Chapter 13 - Electrical Properties

Introduction to

Materials Science and Engineering

Chapter 13:

Electrical Properties

Introduction to Materials Science and Engineering

Chapter 13: Electrical Properties


1. Electrical conduction • How many moveable electrons are there in a material (carrier density)? • How easily do they move (mobility) ?

2. Semiconductivity• Electrons and holes• Intrinsic and extrinsic carriers• Semiconductor devices: p-n junctions and transistors

3. Conduction in polymers and ionic materials

4. Dielectric behavior



Basic laws and electrical properties of metals (I)

When an electrical potential V [volts, J/C] is applied across a piece of material, acurrent of magnitude I [amperes, C/s] flows. In most metals, at low values of V, thecurrent is proportional to V, and can be described by Ohm's law:

I = V/Rwhere R is the electrical resistance [ohms, Ω]. R depends on the intrinsic resistivity ρof the material [Ω-m] and on the geometry (length l and area A through which thecurrent passes):

R = ρl/A

In most materials (e.g. metals), thecurrent is carried by electrons(electronic conduction).

In ionic crystals, the charge carriersare ions (ionic conduction).



The electrical conductivity (the ability of a substance to conduct an electric current) is the inverse of the resistivity:

σ = 1/ρSince the electric field intensity in the material is E = V/l, Ohm's law can be rewritten in terms of the current density J = I/A as:

J = eElectrical conductivity varies between different materials by over 27 orders of magnitude, the greatest variation of any physical property

Basic laws and electrical properties of metals (II)



Energy Band Structures in Solids (I)

In an isolated atom electrons occupy well defined energy states.

When atoms come together to form a solid, their valence electrons interactwith each other and with nuclei due to Coulomb forces. In addition, twospecific quantum mechanical effects happen. First, by Heisenberg'suncertainty principle, constraining the electrons to a small volume raisestheir energy, this is called promotion. The second effect, due to the Pauliexclusion principle, limits the number of electrons thatcan have the same energy.

As a result of these effects, the valence electrons of atoms form wideelectron energy bands when they form a solid. The bands are separated bygaps, where electrons cannot exist.



Energy Band Structures in Solids (I)



Energy Band Structures and ConductivityFermi Energy (EF) - highest filled state at 0 KConduction band - a partially filled or empty energy bandValence band – the highest partially or completely filled band

In semiconductors and insulators, the valence band is filled, and no more electrons can be added (Pauli's principle).

Electrical conduction requires that electrons be able to gain energy in an electric field. This is not possible in these materials because that would imply that the electrons are promoted into the forbidden band gap.



Energy Band Structures and Conductivity (semiconductors and insulators)

In semiconductors and insulators, electrons have to jump across the bandgap into conduction band to find conducting states above Ef. The energy needed for the jump may come from heat, or from irradiationat sufficiently small wavelength. The difference between semiconductors and insulators is that insemiconductors electrons can reach the conduction band at ordinarytemperatures, where in insulators they cannot. The probability that an electron reaches the conduction band is aboutexp(-Eg/2kT) where Eg is the band gap. If this probability is < 10-24 onewould not find a single electron in the conduction band in a solid of 1 cm3.This requires Eg/2kT > 55. At room temperature, 2kT = 0.05 eV Eg > 2.8eV corresponds to an insulator. An electron promoted into the conduction band leaves a hole (positivecharge) in the valence band, that can also participate in conduction. Holesexist in metals as well, but are more important in semiconductors andinsulators.



Energy Band Structures and Conductivity (metals)

In metals, highest occupied band is partiallyfilled or bands overlap.

Conduction occurs by promoting electronsinto conducting states, that starts rightabove the Fermi level. The conducting statesare separated from the valence band by aninfinitesimal amount.

Energy provided by an electric field issufficient to excite many electrons intoconducting states High conductivity.



Energy Band Structures and Bonding (metals, semiconductors, insulators)

Relation to atomic bonding:

Insulators – valence electrons are tightly bound to (or shared with) theindividual atoms – strongest ionic (partially covalent) bonding.

