Chapter 1: Crystal Structure of Solids
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Semiconductor Materials
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Semiconductor Materials
• Elemental semiconductors
– silicon (Si)
– germanium (Ge)
– diamond (C)
• Binary (two-element) compound semiconductors
– III-V
• one element from group III (e.g., aluminum (Al), gallium (Ga), indium (In))
• one element from group V (e.g., nitrogen (N), arsenic (As), phosphorous (P),
antimony (Sb))
• e.g., gallium arsenide (GaAs), indium phosphide (InP)
– II-VI
• e.g., zinc selenide (ZnSe), cadmium telluride (CdTe)
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Semiconductor Materials
• Ternary (three-element) compound semiconductors
– aluminum gallium arsenide (AlxGa1-xAs)
• x is fraction (0<x<1) of Al in group III elements (still equal numbers overall of
group III and group V elements)
– mercury cadmium telluride (Hg1-xCdxTe)
• Quaternary (four-element) compound semiconductors
– indium gallium aluminum arsenide (In1-x-yGaxAlyAs)
– indium gallium arsenide phosphide (In1-xGaxAs1-yPy)
• In general, it is not easy to predict the properties of these
various compound materials
– e.g., not all group IV materials are semiconductors (e.g., tin (Sn), lead
(Pb))
– e.g., some III-V materials are insulators (at room temperature) (e.g., AlN,
BN)
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Types of Solids
Amorphous
Poly-crystalline
no discernible
repetitive order in the
structure
formed from a collection
of crystals, though the
different crystals may
have different orientations
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Single-crystalline
formed from exact repetition
of identical structural units
(totally ordered)
Modern Devices are mostly made of
3 Solid Material Types
Polycrystalline Si
Amorphous SiO2
Crystalline Si
(Source: Intel Corporation)
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Crystal Structure: Unit Cells
• Unit cell
– portion of a crystal that can be used to reproduce the crystal
– identical “building blocks” that can completely construct the crystal when
used repetitively
– not necessarily unique. unit cells used in practice not always the
“primitive” or smallest unit cell possible
• Primitive cell
– smallest unit cell that can be repeated to form the lattice
Lattice point
Unit cells
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Crystal Structure: Basis Vector
• a – vector of length a parallel to a-side of unit cell, where
a is the “repeat distance”
• b – vector of length b parallel to the b-side
• The entire two-dimensional lattice can be constructed by
the translations of the unit cells, and this can be
represented by the combination of basis vectors
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Crystal Structure
• Every equivalent lattice point in the 3D crystal can be found
using
𝒓 = 𝒑𝒂 + 𝒒𝒃 + 𝒓𝒄
Unit cell: Three-dimensional generalization
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Crystal Structure: An example
𝒓
c
b
a
𝒓 = 𝒑𝒂 + 𝒒𝒃 + 𝒓𝒄
𝒂, 𝒃, 𝒄 : primitive vectors.
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p, q, r : integers
Basic Crystal Structures
Simple cubic unit cell
Body-centered cubic (bcc) Face-centered cubic (fcc)
atom at each corner of cube
add one atom to middle of
simple cube (now two
atoms per unit cell)
note only 1/8th of each atom
is in the unit cell, and a total
of 1 atom per unit cell
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add atom to middle of each
face of simple cube (now
four atoms per unit cell)
Basic Crystal Structures: Primitive Vectors
Simple Cubic
z
Body Centered Cubic
(BCC)
Face Centered Cubic
(FCC)
y
x
S
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Alternatively,
𝑎
𝑎⃗ = 1,1, −1
2
𝑎
𝑏 = 1, −1,1
2
𝑎
𝑐⃗ = −1,1,1
2
Unit Cell vs. Primitive Cell of BCC
0.5Ö3a
= 0.87a
primitive cell
a
Unit cell
109o28’
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Unit Cell vs. Primitive Cell of FCC
Unit cell
primitive cell
a1
a2
a3
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Dense Packing of Spheres (FCC)
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Diamond and Zinc-Blende Structure
• Most important lattices for semiconductor-based electronic
and optoelectronic devices
– diamond lattice – silicon, germanium, diamond
– zinc-blende lattice – GaAs, InP, most III-V
• 8 atoms per unit cell
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Diamond Structure
• We can construct the diamond structure by overlapping
two fcc lattices translated each other by the vector
𝒂
(𝟏, 𝟏, 𝟏).
