5.4 Energy bands and bandgaps

Energy bands in solids

Energy bands and bandgaps explain how electrons behave in crystalline solids. They determine whether a material conducts electricity, acts as a semiconductor, or insulates. These concepts are the foundation for understanding electronic devices and for engineering new materials with specific properties.

This guide covers band formation, Brillouin zones, solid classification by bandgap, direct vs. indirect gaps, effective mass, density of states, and bandgap modification techniques.

Formation of energy bands

When atoms are isolated, their electrons occupy discrete energy levels. As you bring many atoms together to form a crystal, the wave functions of electrons on neighboring atoms begin to overlap and interact. This overlap causes each discrete atomic level to split into a huge number of closely spaced levels, forming a quasi-continuous energy band.

The width of each band depends on two things: how much the atomic orbitals overlap and how strong the interatomic interactions are. Inner-shell (core) electrons have tightly bound orbitals with little overlap, so they form narrow bands. Outer-shell (valence) electrons overlap significantly, producing wide bands. This is why valence and conduction bands dominate the electronic behavior of solids.

Allowed and forbidden energy states

Within each energy band, there is a continuous range of allowed energy states that electrons can occupy. Between these bands lie forbidden energy gaps (bandgaps) where no electron states exist.

• Electrons in a crystal can only have energies within the allowed bands.
• The bandgap is the energy range between the top of one allowed band (the valence band) and the bottom of the next (the conduction band).
• The presence and width of this gap directly determine a solid's electrical conductivity.

Brillouin zones

Brillouin zones describe the allowed wave vectors (k-points) for electrons in a periodic crystal, defined in reciprocal lattice space.

• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It contains all the unique k-values needed to describe every electronic state.
• The boundaries of Brillouin zones are defined by Bragg planes, the planes in reciprocal space where electron diffraction occurs. At these boundaries, standing waves form and energy gaps open up.
• High-symmetry points within the Brillouin zone (like $\Gamma$, $X$, $L$, $K$) are labeled by convention and are where band structure features are most commonly plotted.

Reduced and extended zone schemes

There are two standard ways to plot energy bands as a function of wave vector k:

• Reduced zone scheme: All bands are folded back into the first Brillouin zone. This emphasizes the periodicity of the lattice and is the most common representation. Each band index $n$ labels a different branch within the same zone.
• Extended zone scheme: Bands are plotted continuously across multiple Brillouin zones without folding. This gives a more intuitive picture of how energy increases with $k$ and makes it easier to see where gaps open at zone boundaries.

Both schemes contain the same physics. The reduced scheme is more compact; the extended scheme is better for visualizing the nearly-free-electron picture and how Bragg reflection opens gaps.

Classification of solids by bandgap

The size of the bandgap and the position of the Fermi level together determine whether a solid behaves as a metal, semiconductor, or insulator. This classification is central to choosing and designing materials for electronic applications.

Metals vs semiconductors vs insulators

• Metals have overlapping valence and conduction bands (no bandgap). Electrons move freely, giving high electrical conductivity. Examples: copper, aluminum.
• Semiconductors have a small bandgap, typically 0.1 to about 4 eV. Thermal or optical energy can excite electrons across the gap. Silicon has a bandgap of 1.12 eV at room temperature; GaAs has 1.42 eV.
• Insulators have a large bandgap, generally greater than 4 eV. Very few electrons gain enough energy to cross the gap at normal temperatures. Diamond, for instance, has a bandgap of about 5.5 eV.

Fermi level and its significance

The Fermi level ($E_F$) is the energy at which the probability of electron occupation is exactly 50% at any finite temperature. At absolute zero, all states below $E_F$ are filled and all states above are empty.

• In metals, $E_F$ lies within a partially filled band, so empty states are immediately available for conduction.
• In semiconductors and insulators, $E_F$ lies within the bandgap. No states exist at that energy, so conduction requires thermal excitation of electrons into the conduction band.

The position of $E_F$ relative to the band edges controls carrier concentration and, therefore, conductivity.

Temperature dependence of Fermi level

In an intrinsic (undoped) semiconductor at absolute zero, $E_F$ sits exactly at the midpoint of the bandgap. As temperature increases:

1. More electrons gain enough thermal energy to cross the gap into the conduction band.
2. The Fermi-Dirac distribution broadens, and the occupation probability near the band edges changes.
3. In an intrinsic semiconductor, $E_F$ stays near mid-gap but shifts slightly depending on the effective masses of electrons and holes: $E_F = \frac{E_c + E_v}{2} + \frac{3}{4}k_BT\ln\left(\frac{m_h^*}{m_e^*}\right)$

For doped semiconductors, $E_F$ shifts toward the conduction band (n-type) or valence band (p-type) at low temperatures, and moves back toward mid-gap as temperature rises and intrinsic carriers dominate.

