Miller indices are a crucial tool in crystallography, providing a standardized system for describing crystal planes. These indices help identify and compare different planes within crystal structures, offering insights into their orientation and symmetry.

Understanding Miller indices is essential for semiconductor physics, as they influence crystal growth, cleavage, and electrical properties. By mastering this concept, we can better grasp how crystal structure impacts device performance and optimize semiconductor materials for specific applications.

Definition of Miller indices

• Provide a system for describing the orientation and symmetry of crystal planes within a lattice structure
• Consist of three integers (h, k, l) that represent the reciprocal of the fractional intercepts made by the plane on the crystallographic axes
• Enable a standardized way to identify and compare different crystal planes across various crystal systems

Representation of crystal planes

Notation for crystal planes

• Denoted by (hkl) where h, k, and l are the Miller indices
• Parentheses are used to enclose the indices, indicating a specific plane
• Examples: (100), (111), (hkl)

Examples of Miller indices

• (100) plane intercepts the x-axis at 1, and is parallel to the y and z axes
• (111) plane intercepts all three axes at 1
• (110) plane intercepts the x and y axes at 1, and is parallel to the z-axis

Significance in crystallography

Relationship to crystal structure

• Miller indices provide information about the orientation of crystal planes relative to the crystallographic axes
• Different planes have varying atomic densities and arrangements, influencing properties such as cleavage, surface energy, and growth rates

Identification of crystal faces

• External faces of a crystal can be identified using Miller indices
• Helps in understanding the morphology and symmetry of crystals
• Useful for characterizing crystal habits and growth patterns

Calculation of Miller indices

Steps to determine indices

1. Find the intercepts of the plane with the crystallographic axes (x, y, z)
2. Take the reciprocals of the intercepts
3. Reduce the reciprocals to the smallest integer ratio
4. Enclose the integers in parentheses (hkl)

Negative and zero values

• Negative indices are denoted with a bar above the number, e.g., ($\bar{1}$00)
• Zero index means the plane is parallel to the corresponding axis

Families of crystal planes

Planes with similar indices

• Planes with the same ratio of h:k:l belong to the same family
• Share similar properties and symmetry
• Examples: {100}, {111}, {hkl}

Bracket notation for families

• Curly brackets {} are used to denote a family of planes
• All permutations of the indices are included in the family
• Example: {100} includes (100), (010), (001), ($\bar{1}$00), (0$\bar{1}$0), (00$\bar{1}$)

Interplanar spacing

Spacing between parallel planes

• Distance between two adjacent parallel planes in a crystal
• Depends on the Miller indices and lattice parameters
• Affects properties such as X-ray diffraction and electron diffraction patterns

Calculation using Miller indices

• For cubic crystals: $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$
  • $d_{hkl}$ is the interplanar spacing
  • $a$ is the lattice parameter
  • $h$, $k$, $l$ are the Miller indices

Direction of crystal planes

Normal direction to plane

• The direction perpendicular to a crystal plane
• Denoted by [hkl] using square brackets
• Example: [100] is the direction normal to the (100) plane

Relationship to crystal axes

• The [hkl] direction is related to the crystallographic axes (x, y, z)
• For cubic crystals, [hkl] is parallel to the vector from the origin to the point (h, k, l) in the crystal lattice

Planes in cubic crystals

Low-index planes

• Planes with small Miller indices, typically (100), (110), and (111)
• Have high atomic density and low surface energy
• Often observed as cleavage planes or growth faces

High-index planes

• Planes with larger Miller indices, such as (311), (511), etc.
• Have lower atomic density and higher surface energy compared to low-index planes
• May exhibit stepped or kinked surfaces

Planes in hexagonal crystals

Miller-Bravais indices

• Four-index notation (hkil) used for hexagonal crystal systems
• The third index i is the negative sum of h and k: i = -(h+k)
• Provides a more symmetrical representation of planes in hexagonal lattices

Conversion from Miller indices

• Miller indices (hkl) can be converted to Miller-Bravais indices (hkil)
• For hexagonal crystals: (hkl) → (hki(−h−k)l)
• Example: (101) in Miller notation becomes (10$\bar{1}$1) in Miller-Bravais notation

Applications in semiconductors

Cleavage planes vs growth planes

• Cleavage planes are preferred for creating smooth surfaces by breaking the crystal along specific planes (e.g., (111) in silicon)
• Growth planes are used for epitaxial growth of semiconductor layers (e.g., (100) for silicon)
• Choice of plane affects surface morphology, defect density, and electronic properties

Impact on electrical properties

• Orientation of crystal planes influences carrier mobility and band structure
• Different planes have varying atomic arrangements, affecting bonding and electronic states
• Proper selection of substrate orientation is crucial for optimizing device performance in semiconductor applications