Semiconductors - mostly covalent bonding somewhat weaker bonding.

Metals – valence electrons form an “electron gas” that are not bound toany particular ion.



Energy Band Structures and Conductivity

Metals



Energy Band Structures and Conductivity

Semiconductors and Insulators



Electron Mobility

The force acting on the electron is -eE, where e is the electric charge.

This force produces a constant acceleration so that, in the absence ofobstacles the electron speeds up continuously in an electric field. This is thecase in vacuum (e.g. inside a TV tube) or in a perfect crystal (this is aconclusion from quantum mechanics).

In a real solid, the electrons scatter by collisions with imperfections and dueto atomic thermal vibrations “frictional forces” resistance a netdrift velocity of electron motion is established:

vd = µeE

where µe – electron mobility [m2/V-s].The “friction” transfers part of theenergy supplied by the electric field intothe lattice as heat. That is how electricheaters work.



Electrical conductivity is proportional to number of free electrons andelectron mobility:

Electron Mobility



Conductivity / Resistivity of MetalsThe resistivity ρ is defined by scattering events due to the imperfections and thermal vibrations. Total resistivity ρtot can be described by the Matthiessen rule:

ρtotal = ρthermal + ρimpurity + ρdeformation

ρthermal - from thermal vibrations,

ρimpurity - from impurities,

ρdeformation - from deformation-- induced defects



Influence of temperature: Resistivity rises linearly with temperature(increasing thermal vibrations and density of vacancies)

Conductivity / Resistivity of Metals



Materials of Choice for Metal Conductors

One of the best material for electricalconduction (low resistivity) is silver, but userestricted due to high cost Most widely used conductor is copper:inexpensive, abundant, high σ, but rather soft –cannot be used in applications where mechanicalstrength is important. Solid solution alloying and cold workinginprove strength but decrease conductivity.Precipitation hardening is preferred, e.g. Cu-BealloyWhen weight is important one uses aluminum,which is half as good as Cu and more resistantto corrosion. Heating elements require low σ (high R), andresistance to high temperature oxidation:nickel-chromium alloy



Semiconductivity

Intrinsic semiconductors - electricalconductivity is defined by the electronicstructure of pure material.

Extrinsic semiconductors - electricalconductivity is defined by impurityatoms.



Intrinsic semiconductors (I)

Number of electrons in conduction band increases exponentially withtemperature:

An electron promoted into the conduction band leaves a hole (positive charge)in the valence band. In an electric field, electrons and holes move in oppositedirection and participate in conduction.

In Si (Eg = 1.1 eV) one out of every 1013 atoms contributes an electron to theconduction band at room temperature.



Since both electrons and holes conduct the conductivity of an intrinsicsemiconductor is

σ = n|e|µe + p|e|µhwhere p is the hole concentration and µh the hole mobility.

Electrons are more mobile than holes, µe > µh

In an intrinsic semiconductor, a hole is produced by the promotion of eachelectron to the conduction band. Therefore, n = p and (only for intrinsicsemiconductors)

σ = n|e|(µe + µh) = p|e|(µe + µh)

n (and p) increase exponentially with temperature, whereas µe and µh

decrease (about linearly) with temperature.

The conductivity of intrinsic semiconductors is increasing withtemperature (different from metals!)

Intrinsic semiconductors (II)



Intrinsic semiconductors (III)



Extrinsic semiconductors

Extrinsic semiconductors - electrical conductivity is defined by impurityatoms.Example: Si is considered to be extrinsic at room T if impurity concentrationis one atom per 1012 (remember our estimation of the number of electronspromoted to the conduction band by thermal fluctuations at 300 K)

Unlike intrinsic semiconductors, an extrinsic semiconductor may havedifferent concentrations of holes and electrons. It is called p-type if p > nand n-type if n > p.

One can engineer conductivity of extrinsic semiconductors by controlledaddition of impurity atoms – doping (addition of a very small concentration ofimpurity atoms).

