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Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
ELEC311(물리전자, Physical Electronics)
Lecture notes are prepared with PPT and available before the class
(http://abeek.knu.ac.kr). The topics in the notes are from Chapter 1, 2, 4,
and 5 in the main text introduced below, but the course covers the
materials not only in the text, but also in various references. The
remaining topics in Chapter 3, 6, 7, and 8 will be discussed in Electronic
Devices in next semester.
Main Text: Modern Semiconductor Devices for Integrated Circuits
by Chenming Calvin Hu ( 2010, UC Berkeley)
References: 1) Solid State Electronic Devices
by Ben G. Streetman and Sanjay Kumar Banerjee
(2006, U of Texas ar Austin)
2) Semiconductor Device Fundamentals
by Robert F. Pierret(1996, Purdue University)
3) An Introduction to Semiconductor Devices
by Donald Neamen (2006, U of New Mexico)
4) Principles of Semiconductor Devices
by Sima Dimitrijev (2006, Griffith University)
Grading: Three Exams (30 % each), Homework (10 %)
by Professor Jung-Hee LeeCourse Outlines:
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Electrons and Holes in
Semiconductors
Chapter 1
OBJECTIVES
1. Provides the basic concepts and terminology for
understanding semiconductors.
2. Understand conduction and valence energy band, and
how bandgap is formed
3. Understand carriers (electrons and holes), and doping in
semiconductor
4. Use the density of states and Fermi-Dirac statistics to
calculate the carrier concentration
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Transistor inventors John Bardeen, William Shockley, and Walter Brattain (left to right) at Bell
Telephone Laboratories. (Courtesy of Corbis/Bettmann.)
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Crystal Lattice
Simple cubic lattice and its unit cell
(a is the lattice constant)
It is possible to analyze the crystal as a whole by investigating a representative volume (e.g. unit cell).
A two-dimensional lattice showing translation of
a unit cell by r = 3a +2b.
periodic atomic arrangement in the crystal, or
symmetric array of points in space
Unit cell
: a small portion of any given crystal that
can be used to reproduce the crystal
Primitive cell
: the smallest unit cell possible
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Simple 3-D Unit Cell
Simple cubic:
• only 1/8 of each corner atom is inside the cell
contains 1 atom in total
Body centered cubic (bcc):
• an atom at the center of the cube
in addition to the atoms at each
corner
contains 2 atoms
Face centered cubic (fcc):
• contains an atom at each face of the
cube in addition to the atoms at each
corner
• ½ of each face atom lies inside the
fcc
contains 4 atoms
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Homework #1:
1. What is the maximum fraction of the FCC lattice volume that can be
filled with atoms by approximating the atoms as hard spheres?
2. Do the same calculation for the simple cubic and body-centered
cubic.
Packing of Hard Spheres in an FCC Lattice
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Miller indices: A set of integers with no common integral divisors that are
inversely proportional to the intercepts of the crystal planes along the crystal axes.
These indices are enclosed in parenthesis (hkl).
Miller Indexing procedure:
1. Determine the intercepts of the face along the crystallographic axes,
in terms of unit cell dimensions. 1, 2, 3
2. Take the reciprocals 1, 1/2, 1/3
3. Clear fractions using an appropriate multiplier 6, 3, 2
4. Reduce to lowest terms (already there) and enclose the whole-number set in
parenthesis (632)
Special facts:
• The plane that is parallel to a coordinate axis is taken to be infinity. Thus, intercepts
at , , 1 , for example, result in (001) plane.
• For a negative axis, a minus sign is placed over the corresponding index number so
that an intercept at 1, -1, 2 is designated a plane.
• A group of equivalent planes is referenced through the use of { }.
Crystallographic Planes and Directions
Example. A (214) crystal plane
(221)
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
[Miller indices for the three most important planes in cubic crystals]
Because there is no crystallographic difference between the (100), (010), (001)
planes,
they are uniquely labeled as {100}.
{110} plane intersects two axes at a and is parallel to the 3rd axis.
{111} plane intersects all the axes at a.
Crystallographic planes
Equivalence of the cube faces ({100} planes) by rotation of the unit cell within the cubic
lattice.
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
[Important directions in cubic crystals]
• The direction perpendicular to (hkl) plane is labeled as [hkl].
• A set of equivalent directions is labeled as <hkl>;
e.g. <100> represents [100], [010], [001], [100] and so on.
Convention Interpretation
(hkl) Crystal plane
{hkl} Equivalent planes
[hkl] Crystal direction
<hkl> Equivalent directions
Miller Convention Summary
Crystallographic Directions
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
(c) Scanning tunneling microscope
view of the individual atoms of
silicon (111) plane.
(a) A system for describing the crystal planes.
Each cube represents the unit cell
(b) Silicon wafers are usually cut along the
(100) plane. This sample has a (011) flat to
identify wafer orientation during device
fabrication.
