# Electrical Properties

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1 Electrical Properties 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 1 3 Goals of this topic: 1. Basic laws and electrical properties of metals 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: pn junctions and transistors Ionic conduction Electronic Properties of Ceramics: Dielectrics, Ferroelectrics and Piezoelectrics 2 Ohm s Law V = IR E = V / L where E is electric field intensity µ = ν/ E where µ = the mobility ν = the drift velocity Resistivity ρ = RA / L (Ω.m) Conductivity σ = 1 / ρ (Ω.m) 1 4

2 Materials Choices for Metal Conductors Electrical conductivity between different materials varies by over 27 orders of magnitude, the greatest variation of any physical property Metals: σ > 10 5 (Ω.m) 1 Semiconductors: 10 6 < σ < 10 5 (Ω.m) 1 Most widely used conductor is copper: inexpensive, abundant, very high σ lver 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 ChemicalMechanical Polishing ) Remember deformation reduces conductivity, so high strength generally means lower σ : tradeoff. Precipitation hardening may be best choice: e.g. CuBe. Heating elements require low σ (high R), and resistance to high temperature oxidation: nichrome. Insulators: σ < 10 6 (Ω.m) Conductivity / Resistivity of Metals High number of free (valence) electrons high σ Defects scatter electrons, therefore they increase ρ (lower σ). ρ total = ρ thermal ρ impurity ρ deformation ρ thermal from thermal vibrations 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: v d, 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 ρ impurity from impurities ρ deformation from deformationinduced point defects Resistivity increases with temperature (increased thermal vibrations and point defect densities) ρ T = ρ o at Additions of impurities that form solid sol: ρ I = Ac i (1c i ) (increases ρ) Two phases, α, β: ρ i = ρ α V α ρ β V β 6 8

3 E Scattering events 2. Band theory of solids: metals, semiconductors and insulators Band Theory of Solids Schroedinger s eqn (quantum mechanical equation for behavior of an electron) Net electron motion Kψ V ψ = E ψ v d = µ e E δ 2 ψ (h 2 /2m) V ψ = ih δx 2 δ ψ δt σ = n e µ e n : number of free or conduction electrons per unit volume Solve it for a periodic crystal potential, and you will find that electrons have allowed ranges of energy (energy s) and forbidden ranges of energy (gaps) Electrons in an Isolated atom (Bohr Model) (m) = Metal (s) = Semicon Mobility (RT) µ (m 2 V 1 s 1 ) Carrier Density N e (m 3 ) Na (m) x Ag (m) x Al (m) x (s) x GaAs (s) x 10 6 InSb (s) 8.00 Electron orbits defined by requirement that they contain integral number of wavelengths: quantize angular momentum, energy, radius of orbit σ metal >> σ semi 10 12

4 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 of allowed energies Each 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 cannot conduct In metals, highest occupied is partially filled or s overlap Highest filled state at 0 Kelvin is the Fermi Energy, E F Semiconductors, insulators: highest occupied filled at 0 Kelvin: electronic conduction requires thermal excitation across gap; σ T (At 0 Kelvin) highest filled : valence ; lowest empty : conduction. E f is in the gap Metals, Semiconductors, Insulators Empty Band gap Empty states Filled states Metals E f Filled Empty Insulators E g > 2 ev Empty conduction Band gap E f E f Ef Filled valence Semiconductors E g < 2 ev Empty conduction Band gap Filled valence 14 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, m l 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 16

5 The Fermi Function Metals 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 E f and equal to 0 above E f Empty states f (E) = [1] / [e (E E f ) / kt 1] Energy E F E F Electron excitation Filled states Semiconductors, Insulators 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 E f very easily (temp or electric field) High conductivity Atomistically: weak metallic bonding of electrons In semiconductors, insulators, electrons have to jump across gap into conduction to find conducting states above E f : requires jump >> kt No. of electrons in CB decreases with higher gap, lower T Relatively low conductivity An electron in the conduction leaves a hole in the valence, that can also conduct Atomistically: strong covalent or ionic bonding of electrons Energy Conduction Band Gap Valence E F Conduction Valence Free electron Electron excitation Hole in valence 18 20

6 E field free electron hole E field hole free electron Electrical conduction in intrinsic, before excitation, and (c) after excitation, see the response of the electronhole 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) Intrinsic Semiconductors: Conductivity Both electrons and holes conduct: σ = n e µ e p e µ h n: number of conduction electrons per unit volume p: number of holes in VB per unit volume In intrinsic semiconductor, n = p: σ = n e (µ e µ h ) = p e (µ e µ h ) Number of carriers (n,p) controlled by thermal excitation across gap: n = p = C exp ( E g /2 kt) C : Material constant E g : Magnitude of the gap Electrical properties of semiconductors Extrinsic Semiconductors 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 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 Engineer conductivity by controlled addition of impurity atoms: Doping Room Temp 22 24

