Electrical Properties of Materials

Conductivity, Bands & Bandgaps

Objectives

To understand: Electronic Conduction in materials Band Structure Conductivity

Metals Semiconductors Ionic conduction in ceramics

Dielectric Behavior Polarization

Definitions

Ohm’s Law V = iR

V - Voltage, i - current, R -Resistance Units

V - Volts (or W/A (Watts/amp) or J/C (Joules/Coulomb))

i - amps (or C/s (Coulombs/second)

R - ohms ()

Definitions

Resistance

ρ= RAl

= VAil

= m2

m = m

Area

Length

i

Consider current moving through a conductor with cross sectional area, A and a length, l

R = V/i

Definitions

Conductivity, :

Conductivity is the “ease of conduction”

Ranges over 27 orders of magnitude!

= 1/ρ (units: (-cm)-1

ConductivityMetals 107 1/cmSemiconductors 10-6 - 104 1/cmInsulators 10-10 -10-20 1/cm

Definitions

Electronic conduction: Flow of electrons, e and electron holes, h

Ionic conduction Flow of charged ions, Ag+

Charge carriers can be electrons or ions

Electronic Conduction

In each atom there are discrete energy

levels occupied by electrons Arranged into:

Shells K, L, M, NSubshells s, p, d, f

In Solid Materials

Each atom has a discrete set of electronic energy levels in which its electrons reside.

As atoms approach each other and bond into a solid, the Pauli exclusion principle dictates that electron energy levels must split.

Each distinct atomic state splits into a series of closely spaced electron states - called an energy band

Electronic Conduction

Energy

interatomic separation

1s

2s

Band

Pauli Exclusion Principle - no two electrons within a system may exist in the same “state” All energy levels (occupied or not) “split” as atoms approach each other

1S1 1S1

E 1S11S1

A1 A2

For two atoms For many atoms

BandingE

nerg

y

1S

3S

3P

2P

2S

4S

3D

Isolated AtomE

nerg

y

1S

3S

3P

2P

2S

4S

3D

Bonded Atoms

Electronic Conduction

Once states are split into bands, electrons fill states starting with lowest energy band. Electrical properties depend on the arrangement of the outermost filled and unfilled electron bands. “boxes of marbles analogy”

Inter-atomic separation

Equilibrium Separation

BandGap

Band Structure

Valence Band Band which contains highest energy electron

Conduction Band The next higher band

Insulator

filled

empty

filled

empty

Metal

filledempty Valence

Band

Conduction Band

Semiconductor

Band Structure

Fermi Energy, EfEnergy corresponding to the highest

filled state

Only electrons above the Fermi level can be affected by an electric field (free electrons)

Ef

E

Conduction in Metals- Band Model

For an electron to become free to conduct, it must be promoted into an empty available energy state

For metals, these empty states are adjacent to the filled states

Generally, energy supplied by an electric field is enough to stimulate electrons into an empty state

Resistivity, ρ in Metals

Resistivity typically increases linearly with temperature: ρt = ρo + T

Phonons scatter electrons

Impurities tend to increase resistivity: Impurities scatter electrons in metals

Plastic Deformation tends to raise resistivity dislocations scatter electrons

Temperature Dependence, Metals

There are three contributions to ρρt due to phonons (thermal)ρi due to impuritiesρd due to deformation (not shown)

ρ = ρi + ρo+ ρd

ρ = ρi + ρo+ ρd

Electrical Conductivity, Metals

For charge transport to occur - must have:- something the carry the charge- the ability to move

= conductivity = 1/ρ

= ne

Electrical Conductivity, Metals

= electrical conductivity n = number of concentration of charge

carriers depends on band gap size and amount of thermal energy

= mobility measure of resistance to electron motion - related to

scattering events - (e.g. defects, atomic vibrations) “highway analogy”

= ne

Temperatures Dependence, Metals

Metals, decreases with T ( = ne Two parameters in Ohm’s law may be T dependent: n and

Metals - number of electrons (in conduction band) does not vary with T. n = number of electrons per unit volume n 1022 cm-3 and

102-103 cm2/Vsec

105-106 (ohm-cm)-1

All of the observed T dependence of in metals arises from

Semiconductors and Insulators

Electrons must be promoted across the energy gap to conduct

Electron must have energy: e.g. heat or light absorptrion

If gap is very large (insulators) no electrons get promoted low electrical conductivity,

