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Homework #1 – Due 09/09/14 • • • • • 2.2 2.3 2.7 2.13 2.20 Matthiessen’s Rule = T + I = effective resistivity, T = resistivity due to scattering by thermal vibrations only, I = resistivity due to scattering of electrons from impurities only. = T + R = overall resistivity, T = resistivity due to scattering from thermal vibrations, R = residual resistivity Residual resistivity shows very little temperature dependence, while T = AT. Therefore, effective resistivity is given by: = AT + B From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Temperature Coefficient of Resistivity The fractional change in the resistivity per unit temperature increase at the reference temperature To 1 o o T T To o = TCR (temperature coefficient of resistivity), = change in the resistivity ( = - o), o = resistivity at reference temperature To , T = small increase in temperature (T = T – To), To = reference temperature Temperature Dependence of Resistivity [1 + o(TTo)] = resistivity, o = resistivity at reference temperature, 0 = TCR (temperature coefficient of resistivity), T = new temperature, T0 = reference temperature From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Temp. Dependence of Resistivity • When the resistivity follows the linear behavior previously mentioned: = AT + B • Then, in 1 o o T T T o • The o is constant over a temperature range To to T and leads to: [1 + o(TTo)] From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Actually dominated by the residual resistivity; thus, relatively insensitive to temp Fig 2.7 Magnetic materials (e.g., iron and nickel) do not follow the expected dependence -Besides lattice vibrations, these materials are also affected by the magnetic interactions between ions in the lattice Expected linear dependence 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 ( ~ 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, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • Resistivity v. temperature behavior of pure metals can be empirically represented as: Fig 2.8 • Obviously, the overly simple linear prediction does not represent resistivity at low temperatures • Lattice vibrations reduce rapidly as temp reduces; therefore less scattering and simple model no longer works The resistivity of copper from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Above about 100 K, T, whereas at low temperatures, T 5 and at the lowest temperatures approaches the residual resistivity R. The inset shows the vs. T behavior below 100 K on a linear plot (R is too small on this scale). From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Power law fit: -Notice near unity for pure metals -Closer to 2 for the magnetic Fig 2.8 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Solid Solutions and Nordheim’s Rule • Isomorphous alloy of two metal (a binary alloy that forms a solid solution) is expected to follow: = T + I – Along with temp-independent I increasing with the concentration of solute atoms • This means that as the alloy concentration increases, the resistivity increases and becomes less temperature dependent as I overwhelms T • This can be an advantage of alloying metals Fig 2.8 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • This table shows how alloying metals can affect the resistivity • “Buried” in this table is that when you alloy 80% nickel with 20% chromium, the resistivity of nickel increases almost 16 times!! • This alloy is known as Nichrome, and it is used as a heater wire in household appliances and industrial furnaces From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) (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 • Both metals are FCC with Cu being only ~3% larger, so the alloy will remain FCC • When Ni is added to Cu, the I in the Matthiessen expression will increase • Obviously, as the alloy becomes more Nirich (i.e., becoming pure Ni), resistivity reduces as it should for pure metals Fig 2.11 The Cu-Ni alloy system. SOURCE: Data extracted from Metals Handbook, 10th ed., 2 and 3 Metals Park, Ohio: ASM, 1991, and M. Hansen and K. Anderko, Constitution of Binary Alloys, New York: McGraw-Hill, 1958. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Nordheim’s Rule for Solid Solutions An important semi-empirical equation that can be used to predict the resistivity of an alloy is Nordheim’s Rule: I = CX(1 X) I = “impurity” resistivity due to scattering of electrons from impurities C = Nordheim coefficient: represents the effectiveness of the solute atom in increasing the resistivity; assumes a dilute alloy mixture X = atomic fraction of solute atoms in a solid solution The Nordheim Rule Eqn. relates impurity resistivity to the atomic fraction X of the solute atoms in a solid solution From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) • C depends on the type of solute and the solvent • A solute atom that is drastically different in size to solvent atom will produce a larger I, and therefore, lead to a larger C • An important assumption in Nordheim’s rule is that alloying does not significantly vary the number of conduction electrons/atom in the alloy • This is true for alloys with the same valency (i.e., same column on the Periodic Table) • However, it is not true for alloys with different valencies • To correct for this, C must change to provide an effective Nordheim coefficient Ceff From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Combined Matthiessen and Nordheim Rules = matrix + CX(1 X) = resistivity of the alloy (solid solution) matrix = 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 C assumes a dilute alloy mixture From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Fig 2.12 Electrical resistivity vs. composition at room temperature in Cu-Au alloys. The quenched sample (dashed curve) is obtained by quenching the liquid, and the Cu and Au atoms are randomly mixed. The resistivity obeys the Nordheim rule. When the quenched sample is annealed or the liquid is slowly cooled (solid curve), certain compositions (Cu3Au and CuAu) result in an ordered crystalline structure in