Outline of this Topic

(1)

Introduction To Materials Science FORENGINEERS, Ch. 19

University of Tennessee, Dept. of Materials Science and Engineering 1

Electrical Properties

Goals of this topic:

• Understand how electrons move in materials: electrical

conduction

• How many moveable electrons are there in a material

(carrier density)

, how easily do they move

(mobility)

• Metals, semiconductors and insulators

• Electrons and holes

• Intrinsic and Extrinsic Carriers

• Semiconductor devices: p-n junctions and transistors

• Ionic conduction

• Electronic Properties of Ceramics: Dielectrics,

Ferroelectrics and Piezoelectrics

Outline of this Topic

• 1. Basic laws and electrical properties of metals

• 2. Band theory of solids: metals, semiconductors

and insulators

• 3. Electrical properties of semiconductors

• 4. Electrical properties of ceramics and polymers

• 5. Semiconductor devices

• Ohm’s Law

V = IR

E

= V / L

where E is

electric field intensity

µ

= / E where

µ

= the mobility

• Resistivity

ρ

= RA / L (

Ω

.m)

• Conductivity

σ

= 1 /

ρ

(

Ω

.m)

-1

ν

= the drift velocity

(2)

• Electrical conductivity between different materials

varies by over 27 orders of magnitude, the greatest

variation of any physical property

Metals:

σ

> 10

5 ₍

Ω

_.m)

-1

Semiconductors:

10 -6

_<

σ

_{< 10}

5 ₍

Ω

_.m)

-1

Insulators:

σ

< 10

-6

₍

Ω

_.m)

-1

Conductivity / Resistivity of Metals

• High number of free (valence) electrons

→

high

σ

• Defects scatter electrons, therefore they

increase

ρ

(lower

σ

).

ρ

total

=

ρ

thermal

+

ρ

impurity

+

ρ

deformation ρ

thermalfrom thermal vibrations

ρ

impurityfrom impurities

ρ

deformationfrom deformation-induced point defects

• Resistivity increases with temperature

(increased thermal vibrations and point

defect densities)

ρ

T

=

ρ

o

+ aT

• Additions of impurities that form solid

sol:

ρ

I =

Ac

i

(1-c

i

) (increases

ρ

)

• Two phases,

α

,

β

:

ρ

i =

ρ

α

V

α

+

ρ

β

V

β

Materials Choices for Metal Conductors

• Most widely used conductor is copper: inexpensive,

abundant, very high

σ

• Silver has highest

σ

of metals, but use restricted due to cost

• Aluminum main material for electronic circuits, transition

to electrodeposited Cu (main problem was chemical

etching, now done by “Chemical-Mechanical Polishing”)

• Remember deformation reduces conductivity, so high

strength generally means lower

σ

: trade-off. Precipitation

hardening may be best choice: e.g. Cu-Be.

• Heating elements require low

σ

(high R), and resistance to

high temperature oxidation: nichrome.

• Electric field causes electrons to accelerate in direction opposite

to field

• Velocity very quickly reaches average value, and then remains

constant

• Electron motion is not impeded by periodic crystal lattice

• Scattering occurs from defects, surfaces, and atomic thermal

vibrations

• These scattering events constitute a “frictional force” that

causes the velocity to maintain a constant mean value: v

d

, the

electron drift velocity

• The drift velocity is proportional to the electric field, the

constant of proportionality is the

mobility,

µ

.

This is a measure

of how easily the electron moves in response to an electric field.

• The conductivity depends on how many free electrons there

(3)

v

_d

=

µ

_e

E

σ

= n

|

e

| µ

_e

n : number of “free” or

conduction electrons per

unit volume

E

Scattering events

Net electron motion

(m) = Metal

(s) = Semicon

Mobility (RT)

µ

(m

2

_V

-1

_s

-1

₎

Carrier Density

N

e

(m

-3

)

Na (m)

0.0053

2.6 x

10

28

Ag (m)

0.0057

5.9 x

10

28

Al (m)

0.0013

1.8 x

10

29

Si (s)

0.15

1.5 x

10

GaAs (s)

0.85 1.8 x 10

6

InSb (s)

8.00 σ

metal

>>

σ

semi

Band Theory of Solids

• Schroedinger’s eqn (quantum mechanical equation for

behavior of an electron)

• Solve it for a periodic crystal potential, and you will find

that electrons have allowed ranges of energy (

energy

bands

) and forbidden ranges of energy (

band-gaps

).

