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Tao Deng, [email protected] 1

1896 1920 1987 2006

Properties of Materials

Chapter 3

Electrical Properties of Materials

Tao Deng


3. The electrical properties

• Electrical conductivity

• Dielectric property

• Thermoelectric

• Pyroelectric

• Piezoelectric

• Ferroelectric

• Photoelectric


3.1 Electrical conductivity

3.1.1 Introduction

3.1.1.1 Characterization of electrical conductivity

Electrical conduction: when a voltage is applied between two

ends of a material , there is a current flowing through the

material.

S

LR Resistance：

L

SRResistivity： 1μ·cm＝10-9·m＝10-6·cm＝10-2·mm2/m

R

VI Ohm's Law


I=SJ（J is electrical current density ）

V=LE（E is electric al field intensity )

1J E

1Electrical conductivity (-1·m-1 or S/m):

J E

Relative electrical conductivity （IACS%）：

If the conductivity of the international standard soft copper (resistivity at 20

0C: 0.01724·mm2/m) is 100%, a material’s relative conductivity is defined as the

percentage of its’ conductivity divided by soft copper's conductivity. For example, the

IACS% of iron is 17%; The IACS% of aluminum is 65%.

3.1 Electrical conductivity


3.1.1 The electrical conductivity of typical materials at

room temperature

Silver 6.8 x 107

Copper 6.0 x 107

Iron 1.0 x 107

METALS conductors

Silicon 4 x 10-4

Germanium 2 x 100

GaAs 10-6

SEMICONDUCTORS

semiconductors

Polystyrene <10-14

Polyethylene 10-15 -10-17

Soda-lime glass 10-10

Concrete 10-9

Aluminum oxide <10-13

CERAMICS

POLYMERS

insulators

- 10-11

• Room temperature values (Ohm-m)-1 = ( - m)-1


3.1.1.2 Materials’ electrical conductivity

Conductor： ρ＜10-5 Ω.m；

Semiconductor： ρ=10-5~ 1010 Ω.m；

Insulator： ρ＞1010 Ω.m；


3.1.1.3 Mechanism of electrical conduction

(1) Carriers

Current is the directional flow of electrical charge in space. In any

matter, as long as there are free particles with charges (carriers), there will be

current under electric field

• In metals, the carriers are free electrons (electronic conduction)

• In inorganic materials, there are two types of carriers：

– Ions, including positive/negative ions and vacancies (ionic conduction)

– Electrons, including negative electrons and holes (electronic conduction)

• In polymers, the carriers are solitons

• In superconducting materials, the carriers are two-electron Cooper pairs


（2）Mobility of carriers

In the conductor with a cross-sectional area of the unit area, n is the number of

carriers in the unit volume and q is the charge carried by each carrier. If the external

electric field (E) is applied along the longitudinal direction, there is a force of qE acting

on each carrier . Under the action of this force, each carrier moves along the direction of

E, and its average speed is v.

J nqv

J nqv

E E

v

E

Mobility of carrier is

defined as ：

It is the average drift velocity of the carriers under the action of a unit

electric field.

E



nq

If there are several carriers that contribute to the conductivity in the

material, then the total conductivity is the sum of all individual carriers:

i i i i

i i

n q

Two key factors for material‘s conductivity:

• The concentration of carriers

• Carrier mobility.

External conditions (such as temperature and pressure), bonding,

composition, and other factors have an impact on the carrier concentration

and carrier mobility.



3.1.1.4 Band theory

（1）Basic concept

－The potential field from positive ions has a

periodicity and it makes the movement of free

electrons not completely free.

－Study the energy distribution of electrons in the

periodic potential field in metals.


3.1.1.4 Band theory

x1

When the distance between

atoms is large (x1),

electrons in atoms are

mutually independent.

Arrangement of metallic atoms


3.1.1.4 Band theory

Arrangement of metallic atoms within Bulk Metals

x3

When the spacing is less than x2,

the originally independent energy

levels of an atom spread into a

band consisting of a discrete

series of energy levels; the energy

difference between these levels is

small (about 10-23eV.)


