Lecture1_PMMD_effectivemass.pptx

BITS Pilani
Pilani Campus
BITS Pilani
presentation
R.K.Tiwary
EEE
Rk.Tiwari@pilani.bits-pilani.ac.in

BITS Pilani
Pilani Campus
MELZG631, PMMD
Lecture No.1(Fundamental of semiconductors; Crystal Structure)

BITS Pilani, Deemed to be University under Section 3 of UGC Act, 1956
PHYSICS OF SEMICONDUCTOR MATERIALS
 Semiconductors are a group of materials having electrical conductivities intermediate
between metals and insulators.
 Conductivity of these materials can be varied over orders of magnitude by changes in
 temperature,
 optical excitation,
 and impurity content.
 This variability of electrical properties makes the semiconductor materials natural
choices for electronic device investigations.
 Semiconductor materials :
 column IV: Si, Ge
 compounds of column III and column V : AlP, AlAs ,GaAs, InAs …….
 Binary II-VI compounds :ZnS , ZnSe , CdS
 IV compounds: SiC , SiGe

A portion of the periodic table
A list of some semiconductor materials

 A wide variety of electronic and optical properties of SCs provides great flexibility in
the design of electronic and optoelectronic functions
 Early stage Ge now Si : (Why so ?)
 Two-element (binary) III-V compounds such as GaN, GaP, and GaAs (LEDs)
 Three-element (ternary) ) and four-element (quaternary) compounds GaAsP &
InGaAsP provide added flexibility in choosing materials properties.
 II-VI compound used as Fluorescent materials in TV screens
 InSb, CdSe, : . Light detectors
 Gunn diode: made of GaAs or InP
 Semiconductor lasers: GaAs, AlGaAs,
 One of the most important characteristics of a semiconductor : Energy band gap
 λ(µm) is λ = 1.24/E. For GaAs, λ = 1.24/1.43 = 0.87 µm (infrared wavelengths)
 For GaP, λ = 1.24/2.3 = 0.539 µm (wavelengths in the green portion of the
spectrum.)

TYPES OF SOLIDS
 A crystalline solid is distinguished by the fact that the atoms making
up the crystal are arranged in a periodic fashion.
 classified according to atomic arrangement
 characterized by the size of an ordered region within the material
1. Amorphous: have order only within a few atomic or molecular
dimensions
2. Polycrystalline: have a high degree of order over many atomic or
molecular dimensions
3. and single crystals: ideally, have a high degree of order, or regular
geometric periodicity, throughout the entire volume of the material

solids
(a) amorphous,
(b) polycrystalline,
(c) single crystal
(crystalline).
1. An amorphous -Si thin-film transistor used as the switching
element in liquid crystal displays (LCDs):
2. polycrystalline Si gates are employed in MOSFETs
3. the active region of he device is crystalline

Cross section of a MOSFET.
The inset shows a magnified view of the three regions, in which individual rows of
atoms in the crystalline silicon can be distinguished. (Photograph courtesy of AT&T
Bell Laboratories.)
 shows the silicon channel
and metal gate separated
by a thin (40 A0, 4 nm)
silicon-dioxide insulator.
 Illustrates the periodic
array of atoms in the
single-crystal silicon of a
transistor channel
compared with the
amorphous Si02 (glass) of
the oxide layer

Primitive and Unit Cell
 The periodicity in a crystal is defined in terms of a symmetric array of
points in space called the lattice
 Atoms are added at each lattice point in an arrangement called a basis
 One atom or a group of atoms added having the same spatial
arrangement, form a crystal
 The lattice contains a volume or cell that represents the entire lattice and
is regularly repeated throughout the crystal
A rhombic lattice, with a primitive
cell ODEF, which is the smallest
such cell

 we can define vectors a and b such that if the primitive cell is translated by
integral multiples of these vectors, a new primitive cell identical to the
original is found (e.g., O'D'E'F')
 These vectors, a and b (and c if the lattice is three dimensional), are called
the primitive vectors for the lattice.
 Points within the lattice are indistinguishable if the vector between the
points is r = pa + qb + sc where p, q, and s are integers
A rhombic lattice, with a primitive
cell ODEF, which is the smallest
such cell

A rectangular (PQRS) with a lattice
point in the center at T (a so-called
centered rectangular lattice)
 A primitive cell has lattice points only at the corners of the cell.
 It is not unique, but the convention is to choose the smallest primitive
vectors.
 In a primitive cell, the lattice points at the corners are shared with adjacent
cells; thus, the effective number of lattice points belonging to the primitive
cell is always unity
 The distances and orientation between atoms can take many forms, it is
the symmetry that determines the lattice, not the magnitudes of the
distances between the lattice points.