7 ntype semiconductors E field In 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. So N As As atoms per unit volume produce n additional conduction electrons per unit volume Impurities which produce extra conduction electrons are called donors, N D = N As ~ n These additional electrons are in much greater numbers than intrinsic hole or electron concentrations, σ ~ n e µ e ~ N D e µ e Typical values of N D ~ cm 3 (Many orders of magnitude greater than intrinsic carrier concentrations at RT) P 5 B 3 hole free electron P 5 hole B 3 ntype ptype ptype semiconductors Substitution of trivalent B (or Al, Ga...) atoms in 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. So N B B atoms per unit volume produce p additional holes per unit volume Impurities which produce extra holes are called acceptors, N A = N B ~ p These additional holes are in much greater numbers than intrinsic hole or electron concentrations, σ ~ p e µ h ~ N A e µ h Typical values of N A ~ cm 3 (Many orders of magnitude greater than intrinsic carrier concentrations at RT) 26 Energy Conduction Band Gap Valence Semiconductors ntype more electrons Donor state Conduction Valence For an ntype material, excitation occurs from the donor state in which a free electron is generated in the conduction. Free electrons in the conduction 28

8 Semiconductors ptype more holes Energy Conduction Band Gap Acceptor state Conduction 4. Electrical properties of ceramics and polymers Valence Valence Hole in the valence For an ptype material, excitation of an electron into the acceptor level, leaving behind a hole in the valence Temperature Dependence of carrier Concentration and Conductivity Dielectric Materials ln p, n Intrinsic Saturation { ln p/ [ (1/T)]} = E g / 2 k 1/T Extrinsic Our basic equation: σ = n e µ e p e µ h Main temperature variations are in n,p rather than µ e, µ h Intrinsic carrier concentration n = p = C exp ( E g /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 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

9 Capacitance Capacitance is the ability to store charge across a potential difference. Examples: parallel conducting plates, semiconductor pn junction Magnitude of the capacitance, C: C = Q / V Units: Farads Parallel plate capacitor, C depends on geometry of plates and material between plates C = ε r ε o A / L A : Plate Area; L : Plate Separation D P ε o : Permittivity of Free Space (8.85x10 12 F/m 2 ) L ε r : Relative permittivity, ε r = ε /ε o Vac, ε r = 1 N 33 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 = ε o ε r ξ D : Electric Displacement (units Coulombs / m 2 ) 35 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 34 Increase in capacitance in dielectric medium compared to vacuum is due to polarization of electric dipoles in dielectric. In absence of applied field, 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 Q t = QQ. So, C = Q t / V has increased 36

10 Surface density charge now D = εξ = ε o ε r ξ = ε o ξ P Electronic P is the polarization of the material (units Coulombs/m 2 ). 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 Ionic From equations at top of page P = ε o (ε r 1)ξ Orientation 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. P tptal = P e P i P o Barium Titanate, BaTiO 3 : Permanent Dipole Moment for T < 120 C (Curie Temperature, T c ). Above T c, unit cell is cubic, no permanent electric dipole moment 38 40

11 Piezoelectricity In some ceramic materials, application of external forces produces an electric (polarization) field and viceversa Applications of piezoelectric materials microphones, strain gauges, sonar detectors Materials include barium titanate, lead titanate, lead zirconate Ionic Conduction in Ceramics Electrical Properties of Polymers 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 σ total = σ electronic σ ionic Overall conductivities, however, remain very low in ceramics. 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 ntype 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 42 44

12 5. Semiconductor Devices and Circuits P V b D V o Applied Voltage D P N N V b Reverse Bias Forward Bias V o V b Ec E c0 V o V o V b E c0 E F0 E c E F E v E v0 E v0 E v 45 Lower Barrier, I Higher Barrier, I 47 The Semiconductor pn Junction Diode P D ξ N n p V h A rectifier or diode allows current to flow in one direction only. pn junction diode consists of adjacent p and ndoped semiconductor regions Electrons, holes combine at junction and annihilate: depletion region containing ionized dopants Electric field, potential barrier resists further carrier flow V e 46 48

13 Transistors MOSFET (MetalOxideSemiconductor Field Effect Transistor) 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 transistors in total: enough for several for every man, woman, and child on the planet Cost to consumer ~ c each. Achieved through submicron engineering of semiconductors, metals, insulators and polymers. Requires ~ \$2 billion for a stateoftheart fabrication facility 49 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 sourcedrain is therefore modulated by gate voltage. Put input signal onto gate, output signal (sourcedrain current) is correspondingly modulated: amplification and switching Stateoftheart gate lengths: 0.18 micron. Oxide layer thickness < 10 nm 51 Bipolar Junction Transistor npn or pnp sandwich structures. Emitterbasecollector. Base is very thin (~ 1 micron or less) but greater than depletion region widths at pn junctions. Emitterbase 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 baseemitter voltage causes a relatively large change in emitterbasecollector 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 50 Take Home Messages Language: Resistivity, conductivity, mobility, drift velocity, electric field intensity, energy s, gap, conduction, valence, 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, pn 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 s, gaps, holes, differences between metals, semiconductors and insulators Semiconductors: Dependence of intrinsic and extrinsic carrier conc. on temperature, gap; dopants acceptors and donors. Capacitance: Dielectrics, polarization and its causes, piezoelectricity Semiconductor devices: basic construction and operation of pn junctions, bipolar transistors and MOSFETs 52

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