Two common methods of doping are diffusion and ion implantation.



n-type extrinsic semiconductors (I) Excess electron carriers are produced by substitutional impurities that have morevalence electron per atom than the semiconductor matrix.Example: phosphorus (or As, Sb..) with 5 valence electrons, is an electron donor in Si sinceonly 4 electrons are used to bond to the Si lattice when it substitutes for a Si atom. Fifthouter electron of P atom is weakly bound in a donor state (~ 0.01 eV) and can be easilypromoted to the conduction band.

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

Elements in columns V and VIof the periodic table are donorsfor semiconductors in the IVcolumn, Si and Ge.



The hole created in donor state is far from the valence band and is immobile.Conduction occurs mainly by the donated electrons (thus n-type).

σ ~ n|e|µe ~ ND|e|µe

(for extrinsic n-type semiconductors)

n-type extrinsic semiconductors (II)



p-type extrinsic semiconductors (I)

Excess holes are produced by substitutionalimpurities that have fewer valence electronsper atom than the matrix.

A bond with the neighbors is incomplete andcan be viewed as a hole weakly bound to theimpurity atom.

Elements in columns III of the periodictable (B, Al, Ga) are donors forsemiconductors in the IV column, Si and Ge.

Impurities of this type are called acceptors,NA =NBoron ~p



The energy state that corresponds to the hole (acceptor state) is close to the top of thevalence band. An electron may easily hop from the valence band to complete the bondleaving a hole behind. Conduction occurs mainly by theholes (thus p-type).

σ ~ p|e|µp ~ NA |e|µp(for extrinsic p-type semiconductors)

p-type extrinsic semiconductors (II)



Temperature variation of conductivity (I)

Basic equation for conductivity:

σ = n|e|µe + p|e|µh

Temperature dependence of mobilities,µe and µh is weak as compared to thestrong exponential dependence ofcarrier concentration in intrinsicsemiconductors (exp(-Eg/2kT) is muchstronger than T3/2):

n = p ≅ A exp (- E /2 kT)≅ C exp (- E /2 kT)



Temperature variation of conductivity (II)

n = p ≅ A exp (- E /2 kT)≅ C exp (- E /2 kT)

Plotting ln(σ), ln(p), or ln(n) vs. 1/Tproduces a straight line of slope Eg/2kfrom which the band gap energy can bedetermined.



Extrinsic semiconductors

• low T: all carriers are due to theextrinsic excitations• mid T: most dopants are ionized(saturation region)• high T: intrinsic generation of carriersdominates

Temperature variation of conductivity (III)



A rectifier or diode allows current flowin one direction only p-n junction diodeconsists of adjacent p- and n-dopedsemiconductor regions

If the positive side of a battery isconnected to the p-side (forward bias) alarge amount of current can flow sinceholes and electrons are pushed into thejunction region, where they recombine(annihilate).

Semiconductor Devices. Diode (I)



If the positive side of a battery isconnected to the p-side (forward bias) alarge amount of current can flow sinceholes and electrons are pushed into thejunction region, where they recombine(annihilate).

If the polarity of the voltage is flipped,the diode operates under reverse bias.Holes and electrons are removed fromthe region of the junction, whichtherefore becomes depleted of carriersand behaves like an insulator.

Semiconductor Devices. Diode (II)



For reverse bias, both holes andelectrons are drawn away from thejunction, leaving the junctionregion depleted of free carriers the current is very small.

Semiconductor Devices. Diode (III)



The asymmetric current-voltagecharacteristics of diodes is used toconvert alternating current into directcurrent (rectification).

Semiconductor Devices. Diode (IV)



Semiconductor Devices. Transistors.

Transistors are used to amplify anelectric signal and as switching devicesin computers.

Two major types of transistors arejunction (or bimodal) transistor andMOSFET transistor.

p-n-p (or n-p-n) junction transistorjunction transistor contains two diodesback-to-back.

The central region (base) is very thin (~1 micron or less) and is sandwiched inbetween emitter and collector regions.



