Chapter 19 Electrical Properties - University of Tennesseeprack/mse201/Chapter 19 Electrical...

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Introduction To Materials Science FOR ENGINEERS, Ch. 19

University of Tennessee, Dept. of Materials Science and Engineering1

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 LawV = IRE = V / Lwhere E is electric field intensityµ = / E where µ = the mobility

• Resistivityρ = RA / L (Ω.m)

• Conductivityσ = 1 / ρ (Ω.m)-1

ν

ν = the drift velocity

1. Basic laws and electrical properties of metals

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• Electrical conductivity between different materials varies by over 27 orders of magnitude, the greatest variation of any physical property

Metals: σ > 105 (Ω.m)-1

Semiconductors: 10-6 < σ < 105 (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). ρtotal = ρthermal+ρimpurity+ρdeformation

ρthermal from thermal vibrations

ρimpurity from impuritiesρdeformation from deformation-induced point defects

• Resistivity increases with temperature (increased thermal vibrations and point defect densities)

ρT = ρo + aT• Additions of impurities that form solid

sol:ρI = Aci(1-ci) (increases ρ)

• Two phases, α, β:ρi = ραVα + ρ βV 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: vd, 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 are, n, and how easily they move

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vd = µeE

σ = n|e| µen : number of “free” or conduction electrons per

unit volume

EScattering events

Net electron motion



(m) = Metal (s) = Semicon

Mobility (RT) µ (m2V-1s-1)

Carrier DensityNe (m-3)

Na (m) 0.0053 2.6 x 1028

Ag (m) 0.0057 5.9 x 1028 Al (m) 0.0013 1.8 x 1029 Si (s) 0.15 1.5 x 1010

GaAs (s) 0.85 1.8 x 106

InSb (s) 8.00

σmetal >> σsemi



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).

δ2 ψ

δx2

δ ψ

δt

Kψ + 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:

quantize angular momentum, energy, radius of orbit

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• 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



MetalsSemiconductors Eg

< 2 eVInsulators Eg > 2 eV

Empty band

Empty conduction

bandEmpty band

Band gap

Empty states

Filled states

Filledband

Filled valenceband

Empty conduction

band

Ef

Ef

Ef

Ef

Band gapBand gap

Filled valenceband



• 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, EF• 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. Ef 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, ml

– 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)– Holes can and do exist in metals, but are more important in

semiconductors and insulators

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The Fermi Function

f (E) = [1] / [e(E - Ef) / 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 Ef and equal to 0 above Ef



• 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 Ef 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 Ef : 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

Empty states

Filled states

(b)(a)

EF

Ene

rgy

Electronexcitation

EF



Semiconductors, Insulators

Val

ence

band

Con

duct

ion

band

Ban

dG

ap

Val

ence

band

Con

duct

ion

band

(b)(a)

Electronexcitation

Freeelectron

Hole in valence

band

Ene

rgy

EF

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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 1010

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|µe + p|e|µh

n: number of conduction electrons per unit volumep: number of holes in VB per unit volume

• In intrinsic semiconductor, n = p:σ = n|e|(µe +µh) = p|e|(µe +µh)

• Number of carriers (n,p) controlled by thermal excitation across band gap:

n = p = C exp (- Eg /2 kT)C : Material constant

Eg : Magnitude of the bandgap



Extrinsic Semiconductors

• Engineer conductivity by controlled addition of impurity atoms: Doping

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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 NAs As atoms per unit volume produce n additional conduction electrons per unit volume

• Impurities which produce extra conduction electrons are called donors, ND = NAs ~ n

• These additional electrons are in much greater numbers than intrinsic hole or electron concentrations, σ ~ n|e|µe ~ ND |e|µe

• Typical values of ND ~ 1016 - 1019 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 NB B atoms per unit volume produce p additional holes per unit volume

• Impurities which produce extra holes are called acceptors, NA = NB ~ p

• These additional holes are in much greater numbers than intrinsic hole or electron concentrations, σ ~ p|e|µh ~ NA |e|µh

