Crystal Structure and Lattice Parameters Fundamentals

Rationale for Studying Crystal Structure in Electronics and Communication

• The electrical, thermal, optical, and mechanical properties of semiconductor materials depend fundamentally on their crystal structure. Consequently, understanding this structure is a primary concern in the high-level design of semiconductor devices.
• Materials are categorized by the availability and mobility of charge carriers:
  • Metals: Contain a vast number of free charge carriers.
  • Insulators: Contain nearly no free charge carriers.
  • Semiconductors: Contain a quantity of charge carriers that is high enough to allow for conduction, but low enough that the quantity can be easily manipulated.

States of Matter and Types of Solid Materials

• States of Matter:
  • Solids: Maintain a fixed shape and a fixed volume.
  • Liquids: Take the shape of their container, possess a free surface, and have a fixed volume.
  • Gases: Take the shape and the total volume of their container.
• Classification of Solid Materials:
  • Crystalline Solid: A form of substance where atoms or molecules are arranged in a definite, periodically repeating pattern across three dimensions.
  • Single Crystal: An atomic structure that repeats periodically across the entire volume of the material. Each atom is related to every other equivalent atom by translational symmetry, even at infinite length scales. Pyrite is an example of a material that can form a single crystal.
  • Polycrystalline Solid: A material composed of an aggregate of many small single crystals, known as grains or crystallites. These materials exhibit a high degree of order over many atomic or molecular dimensions within individual grains. The boundaries between these grains are called grain boundaries.
  • Amorphous (Non-Crystalline) Solid: Materials that lack a long-range periodic arrangement of atoms or molecules.

Fundamental Concepts of Crystal Architecture

• Crystal Structure Composition: A crystal structure is defined by the combination of a crystal lattice and a basis (or motif).
  • $\text{Crystal Structure} = \text{Crystal Lattice} + \text{Basis}$
• Crystal Lattice: The symmetrical three-dimensional structural arrangement of atoms, ions, or molecules inside a crystalline solid, represented as points in space.
• Basis (Motif): The atoms, groups of atoms, or molecules that are attached to the lattice sites or lattice points to create the final crystal structure.
• Dimensions of Lattices:
  • One-Dimensional Lattice: Particles lie along a straight line at equal or periodically repeating distances.
  • Two-Dimensional Lattice: Particles lie in a single plane at equal or periodically repeating distances.
  • Three-Dimensional Lattice: Particles are arranged geometrically as points in three-dimensional space.
• Bravais Lattice (BL) vs. Non-Bravais Lattice (non-BL):
  • Bravais Lattice: Consists of atoms of the same kind where all lattice points are equivalent.
  • Non-Bravais Lattice: Can contain different kinds of atoms. Certain lattice points are not equivalent to others. It can be viewed as a combination of two or more Bravais lattices.

Lattice Parameters and Unit Cells

• Lattice Parameters: These are the physical dimensions and angles that determine the geometry, actual size, and shape of a unit cell in a crystal lattice.
  • $a, b, c$: The distances between any two consecutive atoms along the $x$, $y$, and $z$ axes of the crystal lattice.
  • $\alpha$: The interplanar angle between the $xy$ and $xz$ planes.
  • $\beta$: The interplanar angle between the $xy$ and $yz$ planes.
  • $\gamma$: The interplanar angle between the $xy$ and $yz$ planes.
• Types of Unit Cells:
  • Primitive (P): Contains only a single lattice point.
  • Body-Centered (Internal - I): Contains an atom at the center of the cell body.
  • Face-Centered (F): Contains atoms on all faces of the planes that compose the cell.
  • End-Centered (C): Contains atoms centered on specific sides of the unit cell.

The 7 Crystal Systems and 14 Bravais Lattices

There are only seven distinct shapes of unit cells that can be stacked to fill 3D space without overlapping. These combine with centering types to form 14 unique Bravais lattices.

• 1. Cubic:
  • Geometry: $a = b = c$ and $\alpha = \beta = \gamma = 90^\circ$.
  • Examples: Pyrite, Cube.
  • Bravais Types: Primitive (P).
• 2. Tetragonal:
  • Geometry: $a = b \neq c$ and $\alpha = \beta = \gamma = 90^\circ$.
  • Example: Zircon.
• 3. Orthorhombic:
  • Geometry: $a \neq b \neq c$ and $\alpha = \beta = \gamma = 90^\circ$.
  • Example: Topaz.
• 4. Hexagonal:
  • Geometry: $a = b \neq c$ and $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$.
  • Example: Corundum.
• 5. Rhombohedral (Trigonal):
  • Geometry: $a = b = c$ and $\alpha = \beta = \gamma \neq 90^\circ$.
  • Example: Tourmaline.
• 6. Monoclinic:
  • Geometry: $a \neq b \neq c$ and $\alpha = \gamma = 90^\circ$, $\beta \neq 90^\circ$.
  • Example: Kunzite.
• 7. Triclinic:
  • Geometry: $a \neq b \neq c$ and \alpha \neq \t\beta \neq \gamma \neq 90^\circ.
  • Example: Amazonite.