Direct and indirect bandgaps

Whether a bandgap is direct or indirect has major consequences for how a material interacts with light. This distinction determines which semiconductors are suitable for LEDs, lasers, and photodetectors versus those better suited for transistors and solar cells.

E-k diagrams

An E-k diagram plots electron energy ($E$) versus crystal momentum (wave vector $k$) along high-symmetry directions in the Brillouin zone. It's the primary tool for visualizing band structure.

• In a direct bandgap semiconductor, the conduction band minimum (CBM) and valence band maximum (VBM) occur at the same $k$-point (usually $\Gamma$, the zone center).
• In an indirect bandgap semiconductor, the CBM and VBM occur at different $k$-points. For silicon, the VBM is at $\Gamma$ while the CBM is near the $X$ point.

Optical transitions in direct bandgaps

In direct bandgap materials, an electron can transition from the VBM to the CBM by absorbing a single photon. No change in crystal momentum is needed because both extrema share the same $k$-value. Photons carry very little momentum compared to electrons, so this transition is momentum-conserving by default.

This makes direct bandgap semiconductors highly efficient at both absorbing and emitting light. That's why GaAs and GaN are the materials of choice for LEDs, laser diodes, and optical communication devices.

Phonon-assisted transitions in indirect bandgaps

In indirect bandgap materials, the CBM and VBM sit at different $k$-values. A photon alone can't supply the large momentum difference, so the transition also requires a phonon (a quantum of lattice vibration) to conserve crystal momentum.

1. An electron absorbs a photon (providing energy).
2. Simultaneously, a phonon is absorbed or emitted (providing the momentum difference $\Delta k$).
3. This two-particle process is far less probable than a single-photon transition.

Because of this, indirect bandgap semiconductors like silicon are poor light emitters. However, silicon still works well for solar cells because absorption, while weaker per unit thickness, can be compensated with thicker material and light-trapping designs.

Examples of direct and indirect bandgap materials

Effective mass of electrons and holes

In a crystal, electrons don't behave like free particles. The periodic potential of the lattice modifies their response to external forces. The effective mass ($m^*$) captures this modification: it lets you treat a crystal electron as a free particle, but with a different mass.

Concept of effective mass

Effective mass quantifies how easily a charge carrier accelerates under an applied force. A smaller $m^*$ means the carrier responds more readily to electric fields, leading to higher mobility.

• $m^*$ accounts for all the complex interactions between the electron and the periodic lattice potential.
• It can differ for electrons and holes in the same material.
• It can also be anisotropic, meaning it varies depending on the crystallographic direction. Silicon, for example, has longitudinal and transverse effective masses for conduction band electrons.

Calculation from E-k diagrams

The effective mass is extracted from the curvature of the E-k diagram:

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

• A band with strong curvature (steep parabola) gives a small effective mass.
• A band with weak curvature (flat parabola) gives a large effective mass.
• Near band extrema, the E-k relation is approximately parabolic, so this formula works well. Far from the extrema, higher-order corrections may be needed.

Light and heavy holes

The valence band in most semiconductors isn't a single parabola. It splits into multiple subbands near the $\Gamma$ point:

• Heavy hole (HH) band: smaller curvature, larger effective mass, lower mobility.
• Light hole (LH) band: larger curvature, smaller effective mass, higher mobility.
• A third band, the split-off band, is separated by spin-orbit coupling.

The distinction matters for optical properties (different transition probabilities) and transport (different contributions to hole conductivity). In strained materials, the degeneracy of HH and LH bands can be lifted, which is exploited in strained quantum well lasers.

Impact on carrier mobility

Mobility $\mu$ describes how fast carriers drift under an electric field. It's related to effective mass by:

$\mu = \frac{e\tau}{m^*}$

where $e$ is the electron charge and $\tau$ is the average scattering time.

• Smaller $m^*$ → higher mobility → faster devices.
• GaAs has a small electron effective mass ($\approx 0.067\,m_e$), giving it much higher electron mobility than silicon ($m^* \approx 0.26\,m_e$ for conductivity effective mass). This is why GaAs is used in high-frequency electronics.
• Hole mobilities are generally lower than electron mobilities because holes tend to have larger effective masses.

Density of states

The density of states (DOS), $g(E)$, tells you how many electronic states are available per unit energy per unit volume at a given energy. It connects the band structure to measurable quantities like conductivity, optical absorption, and heat capacity.

Definition and significance

$g(E)$ determines how many electrons can occupy a narrow energy window around energy $E$. A high DOS at some energy means many states are packed closely together there.

• The DOS, combined with the Fermi-Dirac distribution, gives you the actual electron (or hole) concentration at any temperature.
• It directly controls optical absorption strength, emission rates, and thermoelectric performance.
• Engineering the DOS is one of the main goals of nanostructure design.

