Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
High-temperature superconductivity wikipedia , lookup
Thermal conduction wikipedia , lookup
State of matter wikipedia , lookup
Thermal conductivity wikipedia , lookup
Superconductivity wikipedia , lookup
Condensed matter physics wikipedia , lookup
Electron mobility wikipedia , lookup
Drude model of electrical conduction Ex Δx A vdx Jx Fig. 2.1: Drift of electrons in a conductor in the presence of an applied electric field. Electrons drift with an average velocity vdx in the x-direction.(Ex is the electric field.) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca J x (t ) = envdx (t ) Drift velocity = average velocity in x-direction 1 v dx = [v x1 + v x 2 + vx 3 + ⋅⋅⋅ + v xN ] N Current density Jx = Δq enAvdx Δt I = = = envdx A AΔt AΔt Jx = current density in the x direction, e = electronic charge, n = electron concentration, vdx = drift velocity in x direction, N = number of conduction electrons Drude model of electrical conduction Ex Δx A vdx Jx Fig. 2.1: Drift of electrons in a conductor in the presence of an applied electric field. Electrons drift with an average velocity vdx in the x-direction.(Ex is the electric field.) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Drift velocity = average velocity in x-direction 1 v dx = [v x1 + v x 2 + vx 3 + ⋅⋅⋅ + v xN ] N Current density Jx = Δq enAvdx Δt I = = = envdx A AΔt AΔt ? J x (t ) = envdx (t ) Jx = current density in the x direction, e = electronic charge, n = electron concentration, vdx = drift velocity in x direction, N = number of conduction electrons Electrical resistance is caused by scattering on “perturbations” of ideal crystal Ex Velocity gained along x vx1°ux1 Present time Last collision Electron 1 time u Δx Vibrating Cu+ ions t1 vx1°ux1 t V Electron 2 (b) (a) Free time t2 t time vx1°ux1 Fig. 2.2 (a): A conduction electron in the electron gas moves about randomly in a metal (with a mean speed u) being frequently and randomly scattered by by thermal vibrations of the atoms. In the absence of an applied field there is no net drift in any direction. (b): In the presence of an applied field, Ex, there is a net drift along the x-direction. This net drift along the force of the field is superimposed on the random motion of the electron. After many scattering events the electron has been displaced by a net distance, Δx, from its initial position toward the positive terminal Fig. 2.3: Velocity gained in the x-direction at time t from the electric field (Ex) for three electrons. There will be N electrons to consider in the metal. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Electron 3 t3 eE x (t − ti ) me t time where uxi=thermal velocity, Full velocity of i-th electron v xi = u xi + Average drift velocity vdx = eE eE 1 [v x1 + v x 2 + v x 3 + ⋅ ⋅ ⋅ + v xN ] = x (t − ti ) = x τ N me me Ex=electric field e=electron charge, me=mass of electron τ = average time between collisions = scattering time = relaxation time Definition of Drift Mobility vdx = eE x τ me vdx = μdEx vdx = drift velocity, μd = drift mobility, Ex = applied field Drift Mobility and Mean Free Time eτ μd = me μd = drift mobility, e = electronic charge, τ = mean scattering time (mean time between collisions) = relaxation time, me = mass of an electron in free space. Drift Mobility, Concentration and Conductivity vdx = μdEx J x = envdx = enμ d E x J x = σE x σ = enμ d σ = conductivity Electron drift mobility in metals (Example 2.2) Conductivity of copper (Cu) σ = 5.9×105 Ω-1cm-1 Density of copper Atomic mass of copper ρ = 8.96 g cm-3 Μat = 63.5 g mol-1 μd = ? τ =? σ = enμ d n ≈ N at ρN A N at = NA μd = 43.4 cm2V-1s-1 M at = 6.0220 × 1023 - an Avogadro's number eτ μd = me τ = 2.5×10-14 s Subshell n Shell l=0 s 1 p 2 d 1 K 2 2 L 2 6 3 M 2 6 10 4 N 2 6 10 3 f Table 1.1. Maximum possible number of electrons in the shells and subshells of an atom Quantum numbers: 14 Principal Orbital Magnetic Spin n = 1, 2, 3... l = 0,1,2 ... n-1 (l < n) ml = -l , -l +1, ...0... l -1, l or (2l +1) ms = + ½ , - ½ [Ar] 3d104s1 Scattering cross-section S = π a2 l=uτ a u Mean Free Time Between Collisions 1 τ= SuNs τ = mean free time, u = mean speed (thermal + drift), Ns = concentration of scatterers, S = cross-sectional area Electron Fig. 2.4: Scattering of an electron from the thermal vibrations of the atoms. The electron travels a mean distance l = u τ between collisions. Since the scattering cross sectional area is S, in the volume Sl there must be at least one scatterer, Ns(Suτ) = 1. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Scattering on Atomic Vibrations 1 τ∝ T or eτ e C 1 = en × ⇒ σ ∝ σ = enμ d = en me me T T C τ= T Resistivity Due to Thermal Vibrations of the Crystal ρT = AT ρT = resistivity of the metal, A = temperature independent constant, T = temperature Scattering from impurities Strained region by impurity exerts a scattering force Scattering time (τΙ) does NOT depend on temperature τI 1 τΤ Fig. 2.5: Two different types of scattering processes involving scattering from impurities alone and thermal vibrations alone. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca τ 1 μ = = 1 τT 1 μT + 1 τI + 1 μI ρ = ρT + ρ I eτ μd = me 1 ρ= nμ Matthiessen’s Rule ρ = ρT + ρI ρ = effective resistivity, ρT = resistivity due to scattering by thermal vibrations, ρI = resistivity due to scattering of electrons from impurities. ρ = ρT + ρ R ρ = ΑΤ + Β ρ = overall resistivity, ρT = resistivity due to scattering from thermal vibrations, ρR = residual resistivity Definition of Temperature Coefficient of Resistivity 1 ⎡ δρ ⎤ ⎢ ⎥ αo = ρo ⎣ δT ⎦ T =To αo = TCR (temperature coefficient of resistivity), δρ = change in resistivity, ρo = resistivity at reference temperature To , δT = small increase in temperature, To = reference temperature Temperature Dependence of Resistivity ρ = ρο [1 + α0(T−T0)] ρ = resistivity, ρo = resistivity at reference temperature, α0 = TCR (temperature coefficient of resistivity), T = new temperature, T0 = reference temperature Temperature dependence of resistivity of some metals and alloys 2000 Inconel-825 NiCr Heating Wire 1000 Iron Tungsten Resistivity (nΩ m) Monel-400 ρ∝T Tin 100 Platinum Copper Nickel Silver 10 100 1000 10000 Temperature (K) Fig. 2.6: The resistivity of various metals as a function of temperature above 0 °C. Tin melts at 505 K whereas nickel and iron go through a magnetic to non-magnetic (Curie) transformations at about 627 K and 1043 K respectively. The theoretical behavior (ρ ~ T) is shown for reference. [Data selectively extracted from various sources including sections in Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals Park, Ohio, 1991)] From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Temperature dependence of resistivity of copper 100 ρ∝T 10 Resistivity (nΩ m) 1 ρ (nΩ m) 3.5 0.1 ρ∝ 0.01 3 2.5 ρ∝T 2 1.5 ρ ∝ T5 1 0.5 ρ = ρR 0 0 20 40 60 0.001 ρ = ρR 0.0001 T5 80 100 T (K) 0.00001 1 10 100 1000 10000 Temperature (K) Fig.2.7: The resistivity of copper from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Above about 100 K, ρ ∝ T, whereas at low temperatures, ρ ∝ T 5 and at the lowest temperatures ρ approaches the residual resistivity ρR . The inset shows the ρ vs. T behavior below 100 K on a linear plot ( ρR is too small on this scale). From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Definition of Temperature Coefficient of Resistivity 1 ⎡ δρ ⎤ ⎢ ⎥ αo = ρo ⎣ δT ⎦ T =To αo = TCR (temperature coefficient of resistivity), δρ = change in resistivity, ρo = resistivity at reference temperature To , δT = small increase in temperature, To = reference temperature Temperature Dependence of Resistivity ρ = ρο [1 + α0(T−T0)] ρ = resistivity, ρo = resistivity at reference temperature, α0 = TCR (temperature coefficient of resistivity), T = new temperature, T0 = reference temperature 2000 Inconel-825 NiCr Heating Wire 1000 Iron Tungsten Monel-400 Resistivity (nΩ m) Temperature dependence of resistivity of some metals and alloys ρ∝T Tin 100 Platinum Copper Nickel Silver 10 100 1000 10000 Temperature (K) Fig. 2.6: The resistivity of various metals as a function of temperature above 0 °C. Tin melts at 505 K whereas nickel and iron go through a magnetic to non-magnetic (Curie) transformations at about 627 K and 1043 K respectively. The theoretical behavior (ρ ~ T) is shown for reference. [Data selectively extracted from various sources including sections in Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals Park, Ohio, 1991)] From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca What is the temperature of the filament? 