1.1 Crystal lattices and bonding

Crystal lattices and bonding form the foundation of semiconductor physics. These concepts explain how atoms arrange themselves in solids, creating the structures that determine a material's electronic and optical properties. This guide covers crystal structure types, lattice parameters, bonding mechanisms, energy bands, Brillouin zones, crystal defects, and growth techniques.

Types of crystal structures

A crystal structure is the regular, repeating arrangement of atoms in a solid. The specific structure a semiconductor adopts has a direct impact on its electronic and optical behavior, so recognizing the major structure types is a key first step.

Unit cells in crystals

A unit cell is the smallest repeating unit that, when stacked in three dimensions, builds the entire crystal lattice. It contains all the symmetry and structural information of the crystal. The geometry and dimensions of the unit cell dictate the crystal's overall properties.

Common unit cell shapes include cubic, tetragonal, orthorhombic, and hexagonal. In this course, you'll mostly encounter cubic and hexagonal types.

Face-centered cubic lattice

In a face-centered cubic (FCC) lattice, atoms sit at each corner of the cube and at the center of each face. This gives 4 atoms per unit cell (each corner atom is shared among 8 cells, each face atom among 2).

• Packing efficiency: 74%, one of the most tightly packed arrangements
• Coordination number: 12 (each atom touches 12 neighbors)
• Many metals like copper and aluminum adopt this structure

Note that Si and Ge are sometimes loosely called "FCC," but their actual structure is the diamond cubic lattice (described below), which is based on the FCC arrangement with extra atoms.

Body-centered cubic lattice

A body-centered cubic (BCC) lattice has atoms at each corner plus one atom at the center of the cube, giving 2 atoms per unit cell.

• Packing efficiency: 68%, less dense than FCC
• Coordination number: 8
• BCC is common in metals like iron and tungsten but rare among semiconductors

Diamond cubic lattice

The diamond cubic lattice is the structure that matters most for mainstream semiconductors. It consists of two interpenetrating FCC lattices, offset by one-quarter of the body diagonal. The basis atoms sit at positions (0,0,0) and (1/4, 1/4, 1/4) relative to each FCC lattice point.

• Each atom is covalently bonded to four neighbors in a tetrahedral arrangement
• Silicon (Si), germanium (Ge), and diamond (C) all crystallize in this structure
• The tetrahedral bonding angle is approximately 109.5°

This structure is central to understanding why Si and Ge behave as semiconductors. The strong, directional covalent bonds create the energy gap between valence and conduction bands.

Hexagonal close-packed lattice

In a hexagonal close-packed (HCP) lattice, atoms are arranged in hexagonal layers that alternate in an ABAB... stacking sequence.

• Packing efficiency: 74%, the same as FCC
• Several important compound semiconductors adopt the wurtzite structure, which is based on HCP. Examples include GaN, AlN, and InN.
• Wurtzite-structure materials are critical for blue/UV LEDs and power electronics

Lattice constants and parameters

Lattice constants describe the size and shape of the unit cell. They're used to calculate quantities like atomic density, interplanar spacing, and strain in thin films.

Lattice constant definition

The lattice constant (commonly denoted $a$) is the edge length of the unit cell. For cubic lattices, all three edges are equal, so a single value $a$ fully describes the cell size. For example, silicon has $a = 5.431 \, \text{Å}$ at room temperature.

In non-cubic lattices (like hexagonal), you need more than one parameter. Hexagonal cells require both $a$ (the in-plane spacing) and $c$ (the height of the cell). For wurtzite GaN, $a = 3.189 \, \text{Å}$ and $c = 5.185 \, \text{Å}$.

Miller indices for planes

Miller indices $(hkl)$ describe the orientation of crystal planes. To determine them:

1. Find where the plane intercepts the three crystal axes (in units of the lattice constants).
2. Take the reciprocals of those intercepts.
3. Reduce to the smallest set of integers.

