Glossary

1.3 Miller indices and crystal planes

4 min read•august 20, 2024

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

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  • 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

  • intercepts the x-axis at 1, and is parallel to the y and z axes
  • 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., (1ˉ\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), (1ˉ\bar{1}00), (01ˉ\bar{1}0), (001ˉ\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: dhkl=ah2+k2+l2d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}
    • dhkld_{hkl} is the
    • aa is the lattice parameter
    • hh, kk, ll 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

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 (101ˉ\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
  • 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

Key Terms to Review (16)

(100) plane: The (100) plane refers to a specific crystallographic plane in a cubic crystal system, denoted by the Miller indices (1, 0, 0). This notation describes a plane that intersects the x-axis at 1 and is parallel to both the y-axis and z-axis. The (100) plane is significant in understanding the orientation of atoms within a crystal lattice and how these planes influence various physical properties, including electron mobility and surface interactions.
(111) plane: (111) plane refers to a specific crystallographic plane in a crystal lattice, identified using Miller indices. These indices provide a shorthand notation to describe the orientation of planes in a crystal structure. The (111) plane intersects the axes of a cubic crystal at one-third of the way along each axis, representing a close-packed arrangement of atoms, which is significant for understanding properties like slip systems and crystal growth.
Angle of incidence: The angle of incidence is the angle formed between an incoming wave or particle and a surface normal at the point of contact. This concept is crucial in understanding how waves, such as light or particles, interact with surfaces, affecting phenomena like reflection, refraction, and transmission. The angle of incidence plays a significant role in determining the efficiency of processes like ion implantation in semiconductor manufacturing and the arrangement of crystal planes in a solid structure.
Band Structure: Band structure refers to the range of energy levels that electrons can occupy in a solid, illustrating how these levels are distributed in relation to the crystal momentum of the electrons. It provides insight into the electronic properties of materials, helping to distinguish between conductors, semiconductors, and insulators based on the arrangement of these energy bands and the existence of band gaps. Understanding band structure is crucial for analyzing how the reciprocal lattice and Brillouin zones affect electron behavior, as well as how Miller indices define crystal planes and their influence on electronic transitions.
Body-centered cubic: The body-centered cubic (BCC) structure is a type of crystal lattice where each unit cell has one atom at each corner and a single atom at the center of the cube. This arrangement leads to a coordination number of 8, meaning each atom is in contact with 8 others, which influences the material properties such as density and strength. BCC lattices are significant in understanding metallic bonding and how crystal structures impact the mechanical behavior of materials.
Crystal lattice: A crystal lattice is a three-dimensional arrangement of atoms, ions, or molecules in a repeating pattern that defines the structure of a crystalline solid. This ordered structure is crucial because it determines the material's properties, such as electrical conductivity and optical behavior. The arrangement of these particles in a crystal lattice is foundational for understanding how different planes and directions within the crystal can impact its overall characteristics.
Dislocation: Dislocation is a type of defect in a crystal lattice that occurs when there is an irregularity in the arrangement of atoms, specifically involving the misalignment of rows of atoms. This misalignment creates a linear defect, which can significantly influence the mechanical properties of materials, including their strength and ductility. Dislocations play a critical role in the processes of plastic deformation, making them essential to understand when examining how materials respond to stress.
Electron mobility: Electron mobility is a measure of how quickly electrons can move through a material when subjected to an electric field. It influences the electrical conductivity and performance of semiconductor devices, which is crucial for understanding various electronic properties, including how materials interact with impurities and crystal structures.
Face-centered cubic: The face-centered cubic (FCC) structure is a type of crystal lattice arrangement where atoms are located at each of the corners and the centers of all the cube faces. This arrangement allows for a high packing efficiency and contributes to the material's properties, such as ductility and strength. The FCC lattice plays a crucial role in understanding crystal bonding, reciprocal lattices, Miller indices, and crystal growth processes.
Hkl notation: hkl notation is a system used to describe the orientation of crystal planes in a crystalline structure, where 'h', 'k', and 'l' are integers that represent the intercepts of the plane with the crystallographic axes. This notation helps in identifying specific planes within a crystal lattice and is crucial for understanding the geometry and symmetry of crystals, as well as their physical properties.
Interplanar spacing: Interplanar spacing refers to the distance between parallel planes of atoms in a crystalline material. This spacing is crucial for understanding the arrangement of atoms in a crystal lattice, as it affects the diffraction patterns observed during techniques like X-ray diffraction, which are used to study crystal structures. The interplanar distance can be calculated using Miller indices, providing insight into the geometric arrangement of planes in the lattice.
Miller indices: Miller indices are a notation system in crystallography used to describe the orientation of crystal planes in a lattice. This system simplifies the identification of specific planes and directions within a crystal structure by using a set of three integers (h, k, l) that represent the reciprocal of the fractional intercepts that the plane makes with the axes of the crystal lattice. Miller indices are fundamental for understanding various properties of materials, such as diffraction patterns and electronic behaviors.
Reciprocal lattice: The reciprocal lattice is a mathematical construct used to describe the periodicity of a crystal in momentum space, representing the Fourier transform of the real-space crystal lattice. It provides a framework for understanding wave vectors associated with electron and phonon states, linking directly to band structures and allowed energy levels in solid materials. By visualizing the reciprocal lattice, one can derive essential properties like Brillouin zones, which help explain the behavior of electrons in crystalline solids.
Unit Cell: A unit cell is the smallest repeating unit in a crystal lattice that shows the full symmetry of the crystal structure. It serves as the building block for the entire crystal, with its dimensions and angles defining the overall structure. Understanding the unit cell is essential for analyzing how atoms are arranged in different materials, which directly influences their physical properties.
Vacancy: A vacancy is a type of point defect in a crystal lattice that occurs when an atom is missing from its regular lattice position. This missing atom creates an empty site, which can significantly influence the physical properties of the material, such as electrical and thermal conductivity. The presence of vacancies is crucial for understanding how defects and impurities affect semiconductor behavior and carrier diffusion.
Zone Axis: The zone axis is a specific direction in a crystal lattice that defines a particular orientation of the crystal and is crucial for understanding the arrangement of atoms within the lattice. It represents a line along which multiple crystal planes intersect, allowing for the characterization of the crystal structure through its geometry. This concept is particularly important when analyzing diffraction patterns in electron microscopy and X-ray diffraction, where identifying the zone axis helps in determining the symmetry and orientation of the crystal.
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