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Diamond lattice unit cell (Si, Ge, C):
• Two interpenetrating FCC lattices
(the 2nd FCC lattice displaced ¼ of a body diagonal along a
body diagonal direction relative to the 1st FCC lattice)
• 8 Si atoms in unit cell (volume=a3, a=5.43Å ) ~5×1022 atoms/cm3
Zincblende lattice unit cell (GaAs, InP…):
• Identical to diamond lattice unit cell, but 2 FCCs are different atoms.
i.e. Ga locates on one of the two interpenetrating FCC sub-lattice
and As populates the other FCC sub-lattice.
Atoms in the diamond and zincblende lattices have 4 nearest neighbors.
Semiconductor Lattice (The diamond lattice)
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Diamond lattice structure: (a) a unit cell of the diamond lattice constructed by placing
atoms ¼ , ¼ , ¼ from each atom in an fcc; (b) top view (along any <100> direction) of an
extended diamond lattice.The colored circles indicate one fcc sublattice and the black circles
indicate the interpenetrating fcc.
Homework #2:
What is the maximum fraction of the
diamond lattice volume that can be filled
with atoms by approximating the atoms
as hard spheres?
Find the number density (atoms/cm3) and
density (g/cm3) of the Si lattice.
3
4a
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
- Every solid has its own characteristic energy band structure.
- The band structure is responsible for electrical characteristics.
Semiconductors, Insulators, and conductors
Ev
Ec
Ev
Ec
The valence band of Si is completely filled with
electrons at 0 K and the conduction band is empty
good insulator at 0 K.
What will happen if temperature increases?
1018 1016 1014 1012 1010 108 106 104 102 1 10 -2 10 -4 10 -6 10 -8
Resistivity [Wm]
Insulator
Semiconductor
Conductor
[Conductivity]
Insulator < Semiconductor < Metal
(Si)(SiO2) (Conductor)
1.1 eV
Ec
9 eV
Elemental solids with odd atomic numbers
(and therefore odd numbers of electrons)
such as Au, Al, and Ag
Elemental solids with even atomic
numbers
(and therefore even numbers of electrons)
such as Zn and Pb
known as “semimetal”
Difference between semiconductor and
Insulator
- Eg,insulator >> Eg,semiconductor
- Semiconductor can be N or P-type with low
resistivity through impurity doping.
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Periodic Table
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Semiconductor Materials
Semiconductor - elemental semiconductor : group Ⅳ
- compound semiconductor : group Ⅲ &Ⅴ, group II & VI etc.
- Binary = two elements
- Ternary = three elements
- Quaternary = four elements (InGaAsP)
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
1) Ionic Bonding in NaCl • Na is surrounded by 6 nearest neighbor Cl atoms.
Na (Z=11): [Ne]3s1
Cl (Z=17): [Ne]3s23p5
• Each Na atom gives up its outer 3s electron to a Cl atom
Crystal is made up of ions with the electronic structures of
the inert atoms, Ne and Ar (Ar (Z18): [Ne]3s23p6)
Ionic bonding
• These Coloumbic forces pull the lattice together until a
balance is reached with repulsive forces.
Bonding Forces in Solids
attraction
Features of ionic solid • Tightly bonded electrons good insulators
• The energy levels in outer orbits are either totally filled or totally empty
• Very stable
Bond Model of Electrons and Holes
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
2) Metallic Bonding
• In a metal atom the outer electronic
shell is only partially filled.
(Na+ has only one electron in the outer shell.)
• These electrons are loosely bound and
are given up easily in ion formation.
The solid is made up of ions with
closed shells immersed in a sea of
free electrons.
Significant number of free electrons
excellent thermal/electrical conductor
Sea of electrons
Na+
Coulombic forces
between Na+ and electron sea
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Covalent bonding in Ge, Si, or C diamond
lattice The bonding forces arise from a quantum
mechanical interaction between the shared
electrons.
Electrons are essentially attached to their
own nuclei but they are being shared by
two nuclei at the same time.
3) Covalent Bonding in Si
Covalent bonding is stable;
• Either insulators or semiconductors
• Sharing the outermost electrons lower excitation energy absorption infrared (IR)
range
• Sensitive to the temperature change (an idealized lattice at 0 K)
The silicon crystal structure in a two-
dimensional representation at 0 K.( no
free electron to conduct electric current at
0 K)
.
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
At elevated temperature, a covalent electron breaks
loose, becomes mobile and can conduct electric
current (conduction electron).
It also creates a void or a hole represented by the
open circle. The hole also move about as indicated
by the arrow and thus conduct electric current.
Doping of a semiconductor is illustrated with the
bond model. (a) As (V) is a donor. (b) B (III) is an
acceptor.
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Shows quite weak boding force
ex. Pairing of inert gas ions, such as He-He
molecules
Solid Bond Type Bond Energy [kJ/mol] Bond Length [nm]
NaCl Ionic 748 0.282
Al Metallic 326 0.152
C-C Covalent 370 0.154
FeO Covalent 509 0.216
Ar van der Waals 1 0.382
4) van der Waals Solid
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
The potential energy has been lowered
because an electron here would be attracted
by two nuclei, rather than just one.
