Electrical Properties ISSUES TO ADDRESS... • How are electrical conductance and resistance characterized? • What are the physical phenomena that distinguish conductors, semiconductors, and insulators? • For metals, how is conductivity affected by imperfections, T, and deformation? • For semiconductors, how is conductivity affected by impurities (doping) and T? 1 1

View of an Integrated Circuit • Scanning electron microscope images of an IC: 0.5 mm (a) (d) 45 m Al Si (doped) • A dot map showing location of Si (a semiconductor): -- Si shows up as light regions. (b) • A dot map showing location of Al (a conductor): -- Al shows up as light regions. (c) Fig. (d) from Fig. 18.27 (a), Callister 7e. (Fig. 18.27 is courtesy Nick Gonzales, National Semiconductor Corp., West Jordan, UT.) Fig. (a), (b), (c) from Fig. 18.0, Callister 7e. 2 2

Electrical Conduction • Ohm's Law: V = I R voltage drop (volts = J/C) C = Coulomb resistance (Ohms) current (amps = C/s) I e - A (cross sect. area) V L • Resistivity,  and Conductivity, : -- geometry-independent forms of Ohm's Law E: electric field intensity resistivity (Ohm-m) J: current density conductivity -- Resistivity is a material property & is independent of sample • Resistance: 3 3

Electrical Conduction 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]. R depends on the intrinsic resistivity ρ of the material [Ω-m] and onthe geometry (length l and area A through which the current passes): R = ρl/A In most materials (e.g. metals), the current is carried by electrons (electronic conduction). In ionic crystals, the charge carriers are ions (ionic conduction). 4 4

Electrical Properties Which will conduct more electricity? Analogous to flow of water in a pipe So resistance depends on sample geometry, etc. D 2D 5 5

Definitions J =   <= another way to state Ohm’s law Further definitions J =   <= another way to state Ohm’s law J  current density   electric field potential = V/ or (V/  ) Electron flux conductivity voltage gradient J =  (V/ ) Current carriers electrons in most solids ions can also carry (particularly in liquid solutions) 6 6

Conductivity: Comparison -1 -1 • Room T values (Ohm-m) = ( - m) METALS conductors Polystyrene <10 -14 Polyethylene 10 -15 -10 -17 Soda-lime glass 10 Concrete 10 -9 Aluminum oxide <10 -13 CERAMICS POLYMERS insulators -11 7 Silver 6.8 x 10 7 Copper 6.0 x 10 7 Iron 1.0 x 10 Up to 27 orders of magnitude, possibly widest variation in materials properties Silicon 4 x 10 -4 Germanium 2 x 10 GaAs 10 -6 SEMICONDUCTORS semiconductors Selected values from Tables 18.1, 18.3, and 18.4, Callister 7e. 7 7

Conductivity / Resistivity 8

Example: Conductivity Problem What is the minimum diameter (D) of the wire so that V < 1.5 V? 100m - - e I = 2.5A + Cu wire V < 1.5V 2.5A 6.07 x 10 (Ohm-m) 7 -1 100m Solve to get D > 1.87 mm 9 9

Energy Band Structures in Solids Electrical conductivity in materials are strongly related to # of electrons available for conduction Not all electrons will move in a material under an applied potential difference ! More detail in Quantum Mechanics ! In an isolated atom electrons occupy well defined energy states, as discussed in Chapter 2. Shells (1,2,3….), subshells (s,p,d,f…..), states in subshells, spin states When atoms come together to form a solid (N =12 atoms), a bonded regular atomic arrangement, their valence electrons interact with each other and with nuclei due to Coulombic 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 at distinct energy states get split into closely spaced electron states and form wide electron energy bands when they form a solid. These bands are separated by gaps, where electrons cannot exist. 10

Energy Band Structures in Solids The extend of energy states splitting depends on inter-atomic distance Outermost shells start forming bands as atoms get closer, however core shells may not form bands as still far from each other. Band gaps, depending upon the inter-atomic distances also start forming 11

Electronic Band Structures So the individual atomic energy levels interact to form molecular energy levels In solids gap between states could be 10-10 eV apart Adapted from Fig. 18.2, Callister 7e. 12 12