Primitive and Unit Cell (Periodic Structures )
 Unit cell: Simpler to deal with a rectangle rather than a rhombus.
 Choose to work with a larger rectangular unit cell, PQRS
 A unit cell allows lattice points not only at the corners, but also at the face
center (and body center in 3-D) if necessary.
 Sometimes used instead of the primitive cell if it can represent the
symmetry of the lattice better
 Replicates the lattice by integer translations of basis vectors
 The importance of the unit cell lies in the fact that we can analyze the
crystal as a whole by investigating a representative volume

Primitive and Unit Cell (application)
 From the unit cell we can find the distances between nearest atoms and
next nearest atoms for calculation of the forces holding the lattice together;
 we can look at the fraction of the unit cell volume filled by atoms and relate
the density of the solid to the atomic arrangement.
 But even more important for our interest in electronic devices, the properties
of the periodic crystal lattice determine the allowed energies of electrons
that participate in the conduction process.
 Thus the lattice determines not only the mechanical properties of the crystal
but also its electrical properties.

Cubic Lattices
 The simplest three-dimensional lattice is one in which the unit cell is a
cubic volume, such as the three cells shown in Fig.
 Simple cubic structure (abbreviated sc) has an atom located at each
corner of the unit cell
 The body centered cubic (bcc) lattice has an additional atom at the
center of the cube
 and the face-centered cubic (fcc) unit cell has atoms at the eight
corners and centered on the six faces
(a) simple cubic,
(b) body-centered cubic,
(c) face-centered cubic.

Body-Centered Cubic Lattice Constants
Using the hard sphere model, which imagines each atom as a discrete sphere, the
BCC crystal has each atom touch along the body diagonal of the cube.

Body-Centered Cubic Atomic Packing Factor
Therefore, if the atoms in a bcc lattice are packed as densely as possible, with no
distance between the outer edges of nearest neighbors, 68% of the volume is filled.

Two-dimensional representation of a
single-crystal lattice
Two-dimensional representation of a
single-crystal lattice showing various
possible unit cells.
 A unit cell is a small volume of the crystal that can be used to reproduce the entire
crystal
 A unit cell is not a unique entity

Crystal Planes and Miller Indices
Describe the plane shown in Figure. (The lattice points in Figure are shown along
the 𝑎 , 𝑏 , and 𝑐 axes only.)
the intercepts of the plane correspond to p =3,
q =2, and s = 1. Now write the reciprocals of the
intercepts, which gives ( 1/3 , 1/2 ,1/1 )
Multiply by the lowest common denominator, which
in this case is 6, to obtain (2, 3, 6).
Fig is then referred to as the (236) plane. The integers are referred to
as the Miller indices. We will refer to a general plane as the ( hkl )
plane.

Crystal Planes and Miller Indices
Three lattice planes: (a) (100) plane, (b) (110) plane, (c) (111) plane
Again, any plane parallel to the one shown in Fig(a) and separated by an integral
number of lattice constants is equivalent and is referred to as the (100) plane
The plane in Fig(a) is parallel to the b and c axes so the intercepts are given as p
=1, q =∞, and s= ∞ . Taking the reciprocal, we obtain the Miller indices as (1, 0, 0)

Bond Model
Fails to account for important
quantum mechanical constraints
on the behavior of electrons in
crystal
The diamond-crystal lattice characterized by four
covalently bonded atoms. The lattice constant,
denoted by ao, is 0.356, 0.543 and 0.565 nm for
diamond, Si, and Ge, respectively. Nearest
neighbors are spaced (√3ao/4) units apart. Of
the 18 atoms shown in the figure, only 8 belong
to the volume a0
3. Because the 8 corner atoms
are each shared by 8 cubes, they contribute a
total of 1 atom; the 6 face atoms are each
shared by 2 cubes and thus contribute 3 atoms,
and there are 4 atoms inside the cube. The
atomic density is therefore 8/a0
3, which
corresponds to 17.7, 5.00, and 4.43 x 1022 cm-3,
respectively.