Emitter-base junction is forward biased and holes are pushed acrossjunction. Some recombine with electrons in the base, but most cross thebase as it is so thin. They are then swept into the collector.

A small change in base-emitter voltage causes a relatively large change inemitter-base-collector current, and hence a large voltage change acrossoutput (“load”) resistor - voltage amplification

Junction transistor



Junction transistor



Conduction in Polymers and Ionic Materials

Ionic Materials

In ionic materials, the band gap is large and only very few electrons can be promoted to the valence band by thermal fluctuations. Cation and anion diffusion can be directed by the electric field and can contribute to the total conductivity: σtotal = σelectronic + σionic

High temperatures produce more Frenkel and Schottky defects which result in higher ionic conductivity.

Polymers

Polymers are typically good insulators but can be made to conduct by doping. A few polymers have very high electrical conductivity - about one quarter that of copper, or about twice that of copper per unit weight.



When a voltage V is applied to two parallel conducting plates, the plates arecharged by +Q, –Q, and an electric field E develops between the plates.

The charge remains on the plates even after the voltage has been removed.

The ability to store charge is called capacitance and is defined as a charge Qper applied voltage V:

C = Q / V [Farads]

For a parallel-plate capacitor, C depends on geometry of plates and materialbetween plates

C = εrεoA / L = εA / Lwhere A is the area of the plates, L is the distance between plates, ε is thepermittivity of the dielectric medium, εo is the permittivity of a vacuum(8.85x10-12 F/m2), and ε r is relative permittivity (or dielectric constant) ofthe material,

εr = ε / εo = C / Cvac

Capacitance



Dielectric Materials

The dielectric constant of vacuum is 1 and is close to 1 for air and manyother gases. But when a piece of a dielectric material is placed between thetwo plates in capacitor the capacitance can increase significantly.

C = εrεoA/L with εr = 81 for water, 20 for acetone, 12 forsilicon, 3 for ice, etc.

A dielectric material is an insulator in which electric dipoles can be inducedby the electric field (or permanent dipoles can exist even without electricfield), that is where positive and negative charge are separated on an atomicor molecular level

Magnitude of electric dipole moment is p = qd




Dipole formation and/or orientation along the external electric field in thecapacitor causes a charge redistribution so that the surface nearest to thepositive capacitor plate is negatively charged and vice versa.

The process of dipole formation/alignment in electric field is calledpolarization and is described by P = Q’/ADipole formation induces additional charge Q’ on plates: total plate charge

Qt = |Q+Q’|.Therefore, C = Qt/ V has increased and dielectric constant of the material

εr = C / Cvac > 1The process of dipole formation/alignment in electric fieldis calledpolarization and is described by P = Q’/A



In the capacitor surface charge density (also called dielectric displacement)is

D = Q/A = εrεoE = εoE + PPolarization is responsible for the increase in charge density above that forvacuum

Mechanisms of polarization (dipole formation/orientation) electronic (induced) polarization: Applied electric field displacesnegative electron “clouds” with respect to positive nucleus. Occurs in allmaterials. ionic (induced) polarization: In ionic materials, applied electric fielddisplaces cations and anions in opposite directions molecular (orientation) polarization: Some materials possesspermanent electric dipoles (e.g. H2O). In absence of electric field, dipolesare randomly oriented.

Applying electric field aligns these dipoles, causing net (large) dipole moment.

Ptotal = Pe + Pi+ Po




Mechanisms of polarization



Dielectric strength

Very high electric fields (>108 V/m) can excite electrons to the conductionband and accelerate them to such high energies that they can, in turn, freeother electrons, in an avalanche process (or electrical discharge). The fieldnecessary to start the avalanche process is called dielectric strength orbreakdown strength.



Piezoelectricity

In some ceramic materials, application of external forces produces anelectric (polarization) field and vice-versa

Applications of piezoelectric materials is based on conversion of mechanicalstrain into electricity (microphones, strain gauges, sonar detectors)

Piezoelectric materials include barium titanate BaTiO3, lead zirconatePbZrO3, quartz.

Chapter 13 - Electrical Properties

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