Band Structure Valence band – filled – highest occupied energy levels Conduction band – empty – lowest unoccupied energy levels Conduction band valence band contains valence electrons from the atoms Adapted from Fig. 18.3, Callister 7e. 13 13

Band Structure defines Electrical Properties insulator conductor >2eV <2eV Metals, single s valence electron N atoms => 4s band capable of 2N electrons => but only one s electron means half filled band structure. Metals with overlaping bands. Eg. Both s valence electrons exists in s subshell. In x’tal structure s and p subshells overlap, but p shell is empty. 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. 14

Energy States: Insulators & Semiconductors -- Higher energy states not accessible due to gap (> 2 eV). • Semiconductors: -- Higher energy states separated by smaller gap (< 2 eV). Energy filled band valence empty filled states GAP ? Energy filled band valence empty filled states GAP 15 15

Semiconductors and Insulators In semiconductors and insulators, electrons have to jump/move across the band gap into conduction band to find conducting states above Ef The energy needed for the jump may originate from heat, or from irradiation at sufficiently small wavelength. The difference between semiconductors and insulators is that in semiconductors electrons can reach the conduction band at ordinary temperatures, where in insulators they cannot. Eg is too large for insulators to have thermally or optically exited electrons promote to the conduction band. The probability that an electron reaches the conduction band is about exp(- Eg/2kT) where Eg is the band gap. If this probability is < 10-24 one would not find a single electron in the conduction band in a solid of 1 cm3. How come ? Remember NAv ~ 1024 This requires Eg/2kT > 55. At room temperature, 2kT = 0.05 eV, Eg > 2.8 eV corresponds to an insulator. An electron promoted into the conduction band leaves a hole (positive charge) in the valence band, that can also participate in conduction. Holes exist in metals as well, but are more important in semiconductors and insulators. 16

Charge Carriers Two charge carrying mechanisms Adapted from Fig. 18.6 (b), Callister 7e. Two charge carrying mechanisms Electron – negative charge Hole – equal & opposite positive charge Move at different speeds - drift velocity Higher temp. promotes more electrons into the conduction band   as T Electrons scattered by impurities, grain boundaries, etc. 17 17

Conduction & Electron Transport • Metals (Conductors): -- Thermal energy puts many electrons into a higher energy state. + - • Energy States: -- for metals nearby energy states are accessible by thermal fluctuations. filled band Energy partly valence empty GAP filled states Energy filled band valence empty filled states 18 18

Conduction in Metals In metals, highest occupied band is partially filled or bands overlap. Conduction occurs by promoting electrons into conducting states, that starts right above the Fermi level. The conducting states are separated from the valence band by an infinitesimal amount (~10-10 eV). Energy provided by an electric field is sufficient to excite many electrons into conducting states. => High conductivity. 19

Summary 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. Remember electro-negativity. Semiconductors - mostly covalent bonding somewhat weaker bonding. Sharing of electrons. Metals – valence electrons form an “electron gas” that are not bound to any particular ion. 20

Electron Mobility The force acting on the electron is -eE, where e is the electric charge and F=ma. As long as the electric field is present, in the absence of obstacles the electron is expected to speed up continuously in an electric field. So, is the case in vacuum (e.g. inside a TV tube) or in a perfect crystal . In a real solid, electrons motion are hindered by defects (dislocations, impurities, vacancies, etc), and even thermal vibrations of atoms. Electrons scatter by collisions with imperfections and due to atomic thermal vibrations. “frictional forces” => resistance => a net drift velocity of electron motion is established: where μe – electron mobility [m2/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. 21

Electron Mobility σ = Ne |e| μe Electrical conductivity is proportional to number of free electrons and electron mobility: σ = Ne |e| μe Ne - number of “free” or conduction electrons per unit volume e is the absolute charge on a free electron 1.609 x 10-19 C 22