Calculation for 3D, 2D, 1D, and 0D structures

The functional form of the DOS changes dramatically with dimensionality:

• 3D (bulk): $g_{3D}(E) \propto \sqrt{E - E_c}$ — a smooth square-root onset above the band edge $E_c$.
• 2D (quantum wells): $g_{2D}(E) \propto \sum_n \Theta(E - E_n)$ — a staircase function. Each quantized subband $E_n$ adds a constant step to the DOS.
• 1D (quantum wires): $g_{1D}(E) \propto \sum_n \frac{1}{\sqrt{E - E_n}}$ — divergent peaks (Van Hove singularities) at each subband edge, falling off above.
• 0D (quantum dots): $g_{0D}(E) \propto \sum_n \delta(E - E_n)$ — discrete delta functions, like an artificial atom.

As dimensionality decreases, the DOS becomes more sharply peaked. This concentration of states at specific energies is what makes low-dimensional structures so useful for lasers and other devices that benefit from sharp spectral features.

Van Hove singularities

Van Hove singularities are points where the DOS has a discontinuity or divergence. They occur at energies where $\nabla_k E(\mathbf{k}) = 0$, i.e., at critical points in the band structure where the band is flat in $k$-space.

• In 3D, Van Hove singularities produce kinks (changes in slope) in the DOS.
• In 2D, they produce logarithmic divergences.
• In 1D, they produce inverse-square-root divergences.

These singularities show up experimentally as peaks in optical absorption spectra and can strongly enhance light-matter interaction at specific energies.

Effect on optical and electronic properties

• A high DOS at a particular energy increases the probability of optical transitions (absorption or emission) at that energy.
• Near the Fermi level, a high DOS enhances electrical conductivity because more states are available for carriers.
• Thermoelectric performance benefits from a sharply peaked DOS near $E_F$, which increases the Seebeck coefficient.
• Band structure engineering (through quantum confinement, strain, or alloying) is used to reshape the DOS for optimized device performance.

Modification of bandgaps

Bandgap engineering is the deliberate tuning of a material's bandgap to achieve desired electronic or optical properties. Several techniques are available, often used in combination.

Doping and its effects

Doping introduces impurity atoms into a semiconductor to control its carrier concentration and shift the Fermi level.

• n-type doping: Adding donor atoms (e.g., phosphorus in silicon, with 5 valence electrons) provides extra electrons near the conduction band edge. $E_F$ shifts upward toward $E_c$.
• p-type doping: Adding acceptor atoms (e.g., boron in silicon, with 3 valence electrons) creates holes near the valence band edge. $E_F$ shifts downward toward $E_v$.

Doping doesn't change the fundamental bandgap significantly, but at very high doping levels (degenerate doping), impurity bands can form and effectively narrow the gap. Doping also introduces localized energy states within the gap, which affect recombination and absorption.

Strain and quantum confinement

Strain modifies the bandgap by physically distorting the crystal lattice:

• Compressive strain generally increases the bandgap.
• Tensile strain generally decreases it.
• Strain also lifts the degeneracy of the heavy and light hole bands, which can improve device performance (e.g., in strained-Si MOSFETs and strained quantum well lasers).

Quantum confinement occurs when one or more dimensions of a structure shrink to the nanoscale (comparable to the de Broglie wavelength of carriers):

• Energy levels become quantized, and the effective bandgap increases as the structure gets smaller.
• This is why CdSe quantum dots emit different colors depending on their size: smaller dots have larger effective gaps and emit bluer light.
• Quantum wells (2D confinement), quantum wires (1D), and quantum dots (0D) each show progressively stronger confinement effects.

Heterostructures and band alignment

Heterostructures are junctions between two different semiconductor materials. The way their bands line up at the interface is critical:

• Type-I (straddling): The smaller-gap material's conduction and valence bands both lie within the larger-gap material's bands. Both electrons and holes are confined in the same layer. Used in quantum well lasers and LEDs.
• Type-II (staggered): The bands are offset so that electrons and holes are confined in different layers. Useful for charge separation in certain solar cells and photodetectors.
• Type-III (broken gap): The conduction band of one material overlaps the valence band of the other. Found in InAs/GaSb systems, used for tunneling devices.

Band alignment is determined by the electron affinities and bandgaps of the constituent materials, and can be further tuned by strain at the interface.

Applications in optoelectronic devices

Bandgap engineering underpins most modern optoelectronic technology:

• LEDs and laser diodes: Heterostructures and quantum wells with specific bandgaps produce light at targeted wavelengths. InGaN alloys cover the blue-to-green range; AlGaInP covers red-to-yellow.
• Solar cells: Multi-junction cells stack semiconductors with different bandgaps to absorb different portions of the solar spectrum, pushing efficiencies above 45%.
• Photodetectors: Bandgap tuning sets the spectral sensitivity. HgCdTe alloys, for example, can be tuned to detect infrared wavelengths by adjusting composition.
• Transistors: Strained silicon and SiGe heterostructures improve carrier mobility and switching speed in modern CMOS technology.