which the Cu and Au atoms are positioned in an ordered fashion in the crystal and the scattering effect is reduced. • In solid solutions, at some concentrations of certain binary alloys, the annealed solid has an orderly structure where the two elements are not randomly mixed, but occupy regular, ordered sites • These can actually be viewed as a pure compound – like Cu3Au and CuAu • Their resistivities will be less than the same composition random alloy due to being quenched from the melt From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Resistivity of Mixtures and Porous Materials Consider a material with two distinct phases and b, as described below: The effective resistivity of a material with a layered structure. (a) Along a direction perpendicular to the layers. (b) Along a direction parallel to the plane of the layers. (c) Materials with a dispersed phase in a continuous matrix. Fig 2.13 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Effective Resistance of Mixtures Example: Effective resistivity for current flow in the x direction L Lb b Reff A A Reff = effective resistance L = total length (thickness) of the -phase layers = resistivity of the -phase layers A = cross-sectional area Lb = total length (thickness) of the b-phase layers b = resistivity of the b-phase layers c = volume fractions of the phases From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Resistivity-Mixture Rule eff = c cbb eff = effective resistivity of mixture, c= volume fraction of the phase, = resistivity of the -phase, cb = volume fraction of the bphase, b= resistivity of the b-phase Conductivity-Mixture Rule eff = c cbb eff = effective conductivity of mixture, c= volume fraction of the -phase, = conductivity of the -phase, cb = volume fraction of the b-phase, b= conductivity of the b-phase From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Non-ideal Case • Those two rules pertain to two special cases • However, in general, a random mixture of the two phases usually exists • Therefore, we would not expect either equation to apply rigorously • If resistivity of one phase is very different than the other, there are two semiempirical rules that are quite useful From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Mixture Rule (d > 10c ) eff 1 (1 cd ) 2 c (1 c d ) Dispersed phase is much more resistive with respect to the continuous phase eff = effective resistivity, c = resistivity of continuous phase, cd = volume fraction of dispersed phase, d = resistivity of dispersed phase Mixture Rule (d < 0.1c ) eff (1 c d ) c (1 2 c d ) Dispersed phase is much less resistive with respect to the continuous phase eff = effective resistivity, c = resistivity of the continuous phase, cd = volume fraction of the dispersed phase, d = resistivity of the dispersed phase From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Two-Phase Alloy Resistivity and Electrical Contacts Eutectic-forming alloys, e.g., Cu-Ag. (a) The phase diagram for a binary, eutectic-forming alloy. (b) The resistivity versus composition for the binary alloy. Fig 2.15 • For most compositions, alloys form a two-phase heterogenous mixture of phases and b as shown in the figure • Between 0 and X1, increases with the concentration by virtue of Nordheim’s rule • At X1, the solid solubility limit of one metal into another is reached. Now, two phases are present, so is given by the resistivity-mixture rule • As with the 0 to X1 region, at the other metal end, between X2 and 100%, the resistivity becomes Nordheim’s rule again because only one phase exists From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Hall Effect Fig 2.16 • “Electron as a particle” allows to us to readily explain the Hall effect • Magnetic field is applied perpendicular to the direction of both the applied field that is driving the current • The result is a transverse field in the sample that is perpendicular to the direction of both the applied field, Ex, and the magnetic field, Bz • The Hall voltage can be obtained • Applied electric field drives a current Jx where the electrons move in the –x direction with drift velocity vdx. Illustration of the Hall effect. The z direction is out of the plane of the paper. The externally applied magnetic field is along the z direction. From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Hall Effect (cont.) • Due to the magnetic field, there is a force, known as the Lorentz force, acting on each electron given by • The direction of this Lorentz force is in the –y direction and governed by the vector product Fig 2.16 • Since Lorentz force is in the –y direction, electrons are forced downward • As a result, negative charge builds up near the bottom and positive at the top due to exposed metal ions https://www.youtube.com/watch?v=R9Jpi2bIiU8 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) F = qv B F = force, q = charge, v = velocity of charged particle, B = magnetic field Fig 2.17 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, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Lorentz Force • Accumulation of electrons near the bottom results in an internal electric field, EH, in the –y direction. This is called the Hall field and give rise to the Hall voltage, VH between the top and bottom of the sample • Electron accumulation continues until the increase in EH is sufficient to stop further accumulation • When this happens, the magnetic force that pushes the electrons down just balances the force eEH, that prevents further accumulation, so at steady state: • However: • Therefore, we can substitute for vdx to obtain: Hall Coefficient From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005) Definition of Hall Coefficient RH Ey J x Bz • Measures the resulting Hall field, along y, per unit transverse applied current and magnetic field • The larger the RH, the greater the field • Therefore, RH is a gauge of the magnitude of the Hall effect RH = Hall coefficient, Ey = electric field in the y-direction, Jx = current density in the x-direction, Bz = magnetic field in the z-direction From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)