3.1.1.4 Band theory

Arrangement of metallic atoms within Bulk Metals

x4

When the atomic spacing

is further reduced (x4) ,

the band is broadening.


Energy band

• The highest energy band partialy occupied by some

electrons is called the valence band

• Core band – band energy is lower than that of the valence

band

• Conduction band – band energy is higher than that of the

valence band

• Band gap ( Eg)- the forbidden energy gap between the

valence band and the conduction band


The energy band of three typical materials

Empty band Valence band

Metallic conductor

Overlapping of Valence

band and the empty band

Valence band is

half full

Ov

erlap

regio

n

Co

nd

uctio

n B

an

d

con

du

ction

ba

nd

For example

Si： Eg＝1.1eV

Ge：Eg＝0.71eV

Semiconductor

Eg ≈ 0.2~2.5eV

Insulator

For example ：

diamond

Eg＝6.0eV

Eg＞2.5eV


3.1.1.5 Theory of the electrical conductivity

(1）The classical electron theory

•Free electrons are considered as "electron gas” - - analyze with

classic gas molecules kinetic theory.

• The interaction within the free electrons and between them and

the positive ions are treated as the mechanical collision.

tm

ne

mv

lne

22

22

l －mean free path of the electron；m －mass of the electron；v －average

speed of the electron；e － charge of the electron ；t － the average time

between twice collisions；n －the number of free electrons per unit volume；


The influence of the temperature and the point

defects on electron movement

The influence of temperature

6 collisions

The model of an electron moving in the lattice

3 collisions

The influence of point defects

8 collisions


（2）Quantum theory of free electrons

Basic assumptions：

• In metals, the electric field formed by the positive ions is

uniform;

• The electrons in the valence band belong to the entire metal;

they have no interaction with ions, and can move freely;

• The electrons in the inner bands of each atom have the same

energy states with those in the original single atoms, while the

electrons in the valence band have different energy states that are

quantalized to different energy levels.

3.1.1.5 Theory of the electrical conductivity


（2）Quantum theory of free electrons

p

h

mv

h

22

2

82

1K

m

hmvE

2K

Wave-Particle Duality:

For monovalent metal, the kinetic

energy of free-electron：

Where， wave frequency (wave number or wave vector)

h

p

h

mv

222


The relationship between electronic energy and wave

vector

-K +K O

E

E - K curve of free electrons

+K -K O

E

+K -K O

E

e

E

The effect of the electric field on the E-K curve


The interference of electronic wave

The electrical resistance is caused by the

scattering of electronic waves through:

• Ion lattice;

• Static lattice distortion generated by defects or

impurities;

• Dynamic lattice distortion generated by thermal

vibration.


3.1.1.6 Ionic conduction

(1) Conductive mechanism of ionic crystals

a) Conduction through intrinsic ions

Due to the increase of thermal vibration, ions leave the lattice site to

form interstitial ions and vacancies (thermal defects). Such thermal defects

can move under the electric field to generate current. Concentration of

thermal defects increases with in temperature, so the ionic intrinsic

conductivity increases with temperature as well:

Where Es - The activation energy of ions.

2) Conduction through impurity ions

kT

EA s

ss exp

T

BAim exp

k

EB s


（2）The conduction mechanism in glass

• Glass is usually an insulator -- at high temperature,

some glasses can become a conductor.

• Glass is also a conductor of the electrolyte. Its

conductivity comes from the movability of ions in the

structure. For example, in the silica network of a soda

glass, a sodium ion jumps from one structural gap to

another to generate the current. This conduction is

similar to the conduction from interstitial ions in the

ionic crystals.

• The composition of the glass has a great impact to the

resistance.


3.1.1.7 The conduction in polymers

• In 1977, Shirakawa from Japan and MacDiamid from the

United States found that the conductivity of polyacetylene,

doped with I2 or AsF5 , increases from 10-9 S / cm to 103 S/cm.