𝟒
𝒂
(𝟏, 𝟏, 𝟏)
𝟒
Lattice 2
Lattice 1
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Diamond Structure: Covalent Bonding in Si
Diamond lattice structure
Si atomic density: 5x1022 cm-3
Tetrahedron bond
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Diamond Structure
• There are four tetrahedral structures inside the
diamond structure unit cell
Top
Half
Bottom
Half
(building block)
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Zinc-Blende Structure
• Interlocking face-centered cubic lattices
– second lattice exactly one quarter down space in zinc-blende, one atom
species (e.g., Ga) occupies one fcc lattice, second (e.g., As) occupies
other
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Bravais Lattices
• How many different kinds of
arrangements of points in space
are there that can constitute
distinct lattices (so-called “point
lattices”)?
– only 14!
– proved by Bravais in 1840’s
• All other systems of points that
can repetitively fill all (3-D) space
can be shown to be equivalent to
one of the 14 Bravais lattices
• http://en.wikipedia.org/wiki/Bravai
s_lattice
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Real Crystals are not Perfect!
Imperfections
Point Defects
vacancy
vacancy
S
interstitial
Line dislocation
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vacancy-interstitial
(Frenkel defect)
Real Crystals are not Perfect!
Impurities
• Intentional addition of impurity atoms is called ‘doping’.
• We will learn more about doping later in this course.
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Miller Indices
• Method of labeling the distinct planes and directions
within a crystalline lattice
• Procedure for planes
– 1. Find the intercepts of the plane on the 3 Cartesian coordinates in
terms of lattice spacing
– 2. Take the reciprocals of these numbers and reduce them to the
smallest 3 integers having the same ratio
– 3. Enclose the result in parentheses (hkl) as the Miller indices for a
single plane
1, 1/3, 1/2
à6, 2, 3
A (623)-crystal plane
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Miller Indices
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Miller Indices
• Planes parallel to a unit cell coordinate axis are viewed as
intercepting the axis at infinity, so have an associated Miller
index in that direction of zero, e.g., (100) plane
• Planes intersecting along the negative axis use a bar over
the index rather than a negative sign, e.g., 1 rather than -1
• Groups of equivalent planes
– (100), (010), (001), (100), (010), (001) are all equivalent because rotation
about the 3 fold axes on the cube diagonals maps the various faces into
one another, making the planes equivalent
• are notated in curly brackets
– {100} for the above set of equivalent planes
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Family of Equivalent Planes
(010)
(100)
….
{100} Family
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(001)
….
Family of Equivalent Planes
(110)
(101)
{110} Family
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(011)
Surface Density of Atoms in a Plane
• What is the surface density of atoms in the (110) plane?
2a
BCC
a
R
a
2a
FCC
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a
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Miller Indices for Directions
• Procedure
– Set up a vector of arbitrary length in the direction of interest (must be a
crystal direction, i.e., connecting two crystal points)
– Decompose the vector into its basis vector components in the a, b, and c
directions
– Convert the resulting numbers to the lowest possible set of integers by
multiplying by an appropriate number
– Directions are notated using square brackets, e.g., [111]
• For cubic crystals
– directions perpendicular to particular crystal planes can be indexed using
the same index as the plane
• Sets of equivalent directions are specified by triangular
brackets
– e.g., <100>
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Miller Indices for Directions
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Summary of Miller Indices
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Convention
Interpretation
(hkl)
Crystal plane
{hkl}
Family of equivalent planes
[hkl]
Crystal direction
<hkl>
Family of equivalent directions
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