For example, a plane that intercepts the x-axis at 1, the y-axis at ∞, and the z-axis at ∞ has reciprocals (1, 0, 0), giving the (100) plane. Common planes you'll encounter in cubic crystals are (100), (110), and (111). The (111) plane in silicon is particularly important because it has the highest atomic density and is a common wafer orientation.

Reciprocal lattice concept

The reciprocal lattice is a mathematical construction representing the Fourier transform of the real-space lattice. Each point in the reciprocal lattice corresponds to a set of crystal planes in real space.

• Reciprocal lattice vectors are perpendicular to the corresponding real-space planes
• Their magnitudes are inversely proportional to the interplanar spacing $d_{hkl}$
• The reciprocal lattice is essential for understanding diffraction experiments (like X-ray diffraction) and for constructing Brillouin zones

Bonding in semiconductor crystals

The type of bonding in a semiconductor crystal determines its band gap, carrier mobility, mechanical hardness, and thermal properties. Most semiconductors involve covalent bonding, ionic bonding, or a mix of both.

Covalent bonding characteristics

Covalent bonding involves atoms sharing valence electrons to achieve a stable configuration. In semiconductors like Si and Ge, each atom has four valence electrons and forms four bonds with its neighbors through $sp^3$ hybrid orbitals, producing the tetrahedral geometry of the diamond cubic structure.

Covalent bonds are strong and directional, which is why covalently bonded semiconductors tend to be hard and have well-defined crystal structures. The shared electrons form the valence band; the energy needed to break these bonds and free an electron corresponds roughly to the band gap.

Ionic bonding characteristics

Ionic bonding occurs when one atom transfers electrons to another, creating positive and negative ions held together by electrostatic attraction. This happens when there's a large electronegativity difference between the atoms.

• II-VI compound semiconductors like ZnSe and CdTe have significant ionic character
• Ionic bonds are non-directional, and the crystal structure is determined largely by the relative sizes of the ions
• Purely ionic semiconductors tend to have wider band gaps than purely covalent ones

Metallic bonding characteristics

Metallic bonding features a "sea" of delocalized electrons shared among positively charged ion cores. This type of bonding is not typical of semiconductors. However, when a semiconductor is very heavily doped (degenerately doped), it can begin to exhibit metallic-like conduction because the impurity states merge with the band edge and the material behaves more like a metal.

Hybrid bonding in semiconductors

Most compound semiconductors have bonding that's a mix of covalent and ionic character. The balance depends on the electronegativity difference between the constituent atoms.

• III-V compounds (GaAs, InP, GaN): mostly covalent with some ionic character. GaAs, for instance, is roughly 31% ionic.
• II-VI compounds (ZnSe, CdTe): more ionic than III-V materials due to the larger electronegativity difference.

The degree of ionicity affects the band gap, optical properties, and piezoelectric behavior of the material.

Energy bands in crystals

Energy bands describe the allowed energy states for electrons in a solid. They arise from the interaction of atoms in a crystal and are the key to understanding why materials are conductors, insulators, or semiconductors.

Formation of energy bands

When isolated atoms are brought close together to form a crystal, their discrete energy levels split and broaden into continuous energy bands. This happens because the electron wave functions of neighboring atoms overlap and interact.

Think of it this way: a single atom has sharp energy levels. Two atoms close together split each level into two. For $N$ atoms in a crystal (where $N$ is on the order of $10^{23}$), each level splits into $N$ closely spaced states, effectively forming a continuous band. Between these bands are forbidden energy gaps where no electron states exist.

Valence and conduction bands

• The valence band is the highest energy band that is fully occupied by electrons at absolute zero (0 K). These electrons are involved in bonding and are not free to move.
• The conduction band is the lowest energy band that is empty at 0 K. Electrons that gain enough energy to reach this band are free to move through the crystal and carry current.
• The energy separation between the top of the valence band and the bottom of the conduction band is the band gap ($E_g$).

Band gap concept

The band gap ($E_g$) is the minimum energy an electron needs to jump from the valence band to the conduction band. It's the single most important parameter for classifying a semiconductor's behavior.