3.1.1.7 The conduction in polymers


Polyacetylene

• Carbon atoms are bonded with each other through double bond → single bond →

double bond to form a quasi-one-dimensional carbon chain,;every hydrogen atom

bonds with a carbon atom located on the carbon chain;

•Each carbon atom is adjacent to two carbon atoms and one hydrogen atom ；

• Each carbon atom has four valence electrons. In polyacetylene, neighboring carbon

atoms using their sp2 hybrid orbitals to form carbon - carbon σ bonds, and at the

same time, they also use their sp2 orbitals to interact with hydrogen’s s orbitals to

form carbon – hydrogen σ bonds;

• Each carbon atom has a valence electron (pz orbitals) and it becomes a π electron in

the molecular bond of polyacetylene.

H atoms on both sides of the double bond H atoms at the same side the double bond.

Trans-form Cis-form


The band structure of one-dimensional carbon chain

of polyacetylene

The energy band of one-

dimensional equidistant

carbon chain

Should be like a metal??

The energy band of one-

dimensional dimerized

carbon chain

Like a semiconductor!


The soliton model

In a certain range of the doping concentration, trans-polyacetylene has a high

electrical conductivity.

The Soliton model:

Trans-polyacetylene has two

lowest dimerization states: A-phase

and B-phase, with the same energy

and symmetrical structure. If the A-

phase and B-phase coexist in the same

molecular chain, there will be a

domain wall between them, which is

called soliton. The alternating single

bond-double bond structure is

destroyed at the soliton.

A phase, the soliton and B phase in trans-polyacetylene

The conductive carriers of polyacetylene carry charge without spinning.


3.1.2 The electrical conductivity of the metal

3.1.2.1 The electrical conductivity of the elements


Analysis of element conductive band theory

3s valence band is half-filled.

High electrical conductivity

The outermost s band is full, but the outermost s overlaps with the

outermost p to form the conduction band.

The conductivity is higher than IA family.

The outermost p is filled with a small amount of electrons. Most of it is empty, with some

overlapping with the main shell s.

Higher conductivity.

Sp hybrid orbital caused by covalent

bond; involving the 2s electrons(4

valence electron) to form two hybrid

bands, where one is filled, and the

other is empty, but there is a band

gap between the two hybrid band.

The conductivity is poor.

The outermost s band is full, partially overlapping with the

half-filled d-band to form a conduction band with less

empty level .

Conductivity is relatively poor.

Similar to the alkali metal, there is non-overlapping band and valence band is half-filled. Higher conductivity

.


3.1.2.2 Matthiessen Law

The resistance of ideal metals are based on only two scatterings- phononic

scattering and electronic scattering. This resistance is reduced to zero at absolute

zero. The third scattering (electrons scattered by impurities and defects) can be

observed in the non-ideal crystal with defects, which still exists even at absolute

zero. The scattering coefficient is composed of two parts:

T

Where, the scattering coefficient vT is proportional to the temperature T, ν is proportional

to the concentration of impurities and independent of temperature.

T

Where, （T） is the basic resistance of pure metals; is the residual

resistance determined by chemical and physical defects, regardless of the

temperature.

Matthiessen law


3.1.2.3 The relationship between electrical resistivity

and temperature in metals

Usually, in the case of temperature

higher than the room temperature: T 10

where，ρ0 － electrical resistivity at 0℃；α－Temperature coefficient of resistance；β、γ－High-order coefficient；

32

0 1 TTTT

5T

T

2T

Generally, ρ increases as T increases.

In the very low temperature:

Electron - electron scattering;

In the higher Temperature:

Electron - phonon scattering

-T＜ΘD时，ρ∝T5；

-T＞ ΘD时，ρ∝T


Temperature coefficient

The average temperature

coefficient of resistance： T

T

0

0

True temperature coefficient

of resistance： dT

d

T

T

1

• Except the transition metals, for all other pure metals α≈4×10-3。

• Transition metals, especially ferromagnetic metals, have a high α. For

example, iron has a α= 6 ×10-3.


3.1.2.4 The impact of stress and deformation

caused by cold processing

Cold processing will increase

the lattice distortion, thus

increase the resistivity:

• Fe、Cu、Al、Mg .etc. ，

may increase by 2～6％；

• W、Mo、Sn .etc., may

increase by 15～90％.

