# Basic laws and electrical properties of metals (I) Electrical properties. Basic laws and electrical properties of metals (II)

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1 Electrical properties Electrical conduction How many moveable electrons are there in a material (carrier density)? How easily do they move (mobility)? Semiconductivity Electrons and holes Intrinsic and extrinsic carriers Semiconductor devices: p-n junctions and transistors Conduction in polymers and ionic materials 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, a current of magnitude I [amperes, C/s] flows. In most metals, at low values of V, the current is proportional to V, and can be described by Ohm's law: I = V/R where R is the electrical resistance [ohms, Ω, V/A]. R depends on the intrinsic resistivity ρ of the material [Ωm] and on the geometry (length l and area A through which the current passes): R = ρl/a Optional reading: 18.14, 18.15, 18.21, In most materials (e.g. metals), the current is carried by electrons (electronic conduction). In ionic crystals, the charge carriers are ions (ionic conduction). 2 Basic laws and electrical properties of metals (II) The electrical conductivity (the ability of a substance to conduct an electric current) is the inverse of the resistivity: σ = 1/ρ Since 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 = σ E Electrical conductivity varies between different materials by over 27 orders of magnitude, the greatest variation of any physical property σ (Ω.cm) -1 Metals: σ > 10 5 (Ω.m) -1 Semiconductors: 10-6 < σ < 10 5 (Ω.m) -1 Insulators: σ < 10-6 (Ω.m)

2 Energy Band Structures in Solids (I) In an isolated atom electrons occupy well defined energy states, as discussed in Chapter 2. When atoms come together to form a solid, their valence electrons interact with each other and with nuclei due to Coulomb forces. In addition, two specific quantum mechanical effects happen. First, by Heisenberg's uncertainty principle, constraining the electrons to a small volume raises their energy, this is called promotion. The second effect, due to the Pauli exclusion principle, limits the number of electrons that can have the same energy. As a result of these effects, the valence electrons of atoms form wide electron energy s when they form a solid. The s are separated by gaps, where electrons cannot exist. Energy Band Structures and Conductivity The highest state at 0 K Fermi Energy (E F ) The two highest energy s are: Valence the highest where the electrons are present at 0 K Conduction - a partially or empty energy where the electrons can increase their energies by going to higher energy levels within the when an electric field is applied un s Energy conduction valence gap s 5 6 Energy Band Structures and Conductivity (metals) In metals (conductors), highest occupied is partially or s overlap. Conduction occurs by promoting electrons into conducting states, that starts right above the Fermi level. The conducting states are separated from the valence by an infinitesimal amount. Energy provided by an electric field is sufficient to excite many electrons into conducting states. Partially Overlapping s Energy Energy empty GAP empty partly states states Energy Band Structures and Conductivity (semiconductors and insulators) In semiconductors and insulators, the valence is, no more electrons can be added (Pauli's principle). Electrical conduction requires that electrons be able to gain energy in an electric field. To become free, electrons must be promoted (excited) across the gap. The excitation energy can be provided by heat or light. Insulators: wide gap (> 2 ev) Energy states empty conduction GAP valence Semiconductors: narrow gap (< 2 ev) Energy? states empty conduction GAP valence Cu 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 Mg 1s 2 2s 2 2p 6 3s

