Electrical Behavior • Askeland and Phule , the Science and Engineering of Materials, 4Th Topic 3 Ed., Chapter 18

Reading assignment

• Chung, Multifunctional cement- based Materials, Ch. 2. Electrical behavior • Askeland and Phule , The Science and Engineering of Materials, 4th Topic 3 Ed., Chapter 18.

Supplementary reading

Shackelford, Materials Science for Engineers, 6th Ed., Ch. 15.

Figure 18.2 (a) Charge carriers, such as Learning electrons, are deflected by atoms or defects and take an irregular path through a conductor. The average rate at which the carriers move is the drift velocity v. (b) Valence electrons in the metallic bond move 2003 Brooks/Cole, a division of Thomson Learning, Inc. easily. (c) Covalent bonds must be broken in © semiconductors and insulators for an electron to be able to move. (d) Entire ions must diffuse to carry charge in many ionically bonded materials.

Mean free path – The average distance that electrons can move without being scattered by other atoms.

2 Current

charge I = . time 1 coulomb 1 ampere = . sec Current density ~ I J = A

Current density – The current flowing through per unit cross- sectional area.

3 Electrical resistance V R = l I R µ l I (I/A) s = = A A V (V / l) 1 l ~ J R = s = s A (V / l) where s is the electrical conductivity

Electric field – The voltage gradient or volts per unit length.

dV V = - Drift velocity dx l

Electric field v µ E dV E = - dx V dV v = m E = - = E l dx where µ is the mobility ~ J s = E

4 • Drift velocity - The average • Current density - The current flowing through per unit rate at which electrons or cross-sectional area. • Electric field - The voltage gradient or volts per unit other charge carriers move length. • Drift velocity - The average rate at which electrons or through a material under the other charge carriers move through a material under influence of an electric or the influence of an electric or magnetic field. • Mobility - The ease with which a charge carrier moves magnetic field. through a material. • Dielectric constant - The ratio of the permittivity of a • Mobility - The ease with material to the permittivity of a vacuum, thus which a charge carrier moves describing the relative ability of a material to polarize and store a charge; the same as relative permittivity. through a material.

I = qnvA v = m ~ I qnvA J = = = qnv E A A ~ s = q n m J qnv s = = E E

Flux dn Flux J n = - D n dp dx J p = -Dp dx Current density ~ Current density Jn = (- q) J n ~ æ dp ö dp ~ æ dn ö Jp = q J p = q ç - D p ÷ = -q D p . J n = (- q) ç - D n ÷ dx dx è dx ø è ø dn = q D n . dx

5 Energy levels of an isolated atom Einstein relationship

D D p kT n = = mn m p q

6 Energy bands of solid sodium is a trademark used herein under license. ™ Learning 2003 Brooks/Cole, a division of Thomson Learning, Inc. ©

Energy bands of an insulator is a trademark used herein under license. ™ Learning 2003 Brooks/Cole, a division of Thomson Learning, Inc. ©

• Valence band - The energy levels filled by electrons in their lowest energy states. • Conduction band - The unfilled energy levels into which electrons can be excited to provide conductivity. • Energy gap (Bandgap) - The energy between the top of the valence band and the bottom of the conduction band that a charge carrier must obtain before it can transfer a charge.

8 • Holes are in the Holes - Unfilled energy valence band. levels in the valence band. Because • Conduction electrons move to fill electrons are in the these holes, the holes conduction band. move and produce a current.

Electrical conduction through a composite material consisting of Radiative recombination - three components (1, 2 and 3) that are in a parallel configuration Recombination of holes and 1 2 3 electrons that leads to - emission of light; this occurs in direct bandgap materials. l V I

9 Resistance due to component i Total resistance l l R i = r i R = r|| A i (A1 + A 2 + A 3 ) Current through component i Total current

VA V V (A 1 + A 2 + A 3 ) i I = = I i = R r ||l r i l

Total current through the composite V æ A A A ö V V æ A A A ö 1 2 3 I = I + I + I = 1 + 2 + 3 ç + + ÷ = (A + A + A ) 1 2 3 ç ÷ ç ÷ 1 2 3 l ç r r r ÷ l r r r r l è 1 2 3 ø è 1 2 3 ø ||