δ

2

_ψ

δ

x

2

δ

ψ

δ

t

K

ψ

+ V

ψ

= E

ψ

(-h’

2

_{/2m) + V}

ψ

_{= ih’}

2. Band theory of solids: metals, semiconductors and

insulators

Electrons in an Isolated atom (Bohr Model)

Electron orbits defined by

requirement that they contain

integral number of wavelengths:

(4)

• When N atoms in a solid

are relatively far apart, they

do not interact, so electrons

in a given shell in different

atoms have same energy

• As atoms come closer

together, they interact,

perturbing electron energy

levels

• Electrons from each atom

then have slightly different

energies, producing a

“band” of allowed energies

University of Tennessee, Dept. of Materials Science and Engineering 14 Metals

Semiconductors Eg < 2 eV Insulators

Eg > 2 eV

Empty band

Empty conduction

band Empty

band

Band gap

Empty states

Filled states

Filled band

Filled valence band Empty conduction

band

Ef Ef

Ef

Band gap Band gap

Filled valence band

• Each band can contain certain number of electrons (xN, where N is the

number of the atoms and x is the number of electrons in a given atomic

shell, i.e. 2 for s, 6 for p etc.).

Note: it can get more complicated than this!

• Electrons in a filled band cannot conduct

• In metals, highest occupied band is partially filled or bands overlap

• Highest filled state at 0 Kelvin is the Fermi Energy, E

F

• Semiconductors, insulators: highest occupied band filled at 0 Kelvin:

electronic conduction requires thermal excitation across bandgap;

σ↑

T

↑

• (At 0 Kelvin) highest filled band:

valence band

; lowest empty band:

conduction band.

E

f

is in the bandgap

Metals, Semiconductors, Insulators

• At 0 Kelvin all available electron states below Fermi energy

are filled, all those above are vacant

• Only electrons with energies above the Fermi energy can

conduct:

–

Remember “Pauli Exclusion Principle” that only two electrons (spin

up, spin down) can occupy a given “state” defined by quantum

numbers n, l, m

l

–

So to conduct, electrons need empty states to scatter into, i.e. states

above the Fermi energy

• When an electron is promoted above the Fermi level (and can

thus conduct) it leaves behind a

hole

(empty electron state)

–

A hole can also move and thus conduct current: it acts as a “positive

electron)

(5)

The

Fermi

Function

f (E) = [1] / [e

(E - E

f

) / kT

+1]

This equation represents the probability that an energy level, E,

is occupied by an electron and can have values between 0 and 1

. At 0K, the f (E) is equal to 1 up to E

f

and equal to 0 above E

f

• In metals, electrons near the Fermi energy see empty states a very small

energy jump away, and can thus be promoted into conducting states above

E

f

very easily (temp or electric field)

• High conductivity

• Atomistically: weak metallic bonding of electrons

• In semiconductors, insulators, electrons have to jump across band gap into

conduction band to find conducting states above E

f

: requires jump >> kT

• No. of electrons in CB decreases with higher band gap, lower T

• Relatively low conductivity

• An electron in the conduction band leaves a hole in the valence band, that

can also conduct

• Atomistically: strong covalent or ionic bonding of electrons

Metals

(6)

Electrical conduction in

intrinsic

Si, (a) before

excitation, (b) and (c) after excitation, see the

response of the electron-hole pairs to the external

field. Note: holes generally have lower mobilities

than electrons in a given material (require

cooperative motion of electrons into previous

hole sites)

E field

Si Si Si Si

Si Si Si Si Si Si Si Si Si Si Si Si

Si Si Si Si Si Si Si Si

hole free electron

E field

Si Si Si Si

Si Si Si Si Si Si Si Si

hole free electron

(b)

(a)

Semiconductors

• Semiconductors are the key materials in the electronics and

telecommunications revolutions: transistors, integrated circuits,

lasers, solar cells….

• Intrinsic semiconductors

are pure (as few as 1 part in 10

10

impurities) with no intentional impurities. Relatively high

resistivities

• Extrinsic semiconductors

have their electronic properties (electron

and hole concentrations, hence conductivity) tailored by

intentional addition of impurity elements

Room

Temp

3. Electrical properties of semiconductors

Intrinsic Semiconductors: Conductivity

• Both electrons and holes conduct:

σ

= n|e|

µ

e

+ p|e|

µ

h

n: number of conduction electrons per unit volume

p: number of holes in VB per unit volume

• In intrinsic semiconductor, n = p:

σ

= n|e|(

µ

e

+

µ

h

) = p|e|(

µ

e

+

µ

h

)

• Number of carriers (n,p) controlled by thermal

excitation across band gap:

n = p = C exp (- E

g

/2 kT)

C : Material constant

E

g

: Magnitude of the bandgap

Extrinsic Semiconductors

(7)

n-type semiconductors

• In Si which is a tetravalent lattice, substitution of

pentavalent As (or P, Sb..) atoms produces extra electrons,

as fifth outer As atom is weakly bound (~ 0.01 eV). Each As

atom in the lattice produces one additional electron in the

conduction band.

