Here, $\mu$ is the chemical potential (often called the Fermi level $E_F$) and $k_B$ is Boltzmann's constant.

At $T = 0$ K, this is a sharp step function: every state below $E_F$ is filled, every state above is empty. As temperature increases, the step softens. Some electrons gain enough thermal energy to occupy states above $E_F$, leaving empty states (holes) below it. The "smearing" region is roughly $\pm$ a few $k_BT$ around the Fermi level (at room temperature, $k_BT \approx 0.026 \text{ eV}$).

The actual carrier concentration at any energy is the product $g(E) \times f(E)$.

Types of solids

The differences between metals, semiconductors, and insulators come down to band structure and where the Fermi level sits.

Metals vs. semiconductors vs. insulators

Conductivity and band structure

In metals, the conduction band is partially filled (or overlaps with the valence band), so plenty of electrons are already free to carry current. No excitation needed.

In semiconductors, the gap is small enough that thermal energy at room temperature can excite some electrons across it. Conductivity can also be dramatically increased by doping (adding impurity atoms) or by applying electric fields. This controllability is what makes semiconductors so useful.

In insulators, the gap is too large for thermal excitation to matter at normal temperatures. Diamond, for instance, has a 5.5 eV gap, so essentially zero electrons reach the conduction band at room temperature.

Fermi level in different solids

• Metals: $E_F$ lies within a band that has available empty states, so electrons near $E_F$ can easily gain energy and contribute to current.
• Intrinsic semiconductors: $E_F$ sits approximately midway in the band gap. The equal probability of exciting electrons upward and creating holes downward keeps the electron and hole concentrations equal.
• Insulators: $E_F$ is also in the gap, but the gap is so wide that the Fermi-Dirac tail at room temperature doesn't reach the conduction band in any meaningful way.

Semiconductor band structure

Valence band characteristics

The valence band maximum defines the energy from which holes are measured. In most common semiconductors (Si, Ge, GaAs), spin-orbit coupling splits the valence band into three subbands:

• Heavy hole band: higher effective mass, lower mobility
• Light hole band: lower effective mass, higher mobility
• Split-off band: shifted down in energy by the spin-orbit splitting energy $\Delta_{SO}$

These different hole masses affect transport properties and optical transition strengths. For device design, you often need to account for all three.

Conduction band characteristics

The conduction band minimum determines the electron effective mass and mobility. Its location in $k$-space matters:

• In GaAs and InP, the minimum is at the $\Gamma$ point (center of the Brillouin zone), giving a direct gap and excellent optical properties.
• In Si, the minimum is near the $X$ point, and there are six equivalent minima. This indirect gap makes Si poor for light emission but doesn't prevent it from being the dominant material for transistors.
• In Ge, the minimum is at the $L$ point, with four equivalent valleys.

Intrinsic vs. extrinsic semiconductors

Intrinsic semiconductors are undoped (pure). Their carrier concentration depends entirely on thermal excitation across the gap. At room temperature, intrinsic Si has about $1.5 \times 10^{10} \text{ cm}^{-3}$ carriers of each type, which is quite low compared to the $\sim 5 \times 10^{22} \text{ cm}^{-3}$ atoms present.

Extrinsic semiconductors are intentionally doped:

• N-type: Donor impurities (e.g., phosphorus in Si) have an extra valence electron that's easily donated to the conduction band. The Fermi level shifts up toward the conduction band.
• P-type: Acceptor impurities (e.g., boron in Si) have one fewer valence electron, creating a hole in the valence band. The Fermi level shifts down toward the valence band.

Doping concentrations typically range from $10^{14}$ to $10^{20} \text{ cm}^{-3}$, giving precise control over conductivity.

Band structure engineering

Band structure engineering is the deliberate modification of a semiconductor's electronic structure to tailor its properties for specific device applications. The three main approaches are alloying, strain, and quantum confinement.

Alloying effects on band structure

Mixing two or more semiconductors creates an alloy whose band gap falls between those of the parent materials. By adjusting the composition ratio, you can continuously tune the gap.

For example, $\text{Al}_x\text{Ga}_{1-x}\text{As}$ has a band gap that increases with aluminum fraction $x$, ranging from 1.42 eV (pure GaAs) to 2.16 eV (pure AlAs). This tunability is essential for designing heterostructure devices like LEDs at specific wavelengths or laser diodes for fiber-optic communication ($\text{In}_{1-x}\text{Ga}_x\text{As}_y\text{P}_{1-y}$ covers the 1.3–1.55 μm telecom window).

Alloying also changes the effective mass and carrier mobility, so device designers must balance optical and transport requirements.

Strain effects on band structure

When a thin semiconductor layer is grown on a substrate with a slightly different lattice constant, the layer deforms to match. This epitaxial strain modifies the band structure:

• Tensile strain tends to reduce the band gap and can increase electron mobility.
• Compressive strain tends to increase the band gap and can improve hole mobility by splitting the heavy-hole and light-hole bands.

Strained layers are widely used in modern transistors (strained Si in CMOS) and in quantum well lasers (strained InGaAs) to boost performance beyond what unstrained materials can achieve.

Quantum confinement effects

When a semiconductor structure shrinks to dimensions comparable to the electron's de Broglie wavelength (typically a few nanometers), the energy levels become quantized rather than continuous. This is quantum confinement.

The three main confined structures:

• Quantum wells (2D confinement in one direction): thin layers sandwiched between wider-gap barriers. Used in most semiconductor lasers and many LEDs.
• Quantum wires (1D confinement in two directions): nanoscale wires where carriers move freely in only one dimension.
• Quantum dots (0D confinement in all three directions): nanoscale boxes with fully discrete energy levels, sometimes called "artificial atoms." Used in quantum dot displays and emerging solar cell technologies.

Confinement increases the effective band gap (smaller structures → larger gaps), modifies the DOS (from the 3D square-root form to step functions in 2D, inverse-square-root peaks in 1D, and delta functions in 0D), and can significantly enhance optical properties like absorption and emission efficiency.