40 W 0.333 A 120 V Power radiated from a light bulb at 2408 °C is equal to the electrical power dissipated in the filament. P = 40 W V = 120 V L = 38.1 cm D = 33 μm ρ (273K) = 5.51×10-8 Ωm ρ (T)~ T1.2 Resistivity of Cu alloyed with small amount of Ni 60 Cu-3.32%Ni Cu-2.16%Ni 40 Cu-1.12%Ni (Deformed) Cu-1.12%Ni ρCW Resistivity (nΩ m) 20 100%Cu (Deformed) 100%Cu (Annealed) ρI ρT 0 0 100 200 Temperature (K) 300 Fig. 2.8: Typical temperature dependence of the resistivity of annealed and cold worked (deformed) copper containing various amount of Ni in atomic percentage (data adapted from J.O. Linde, Ann. Pkysik, 5, 219 (1932)). From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca ρ = ΑΤ + ρ r Scattering on grain boundaries Grain 1 Grain 2 Grain Boundary (a) (b) Fig. 2.31: Grain boundaries cause scattering of the electron and therefore add to the resistivity by Matthiessen's rule. For a very grainy solid, the electron is scattered from grain boundary to grain boundary and the mean free path is approximately equal to the mean grain diameter. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Scattering on surface Jx D Fig. 2.32: Conduction in thin films may be controlled by scattering from the surfaces. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Scattering on grain boundaries Scattering on surface 300 35 (b) (a) 30 As deposited Annealed at 100 C Annealed at 150 C 100 25 50 20 ρbulk = 16.7 n m 10 15 0 0.05 0.01 0.015 0.02 0.025 1/d (1/micron) 5 10 50 100 Film thickness (nm) (a) ρfilm of the Cu polycrystalline films vs. reciprocal mean grain size (diameter), 1/d. Film thickness D = 250 nm- 900 nmdoes not affect the resistivity. The straight line is ρfilm = 17.8 n m+ (595 n mnm)(1/d), (b) ρfilm of the Cu thin polycrystalline films vs. filmthickness D. In this case, annealing (heat treating) the films to reduce the polycrystallinity does not significantly affect the resistivity because ρfilm is controlled mainly by surface scattering. |SOURCE: Data extracted from (a) S. Riedel et al, Microelec. Engin. 33, 165, 1997 and (b). W. Lim et al, Appl. Surf. Sci., 217, 95, 2003) 500 300 35 (b) (a) 30 As deposited Annealed at 100 C Annealed at 150 C 100 25 50 20 ρbulk = 16.7 n m 10 15 0 0.05 0.01 0.015 0.02 0.025 5 10 ρ ρ crystal ≈ 1 + 1.33β β= λ⎛ R ⎞ ⎜ ⎟ d ⎝1− R ⎠ R = probability of reflection on the grain boundary 100 500 Film thickness (nm) 1/d (1/micron) Mayadas and Shatzkes formula 50 ρ ρ bulk 3λ = 1+ (1 − p) 8D D λ > 0.3 p = fraction of elastic collisions Resistivities of Cu – Ni and Ni – Cr alloys. Nordheim rule Temperature (°C) 1500 1400 LIQUID PHASE 1300 S L+ 1200 1100 1000 ρ(200C) Ni 69 Cr 129 Nichrome 1120 US UID Q I L S IDU L SO SOLID SOLUTION 0 20 100% Cu 40 60 at.% Ni 80 α×103(200C) 6 3 0.3 100 100% Ni (a) Resistivity (nΩ m) 600 500 Cu-Ni Alloys 400 Nordheim’s Rule for Solid Solutions 300 200 100 0 0 20 100% Cu 40 60 at.% Ni 80 100 100% Ni (b) Fig. 2.10(a) Phase diagram of the Cu-Ni alloy system. Above the liquidus line only the liquid phase exists. In the L + S region, the liquid (L) and solid (S) phases coexist whereas below the solidus line, only the solid phase (a solid solution) exists. (b) The resistivity of the Cu-Ni alloy as a function of Ni content (at.