• Typical values of NA ~ 1016 - 1019 cm-3 (Many orders of magnitude greater than intrinsic carrier concentrations at RT)



n-type

p-type

Si4+

(a)

Si4+

Si4+

Si4+

Si4+

Si4+

Si4+

P5+

Si4+

Si4+

Si4+

Si4+

Si4+

(a)

Si4+

Si4+

Si4+

Si4+

Si4+

B3+

Si4+

holeSi4+

Si4+

Si4+

Si4+

(b)

Si4+

Si4+

Si4+

Si4+

Si4+

Si4+

B3+

Si4+

hole

Si4+

Si4+

Si4+

Si4+

Si4+

(b)

Si4+

Si4+

Si4+

Si4+

Si4+

Si4+

P5+

E field

free electron

Si4+

Si4+

Si4+

Si4+



Semiconductors

Val

ence

band

Con

duct

ion

band

Ban

dG

ap

Val

ence

band

Con

duct

ion

band

(b)(a)

Ene

rgy Donor state

n-type “more electrons”

Freeelectrons in the conduction band

For an n-type material, excitation occurs from the donor state in which a free electron is generated in the conduction band.

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Semiconductors

Val

ence

band

Con

duct

ion

band

Ban

dG

ap

Val

ence

band

Con

duct

ion

band

(b)(a)

Ene

rgy

p-type “more holes”

Hole in the valenceband

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|µe + p|e|µh

• Main temperature variations are in n,p rather than µe ,µh

• Intrinsic carrier concentrationn = p = C exp (- Eg /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

Saturation

Intrinsic

1/T

Extrinsic

ln p

, n

∆ln p/ [∆(1/T)]

= Eg / 2 k



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

• When an electric field is applied, these dipoles align to the field, causing a net dipole moment that affects the material properties.

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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 / VUnits: Farads

• Parallel- plate capacitor, C depends on geometry of plates and material between plates

C = εr εo A / LA : Plate Area; L : Plate Separation

ε o : Permittivity of Free Space (8.85x10-12 F/m2)ε r : Relative permittivity, εr = ε /εo

Vac, εr = 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 = εoεrξD : Electric Displacement

(units Coulombs / m2)



• 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 Qt = |Q+Q’|.

• So, C = Qt / V has increased

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• Surface density charge now D = εξ = εoεrξ = εoξ + P

• P is the polarization of the material (units Coulombs/m2). 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 pageP = εo(εr-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.

Ptptal = Pe + Pi + Po



Electronic

Ionic

Orientation



Barium Titanate, BaTiO3 : Permanent Dipole Moment for T < 120 C (Curie Temperature, Tc). Above Tc, unit cell is cubic, no permanent electric dipole moment

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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

σtotal = σelectronic + σionic

• 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• Polymeric light emitting diodes are also becoming a very important

research field

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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

+++

++++++

--

--

-

---

-

D

p

n

Vh

Ve

ξ



Applied Voltage

P N

+++

++++++

--

--

-

---

-

D

+ -

Forward BiasVb

Ev0

Ec0

Vo Ec+

Ev+

Vo-Vb

Lower Barrier , I ↑ Higher Barrier, I ↓

P N

+++

---

--

---

- ++++++

D

+-Reverse BiasVb

EF0

Ev0

Ec0

Vo

Ec-

Ev-

EF-

Vo+|Vb|



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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

1010 - 1011 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, ionic, orientational) polarization, electric displacement, piezoelectric, ionic conduction, p-n junction, rectification, depletion region, (forward, reverse) bias, transistors, amplification.

• Fundamental concepts of electronic motion: Conductivity, drift velocity, mobility, electric field

• Band theory of solids: Energy bands, band gaps, holes, differences between metals, semiconductors and insulators

• Semiconductors: Dependence of intrinsic and extrinsic carrier conc. on temperature, band gap; dopants - acceptors and donors.

• Capacitance: Dielectrics, polarization and its causes, piezoelectricity• Semiconductor devices: basic construction and operation of p-n junctions,

bipolar transistors and MOSFETs

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