• So N

As

As atoms per unit volume produce n additional

conduction electrons per unit volume

• Impurities which produce extra conduction electrons are

called

donors

, N

D

= N

As

~ n

• These additional electrons are in much greater numbers

than intrinsic hole or electron concentrations,

σ

~ n|e|

µ

e

~

N

D

|e|

µ

e

• Typical values of N

D

~ 10

16

- 10

19

cm

-3

(Many orders of

magnitude greater than intrinsic carrier concentrations at

RT)

p-type semiconductors

• Substitution of trivalent B (or Al, Ga...) atoms in Si

produces extra holes as only three outer electrons exist to

fill four bonds. Each B atom in the lattice produces one

hole in the valence band.

• So N

B

B atoms per unit volume produce p additional holes

per unit volume

• Impurities which produce extra holes are called acceptors,

N

A

= N

B

~ p

• These additional holes are in much greater numbers than

intrinsic hole or electron concentrations,

σ

~ p|e|

µ

h

~ N

A

|e|

µ

h

• Typical values of N

A

~ 10

16

- 10

19

cm

-3

(Many orders of

magnitude greater than intrinsic carrier concentrations at

RT)

n-type

Semiconductors

y

Donor state

n-type

“more electrons”

Free electrons in the conduction band

(8)

Semiconductors

p-type

“more holes”

Hole in the valence band Acceptor state

For an p-type material, excitation of an electron into the acceptor level, leaving

behind a hole in the valence band.

Temperature Dependence of carrier Concentration and

Conductivity

• Our basic equation:

σ

= n|e|

µ

e

+ p|e|

µ

h

• Main temperature variations

are in n,p rather than

µ

e

,

µ

h

• Intrinsic carrier concentration

n = p = C exp (- E

g

/2 kT)

Extrinsic carrier concentration

–

low T (< room temp)

Extrinsic

regime:

ionization of dopants

–

mid T (inc. room temp)

Saturated

regime:

most dopants ionized

–

high T

Intrinsic regime

: intrinsic

generation dominates

4. Electrical properties of

ceramics and polymers

Dielectric Materials

• A

dielectric material

is an insulator which contains electric

dipoles, that is where positive and negative charge are

separated on an atomic or molecular level

(9)

Capacitance

• Capacitance

is the ability to store

charge across a potential difference.

• Examples: parallel conducting plates,

semiconductor p-n junction

• Magnitude of the capacitance, C:

C = Q / V

Units: Farads

• Parallel- plate capacitor, C depends on

geometry of plates and

material

between plates

C =

ε

r

ε

o

A / L

A : Plate Area; L : Plate Separation

ε

o

: Permittivity of Free Space (8.85x10

-12

F/m

2

)

ε

r

: Relative permittivity,

ε

r

=

ε

/

ε

o

Vac,

ε

r

= 1

+ + + + +

-P

N

+

-D

L

• Magnitude of dielectric constant depends upon frequency

of applied alternating voltage (depends on how quickly

charge within molecule can separate under applied field)

• Dielectric strength (breakdown strength)

: Magnitude of

electric field necessary to produce breakdown

Polarization

• Magnitude of electric dipole moment

from one dipole:

p = q d

• In electric field, dipole will rotate in

direction of applied field:

polarization

• The surface charge density of a

capacitor can be shown to be:

D =

ε

o

ε

r

ξ

D :

Electric Displacement

(units Coulombs / m

2

₎

• Increase in capacitance in dielectric

medium compared to vacuum is due

to polarization of electric dipoles in

dielectric.

• In absence of applied field (b), these

are oriented randomly

• In applied field these align according

to field (c)

• Result of this polarization is to create

opposite charge Q’ on material

adjacent to conducting plates

• This induces additional charge (-)Q’

on plates: total plate charge Q

t

=

|Q+Q’|.

(10)

• Surface density charge now

D =

εξ

=

ε

o

ε

r

ξ

=

ε

o

ξ

+ P

• P is the

polarization

of the material

(units Coulombs/m

2

_{). It represents}

the total electric dipole moment

per unit volume of dielectric, or the

polarization electric field arising

from alignment of electric dipoles

in the dielectric

• From equations at top of page

P =

ε

o

(

ε

r

-1)

ξ

Origins of Polarization

• Where do the electric dipoles come from?