3 Energy Band Structures and Conductivity (semiconductors and insulators) In semiconductors and insulators, electrons have to jump across the gap into conduction to find conducting states above E f The energy needed for the jump may come from heat, or from irradiation at sufficiently small wavelength (photoexcitation). The difference between semiconductors and insulators is that in semiconductors electrons can reach the conduction at ordinary temperatures, where in insulators they cannot. The probability that an electron reaches the conduction is about exp(-e g /2kT) where E g is the gap. If this probability is < one would not find a single electron in the conduction in a solid of 1 cm 3. This requires E g /2kT > 55. At room temperature, 2kT = 0.05 ev E g > 2.8 ev corresponds to an insulator. An electron promoted into the conduction leaves a hole (positive charge) in the valence, that can also participate in conduction. Holes exist in metals as well, but are more important in semiconductors and insulators. 9 Energy Band Structures and Bonding (metals, semiconductors, insulators) Relation to atomic bonding: Insulators valence electrons are tightly bound to (or shared with) the individual atoms strongest ionic (partially covalent) bonding. Semiconductors - mostly covalent bonding somewhat weaker bonding. Metals valence electrons form an electron gas that are not bound to any particular ion. 10 Energy Band Structures and Conductivity (metals, semiconductors, insulators) Metals 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 of obstacles the electron speeds up continuously in an electric field. This is the case in vacuum (e.g. inside a TV tube) or in a perfect crystal (this is a conclusion from quantum mechanics). In a real solid, the electrons scatter by collisions with imperfections and due to atomic thermal vibrations. frictional forces resistance a net drift velocity of electron motion is established: v d = μ e E where μ e electron mobility [m 2 /V-s]. The friction transfers part of the energy supplied by the electric field into the lattice as heat. That is how electric heaters work. E Scattering events Net electron motion

4 Electron Mobility Electrical conductivity is proportional to number of free electrons and electron mobility: σ = n e μ e n - number of free or conduction electrons per unit volume (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 Si (s) x GaAs (s) x 10 6 InSb (s) 8.00 Conductivity / Resistivity of Metals The resistivity ρ is defined by scattering events due to the imperfections and thermal vibrations. Total resistivity ρ tot can be described by the Matthiessen rule: ρ total = ρ thermal +ρ impurity +ρ deformation where ρ thermal - from thermal vibrations, ρ impurity - from impurities, ρ deformation - from deformation-induced defects resistivity increases with temperature, with deformation, and with alloying. n metal >> n semi μmetal < μ semi σ metal >> σ semi Conductivity / Resistivity of Metals Influence of temperature: Resistivity rises linearly with temperature (increasing thermal vibrations and density of vacancies) ρ T = ρ o + at Conductivity / Resistivity of Metals Influence of plastic deformation: Influence of impurities: Impurities that form solid solution ρ i = Ac i (1-c i ) where c i is impurity concentration, A composition independent constant Two-phase alloy (α and β phases) rule-of-mixtures: ρ i = ρ α V α + ρ β V β Resistivity, ρ (10-8 Ohm-m) Cu-Ni alloy Composition, wt% Ni 15 Normally, the influence of plastic deformation on electrical resistivity is weaker than the influence of temperature and impurities In general, presence of any imperfections (crystal defects) increases resistivity -- grain boundaries -- dislocations -- impurity atoms -- vacancies 16 4

5 Materials of Choice for Metal Conductors One of the best material for electrical conduction (low resistivity) is silver, but its use is restricted due to the high cost Most widely used conductor is copper: inexpensive, abundant, high σ, but rather soft cannot be used in applications where mechanical strength is important. Solid solution alloying and cold working inprove strength but decrease conductivity. Precipitation hardening is preferred, e.g. Cu-Be alloy When weight is important one uses aluminum, which is half as good as Cu and more resistant to corrosion. Heating elements require low σ (high R), and resistance to high temperature oxidation: nickelchromium alloy Semiconductivity Intrinsic semiconductors - electrical conductivity is defined by the electronic structure of pure material. Extrinsic semiconductors - electrical conductivity is defined by impurity atoms Intrinsic semiconductors (I) Examples: Si, Ge, GaP, GaAs, InSb, CdS, ZnTe Number of electrons in conduction increases exponentially with temperature: n= C T 3/2 C is a material constant exp(-e g /2kT) E g is the gap width Conduction Valence T = 0 K E g T = 300 K Conducting Electrons Holes (positive charge carriers An electron promoted into the conduction leaves a hole (positive charge) in the valence. In an electric field, electrons and holes move in opposite direction and participate in conduction. In Si (E g = 1.1 ev) one out of every atoms contributes an electron to the conduction at room temperature. Intrinsic semiconductors (II) Since both electrons and holes are charge carriers in an intrinsic semiconductor, the conductivity is σ = n e μ e + p e μ h where p is the hole concentration and μ h the hole mobility. Electrons are more mobile than holes, μ e > μ h In an intrinsic semiconductor, a hole is produced by the promotion of each electron to the conduction. Therefore, n = p and σ = n e (μ e + μ h ) = p e (μ e + μ h ) (only for intrinsic semiconductors) n (and p) increase exponentially with temperature, whereas μ e and μ h decrease (about linearly) with temperature. The conductivity of intrinsic semiconductors is increasing with temperature (different from metals!)