• A larger band gap means the material requires more energy to excite carriers, resulting in lower intrinsic conductivity at room temperature.
• Band gap also determines which wavelengths of light a material can absorb or emit.

Some reference values:

Direct vs. indirect band gaps

This distinction is critical for optoelectronic applications.

• In a direct band gap semiconductor, the conduction band minimum and valence band maximum occur at the same wave vector ($k$-value) in the Brillouin zone. An electron can transition between bands by absorbing or emitting a photon alone.
• In an indirect band gap semiconductor, the conduction band minimum and valence band maximum occur at different $k$-values. A transition requires both a photon (for energy) and a phonon (a lattice vibration, for momentum), making the process much less probable.

This is why GaAs (direct gap) is used for LEDs and laser diodes, while Si (indirect gap) is not efficient for light emission but works well for solar cells and transistors.

Brillouin zones and band structure

Brillouin zones provide the framework for understanding how electron energy depends on momentum in a crystal. Band structure diagrams, plotted within these zones, reveal the electronic properties that govern device behavior.

First Brillouin zone definition

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It's the region in $k$-space that contains all unique electronic states of the crystal. Its boundaries are defined by the perpendicular bisectors of the reciprocal lattice vectors.

For an FCC real-space lattice (like Si or GaAs), the first Brillouin zone is a truncated octahedron. High-symmetry points within this zone are labeled with standard letters: $\Gamma$ (the zone center, $k = 0$), $X$, $L$, $K$, and others. You'll see band structures plotted along paths connecting these points.

Reduced and extended zone schemes

There are two common ways to display band structure:

• Reduced zone scheme: All bands are "folded back" into the first Brillouin zone. This is the most common representation and is compact and easy to read.
• Extended zone scheme: Bands are shown extending across multiple Brillouin zones without folding. This gives a more complete spatial picture but is harder to interpret.

For this course, you'll mostly work with the reduced zone scheme.

E-k diagrams for band structure

An E-k diagram plots electron energy ($E$) versus wave vector ($k$) along high-symmetry directions in the Brillouin zone. These diagrams reveal:

• The shape and width of the valence and conduction bands
• The size and type (direct or indirect) of the band gap
• The curvature of the bands, which determines effective mass
• Where optical transitions are most likely to occur

Reading E-k diagrams is a core skill in semiconductor physics. The curvature at a band extremum tells you about carrier mobility: a sharply curved band means a small effective mass and high mobility.

Effective mass concept

Electrons and holes in a crystal don't respond to forces the same way free electrons do. The effective mass ($m^*$) accounts for the influence of the periodic lattice potential on carrier motion.

It's defined from the band curvature:

$m^* = \hbar^2 \left(\frac{d^2E}{dk^2}\right)^{-1}$

• A band with strong curvature (wide parabola) gives a small effective mass and high carrier mobility.
• A flat band gives a large effective mass and low mobility.

Some representative values (in units of free electron mass $m_0$):

The very small electron effective mass in GaAs is why it's preferred for high-speed electronics.

Defects and impurities in crystals

No real crystal is perfect. Defects and impurities break the ideal periodicity of the lattice, and they can dramatically alter a semiconductor's electrical, optical, and mechanical properties. Some defects are harmful; others (like intentional doping) are essential.

Point defects: vacancies and interstitials

Point defects involve one or a few atoms:

• A vacancy is a missing atom at a regular lattice site. It leaves behind dangling bonds that can trap carriers.
• An interstitial is an extra atom squeezed into a position between regular lattice sites.
• A Frenkel defect is a vacancy-interstitial pair, where an atom has moved from its site to an interstitial position.

Point defects can introduce energy levels within the band gap, acting as traps or recombination centers that degrade device performance.

Line defects: dislocations

Dislocations are line defects where the regular arrangement of atoms is disrupted along a line through the crystal. The two main types are:

• Edge dislocations: an extra half-plane of atoms is inserted into the lattice
• Screw dislocations: atoms are arranged in a helical pattern around the dislocation line

Dislocations affect both mechanical properties (they allow plastic deformation) and electronic properties (they introduce states in the band gap and act as non-radiative recombination centers). In optoelectronic devices, high dislocation densities severely reduce efficiency.