–

Electronic Polarization

: Displacement of negative

electron “clouds” with respect to positive nucleus.

Requires applied electric field. Occurs in all materials.

–

Ionic Polarization

: In ionic materials, applied electric

field displaces cations and anions in opposite directions

–

Orientation Polarization:

Some materials possess

permanent electric dipoles, due to distribution of charge

in their unit cells. In absence of electric field, dipoles

are randomly oriented. Applying electric field aligns

these dipoles, causing net (large) dipole moment.

P

tptal

= P

e

+ P

i

+ P

o

Electronic

Ionic

Orientation

(11)

Piezoelectricity

• In some ceramic materials, application of external forces

produces an electric (polarization) field and vice-versa

• Applications of

piezoelectric

materials microphones, strain

gauges, sonar detectors

• Materials include barium titanate, lead titanate, lead

zirconate

Ionic Conduction in Ceramics

• Cations and anions possess electric charge (+,-) and

therefore can also conduct a current if they move.

• Ionic conduction in a ceramic is much less easy than

electron conduction in a metal (“free” electrons can move

far more easily than atoms / ions)

• In ceramics, which are generally insulators and have very

few free electrons, ionic conduction can be a significant

component of the total conductivity

σ

total

=

σ

electronic

+

σ

ionic

• Overall conductivities, however, remain very low in

ceramics.

Electrical Properties of Polymers

• Most polymeric materials are relatively poor conductors of electrical

current - low number of free electrons

• A few polymers have very high electrical conductivity - about one

quarter that of copper, or about twice that of copper per unit weight.

• Involves doping with electrically active impurities, similar to

semiconductors: both p- and n-type

• Examples: polyacetylene, polyparaphenylene, polypyrrole

• Orienting the polymer chains (mechanically, or magnetically) during

synthesis results in high conductivity along oriented direction

• Applications: advanced battery electrodes, antistatic coatings,

electronic devices

(12)

5. Semiconductor Devices and Circuits

The Semiconductor p-n Junction Diode

• A

rectifier

or

diode

allows

current to flow in one

direction only.

• p-n junction diode consists of

adjacent p- and n-doped

semiconductor regions

• Electrons, holes combine at

junction and annihilate:

depletion region

containing

ionized dopants

• Electric field, potential barrier

resists further carrier flow

P

N

Applied Voltage

-

Reverse Bias

V

b

(13)

Transistors

• The basic building block of the microelectronic revolution

• Can be made as small as 1 square micron

• A single 8” diameter wafer of silicon can contain as many as

10

_{- 10}

11

_{transistors in total: enough for several for every}

man, woman, and child on the planet

• Cost to consumer ~ 0.00001c each.

• Achieved through sub-micron engineering of semiconductors,

metals, insulators and polymers.

• Requires ~ $2 billion for a state-of-the-art fabrication facility

Bipolar Junction Transistor

• n-p-n or p-n-p sandwich structures. Emitter-base-collector. Base is very thin (~ 1

micron or less) but greater than depletion region widths at p-n junctions.

• Emitter-base junction is forward biased; holes are pushed across junction. Some of

these recombine with electrons in the base, but most cross the base as it so thin. They

are then swept into the collector.

• A small change in base-emitter voltage causes a relatively large change in

emitter-base-collector current, and hence a large voltage change across output (“load”)

resistor:

voltage amplification

• The above configuration is called the “common base” configuration (base is common

to both input and output circuits). The “common emitter” configuration can produce

both amplification (V,I) and very fast switching

MOSFET (Metal-Oxide-Semiconductor Field Effect

Transistor)

• Nowadays, the most important type of transistor.

• Voltage applied from source to drain encourages carriers (in the above case

holes) to flow from source to drain through narrow channel.

• Width (and hence resistance) of channel is controlled by intermediate gate

voltage

• Current flowing from source-drain is therefore modulated by gate voltage.

• Put input signal onto gate, output signal (source-drain current) is

correspondingly modulated: amplification and switching

• State-of-the-art gate lengths: 0.18 micron. Oxide layer thickness < 10 nm

Take Home Messages

• Language:

Resistivity, conductivity, mobility, drift velocity, electric field

intensity, energy bands, band gap, conduction band, valence band, Fermi

energy, hole, intrinsic semiconductor extrinsic semiconductor, dopant,

donor, acceptor, extrinsic regime, extrinsic regime, saturated regime,

dielectric, capacitance, (relative) permittivity, dielectric strength, (electronic,