Simple Cubic Crystal (SCC)

• Coordination Number: The number of atoms, ions, or molecules that a central atom holds as its nearest neighbors. For SCC, it is $6$.
• Atoms per Unit Cell:
  • There are $8$ corners, each contributing $1/8$ of an atom.
  • $8 \times \frac{1}{8} = 1$
• Relation between Lattice Vector ($a$) and Atomic Radius ($R$):
  • $a = 2R$
• Atomic Packing Factor (APF): Defined as the ratio of the volume of atoms in a unit cell to the total volume of the unit cell.
  • $\text{APF} = \frac{\text{Number of atoms} \times \text{Volume of one atom}}{\text{Volume of unit cell}}$
  • $\text{APF} = \frac{1 \times \frac{4}{3} \pi (0.5a)^3}{a^3} = 0.52$

Body-Centered Cubic Structure (BCC)

• Coordination Number: $8$
• Atoms per Unit Cell:
  • $1 \text{ (center)} + (8 \times \frac{1}{8}) \text{ (corners)} = 2$
• Relation between Lattice Vector ($a$) and Atomic Radius ($r$):
  • In triangle $ABC$, $AC^2 = AB^2 + BC^2 = a^2 + a^2$, so $AC = \sqrt{2}a$.
  • In triangle $ACD$, $AD^2 = AC^2 + CD^2 = 2a^2 + a^2 = 3a^2$, so $AD = \sqrt{3}a$.
  • The body diagonal $AD$ equals $4r$.
  • $4r = \sqrt{3}a$
  • $r = \frac{\sqrt{3}}{4} a$
• Atomic Packing Factor (APF):
  • $\text{APF} = \frac{2 \times \frac{4}{3} \pi (\frac{\sqrt{3}a}{4})^3}{a^3} = 0.68$

Face-Centered Cubic Structure (FCC)

• Coordination Number: $12$
• Atoms per Unit Cell:
  • $6 \text{ (faces)} \times \frac{1}{2} + 8 \text{ (corners)} \times \frac{1}{8} = 3 + 1 = 4$
• Relation between Lattice Vector ($a$) and Atomic Radius ($R$):
  • The face diagonal accounts for $4R$.
  • In triangle $ABC$, $AC^2 = AB^2 + BC^2 = a^2 + a^2$, so $AC = \sqrt{2}a$.
  • $4R = \sqrt{2}a$
  • $R = \frac{\sqrt{2}}{4} a$
• Atomic Packing Factor (APF):
  • $\text{APF} = \frac{4 \times \frac{4}{3} \pi (\frac{\sqrt{2}a}{4})^3}{a^3} = 0.74$

Crystallographic Planes and Miller Indices

• Crystallographic Planes: Any set of parallel and equally spaced planes that pass through the centers of atoms in a crystal.
• Miller Indices: A group of three numbers $(hkl)$ that indicates the orientation of a plane or set of parallel planes. These are the reciprocals of the three axial intercepts, cleared of fractions and common multiples.
  • All parallel planes share the same Miller indices.
• Procedure to Determine Miller Indices:
  1. Determine the intercepts of the plane with the axes in terms of lattice constants $a$, $b$, and $c$.
  2. Take the reciprocals of these intercepts.
  3. Reduce the values to the smallest integers while maintaining the same ratio.
  4. Enclose the resulting integers in parentheses: $(hkl)$. Do not use commas.

Miller Indices Calculation Examples

• Example 1:
  • Intercepts: $x = a/2$, $y = b$, $z = 2c/3$ (simplified for demonstration from raw data).
  • Assuming $a = b = c = 1$ (Unit lattice).
  • The example calculation provided leads to indices $(213)$.
• Example 2:
  • Intercepts: $x = -a$, $y = -b/3$, $z = c$.
  • Reciprocals: $h = 1/(-1) = -1$, $k = 1/(-1/3) = -3$, $l = 1/1 = 1$.
  • Indices: $(311)$ as derived in notes context.
• Practice Intercept Sets:
  1. Intercepts: $x = 2a$, $y = b$, $z = -c/4$
  2. Intercepts: $x = -a$, $y = 3b$, $z = c$
  3. Intercepts: $x = a/2$, $y = -b/3$, $z = 2c$
  4. Intercepts: $x = -a$, $y = -2b$, $z = -3c$

Eight Octant Sign Notation

Reference coordinates for points within the lattice space are identified by signs across the eight octants:

• $(+ + +)$, $(- - +)$, $(- - -)$, $(+ - -)$, $(+ - +)$, $(+ + -)$, $(- + -)$, $(- + +)$.