6 Intrinsic semiconductors (III) Intrinsic semiconductors (IV) Let s calculate carrier concentration for Si at 300 K σ = n e (μ e + μ h ) n = n = e ( Ω m 2 C ( )m V s σ μ 1 1 e + μ h ) = V molar volume of Si 12 cm 3 /mol Ω = A N A atoms/mol / = atoms/m 3 fraction of excited electrons per atom ~ m 3 C A = s Extrinsic semiconductors Extrinsic semiconductors - electrical conductivity is defined by impurity atoms. Example: Si is considered to be extrinsic at room T if impurity concentration is one impurity per lattice sites (remember our estimation of the number of electrons promoted to the conduction by thermal fluctuations at 300 K) Unlike intrinsic semiconductors, an extrinsic semiconductor may have different concentrations of holes and electrons. It is called p-type if p > n and n-type if n > p. One can engineer conductivity of extrinsic semiconductors by controlled addition of impurity atoms doping (addition of a very small concentration of impurity 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 more valence electron per atom than the semiconductor matrix. Example: phosphorus (or As, Sb..) with 5 valence electrons, is an electron donor in Si since only 4 electrons are used to bond to the Si lattice when it substitutes for a Si atom. Fifth outer electron of P atom is weakly bound in a donor state (~ 0.01 ev) and can be easily promoted to the conduction. Impurities which produce extra conduction electrons are called donors, N D = N Phosphorus ~ n Elements in columns V and VI of the periodic table are donors for semiconductors in the IV column, Si and Ge

7 n-type extrinsic semiconductors (II) p-type extrinsic semiconductors (I) Excess holes are produced by substitutional impurities that have fewer valence electrons per atom than the matrix. A bond with the neighbors is incomplete and can be viewed as a hole weakly bound to the impurity atom. Elements in columns III of the periodic table (B, Al, Ga) are donors for semiconductors in the IV column, Si and Ge. Impurities of this type are called acceptors, N A = N Boron ~p The hole created in donor state is far from the valence and is immobile. Conduction occurs mainly by the donated electrons (thus n-type). σ ~ n e μ e ~ N D e μ e (for extrinsic n-type semiconductors) p-type extrinsic semiconductors (II) Temperature variation of conductivity Basic equation for conductivity: σ = n e μ e + p e μ h Therefore, the temperature dependence of thermal conductivity is defined by the temperature dependences of carrier concentration and mobility Carrier concentration vs T: Intrinsic semiconductors The energy state that corresponds to the hole (acceptor state) is close to the top of the valence. An electron may easily hop from the valence to complete the bond leaving a hole behind. Conduction occurs mainly by the holes (thus p-type). σ ~ p e μ p ~ N A e μ p (for extrinsic p-type semiconductors) For intrinsic semiconductors n = p ~ exp(-e g /2kT) E g = 0.57 ev for Ge E g = 1.11 ev for Si

8 Carrier concentration vs T: Extrinsic semiconductors Carrier mobility Dopants are impurities and, like in metals, the mobility of carriers decreases with impurity concentration n-type Si doped with m -3 P At dopant levels less than m -3 the effect of dopants on mobility is negligible Extrinsic region: All P donor state electrons are excited Freeze-out region: Thermal energy is too low for exciting the electrons from P donor states to the conduction Intrinsic region: Excitations across the gap dominate Carrier mobility For dopant concentrations below m -3, mobility of both electrons and holes decreases with increasing temperature due to the enhanced thermal scattering. Temperature variation of conductivity Basic equation for conductivity: σ = n e μ e + p e μ h In intrinsic semiconductors, the temperature dependence of mobilities, μ e and μ h is weak as compared to the strong exponential dependence of carrier concentration