Recrystallization annealing

may cancel the increase of the

resistance

99.8%

97.8%

93.5%

80%

44%

Th

e am

ou

nt o

f defo

rma

tion

du

ring

cold

pro

cessing

Annealing temperature/oC


3.1.2.5 The conductive property of the alloys

(1) The resistivity of the solid solution

Generally, with the increase of the concentration

of the solute, the resistivity also increases because of

the lattice distortion. When solute concentration is

small, the resistivity follows Matthiessen's law:

rT • ρT－ Base resistance of pure base metal

•ρr －The additional resistance caused by the

concentration of solute (impurity) 、 point defects,

dislocations, etc., independent of temperature.


（2）The resistivity of ordered alloys

With the increase of order in the solid

solution

• The chemical interaction among the

alloy components is strengthened, and

number of conductive electrons decreases,

so the residual resistance increases;

• The ionic potential field become more

symmetrical, so that the probability of the

electron scattering is greatly reduced and

the residual resistivity decreases;

The second factor is usually dominant, so

the resistivity of the alloy is normally

reduced with increased structural order

Quenching state

Annealed

state

Au, % (atom)


3.1.2.6 The influence of pressure on the metal conductivity

• Normal metal ：

The resistivity decreases as

the pressure increases:

iron, cobalt, nickel, copper,

silver, gold, niobium,

vanadium, lead, etc.

• Abnormal metal ：

The resistivity doesnot

follow the normal trend as

the pressure increases:

alkali metal, alkaline earth

metal and rare earth metals;


The effect of pressure on the conductivity of non-

conductive materials High pressure can often lead to the metallization of the substance, causing

the changes of the conductivity type, enabling the transform of insulator→

semiconductor → metal → superconductors.

Element PC /MPa ρ/(μΩ.cm) Element PC /MPa ρ/(μΩ.cm)

S 40,000 － H 200,000 －

Se 12,500 － Diamond 60,000 －

Si 16,000 － P 20,000 60±20

Ge 12,000 － AgO 20,000 70±20

I 22,000 500 －－－

The critical pressure required for certain semiconductor and

dielectric materials’ transformation into the metallic state


3.1.2.7 The influence of geometric dimensions on

the electronic resistivity

When the size of material is reduced to the same

order of magnitude of the free path of the

conduction electron, the scattering of electrons in

the surface of the sample creates a new additional

resistance. In this case, the effective scattering

coefficient Leff is

Where， L、Ld , respectively, is the free path of the

electrons scattered in the bulk sample and the surface .

deff LLL

111

The resistivity of thin film is：

d

Ld 10

where，ρ0 －The resistivity of bulk sample ；

d － The thickness of film；


3.1.3 The electrical properties of the semiconductors


3.1.3 The electrical properties of the semiconductors

• Crystalline semiconductor

– Elemental semiconductor：Si, Ge, Se, Te and so on

– solid solution semiconductor：Ge-Si, Bi-Sb, GaAs-GaP

and so on ；

– compound semiconductor ：GaAs, CdS, SiC, GeS, AsSe3

and so on ；

• Noncrystalline semiconductor

– noncrystalline silicon（α –Si）、polycrystalline silicon ；

– chalcogenide glass ；

• Organic semiconductor

– Polymer semiconductor


The electronic energy states in semiconductor

1s 2s 2p

3s

3p

n=1

n=2

n=3 Forbidden band

Empty band

(conduction band)

Energy

The distance between atoms

The evolutional process of silicon covalent crystals

The sharing of valence electrons in semiconductor crystals splits the

original atomic electron energy states into a series of levels with very

small difference of energy between them to form a energy band.

single atom

The distance between atoms

in Covalent Crystals

Filled band

(valence band)


3.1.3.1 Intrinsic semiconductor

Intrinsic semiconductor: pure, single crystal, no structural defects.

