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Chapter 2: Semiconductor Fundamentals

2.2. Crystals and crystal structures

2.2.1. Bravais lattices

2.2.2. Miller indices, crystal planes and directions

2.2.3. Common semiconductor crystal structures2.2.4. Growth of semiconductor crystals

Solid materials are classified by the way the atoms are arranged within the solid. Materials in which a

are placed at random are called amorphous. Materials in which atoms are placed in a high ordered strare called crystalline. Poly-crystalline materials are materials with a high degree of short-range order

long-range order. These materials consist of small crystalline regions with random orientation called

separated by grain boundaries.

Of primary interest in this text are crystalline semiconductors in which atoms are placed in a highly o

structure. Crystals are categorized by their crystal structure and the underlying lattice. While some cryhave a single atom placed at each lattice point, most crystals have a combination of atoms associated

each lattice point. This combination of atoms is also called the basis.

The classification of lattices, the common semiconductor crystal structures and the growth of single-c

semiconductors are discussed in the following sections.

2.2.1 Bravais lattices

The Bravais lattices are the distinct lattice types, which when repeated can fill the whole space. The la

can therefore be generated by three unit vectors, and a set of integers k , l and m so that

lattice point, identified by a vector , can be obtained from:

(2.The construction of the lattice points based on a set of unit vectors is illustrated by Figure 2.2.1.

Figure 2.2.1: The construction of lattice points using unit vectors

In two dimensions, there are five distinct Bravais lattices, while in three dimensions there are fourteen

lattices in two dimensions are the square lattice, the rectangular lattice, the centered rectangular lattic

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Table 2.2.2.: Bravais lattices of three-dimensional crystals

The cubic lattices are an important subset of these fourteen Bravais lattices since a large number of 

semiconductors are cubic. The three cubic Bravais lattices are the simple cubic lattice, the body-centecubic lattice and the face-centered cubic lattice as shown in Figure 2.2.3. Since all unit vectors identif

the traditional unit cell have the same size, the crystal structure is completely defined by a single num

This number is the lattice constant, a.

Figure 2.2.3.: The simple cubic (a), the body-centered cubic (b) and the face centered cubic (c) latti

2.2.2 Miller indices, crystal planes and directions

Crystal planes of a crystal are characterized by their Miller indices. The Miller indices are defined as

smallest possible integers, which have the same ratios as the inverse of the intersections of a given plwith a set of axis defined by the unit vectors of that crystal. This definition is further illustrated with F

2.2.4.The intersections between the plane and the axis occur at p, q, and r. The corresponding Miller

are therefore , where A is an integer chosen such that the Miller indices are the smallest possible

integers. It should be clear to the reader that the resulting Miller indices are the same for all parallel p

of atoms in a crystal.

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Figure 2.2.4.: Intersections of a plane and the  x,  y and  z axis, as used to determine the Miller indices

 plane.

Example 2.0 The intercepts of a crystal plane with the axis defined by a set of unit vectors are at 2aand 4c. Find the corresponding Miller indices of this and all other crystal planes paral

this plane.Solution The Miller indices are obtained in the following three steps:

1. Identify the intersections with the axis, namely 2, -3 and 42. Calculate the inverse of each of those intercepts, resulting in 1/2, -1/3 and 1/4.

3. Find the smallest integers proportional to the inverse of the intercepts. Multipl

each fraction with the product of each of the intercepts (24 = 2 x 3 x 4) does re

integers, but not always the smallest integers. These are obtained in this case bmultiplying each fraction by 12.

The Miller indices of this plane and all parallel planes are therefore . Note that thnegative indices are indicated with a bar above the integer for a more compact notatio

2.2.3 Common semiconductor crystal structures

The most common crystal structure among frequently used semiconductors is the diamond lattice, shoFigure 2.2.5. Each atom in the diamond lattice has a covalent bond with four adjacent atoms, which t

form a tetrahedron. This lattice can also be formed from two face-centered-cubic lattices, which are

displaced along the body diagonal of the larger cube in Figure 2.2.5 by one quarter of that body diago

The diamond lattice therefore is a face-centered-cubic lattice with a basis containing two identical ato

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cell so that the number of atoms equals one per unit cell. The packing density is then o

from:

or about half the volume of the unit cell is occupied by the atoms.

The packing density of four cubic crystals is listed in the table below.

2.2.4 Growth of semiconductor crystals

Like all crystals, semiconductor crystals can be obtained by cooling the molten semiconductor materiHowever, this procedure yields poly-crystalline material since crystals start growing in different locat

with a different orientation. Instead when growing single-crystalline silicon one starts with a seed crydips one end into the melt. By controlling the temperature difference between the seed crystal and the

molten silicon, the seed crystal slowly grows. The result is a large single-crystal silicon boule. Such b

have a cylindrical shape, in part because the seed crystal is rotated during growth and in part because cylindrical shape of the crucible containing the melt. The boule is then cut into wafers with a diamon

and further polished to yield the starting material for silicon device fabrication.