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Chapter 1. Electrons and Holes in Semiconductors Modern 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 Lee Course Outlines:

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Page 1: Modern Semiconductor Devices for Integrated Circuits ...contents.kocw.or.kr/document/region/2010/03/01/03_01_01_phy.pdf · Modern Semiconductor Devices for Integrated Circuits Chapter

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:

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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

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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.)

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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

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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

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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

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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)

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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.

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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

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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.

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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)

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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

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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.

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Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits

Periodic Table

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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)

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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

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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

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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)

.

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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.

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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

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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+

+ +

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

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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

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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

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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

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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

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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

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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