Stimulated by the electric field,

temperature, or light

The process of intrinsic excitation

The number of free electrons in the

conduction band=the number of holes in

the valence band


（1） The concentration of the intrinsic carrier

kT

ETKpn

g

ii2

exp23

1

Where ni - the concentration of free electrons; Pi - The concentration of

holes; T-The absolute temperature; k - Boltzmann constant；K1 =

4.82×1015K-3/2;

•The concentration of the carriers increases with increasing temperature;

•The concentration of carriers decreases when the forbidden band becomes

wider.

For example：when T =300K，EgSi =1.1eV, ni

Si =1.5×1010 cm-3；

EgGe =0.72eV, ni

Ge =2.4×1013 cm-3；

Based on the probability of the intrinsic carrier occupying the energy level:


（2）The mobility and the electrical resistivity of

intrinsic semiconductors

Under the external electric field, the average speed of the directional

drifting of carriers is a constant value that is proportional to the electric

field strength ε:

nnv ppv where，μn and μp , respectively, are the average drifting velocities (cm / s) of free

electrons and holes under the unit field strength (V / cm); they are called the mobility

The resistivity of intrinsic semiconductor is：

pnipini

iqnqnqnj

1

where，q －The absolute value of the electronic charge.

For Ge：μn ＝3900 cm2/Vs；μp ＝1900 cm2/Vs

For Si： μn ＝1400 cm2/Vs；μp ＝500 cm2/Vs


3.1.3.2 The electrical properties of the semiconductor

containing impurities

（1）N-type semiconductor (extra electrons)

N-type semiconductor – doping the intrinsic semiconductor (with 4 valence

electrons) with the pentavalent elements, such as P, As, Sb, etc.

EC－ED＜＜Eg，So, it is easy to excite the free electron.

ni＞＞np

nD

nqN

1

where，ND －The dopant concentration

The energy band and the Fermi distribution

of the N-type semiconductor


（2）The P-type semiconductor (extra holes)

P-type semiconductor – doping the intrinsic semiconductor (with 4 valence

electrons) with the trivalent elements, such as Al, Ga, etc.

ni＜＜np

EA－EV＜＜Eg，Electrons in the valence band can enter the EA level at room

temperature, as a result of some vacancy generated in the valence band.

The structure of the P-type semiconductor The energy band and the Fermi distribution

of the N-type semiconductor

+3


3.1.3.3 The impact of the temperature on the

resistance of semiconductor

The influence of temperature depends on the competition of the

resulting change in concentration of carriers and the mobility of carriers.

The dependence of the resistivity of N-type

semiconductor on temperature

Phonon scattering is

weak, but number

of the ionic

impurity donors

increases with

temperature, and

therefore the

resistivity decreases.

All Impurities are ionized. The intrinsic excitation has not yet

started, the concentration of carriers almost remains constant,

and the phonon scattering dominates, so the resistivity increases.

The intrinsic

excitation starts with

a temperature rise,

the carrier

concentration

increases dramatically,

far more than the

phonon scattering,

and therefore the

resistivity decreases.


3.1.3.4 Conduction in semiconductor

T

B

e

0 T

B

e0

Under the normal circumstances, the dependence of the electrical

conductivity (resistivity) of temperature is

Where, B - The conductive activation energy of material. The

higher B, the greater the change of resistivity with temperature.

（1）The effect of temperature


（2) The effect of light

Photoconduction: The irradiation of the light makes the

resistance of some semiconductors decrease.

The energy of photons is transferred to the electrons in

valence band, and the excited electrons jump to the empty

band.

Application of photosensitive effect ：

Photosensitizing effects: automatic control systems, lighting

automation.


（3）The effect of voltage

The relationship between the current and voltage of some

semiconductors (such as the ceramic semiconductor of zinc

oxide ) is not linear, i.e. the resistance varies with the change

of voltage . This effect can be used to make varistors, which

can be applied for voltage absorption, high-pressure

regulator, and surge arresters.


（4）The effect of pressure

– In addition to generating the structural deformation, some

semiconductors under pressure have a change of the

energy band structure., which results in the change of

electrical resistivity. The relationship of semiconductor’

piezoresistive effect with stress is:

T

0

where，ρ0－ the resistivity without stress；Δρ－The change of

resistivity when stress is added; β－Piezoresistive coefficient；T－the applied stress (the tension is positive; the compress is negative).