ATOMIC BONDING
Representation of (a) hydrogen
valence electrons and (b)
covalent bonding in a
hydrogen molecule.
Ionic bond: NaCl (I & VII) Z= 11 & 17 respectively
Van der Waals bond: HF formed by Ionic bond (I & VII) Z= 1& 9 respectively
covalent bonding in the silicon crystal.

IMPERFECTIONS AND IMPURITIES IN SOLIDS
Imperfections in Solids
 Lattice vibrations: Random thermal motion due to thermal energy causes lattice
to vibrate resulting in the distance between atoms to randomly fluctuate
 Point defect : These defects mainly happen due to deviation in the arrangement
of constituting particles. In a crystalline solid, when the ideal arrangement of
solids is distorted around a point/ atom it is called a point defect.
 An atom may be missing from a particular lattice site. This defect is referred
to as a vacancy
 In another situation, an atom may be located between lattice sites. This
defect is referred to as an interstitial

 Frenkel defect: When vacancy and interstitial are in close enough proximity to exhibit a
interaction between the two point defects. produces different effects than the simple
vacancy or interstitial
 . A line defect, for example, occurs when an entire row of atoms is missing from
its normal lattice site and referred to as a line dislocation
a line dislocation disrupts both the normal geometric periodicity of the lattice and
the ideal atomic bonds in the crystal

Impurities in Solids
 In the process of Ion implantation the incident impurity atoms collide with the
crystal atoms, causing lattice-displacement damage
 Sol: Annealing
Breaking of an atom.to-atom bond and
freeing of an electron

The Uncertainty Principle
 Heisenberg uncertainty principle: cannot describe with absolute accuracy
the behavior of subatomic particles.
 impossible to simultaneously describe with absolute accuracy the
position and momentum of a particle
 impossible to simultaneously describe with absolute accuracy the
energy of a particle and the instant of time the particle has this energy.
∆𝐸∆𝑡 ≥ ℎ
∆𝑝∆𝑥 ≥ ℎ
 Implication: we cannot, for example, determine the exact position of an
electron. We will, instead, determine the probability of finding an electron at a
particular position

Materials (Conductivity & Resistivity)
𝜌 = 𝑅
𝐴
𝐿
Aluminum and copper, 10-6 Ω-cm
The resistivity of silicon dioxide is about 1016 Ω-cm -22 orders of magnitude higher
The resistivity of the plastics often used to encapsulate integrated circuits can be as
high as 1018 Ω-cm
Materials with 𝜌 (resistivity) less than 10-2 Ω-cm are conductors and those with
greater than 105 Ω-cm are insulators
Semiconductor lies between them (intermediate region) : Resistivity can be varied
and precisely controlled
Can be made to conduct by one of two types of carriers
Two models: Energy band model & Crystal- bonding model

Energy Band model of solids

Isolated silicon atom electronic structure

Energy levels in Si
The core levels (n = 1,2) in Si are
completely filled with electrons. At
the actual atomic spacing of the
crystal, the 2N electrons in the 3s
subshell and the 2N electrons in
the 3p subshell undergo sp3
hybridization, and all end up in the
lower 4N stales (valence band),
while the higher-lying 4N states
(conduction band) are empty,
separated by a band gap.

Formation of Energy bands
Diamond lattice crystal
is formed by bringing
isolated si atoms
together
Ec is the lowest possible
conduction hand energy,
Ev is the highest possibte
valence hand energy, and
EG = Ec — Ev iv the band
gap energy.

Change in energy
Calculate the change in kinetic energy of an electron when the velocity
changes by a small amount

Effective Mass concept
The movement of an electron in a lattice will, in general, be different from
that of an electron in free space
An electron moving in response to an applied electric field (a) within a vaccum, and
(b) with in a semiconductor crystal.

Effective Mass concept

Direct and Indirect Semiconductors
Energy-band structures of (a) GaAs and (b) Si
Laser

Energy momentum diagram

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