9 Conduction in Polymers and Ionic Materials Ionic Materials In ionic materials, the gap is large and only very few electrons can be promoted to the valence by thermal fluctuations. Cation and anion diffusion can be directed by the electric field and can contribute to the total conductivity: σ total = σ electronic + σ ionic 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. Capacitance When a voltage V is applied to two parallel conducting plates, the plates are charged 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 Q per applied voltage V: C = Q / V [Farads] For a parallel-plate capacitor, C depends on geometry of plates and material between plates C = ε r ε o A / L = ε A / L where A is the area of the plates, L is the distance between plates, ε is the permittivity of the dielectric medium, ε o is the permittivity of a vacuum (8.85x10-12 F/m 2 ), and ε r is relative permittivity (or dielectric constant) of the material, ε r = ε / ε o = C / C vac Dielectric Materials The dielectric constant of vacuum is 1 and is close to 1 for air and many other gases. But when a piece of a dielectric material is placed between the two plates in capacitor the capacitance can increase significantly. Dielectric Materials Dipole formation and/or orientation along the external electric field in the capacitor causes a charge redistribution so that the surface nearest to the positive capacitor plate is negatively charged and vice versa. C = ε r ε o A / L with ε r = 81 for water, 20 for acetone, 12 for silicon, 3 for ice, etc. A dielectric material is an insulator in which electric dipoles can be induced by the electric field (or permanent dipoles can exist even without electric field), that is where positive and negative charge are separated on an atomic or molecular level P Q 0 + Q Q 0 -Q region of no net charge net negative charge at the surface, -Q net negative charge at the surface, Q = PA + _ + _ + d _ Dipole formation induces additional charge Q on plates: total plate charge Q t = Q+Q. _ + Therefore, C = Q t / V has increased and dielectric constant of the material ε r = C / C vac > 1 Magnitude of electric dipole moment is p = q d The process of dipole formation/alignment in electric field is called polarization and is described by P = Q /A

10 Dielectric Materials Mechanisms of polarization In the capacitor surface charge density (also called dielectric displacement) is D = Q/A = ε r ε o E= ε o E+ P Polarization is responsible for the increase in charge density above that for vacuum Mechanisms of polarization (dipole formation/orientation) electronic (induced) polarization: Applied electric field displaces negative electron clouds with respect to positive nucleus. Occurs in all materials. ionic (induced) polarization: In ionic materials, applied electric field displaces cations and anions in opposite directions molecular (orientation) polarization: Some materials possess permanent electric dipoles (e.g. H 2 O). In absence of electric field, dipoles are randomly oriented. Applying electric field aligns these dipoles, causing net (large) dipole moment. P total = P e + P i + P o electronic polarization ionic polarization molecular (orientation) polarization Dielectric strength Very high electric fields (>10 8 V/m) can excite electrons to the conduction and accelerate them to such high energies that they can, in turn, free other electrons, in an avalanche process (or electrical discharge). The field necessary to start the avalanche process is called dielectric strength or breakdown strength. Piezoelectricity In some ceramic materials, application of external forces produces an electric (polarization) field and vice-versa Applications of piezoelectric materials is based on conversion of mechanical strain into electricity (microphones, strain gauges, sonar detectors) Examples: barium titanate BaTiO 3 lead zirconate PbZrO 3 quartz. Summary Make sure you understand language and concepts: Acceptor state Capacitance Conduction Conductivity, electrical Dielectric constant Dielectric strength Donor state Doping Electrical resistance Electron energy Energy gap Extrinsic semiconductor Free electron Hole Insulator Intrinsic semiconductor Matthiessen s rule Metal Charge carrier mobility Ohm s law Permittivity Piezoelectric Polarization Resistivity, electrical Semiconductor Valence stress-free with applied stress

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