Electrical conduction through a composite material consisting of three components (1, 2 and 3) that are in a series configuration - 1 1 A 1 A = 1 + 2 + r r (A + A + A ) r (A + A + A ) || 1 1 2 3 2 1 2 3 1 1 A3 2 V I r3 (A1 + A 2 + A3 ) 3 1 1 1 = f1 + f2 + f3 , + r1 r2 r3 Rule of Mixtures Area A

Total resistance Vi = IR i L + L + L L ( 1 2 3 ) i R = r = Ir i ^ A A

Total voltage drop Total voltage drop I V = IR V = (r L + r L + r L ) A 1 1 2 2 3 3 (L + L + L ) = Ir 1 2 3 ^ A

10 Total voltage drop I I V = (r1 L1 + r 2 L 2 + r 3 L 3 ) (r L + r L + r L ) A A 1 1 2 2 3 3

V = IR (L + L + L ) 1 2 3 L + L + L = Ir ( ) ^ A = Ir 1 2 3 ^ A

çæ r L + r L + r L ÷ö è 1 1 2 2 3 3 ø r = Conduction through a ^ L + L + L composite material with an ( 1 2 3 ) insulating matrix and short conductive fibers = ? 1f 1 + ? 2 f 2 + ? 3 f 3

Rule of Mixtures

Percolation threshold is a trademark used herein under license. ™ Minimum volume fraction of Learning conductive fibers (or particles) for adjacent fibers (or particles) to touch each other and form a 2003 Brooks/Cole, a division of Thomson Learning, Inc. © continuous conductive path.

11 Contact resistance Conduction through 1 Rc µ an interface A

rc Rc = A

where ?c is the contact resistivity

Energy bands of an intrinsic semiconductor Intrinsic silicon

Without thermal With thermal excitation excitation Without thermal With thermal excitation excitation

Electrical conductivity of a semiconductor

s = q n mn + q p mp , For an intrinsic semiconductor (n = p), where q = magnitude of the charge of an electron, n = number of conduction electrons per unit volume, p = number of holes per unit volume, m = mobility of conduction electrons, s = q n (m + m ) . n n p and mp = mobility of conduction holes.

12 • Intrinsic semiconductor - A semiconductor in Current density due to both which properties are controlled by the element an electric field or compound that makes the semiconductor and a concentration gradient and not by dopants or impurities. ~ dn • Extrinsic semiconductor - A semiconductor J n = qn m n E + qD n . prepared by adding dopants, which determine dx the number and type of charge carriers. ~ dp • Doping - Deliberate addition of controlled J p = qp m p E - qD p . dx amounts of other elements to increase the number of charge carriers in a semiconductor. ~ ~ ~ J = Jn + Jp .

Extrinsic semiconductor (doped with an electron donor)

Without thermal With thermal excitation excitation

Extrinsic semiconductor Energy bands (doped with an electron acceptor)

Extrinsic semiconductor Intrinsic (doped with an electron semiconductor donor) Without thermal With thermal excitation excitation

13 Energy bands

Extrinsic semiconductor Intrinsic (doped with an electron

Defect semiconductor

(excess semiconductor Zn1+xO)

Zn+ ion serves as an electron donor.

Energy bands of Zn1+xO Defect semiconductor

(deficit semiconductor Ni 1-xO)

Ni3+ ion serves as an electron acceptor.

14 Energy bands of Ni1-xO

c Consider an n-type semiconductor being n = illuminated. l Illumination increases conduction E µ n electrons and holes by equal number, E = h n (Photon energy) since electrons and holes are generated where h = Planck’s constant in pairs. = 6.6262 x 10-34 J.s Thus, the minority carrier concentration

(pn) is affected much more than the Photon energy must be at least equal to the energy majority carrier concentration (nn). bandgap in order for electrons to be excited from the valence band to the conduction band.

15 • Temperature Effect - When the temperature of a metal increases, thermal energy causes the atoms to vibrate • Effect of Atomic Level Defects -

©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™is a trademark used herein under license. Imperfections in crystal structures scatter electrons, reducing the mobility and conductivity of the metal

Change of resistivity with temperature for a metal Dr = aDT r

where a = temperature coefficient of electrical resistivity

16 Matthiessen’s rule – The resistivity of a metallic material is given by the addition of a base resistivity that accounts for the effect of temperature, and a temperature independent term that reflects the effect of atomic level defects, including impurities forming solid solutions.