Planar defects: grain boundaries

Planar defects are two-dimensional boundaries within the crystal. The most common type is a grain boundary, where two regions of crystal with different orientations meet.

• Grain boundaries scatter charge carriers, reducing mobility
• They introduce interface states that can trap carriers
• They also affect mechanical strength and fracture behavior

Single-crystal semiconductors (used in most high-performance devices) are grown specifically to avoid grain boundaries. Polycrystalline materials, like those used in some solar cells, contain many grain boundaries that limit efficiency.

Substitutional vs. interstitial impurities

Impurities are foreign atoms in the crystal, either introduced intentionally (doping) or present as contamination.

• Substitutional impurities replace a host atom at a regular lattice site. This is how most dopants work. For example, phosphorus (group V) substituting for silicon (group IV) provides an extra electron (n-type doping), while boron (group III) creates a missing electron, or hole (p-type doping).
• Interstitial impurities sit between lattice sites. Small atoms like hydrogen and lithium can occupy interstitial positions in silicon.

Controlled doping with substitutional impurities is the foundation of all semiconductor device engineering.

Crystal growth techniques

Producing high-quality single crystals is essential for semiconductor manufacturing. The growth technique determines crystal purity, defect density, size, and cost.

Czochralski method for bulk growth

The Czochralski (CZ) method is the dominant technique for producing silicon wafers used in the electronics industry.

1. High-purity polycrystalline semiconductor material is melted in a crucible.
2. A small seed crystal with the desired orientation is dipped into the melt.
3. The seed is slowly pulled upward (typically 1-2 mm/min) while rotating.
4. The melt crystallizes onto the seed, forming a large cylindrical single-crystal ingot (boule).
5. The ingot is then sliced into thin wafers for device fabrication.

CZ-grown silicon ingots can exceed 300 mm (12 inches) in diameter. The method is also used for germanium and some III-V materials.

Bridgman method for bulk growth

The Bridgman method grows crystals by directional solidification:

1. The semiconductor material is melted inside a sealed crucible.
2. The crucible is slowly moved through a temperature gradient (or the gradient is moved past the crucible).
3. Crystallization begins at the cooler end and progresses through the melt.

This method is commonly used for compound semiconductors like GaAs and CdTe, which are difficult to grow by the Czochralski method due to high vapor pressures or decomposition at the melting point.

Epitaxial growth techniques

Epitaxy is the growth of a crystalline layer on top of a crystalline substrate, where the deposited layer inherits the substrate's crystal structure and orientation. Epitaxial growth enables:

• Precise control of layer thickness (down to single atomic layers)
• Controlled composition and doping profiles
• Fabrication of heterostructures (layers of different materials stacked together)

Common epitaxial techniques include:

• Liquid-phase epitaxy (LPE): growth from a supersaturated melt; simple but limited in precision
• Vapor-phase epitaxy (VPE): growth from gaseous precursors; good for thicker layers
• Metal-organic chemical vapor deposition (MOCVD): uses metal-organic precursors; widely used for III-V and III-nitride devices
• Molecular beam epitaxy (MBE): the most precise technique (see below)

Molecular beam epitaxy (MBE)

MBE offers the highest level of control over thin-film semiconductor growth.

1. Ultra-pure source elements are heated in separate effusion cells inside an ultra-high vacuum chamber ($< 10^{-10}$ torr).
2. Atomic or molecular beams are directed at a heated crystalline substrate.
3. Atoms arrive at the surface and migrate to lattice sites, building the crystal one atomic layer at a time.
4. Shutters on each cell allow rapid switching between materials, enabling atomically abrupt interfaces.

Growth rates are slow (typically 0.1-1 μm/hour, or roughly 1 Å/s), but this slowness is what gives MBE its precision. MBE is widely used for growing III-V heterostructures for laser diodes, high-electron-mobility transistors (HEMTs), and quantum well devices.