Energy Band Model
Isolated atoms
Ψ2Ψ1 wave
function
1s2s
2p3s
3p
Ψ1 Ψ2
antibonding energy level
bonding energy level
odd or antisymmetry
combination
even or symmetry
combination
)LCAO(21
antibonding
orbital+
ㅡ
+ +
bonding
orbital
…..
…..No interaction between electron wave function
Diatoms: two atoms close to each other
LCAO: linear combinations of
the individual atomic orbitals
Energy level splitting
due to exclusion principle( No two electrons in a given interacting
system may have the same quantum state)
Quantum numbers:
n = 1, 2, 3, 4,……
l = 0, 1, 2, 3,…….n-1
s, p, d, f, g,……
m = -l, -(l-1),….0, 1,….l
s = +1/2, -1/2
Atomic configuration of Si
atom: 14electrons
1s2, 2s2, 2p6, 3s2, 3p2
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
3pE3
3p
E1
E2
3s Energy level splitting
due to Pauli exclusion
principle
3s
There must be at most one electron per level after there is a splitting of discrete energy levels of the
isolated atoms into new levels belonging to the pair rather than no individual atoms
If, instead of 2 atoms, one brings together N atoms, there will be N distinct LCAO and
N closely-spaced energy levels in a band. In solids, where N is very large,
so that the split energy levels form essentially continuous band of energies.
Eg
1s
2s+2p
3p
+
3s
E1
E2
E3conduction band
valence band
core band
N-Atoms
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
If N atoms are brought into close proximity,
there is no significant change in the core, but the energy state of the valence electrons
changes
Ec: lowest conduction band
energy
Ev: highest valence band
energy
Eg = Ec - Ev : Band gap energy
Energy band for Si
Ec
Ev
core
completely filled
sp3 hybridization
1s
2s
2p
3s
3p
Lattice constant of Si atom at equilibrium
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
E-k Diagram for a Free Electron
Plot the E-k dependence
The wave function of free electron (V = 0)
satisfies the Schrödinger wave equation,
22
2
2 from
2
mE k E k
m
E–k diagram for a free electron
Relate the E-k dependence to the classical kinetic energy, Ekin = mv2/2
Because free electron has no potential energy E = Ekin
2 2 22
2 2 2
p mvE k
m m
• The E-k dependence of a free electron is identical to the classical dependence of
kinetic energy on velocity.
The wave function of plane wave satisfies the wave equation1 2( ) jkx jkxx c e c e
2:k
p = k
2
2 2
2( ) 0
d mE V
dx
when
wave number (or wave vector or propagation constant).
22
20,
dk
dx
The wave function of free electron is exactly same as the wave function of the plane
wave;1 2( ) jkx jkxx c e c e
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Energy Gap and Energy Bands in Semiconductors and
Insulators2
2
2E k
m
E–k diagram.
Energy band for free electron: Parabolic E-k
dependence.
Energy band for semiconductor and insulator: still similar to the free electron energy band, but two
slightly modified parabolic bands, conduction band and valence band, with energy gap, Eg.
Free electron approximation
E–x diagram.
Tight binding approximation
( ) ( , ) xj x
xx U x e k
k kmodulates the wave function according
to the periodicity of the lattice
Wave function of electrons in the crystal can be modified by the
periodic crystal potential, U(kx, x), as
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
At T ≈ 0 K,
there is no broken covalent bonds and the valence band is full and the conduction
band is empty.
some of covalent bonds are broken because a sufficient thermal energy is delivered to a valence electrons, these electrons jumps up into the conduction band, leaving empty states behind in the valence band, called holes.
The electrons at higher energy levels in EC will
have kinetic energies according to the upper E-
k branch
Kinetic energy of the holes as current
carriers
The semiconductors becomes insulator, because
there are no electrons in conduction band and the
electrons in the valence band are immobile – they
are tied in the covalent bonds.
At elevated temperature,
The semiconductors becomes conductive.
Insulator: large Eg
Semiconductor: small Eg
Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits
Allowed values of energy can be plotted vs. the propagation constant, k.
Since the periodicity of most lattices is different in various direction, the E-k diagram must be plotted for
the various crystal directions (complex).
Si, Ge, GaP, AlAs :
indirect band gap semiconductor
A transition must necessarily include and
interaction with the crystal so that crystal
momentum is conserved.
GaAs :
the minimum conduction band energy and
maximum valence band energy occur at the same
k-value.
direct band gap semiconductor
semiconductor lasers and other optical devices
-Direct bandgap: a minimum in the conduction band and a maximum in the
valence band for the same k value
-Indirect bandgap: a minimum in the conduction band and a maximum in the
valence band at a different k value
Direct and Indirect Semiconductor