17 Effect of Processing and Strengthening

s = q n m

For a metal, s decreases with increasing temperature is a trademark used herein under license. ™ because µ decreases with increasing temperature. Learning For a semiconductor, s increases with increasing temperature because n and/or p increases with increasing temperature. 2003 Brooks/Cole, a division of Thomson Learning, Inc. ©

18 For a semiconductor Taking natural logarithms, - E /2kT -E /2kT n µ e g , g s = s oe . Eg where Eg = energy band gap between conduction and ln s = ln s o - . valence bands, 2kT k = Boltzmann's constant, Changing the natural logarithms to and T = temperature in K. logarithms of base 10,

The factor of 2 in the exponent is because the E g excitation of an electron across Eg produces an logs = logs o - . intrinsic conduction electron and an intrinsic hole. (2.3)2kT

Thermistor – Conductivity of an ionic solid A semiconductor device that s = q n m + q n m = q n (m + m ) , is particularly sensitive to C A C A changes in temperature, where n = number of Schottkydefects permitting it to serve as an per unit volume accurate measure of mC = mobility of cations, temperature. mA = mobility of anions.

+ - An n-type semiconductor D ® D +e , ne = ND + ,

n=ni +ne , -Eg / 2 kT n i µ e .

- /kT where n = total concentration of conduction µ ED . electrons, ne e n = concentration of intrinsic conduction i ni < < n e . electrons,

ne = concentration of extrinsic conduction p = p i . electrons.

19 Before donor exhaustion No extrinsic holes, thus

ni < < n e . p = pi .

However,

pi = ni

Thus,

p = ni

At high temperatures n @ n e (i.e., donor exhaustion), p @ 0 . n @ ni s = qn + qp . m n m p

s @ qn mn

Arrhenius plot of log conductivity vs. 1/T, where T is temperature in K.

Extrinsic semiconductor (doped with an electron donor)

20 Arrhenius plot of log conductivity vs. 1/T, A p-type semiconductor where T is temperature in K

p = pi + pe , where p = total concentration of conduction holes pi = concentration of intrinsic holes, pe = concentration of extrinsic holes. Extrinsic semiconductor (doped with an electron acceptor)

- - A + e ® A , - + A ® + , A h pi < < pe pe = NA- ,

-Eg / 2 kT pi µ e , before acceptor saturation -EA /kT pe µ e .

n = n i .

n = pi .

p @ pe n @0 . The mass-action law

Product of n and p is a before acceptor saturation constant for a particular semiconductor at a particular temperature

21 Intrinsic semiconductor Consider an n-type semiconductor.

n = n i = p i = p . n @ne = ND+ 2 N = N (Donor exhaustion) np = n i . D+ D

10 -3 n @ N D . n i = 1.5´10 cm forSi 2 2 n = 2.5´1013 cm -3for Ge . n n i p = i = i . This equation applies n ND whether the semiconductor is doped or not.

The pn junction Rectification

A pn junction at bias voltage V=0

22 • Diodes, transistors, lasers, and LEDs are made using semiconductors. Silicon is the workhorse of very large n-p-n bipolar junction transistor scale integrated (VLSI) circuits. • Forward bias - Connecting a p-n junction device so that the p-side is connected to positive. Enhanced diffusion occurs as the energy barrier is lowered, permitting a considerable amount of current can flow under forward bias. • Reverse bias - Connecting a junction device so that the p-side is connected to a negative terminal; very little current flows through a p-n junction under reverse bias. • Avalanche breakdown - The reverse-bias voltage that causes a large current flow in a lightly doped p-n junction. • Transistor - A semiconductor device that can be used to amplify electrical signals.

• Superconductivity - Flow of current through a Superconductivity material that has no resistance to that flow. • Applications of Superconductors - Electronic

is a trademark used herein under license. circuits have also been built using ™

Learning superconductors and powerful superconducting electromagnets are used in magnetic resonance imaging (MRI). Also, very low electrical-loss components, known 2003 Brooks/Cole, a division of Thomson Learning, Inc.

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