Semiconductors

For conduction to occur, electrons must be promoted across the band gap

E

EF

Full

Empty

Energy is usually supplied by heat or light

Note - electrons cannot reside in gap

Thermal Stimulation

P=exp

−ΔEkBT ⎛ ⎝ ⎜ ⎞

⎠ ⎟

Suppose the band gap is Eg = 1.0 eV

P = number of electrons promoted to conduction band

T(°K) kBT (eV) ΔE/kBT

exp

ΔEkBT

0 0 0100 0.0086 58 0.06x10

-24

200 0.0172 29 0.25x10-12

300 0.0258 19.4 3.7 x10-9

400 0.0344 14.5 0.5x10-6

Stimulation of Electrons by Photons

Photoconductivity

Eg ω ≥ Eg h

Conductivity is dependent on the intensity of the incident electromagnetic radiation

E = h = hc c m(sec -

1)

h Eg

Stimulation of Electrons by Photons

Provided

Band Gaps: Si - 1.1 eV (Infra red)Ge 0.7 eV (Infra red)GaAs1.5 eV (Visible red)SiC 3.0 eV (Visible blue)

(If incident photons have lower energy, nothing happens when the semiconductor is exposed to light.)

h Eg

Intrinsic Semiconductors

Intrinsic Semiconductors Once an electron has been excited to the

conduction band, a “hole” is left behind in the valence band

Since neither band is nowcompletely full or empty,both electron and hole canmigrate

Conductivity of Intrinsic S.C.

Intrinsic semiconductor pure material

For every electron, e, promoted to the conduction band, a hole, h, is left in the valence band (+ charge)

Band GapSilicon - 1.1 eVGermanium - 0.7 eV

Total conductivity = e + h = nee + neh

For intrinsic semiconductors: n = p & = ne(e + h)

Extrinsic Semiconductors

Extrinsic semiconductors impurity atoms dictate the properties

Almost all commercial semiconductors are extrinsic

Impurity concentrations of 1 atom in 1012 is enough to make silicon extrinsic at room T!

Impurity atoms can create states that are in the bandgap.

Types of Extrinsic Semiconductors

In most cases, the doping of a semiconductor leads either to the creation of donor or acceptor levels

n-Type

p-Type semiconductorsIn these, the charge carriers are positive

p-Type

n-Type semiconductorsIn these, the charge carriers are negative

Silicon

Diamond cubic lattice Each silicon atom has one s and 3p orbitals that hybridize

into 4 sp3 tetrahedral orbitals Silicon atom bond to each other covalently, each sharing 4

electrons with four, tetrahedrally coordinated nearest neighbors.

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Silicon

n-type semiconductors: Bonding model description:

Element with 5 bonding electrons. Only 4 electrons participate in bonding the extra e- can easily become a conduction electron

p-type semiconductors: Bonding model description:

Element with 3 bonding electrons. Since 4 electrons participate in bonding and only 3 are available the left over “hole” can carry charge

Si Si

Si Si

Si P

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

Si Si

B Si

Si Si

In order to get n-type semiconductors, we must add elements which donate electrons i.e. have 5 outer electrons. Typical donor elements which are added to Si or Ge:

Phosphorus

Arsenic

Antimony

Typical concentrations are ~ 10-6

Doping Elements, n-Type

Group V elements

Doping Elements, p-type

To get p-type behavior, we must add acceptor elements i.e. have 3 outer electrons. Typical acceptor elements are:

Boron

Aluminum

Gallium

Indium

Group III elements

Location of Impurity Energy Levels

Typically, ΔE ~ 1% Eg

Eg

ΔE

ΔE

Conductivity of Extrinsic S.C.

There are three regimes of behavior:

Temperature

impurity excitation

all impuritiesionized

Excitation across band gap

It is possible that one or more regime will not beevident experimentally

n-Type Semiconductors

Band Model description: The dopant adds a donor state in the band gap

Band GapDonor State

If there are many donors n>>p (many more electrons than holes)

Electrons are majority carriers“n-type” - (negative) semiconductor

= e + h = nee + neh

≈ neu

p-Type Semiconductors

Band Model description: The dopant adds a acceptor state in the band gap

Band GapAcceptor State

If there are many acceptors p>>n (many more electrons than holes)

holes are majority carriers“p-type” - (negative) semiconductor

= e + h = nee + neh

≈ peu

III-V, IV-VI Type Semiconductors

Actually, Si and Ge are not the only usuable Semiconductors Any two elements from groups III and Vor II and VI, as long as

the average number of electrons = 4 and have sp3-like bonding, can act as semiconductors. Example: Ga(III), As(V) GaAs

Zn(II), Se(VI) ZnSe Doping, of course, is accomplished by substitution, on either site,

by a dopant with either extra or less electrons. In general, “metallic” dopants will substitute on the “metal” sites and “non-metallic” dopants will substitute on non-metal sites. For the case where the dopant is between the two elements in the compound, substitution can be amphoteric (i.e. on both sites)

Question: Give several p-type and n-type dopant for GaAs and ZnSe. What kind of dopant is Si in InP?