Metals: Resistivity vs T, Impurities Read pages 621- 623 • Imperfections increase resistivity -- grain boundaries -- dislocations -- impurity atoms -- vacancies These act to scatter electrons so that they take a less direct path. deformed Cu + 1.12 at%Ni T (°C) -200 -100 Cu + 3.32 at%Ni Cu + 2.16 at%Ni 1 2 3 4 5 6 Resistivity,  (10 -8 Ohm-m) Cu + 1.12 at%Ni “Pure” Cu • Resistivity increases with: -- temperature -- wt% impurity -- %CW  = thermal + impurity + deformation Adapted from Fig. 18.8, Callister 7e. (Fig. 18.8 adapted from J.O. Linde, Ann. Physik 5, p. 219 (1932); and C.A. Wert and R.M. Thomson, Physics of Solids, 2nd ed., McGraw-Hill Book Company, New York, 1970.) 23 23

Estimating Conductivity • Question: -- Estimate the electrical conductivity  of a Cu-Ni alloy that has a yield strength of 125 MPa. Adapted from Fig. 18.9, Callister 7e. wt. %Ni, (Concentration C) Resistivity,  (10 -8 Ohm-m) 10 20 30 40 50 Yield strength (MPa) wt. %Ni, (Concentration C) 10 20 30 40 50 60 80 100 120 140 160 180 125 30 21 wt%Ni Adapted from Fig. 7.16(b), Callister 7e. CNi = 21 wt%Ni From step 1: 24 24

Pure Semiconductors: Conductivity vs T • Data for Pure Silicon: --  increases with T -- opposite to metals electrons can cross gap at higher T material Si Ge GaP CdS band gap (eV) 1.11 0.67 2.25 2.40 Selected values from Table 18.3, Callister 7e. Energy filled band valence empty filled states GAP ? electrical conductivity,  (Ohm-m) -1 50 10 1 000 -2 2 3 4 pure (undoped) T(K) Adapted from Fig. 19.15, Callister 5e. (Fig. 19.15 adapted from G.L. Pearson and J. Bardeen, Phys. Rev. 75, p. 865, 1949.) 25 25

Semiconductivity Semiconductor do have a lower conductivity than metals but unique properties of semiconductors make them very useful materials. Electrical properties of semiconductors are very sensitive to the presence of impurities: Intrinsic semiconductors - electrical conductivity is based on the electronic structure of pure material. Extrinsic semiconductors - electrical conductivity is dictated by impurity atoms. WHY ? 26

Conduction in Terms of Electron and Hole Migration • Concept of electrons and holes: + - electron hole pair creation no applied applied valence Si atom pair migration electric field electric field electric field • Electrical Conductivity given by: # electrons/m 3 electron mobility # holes/m hole mobility Adapted from Fig. 18.11, Callister 7e. 27 27

Intrinsic Semiconductors Number of electrons in conduction band increases exponentially with temperature: C is a material constant Eg is the bandgap width Thermally excited conductions electrons Eg Holes left behind 0 K RT ~ 300 K 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 opposite direction and participate in conduction In Si (Eg = 1.1 eV) one out of every 1013 atoms contributes an electron to the conduction band at room temperature. 28

Intrinsic Semiconductors Since both electrons and holes conduct the conductivity of an intrinsic semiconductor is Electrons are more mobile than holes, μe > μh In an intrinsic semiconductor, a hole is produced by the promotion of each electron to the conduction band. Therefore, n = p and n (and p) increase exponentially with temperature, whereas μe and μh decrease (about linearly) with temperature. Conductivity of intrinsic semiconductors increase with temperature (different from metals!) 29

Intrinsic Semiconductors 30

Extrinsic Semiconductors Extrinsic semiconductors - electrical properties (conductivity) is dictated by impurity atoms. Example: Si is considered to be extrinsic at room T if impurity concentration is one atom per 1012 (remember our estimation of the number of electrons promoted to the conduction band 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. 31

n-type Extrinsic SMs When 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 thermally promoted to the conduction band. Impurities which produce extra conduction electrons are called donors, ND = NPhosphorus ~ n Elements in columns V and VI of the periodic table are donors for semiconductors in the IV column such as Si and Ge. 32

n-type Extrinsic SMs n >>p The hole created in donor state is far from the valence band and is immobile upon the promotion of the donated electron to the conduction band. n >>p Conduction occurs mainly by the donated electrons (thus n-type). σ ~ ND |e| μe 33