%) at room temperature. [Data extracted from Metals Handbook-10th Edition, Vols 2 and 3, ASM, Metals Park, Ohio, 1991 and Constitution of Binary Alloys, M. Hansen and K. Anderko, McGraw-Hill, New York, 1958] From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca ρI = CX(1 − X) ρI = resistivity due to scattering of electrons from impurities, C = Nordheim coefficient, X = atomic fraction of solute atoms in a solid solution ρI = CX(1 − X) Au Mn Cu Zn Violation of Nordheim’s Rule due to formation of chemical compounds Resistivity (nΩ m) 160 140 Quenched ρI = CX(1 − X) 120 100 80 60 Annealed 40 20 0 Cu 3 Au CuAu 0 10 20 30 40 50 60 70 80 90 100 Composition (at.% Au) Fig. 2.11: Electrical resistivity vs. composition at room temperature in Cu-Au alloys. The quenched sample (dashed curve) is obtained by quenching the liquid and has the Cu and Au atoms randomly mixed. The resistivity obeys the Nordheim rule. On the other hand, when the quenched sample is annealed or the liquid slowly cooled (solid curve), certain compositions (Cu3Au and CuAu) result in an ordered crystalline structure in which Cu and Au atoms are positioned in an ordered fashion in the crystal and the scattering effect is reduced. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Combined Matthiessen and Nordheim Rules ρ = ρmatrix + CX(1 − X) ρ = resistivity, ρmatrix = resistivity of the matrix due to scattering from thermal vibrations and other defects, C = Nordheim coefficient, X = atomic fraction of solute atoms in a solid solution Predictions using Nordheim's rule Alloy Ag-Au Au-Ag Cu-Pd Ag-Pd Au-Pd Pd-Pt Pt-Pd Cu-Ni X (at.%) 8.8% Au 8.77% Ag 6.2% Pd 10.1% Pd 8.88% Pd 7.66% Pt 7.1% Pd 2.16% Ni ρ0 (nΩ m) 16.2 22.7 17 16.2 22.7 108 105.8 17 ρ at X (nΩ m) 44.2 54.1 70.8 59.8 54.1 188.2 146.8 50 X′ 15.4% Au 24.4% Ag 13% Pd 15.2% Pd 17.1% Pd 15.5% Pt 13.8% Pd 23.4% Ni ρ′ at X′ (nΩ m) ?? ?? ?? ?? ?? ?? ?? ?? Ceff ρ′ at X′ (nΩ m) Experimental Percentage Difference ρ alloy = ρ 0 + Ceff X (1 − X ) Ceff = ρ alloy − ρ 0 X (1 − X ) ρ ( X ' ) alloy = ρ 0 + Ceff X ' (1 − X ' ) Predictions using Nordheim's rule Alloy Ag-Au Au-Ag Cu-Pd Ag-Pd Au-Pd Pd-Pt Pt-Pd Cu-Ni X (at.%) 8.8% Au 8.77% Ag 6.2% Pd 10.1% Pd 8.88% Pd 7.66% Pt 7.1% Pd 2.16% Ni ρ0 (nΩ m) 16.2 22.7 17 16.2 22.7 108 105.8 17 ρ at X (nΩ m) 44.2 54.1 70.8 59.8 54.1 188.2 146.8 50 Ceff 348.88 392.46 925.10 480.18 388.06 1133.85 621.60 1561.51 X′ 15.4% Au 24.4% Ag 13% Pd 15.2% Pd 17.1% Pd 15.5% Pt 13.8% Pd 23.4% Ni ρ′ at X′ (nΩ m) ?? ?? ?? ?? ?? ?? ?? ?? ρ′ at X′ (nΩ m) Experimental Percentage Difference ρ alloy = ρ 0 + Ceff X (1 − X ) ρ ( X ) alloy = ρ 0 + ' Ceff = ρ ( X ) alloy − ρ 0 X (1 − X ) ρ alloy − ρ 0 X (1 − X ) X ' (1 − X ' ) Predictions using Nordheim's rule Alloy Ag-Au Au-Ag Cu-Pd Ag-Pd Au-Pd Pd-Pt Pt-Pd Cu-Ni X (at.%) 8.8% Au 8.77% Ag 6.2% Pd 10.1% Pd 8.88% Pd 7.66% Pt 7.1% Pd 2.16% Ni ρ0 (nΩ m) 16.2 22.7 17 16.2 22.7 108 105.8 17 ρ at X (nΩ m) 44.2 54.1 70.8 59.8 54.1 188.2 146.8 50 Ceff 348.88 392.46 925.10 480.18 388.06 1133.85 621.60 1561.51 X′ 15.4% Au 24.4% Ag 13% Pd 15.2% Pd 17.1% Pd 15.5% Pt 13.8% Pd 23.4% Ni ρ′ at X′ (nΩ m) 61.65 95.09 121.63 78.09 77.71 256.51 179.74 296.89 ρ′ at X′ (nΩ m) 66.3 107.2 121.6 83.8 82.2 244 181 300 7.01% less 11.29% less 0.02% more 6.81% less 5.46% less 4.88% more 0.69% less 1.04% less Experimental Percentage Difference ρ alloy = ρ 0 + Ceff X (1 − X ) ρ ( X ' ) alloy = ρ 0 + Ceff = ρ ( X ) alloy − ρ 0 X (1 − X ) ρ alloy − ρ 0 X (1 − X ) X ' (1 − X ' ) Phase diagram of 60% Pb – 40% Sn alloy 400 L L 200 α+L C US M N O VUS P 100 E 183°C F Q L (61.9%Sn) Eutectic US B UID Q I L SOLIDUS L+β D 61.9% β α+β SO L Primary α Eutectic α LIQUID UI D SO LVUS Eutectic L (61.9%Sn) α (19%Sn) LI Q S IDU α L SOL Temperature (oC) A 300 L (61.9%Sn) L Eutectic α (light) and β (dark) G 0 Pb 20 40 60 80 Sn Composition in wt.% Sn T 235°C 183°C Cooling of a 60%Pb40%Sn alloy L L+α M N O L+β+α P β +α Q T Cooling of 38.1%Pb61.9%Sn alloy L 183 °C t L + Eutectic solid (β+α) E F Eutectic solid (β+α) G t Fig. 1.69: The alloy with the eutectic composition cools like a pure element exhibiting a single solidification temperature at 183°C. The solid has the special eutectic structure. The alloy with the composition 60%Pb-40%Sn when solidified is a mixture of primary α and eutectic solid. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca The simplest models of non-isomorphous alloys Continuous phase Dispersed phase y Jy L A A x Jx Jx L α β (a) A (b) L (c) Fig. 2.12: The effective resistivity of a material having a layered structure. (a) Along a direction perpendicular to the layers. (b) Along a direction parallel to the plane of the layers. (c) Material with a dispersed phase in a continuous matrix. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Resistivity-Mixture Rule Reff Lα ρα Lβ ρ β = + A A Reff = ρeff = χαρα + χβρβ ρeff = effective resistivity, χα = volume fraction of the α-phase, ρα = resistivity of the α-phase, χβ = volume fraction of the β-phase, ρβ = resistivity of the β-phase L ρ eff A Conductivity-Mixture Rule σeff = χασα + χβσβ σeff = effective conductivity, χα = volume fraction of the α-phase, σα = conductivity of the α-phase, χβ = volume fraction of the β-phase, σβ = conductivity of the β-phase Resistivity-Mixture Rule (ρα ≈ ρβ) L A (a) α β α β A/N (b) L Fig. 2.13: (a) A two phase solid. (b) A thin fiber cut out from the solid. ρeff = χαρα + χβρβ ρeff = effective resistivity, χα = volume fraction of the α-phase, ρα = resistivity of the α-phase, χβ = volume fraction of the β-phase, ρβ = resistivity of the β-phase Mixture Rule (ρd > 10ρc ) ρeff 1 (1 + χ d ) 2 = ρc (1− χ d ) Mixture Rule (ρd < 0.1ρc ) ρeff ρeff = effective resistivity, ρc = resistivity of the continuous phase, χd = volume fraction of the dispersed phase, ρd = resistivity of the dispersed phase (1 − χ d ) = ρc (1 + 2 χ d ) General case: Reynolds and Hough rule σ eff − σ c σd −σc = χd σ eff + 2σ c σ d + 2σ c σeff = effective resistivity, σc = resistivity of the continuous phase, χd = volume fraction of the dispersed phase, σd = resistivity of the dispersed phase Resisitivity of bronze (95% Cu + 5% Sn) containing 15 vol. % of pores Mixture Rule (ρd > 0.1ρc ) Bronze Air ρeff 1 (1 + χ d ) 2 = ρc (1− χ d ) ρc = 1 ×10-7 Ωm ρeff = 1.27 ×10-7 Ωm Resistivity of Ag-Ni contact alloys for switches Ni% in Ag-Ni 0 10 15 20 30 100 d, g cm-3 10.5 10.3 9.76 9.4 9.47 8.9 Hardness, VHN 30 50 55 60 65 80 χNi 0 0.12 0.16 0.21 0.32 100 Res Mix Rule 23.21 25.86 28.41 34.30 Cond Mix Rule 18.54 19.33 20.15 22.34 Reynolds&Hough 19.07 20.11 21.17 23.96 20.90 23.60 25.00 31.10 (volume fraction) ρexp, nΩ m 16.9 wNi d χ Ni = d Ni 71.4 Electrical resistivity and structure of non-isomorphous alloy 400 L L A L (61.9%Sn) L TB LIQUID (a) UI Liquid, L DU One phase L (61.9%Sn) S β +L β Eutectic αα +L+L M TE αα DUS Bregion: β I U Q TwoN regionLI L+β D only SOLIDUS 200 Ophase E C α 183°C +β 61.9% β T1 P F Temperature (oC) Temperature Primary α Eutectic VU S α+β SO L 100 100%A SOLVUS Eutectic L (61.9%Sn) α (19%Sn) LIQ S IDU α L SOL 300TA X2 100%B X (% B) X1 Q G Eutectic α (light) and β (dark) 0 T 235°C 183°C Resistivity Pb Cooling of a 60%Pb40%Sn alloy L L+α M N O L+β+ρα PA β+α 0 X1 Q t 20 40 (b) 60 80 Sn Composition in wt.% Sn Mixture Rule T Nordheim's Rule 183 °C Composition X (% B) Cooling of 38.1%Pb61.9%Sn alloy L ρB L + Eutectic solid (β+α) E X F100%B Eutectic solid (β+α) 2 G t Fig. 2.14 (a) The phase diagram for a binary, eutectic forming alloy. Fig. 1.69: The(b) alloy the eutectic composition for cools a pure element exhibiting a Thewith resistivity vs composition thelike binary alloy. single solidification temperature at 183°C. The solid has the special eutectic structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) The alloy with the composition 60%Pb-40%Sn when solidified is a mixture of primary http://Materials.Usask.Ca α and eutectic solid. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Temperature Electrical resistivity of Ag – Ni alloy (a) Liquid, L TA TE α T1 100%A X1 α+L TB β+L Two phase region α+β X (% B) β One phase region: β only X2 100%B Resistivity (b) Mixture Rule Nordheim's Rule ρA 0 X1 Composition X (% B) ρB X2 100%B Fig. 2.14 (a) The phase diagram for a binary, eutectic forming alloy. (b) The resistivity vs composition for the binary alloy. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Resistivity of Ag-Ni contact alloys for switches 0 10 15 20 30 100 d, g cm-3 10.5 10.3 9.76 9.4 9.47 8.9 Hardness, VHN 30 50 55 60 65 80 χNi 0 0.12 0.16 0.21 0.32 100 Res Mix Rule 23.21 25.86 28.41 34.30 Cond Mix Rule 18.54 19.33 20.15 22.34 Reynolds&Hough 19.07 20.11 21.17 23.96 20.90 23.60 25.00 31.10 wNi d χ Ni = d Ni 16.9 (a) Liquid, L TA TE α T1 100%A X1 α+L 71.4 TB β+L Two phase region α+β X (% B) β One phase region: β only X2 100%B (b) Resistivity ρexp, nΩ m Temperature Ni% in Ag-Ni Mixture Rule Nordheim's Rule ρA 0 X1 Composition X (% B) ρB X2 100%B Fig. 2.14 (a) The phase diagram for a binary, eutectic forming alloy. (b) The resistivity vs composition for the binary alloy. Classification of conductors Insulators Semiconductors Conductors Many ceramics Superconductors Alumina Diamond Inorganic Glasses Metals Mica Polypropylene PVDF Soda silica glass Borosilicate Pure SnO2 PET Intrinsic Si Amorphous SiO2 Intrinsic GaAs As2Se3 10-18 10-15 10-12 10-9 10-6 10-3 Degenerately Doped Si Alloys Te Graphite NiCr Ag 100 103 106 Conductivity (Ωm)-1 Fig. 2.24 Range of conductivites exhibited by various materials From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca 109 1012 Conductivity of ceramic and glass insulators ⎛ Eσ ⎞ σ = σ o exp⎜ − ⎟ ⎝ kT ⎠ E E Vacancy aids the diffusion of positive ion O2– Si4+ Na+ Anion vacancy acts as a donor Interstitial cation diffuses (a) (b) Fig. 2.27: Possible contributions to the conductivity of ceramic and glass insulators (a) Possible mobile charges in a ceramic (b) A Na+ ion in the glass structure diffuses and therefore drifts in the direction of the field. (E is the electric field.) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Temperature Dependence of Conductivity 1×10-1 Conductivity 1/(Ωm) 1×10-3 24%Na2O-76%SiO2 As3.0Te3.0Si1.2Ge1.0 glass Pyrex ⎛ Eσ ⎞ σ = σ o exp⎜ − ⎟ ⎝ kT ⎠ 1×10-5 1×10-7 1×10-9 1×10-11 PVAc SiO2 conductivity, k = Boltzmann constant, T = temperature PVC 1×10-13 1×10-15 1.2 12%Na2O-88%SiO2 1.6 2 2.4 2.8 103/T (1/K) 3.2 σ = conductivity, σο = constant, Εσ = activation energy for 3.6 4 Fig. 2.28: Conductivity vs reciprocal temperature for various low conductivity solids. (PVC = Polyvinyl chloride; PVAc = Polyvinyl acetate.) Data selectively combined from numerous sources. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Thermally activated diffusion A A* B Arrenius type behavior ∼exp(EA/kT) U = PE(x) U A* ϑ= Avο exp(−EA/kT) A* EA UA= UB B A Displacement X Fig. 1.28: Diffusion of an interstitial impurity atom in a crystal from one void to a neighboring void. The impurity atom at position A must posses an energy EA to push the host atoms away and move into the neighboring void at B. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Unstable (Activated State) U(X) = PE = mgh Metastable Stable E A ΔU UA* UA UB A* A B X XA XA* XB System Coordinate, X = Position of Center of Mass Fig. 1.27: Tilting a filing cabinet from state A to its edge in state A* requires an energy EA. After reaching A*, the cabinet spontaneously drops to the stable position B. PE of state B is lower than A and therefore state B is more stable than A. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca where EA = UA* − UA ϑ = frequency of jumps, A = a dimensionless constant that has only a weak temperature dependence, vo = vibrational frequency, EA = activation energy, k = Boltzmann constant, T = temperature, UA* = potential energy at the activated state A*, UA = potential energy at state A. Classification of conductors Insulators Semiconductors Conductors Many ceramics Superconductors Alumina Diamond Inorganic Glasses Metals Mica Polypropylene PVDF Soda silica glass Borosilicate Pure SnO2 PET Intrinsic Si Amorphous SiO2 Intrinsic GaAs As2Se3 10-18 10-15 10-12 10-9 10-6 10-3 Degenerately Doped Si Alloys Te Graphite NiCr Ag 100 103 106 Conductivity (Ωm)-1 Fig. 2.24 Range of conductivites exhibited by various materials From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca 109 1012 Silicon unit cell C a a a Fig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Each atom has four neighbors and all bonds are saturated Conductivity of Semiconductor E C a a a Fig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Conductivity of Semiconductor E C a a a eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca σ = enμe + epμh n = concentration of electrons p = concentration of holes μe = drift mobility of the electrons μh = drift mobility of the electrons e = electronic charge Conductivity of Semiconductor E C a a a eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca σ = enμe + epμh n = concentration of electrons p = concentration of holes μe = drift mobility of the electrons μh = drift mobility of the electrons e = electronic charge Conductivity of Semiconductor E C a a a eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca σ = enμe + epμh n = concentration of electrons p = concentration of holes μe = drift mobility of the electrons μh = drift mobility of the electrons e = electronic charge Conductivity of Semiconductor E C a a a eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca σ = enμe + epμh n = concentration of electrons p = concentration of holes μe = drift mobility of the electrons μh = drift mobility of the electrons e = electronic charge Conductivity of Semiconductor E C a a h+ a eFig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-Sn) and the elemental semiconductors Ge and Si have this crystal structure. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca σ = enμe + epμh n = concentration of electrons p = concentration of holes μe = drift mobility of the electrons μh = drift mobility of the electrons e = electronic charge Temperature Dependence of Conductivity ⎛ Eσ ⎞ σ = σ o exp⎜ − ⎟ ⎝ kT ⎠ Conductivity of semiconductor ∞ σ = conductivity, σο = constant, Εσ = activation energy for conductivity, k = Boltzmann constant, T = temperature σ = enμe + epμh EA ) probability (E>EA) = ∫ nE dE = A exp(− kT EA General Conductivity σ = Σqi ni μi σ = conductivity, qi = charge carried by the charge carrier species i (for electrons and holes qi = e), ni = concentration of the charge carrier, μi = drift mobility of the charge carrier of species i Hall effect. Monopolar conduction Lorentz Force F = qv × B Jy = 0 VH Bz V + + + + eEH Jx EH + Ex vdx y evdxBz Jx z x A Bz V + Fig. 2.15: Illustration of the Hall effect. The z-direction is out from the plane of paper. The externally applied magnetic field is along the zdirection. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca F = force, q = charge, v = velocity of charged particle, B = magnetic field eEH = evdxBz Jx=envdx ⎛ 1⎞ E H = ⎜ ⎟ J x Bz ⎝ en ⎠ Hall coefficient EH ⎛ 1⎞ RH = = −⎜ ⎟ J x Bz ⎝ en ⎠ ⎛ 1⎞ RH = −⎜ ⎟ ⎝ en ⎠ σ = enμ μ H = − RH σ Face Centered Cubic (FCC) FCC Unit Cell (a) 2R a a a (b) a (c) Fig. 1.30: (a) The crystal structure of copper is Face Centered Cubic (FCC). The atoms are positioned at well defined sites arranged periodically and there is a long range order in the crystal. (b) An FCC unit cell with closed packed spheres. (c) Reduced sphere representation of the FCC unit cell. Examples: Ag, Al, Au, Ca, Cu, γ-Fe (>912 C), Ni, Pd, Pt, Rh From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Simple wattmeter using Hall effect IL Wattmeter IL Load Source VL RL Bz ∝ I L VL IL VH ∝ E H ∝ RH J x Bz ∝ I L2 ∝ W IL C C V VH Bz w R VL Ix = VL/R Fig. 2.17: Wattmeter based on the Hall effect. Load voltage and load current have L as subscript. C denotes the current coils. for setting up a magnetic field through the Hall effect sample (semiconductor) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca Hall effect. Ambipolar conduction Jy = 0 + Ex Jx vhx + eEy + + Jx z x B v B B F = qv×B F = qv×B (a) vex evhxBz q = –e v y + Ey eEy q = +e Bz evexBz + A Bz Fig. 2.16 A moving charge experiences a Lorentz force in a magnetic field. (a) A positive charge moving in the x direction experiences a force downwards. (b) A negative charge moving in the -x direction also experiences a force downwards. