1.3 Miller indices and crystal planes

Miller indices are a standardized system for describing the orientation of planes within a crystal lattice. They give you a precise way to label any plane using just three integers, which turns out to be critical for semiconductor physics: the crystal plane you're working with affects everything from how a wafer cleaves to carrier mobility in a finished device.

Definition of Miller indices

Miller indices are a set of three integers (h, k, l) that describe a crystal plane's orientation relative to the lattice axes. Rather than specifying where a plane sits in absolute terms, the indices capture how the plane is tilted by encoding the reciprocals of its axis intercepts.

This reciprocal approach is what makes the system powerful. A plane that runs parallel to an axis (intercept at infinity) simply gets an index of zero, and planes that cut the axes close to the origin get larger index values. The result is a compact notation that lets you compare planes across any crystal system.

Representation of crystal planes

Notation for crystal planes

• A specific plane is written in parentheses: (hkl)
• Common examples: (100), (110), (111)

Examples of Miller indices

• (100): intercepts the x-axis at 1 and is parallel to both the y- and z-axes (intercepts at infinity, giving indices of 0)
• (110): intercepts the x- and y-axes at 1, parallel to the z-axis
• (111): intercepts all three axes at 1, cutting diagonally through the unit cell

Visualizing these in a cubic unit cell helps a lot. The (100) plane is just one face of the cube. The (111) plane slices corner-to-corner, creating a triangle.

Calculation of Miller indices

Steps to determine indices

1. Identify where the plane intercepts the three crystallographic axes (x, y, z). Express these as multiples of the lattice parameters. If the plane is parallel to an axis, the intercept is $\infty$.
2. Take the reciprocal of each intercept.
3. Clear fractions by multiplying through to get the smallest set of whole numbers.
4. Enclose the result in parentheses: (hkl).

Worked example: A plane intercepts the axes at $x = 2a$, $y = 3a$, $z = \infty$.

• Fractional intercepts: 2, 3, $\infty$
• Reciprocals: $\frac{1}{2}$, $\frac{1}{3}$, 0
• Multiply by 6 (the LCD): 3, 2, 0
• Miller indices: (320)

Negative and zero values

• A zero index means the plane is parallel to that axis (never intercepts it).
• A negative intercept produces a negative index, written with a bar over the number: $(\ \bar{1}00)$ means the plane intercepts the negative x-axis.

Significance in crystallography

Relationship to crystal structure

Different planes slice through the lattice at different angles, so they expose different arrangements of atoms. A (111) plane in an FCC crystal, for instance, is the most densely packed plane, while (100) is less dense. These differences in atomic density directly affect surface energy, chemical reactivity, and how easily a crystal breaks along that plane.

Identification of crystal faces

The external faces you see on a well-formed crystal correspond to specific Miller indices. Identifying these faces tells you about the crystal's symmetry and growth history, which is useful for quality control in semiconductor wafer production.

Families of crystal planes

Planes with similar indices

In a crystal with high symmetry (like cubic), several planes are physically equivalent even though their indices look different. For example, (100), (010), and (001) in a cubic crystal are all identical by symmetry.

Bracket notation for families

• Curly brackets {} denote a family of equivalent planes.
• {100} includes all six permutations: (100), (010), (001), $(\bar{1}00)$, $(0\bar{1}0)$, $(00\bar{1})$
• Every plane in a family shares the same interplanar spacing, atomic density, and physical properties.

Interplanar spacing

Spacing between parallel planes

The interplanar spacing $d_{hkl}$ is the perpendicular distance between adjacent parallel planes with the same Miller indices. This spacing determines where diffraction peaks appear in X-ray diffraction (via Bragg's law: $n\lambda = 2d\sin\theta$), making it experimentally measurable.

Calculation using Miller indices

For cubic crystals, the formula is straightforward:

$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

• $a$ = lattice parameter (the side length of the cubic unit cell)
• $h, k, l$ = Miller indices

For silicon ($a = 5.43 \text{ Å}$), the (111) spacing is:

$d_{111} = \frac{5.43}{\sqrt{1+1+1}} = \frac{5.43}{1.732} \approx 3.14 \text{ Å}$

Higher-index planes always have smaller $d$-spacing, since the denominator grows.

Direction of crystal planes

Normal direction to plane

Crystal directions use square brackets: [hkl]. In a cubic crystal, the direction [hkl] is perpendicular to the plane (hkl). This convenient relationship is specific to cubic systems and does not hold in general for lower-symmetry crystals.

Relationship to crystal axes

The direction [hkl] represents a vector from the origin to the point at coordinates $(h, k, l)$ in the lattice. A family of equivalent directions is written with angle brackets: $\langle hkl \rangle$. For example, $\langle 100 \rangle$ includes [100], [010], [001], and their negatives.

Planes in cubic crystals

Low-index planes

The three most important planes in cubic semiconductors are:

• (100): a flat face of the cube. Most common substrate orientation for silicon MOSFET fabrication.
• (110): cuts diagonally across two axes. Exposes a rectangular arrangement of atoms.
• (111): the most densely packed plane in FCC structures. Preferred cleavage plane in silicon and many III-V semiconductors.

These low-index planes have high atomic density and low surface energy, which is why crystals naturally tend to form and cleave along them.

High-index planes

Planes like (311) or (511) have lower atomic density and higher surface energy. At the atomic level, they often look like a staircase of low-index terraces separated by steps or kinks. They're less common in device fabrication but sometimes chosen for specialized applications where a particular surface reconstruction is needed.

Planes in hexagonal crystals

Miller-Bravais indices

Hexagonal crystals have three-fold symmetry in the basal plane, but standard three-index Miller notation doesn't make that symmetry obvious. The Miller-Bravais system uses four indices (hkil) with three in-plane axes ($a_1, a_2, a_3$) at 120° to each other, plus the c-axis.

The third index is redundant but useful:

$i = -(h + k)$

This constraint ensures that symmetrically equivalent planes have indices that look equivalent, which doesn't happen with three-index notation.

Conversion from Miller indices

To convert three-index (hkl) to four-index Miller-Bravais notation:

1. Keep h and k the same.
2. Calculate $i = -(h+k)$.
3. Keep l the same.

Example: (101) becomes $(10\bar{1}1)$ since $i = -(1+0) = -1$.

Applications in semiconductors

Cleavage planes vs growth planes

• Cleavage: Silicon cleaves most easily along {111} planes because they have the widest spacing between adjacent planes in the diamond cubic structure, meaning fewer bonds per unit area need to be broken.
• Growth: Silicon wafers for CMOS are typically cut along (100) because this orientation produces the lowest density of dangling bonds at the $\text{Si/SiO}_2$ interface, reducing interface trap density.

The choice of crystal plane directly affects surface roughness, defect density, and the quality of epitaxial layers grown on top.

Impact on electrical properties

Carrier mobility depends on crystal orientation because the effective mass of electrons and holes varies with direction in the band structure. For example, electron mobility in silicon is highest along $\langle 100 \rangle$ directions, which is one reason (100) wafers dominate MOSFET manufacturing. Hole mobility, on the other hand, can be higher on (110) surfaces, which is exploited in some advanced device architectures.

Selecting the right substrate orientation is one of the first design decisions in semiconductor device engineering, and Miller indices are the language used to make that choice precise.