Intrinsic vs Extrinsic Conduction # electrons = # holes (n = p) --case for pure Si • Extrinsic: --n ≠ p --occurs when impurities are added with a different # valence electrons than the host (e.g., Si atoms) • n-type Extrinsic: (n >> p) no applied electric field 5+ 4 + Phosphorus atom valence electron Si atom conduction hole • p-type Extrinsic: (p >> n) no applied electric field Boron atom 3 + 4 Adapted from Figs. 18.12(a) & 18.14(a), Callister 7e. 34 34

Intrinsic Semiconductors Pure material semiconductors: e.g., silicon & germanium Group IVA materials Compound semiconductors III-V compounds Ex: GaAs & InSb II-VI compounds Ex: CdS & ZnTe The wider the electronegativity difference between the elements the wider the energy gap. 35 35

Doped Semiconductor: Conductivity vs. T • Data for Doped Silicon: --  increases doping -- reason: imperfection sites lower the activation energy to produce mobile electrons. • Comparison: intrinsic vs extrinsic conduction... -- extrinsic doping level: 1021/m3 of a n-type donor impurity (such as P). -- for T < 100 K: "freeze-out“, thermal energy insufficient to excite electrons. -- for 150 K < T < 450 K: "extrinsic" -- for T >> 450 K: "intrinsic" Adapted from Fig. 18.17, Callister 7e. (Fig. 18.17 from S.M. Sze, Semiconductor Devices, Physics, and Technology, Bell Telephone Laboratories, Inc., 1985.) conduction electron concentration (1021/m3) T (K) 600 400 200 1 2 3 freeze-out extrinsic intrinsic doped undoped doped 0.0013at%B 0.0052at%B electrical conductivity,  (Ohm-m) -1 50 10 1 000 -2 2 3 4 pure (undoped) T(K) Adapted from Fig. 19.15, Callister 5e. (Fig. 19.15 adapted from G.L. Pearson and J. Bardeen, Phys. Rev. 75, p. 865, 1949.) 36 36

Number of Charge Carriers Intrinsic Conductivity  = n|e|e + p|e|e for intrinsic semiconductor n = p   = n|e|(e + n) Ex: GaAs For GaAs n = 4.8 x 1024 m-3 For Si n = 1.3 x 1016 m-3 37 37

p-n Rectifying Junction • Allows flow of electrons in one direction only (e.g., useful to convert alternating current to direct current. • Processing: diffuse P into one side of a B-doped crystal. • Results: + - p-type n-type Adapted from Fig. 18.21, Callister 7e. --No applied potential: no net current flow. --Forward bias: carrier flow through p-type and n-type regions; holes and electrons recombine at p-n junction; current flows. + - p-type n-type --Reverse bias: carrier flow away from p-n junction; carrier conc. greatly reduced at junction; little current flow. + - p-type n-type 38 38

Properties of Rectifying Junction Fig. 18.22, Callister 7e. Fig. 18.23, Callister 7e. 39 39

Transistor MOSFET MOSFET (metal oxide semiconductor field effect transistor) Fig. 18.24, Callister 7e. 40 40

Integrated Circuit Devices Fig. 18.26, Callister 6e. Integrated circuits - state of the art ca. 50 nm line width 1 Mbyte cache on board > 100,000,000 components on chip chip formed layer by layer Al is the “wire” 41 41

Ferroelectric Ceramics Ferroelectric Ceramics are dipolar below Curie TC = 120ºC cooled below Tc in strong electric field - make material with strong dipole moment Fig. 18.35, Callister 7e. 42 42

Piezoelectric Materials Piezoelectricity – application of pressure produces current at rest compression induces voltage applied voltage induces expansion Adapted from Fig. 18.36, Callister 7e. 43 43

Summary • Electrical conductivity and resistivity are: -- material parameters. -- geometry independent. • Electrical resistance is: -- a geometry and material dependent parameter. • Conductors, semiconductors, and insulators... -- differ in accessibility of energy states for conductance electrons. • For metals, conductivity is increased by -- reducing deformation -- reducing imperfections -- decreasing temperature. • For pure semiconductors, conductivity is increased by -- increasing temperature -- doping (e.g., adding B to Si (p-type) or P to Si (n-type). 44 44

ANNOUNCEMENTS Reading: Core Problems: Self-help Problems: 45 45