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca pμ h − nμ e p − nb 2 RH = = 2 e ( pμ h + nμ e ) e( p + nb) 2 2 V Fig. 2.26: Hall effect for ambipolar conduction as in a semiconductor where there are both electrons and holes. The magnetic field Bz is out from the plane of the paper. Both electrons and holes are deflected toward the bottom surface of the conductor and consequently the Hall voltage depends on the relative mobilities and concentrations of electrons and holes. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) http://Materials.Usask.Ca (b) 2 RH = Hall coefficient, p = concentration of the holes, μh = hole drift mobility, n = concentration of the electrons, μe = electron drift mobility, b = μe/μh where μe is the electron drift mobility and μh is the hole drift mobility, e = electronic charge Skin Depth for Conduction ωL R δ= δ = Skin depth 2a At high frequencies, the core region exhibits more inductive impedance than the surface region, and the current flows in the surface region of a conductor defined approximately by the skin depth, δ. 1 1 ωσμ 2 δ = skin depth, ω = angular frequency of current, σ = conductivity, μ = magnetic permeability of the medium HF Resistance per Unit Length Due to Skin Effect ρ ρ rac = ≈ A 2πaδ rac = ac resistance , ρ = resistivity, A = cross-sectional area, a = radius, δ = skin depth Skin effect ωL R δ = Skin depth 2a At high frequencies, the core region exhibits more inductive impedance than the surface region, and the current flows in the surface region of a conductor defined approximately by the skin depth, δ. B2 (a) Total current into paper is I B1 (b) Current in hollow outer cylinder is I/2 B2 B1 B2 B1 (c) Current in inner cylinder is I/2 Illustration of the skin effect. A hypothetical cut produces a hollow outer cylinder and a solid inner cylinder. Cut is placed where it would give equal current in each section. The two sections are in parallel so that the currents in (b) and (c) sum to that in (a). Skin effect rf Copper wire with radius1 mm σdc = 5.9×107 Ω-1cm-1 ρ dc rdc = 2 πa ρ dc rf = 2πaδ rdc =? rf a = rdc 2δ δ= 1 1 ωσμ 2 r10 MHz ≈ 24 rdc r1GHz ≈ 240 rdc Interconnects in microelectronics M7 M6 Low permittivity dielectric M5 M3 Cu interconnects M2 M4 M3 M2 M1 M1 Silicon Metal interconnects wiring devices on a silicon crystal. Three different metallization levels M1, M2, and M3 are used. The dielectric between the interconnects has been etched away to expose the interconnect structure. Cross section of a chip with 7 levels of metallization, M1 to M7. The image is obtained with a scanning electron microscope (SEM). |SOURCE: Courtesy of IBM |SOURCE: Courtesy of Mark Bohr, Intel. Effective multilevel interconnect capacitance C H = ε 0ε r TL X WL CV = ε 0ε r H ρL R= Ceff ⎛T W⎞ = 2(CV + C H ) = 2ε 0ε r L⎜ + ⎟ ⎝X H⎠ T Ar ≡ = 1÷ 2 = aspect ratio W L = 10 mm X ≈W ≈ P T = 1 μm ρ = 17 nΩ m (Cu) RCeff 2 P = 1 μm εr = 3.6 (FSG) TW L2 = 2ε 0ε r ρ TW RCeff ⎛T W⎞ ⎜ + ⎟ ⎝X H⎠ 1 ⎞ ⎛ 4 = 2ε 0ε r ρL ⎜ 2 + 2 ⎟ ⎝P T ⎠ RCeff = 0.43 ns 2 Electromigration (a) (b) Void and failure Hillock Grain boundary Void Electron Hillock Current Interconnect In a ce terf Cold (c) Hot Hillocks Cold Grain boundary Current (a) Electrons bombard the metal ions and force themto slowly migrate (b) Formation of voids and hillocks in a polycrystalline metal interconnect by the electromigration of metal ions along grain boundaries and interfaces. (c) Accelerated tests on 3 mmCVD(chemical vapor deposited) Cu line. T = 200 oC, J = 6 MAcm-2: void formation and fatal failure (break), and hillock formation. |SOURCE: Courtesy of L. Arnaud et al, Microelectronics Reliability, 40, 86, 2000. Black’s equation t MTF = A B J −n ⎛ EA ⎞ exp⎜ ⎟ ⎝ kT ⎠ tMTF = mean time to 50% failure AB = constant J = current density n = integer EA = activation energy