（5）Magnetic effect

Hall Effect

When a semiconductor

with current is placed in the

uniform magnetic field, a

lateral electric field

perpendicular to the direction

of the external electric field

and magnetic field will be

generated.

The magnetoresistant effect

The current density is reduced as a magnetic field, perpendicular to

the current inside the semiconductor, is applied. Due to the presence of the

magnetic field, the resistance of the semiconductor increases.


3.1.4 Superconductivity

• Phenomenon

At a temperature below a certain critical

temperature Tc, the specific resistance of a

material suddenly drops to very low (＜10-25

Ω. cm).

• Potential applications

Superconducting thermonuclear reaction

Lossless superconducting transmission

Super electromagnet

Superconducting maglev train

Magnetic resonance imaging

The dependence of the resistivity of

mercury on temperature(1911-Onnes)


3.1.4.3 The physics of the superconductivity

e2 e1

Positive ion

• In the superconducting state, there are attractive force between the electrons near

the Fermi surface (rather than the electrostatic repulsion in the normal state), the

electrons with opposite momentum and spin are paired together to generate Cooper

pairs . It is the result of the interaction between the electrons and crystal lattice.

• The total momentum and the average speed of Cooper pairs remain constant

during their movement. They don't consume energy and can move through the

lattice with no resistance.


3.1.4.4 The superconducting tunneling effect

(Josephson effect)

In the 1960s, Josephson effect in weakly

connected superconductors is one of the

significant breakthroughs in the research of

superconductivity.

Weakly connected superconductors have a

sandwich structure of superconducting -

insulator - superconductor (SIS) with a

nanometer insulating film in the middle of the

two superconductors.

Josephson effect：For the S-I-S structure with current < IC,there is no

voltage through the dielectric layer. The weakly connected superconductors

have a zero resistance, i.e. the insulating (vacuum, normal) layer between

two superconductors can also pass the superconducting current.

Josephson junction


Josephson junctions

Possible Josephson junctions:

(1) Superconductor- insulator -

Superconductor;

(2) Superconductor - normal metal -

Superconductor;

(3) Superconductor - vacuum -

Superconductor (STM);

(4) two superconductors contacted

by point;

(5) Two superconductor contacted

by microbridge;

(a) Tunnel junction (b) Proximity effect bridge (c)

One-dimensional micro-bridge (d) Two-

dimensional micro-bridge (e) Three-dimensional

micro-bridge (f) the micro-bridge with thickness

changing (g) Point contact


3.1.5 Measurement of electrical resistance

3.1.5.1

(1) Measurement of resistance in

metals

Adjusting the current and four variable resistors,

so that the electric potentials of point f and point c in

the bridge circuit are equal. The bridge is in

equilibrium:

4221

3211

RIRI

RIRIRR nx

2

1

R

RRR nx

If R1 = R3, then R2 and R4 can be adjusted with R2 = R4 in the double bridge

measurement. In other words, the equilibrium of the bridge can be achieved by

adjusting R3 and R4 only:

Measurement through double bridge


（2）The resistance measurements of semiconductors

V

I

S

l

322131

1111

2 llllllV

I

lV

I

2

The two-probe method

The four-probe method


（3） The resistance measurements of insulators

x

UtR

Q

b mQ C

x

b m

UtR

C

Ballistic galvanometer can be used in

the measurement of the resistance of an

insulator. When the switch K is switched to 1

and after a time of t:

Where U is the voltage of DC supply ; t is the charging time; Q is the electric charge on the

capacitor after a charging time t, which can be measured by ballistic galvanometer. When

the switch K is switched to 2, we have

where Cb is the impact constant of ballistic galvanometer; αm is the maximum offset

of the galvanometer (direct readout). Therefore :


3.1.5.2 The application of resistance measurement

Measuring the solubility curve of the solid

solution

Studying of alloy aging

Studying the order - disorder transition in

alloys

Investigating material fatigue process

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