Electrical and Thermal Conduction Chapter 2. Electrical and Thermal Conduction in Solids

An Overview In this chapter, we will treat conduction ‘e’ in metal as “free charges” that can be accelerated by an applied electric field, to explain the electrical and thermal conduction in a solid. Electrical conduction involves the motion of charges in a material under the Influence of an applied electric field. By applying Newton’s 2nd law to ‘e’ motion & using a concept of “mean free time” between ‘e’ collisions with lattice vibrations, crystal defects, impurities, etc., we will derive the fundamental equations that govern electrical conduction in solids. Thermal conduction,i.e., the conduction of thermal E from higher to lower temperature regions in a metal, involves the conduction ‘e’ carrying the energy. Therefore, the relationship between the electrical conductivity and thermal conductivity will be reviewed in this textbook.

CONTENTS Electrical Conductivity of Metals: 2.1 Classical theory : The DRUDE model 2.2 Temperature dependence of resistivity 2.3 MATTHIESSEN’s and NORDHEIM’s Rules. 2.4 Resistivity of mixtures and porous materials 2.5 The Hall Effect and Hall Devices Thermal Conductivity: 2.6 Thermal conduction Electrical Conductivity of Nonmetals: 2.7 Electrical conductivity of nonmetals Additional Issues: 2.9 Thin metal films 2.10 Interconnects in microelectronics 2.11 Electromigration and Black’s equations

2.1 Classical theory : The DRUDE model Goal: To find out the relation between the conductivity (or resistivity) and drift velocity , and thereby its relation to mean free time and drift mobility, from the description of the current density In a conductor where ‘e’ drift in the presence of an electric field, current density is defined as the net amount of charge flowing across a unit area per unit time J : current density q : net quantity of charge flowing through an area A at Ex In this system, electrons drift with an average velocity vdx in the x-direction, called the drift velocity. (Here Ex is the electric field.) Drift velocity is defined as the average velocity of electrons in the x direction at time t, denote by vdx(t) vxi : x direction velocity of the ith electrons N : # of conduction electrons in the metal [2.1]

2.1.1 Metals and conduction by electrons Current density in the x direction can be rewritten as a function of the drift velocity : In time Δt, the total charge Δq crossing the area A is enAΔx, where Δx=vdxΔt and n is assumed to be the # of ‘e’ per unit volume in the conductor (n=N/V). [2.2] : time dependent current density is useful since the average velocity at one time is not the same as at another time, due to the change of Ex Think of motions of a conduction ‘e’ in metals before calculating Vdx. 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 A conduction ‘e’ in the electron gas moves about randomly in a metal (with a mean speed u) being frequently and randomly scattered by thermal vibrations of the atoms. In the absence of an applied field there is no net drift in any direction.

2.1.1 Metals and conduction by electrons To calculate the drift velocity vdx of the ‘e’ due to applied field Ex, we first consider the velocity vxi of the ith ‘e’ in the x direction at t. Since is the acceleration a of the ‘e’ [F=qE=ma], vxi in the x direction at t is given by Let uxi be the initial velocity of ‘e’ i in the x direction just after the collision. Vxi is written as the sum of uxi and the acceleration of the ‘e’ after the collision. Here, we suppose that its last collision was at time ti; therefore, for time (t-ti), it accelerated free of collisions, as shown in Fig.2.3. uxi is velocity of ith‘e’ in the x direction after the collision However, this is only for the ith electron. We need the average velocity vdx for all such electrons along x as the following eqn. (t-ti) : average free time for N electrons between collision (~ τ = mean free time or mean scattering time) Fig 2.3 Velocity gained in the x direction at time t from the electric field ( Ex) for three different electrons. There will be N electrons to consider in the metal.

2.1.1 Metals and conduction by electrons Drift mobility (vs. mean free time) : widely used electronic parameter in semiconductor device physics. Suppose that τ is the mean free time or mean scattering time. Then, for some electrons, (t-ti) will be greater than ,and for others, it will be shorter, as shown in Fig 2.3. Averaging (t-ti) for N electrons will be the same as . Thus we can substitute for (t-ti) in the previous expression to obtain Equation 2.3 shows that the drift velocity increases linearly with the applied field. The constant of proportionality has been given a special name and symbol, called drift mobility , which is defined as , which is often called the relaxation time, is directly related to the microscopic processes that cause the scattering of the electrons in the metal; that is, lattice vibration, crystal imperfections, and impurities.

2.1.1 Metals and conduction by electrons From the expression for the drift velocity vdx the current density Jx follows immediately… by substituting Equation 2.4 into 2.2, that is, Therefore, the current density is proportional to the electric field and the conductivity term is given by Then, let’s find out temperature dependence of conductivity (or resistivity) of a metal by considering the mean time . The mean time between collisions has further significance. Its 1/ represents the mean frequency of collisions or scattering events; that is 1/ is the mean probability per unit time that the electron will be scattered. Therefore, during a small time interval , the probability of scattering will be .

2.2 Temperature dependence of resistivity To find the temperature dependence of , let’s consider the temperature dependence of the mean free time , since this determines the drift velocity. Fig 2.5 scattering of an electron from the thermal vibration of the atoms. The electron travels a mean distance between collisions. Since the scattering cross-sectional area is S, in the volume Sl there must be at least one scatterer as volume Ns : concentration of scattering centers S : cross-sectional area u : mean speed a : amplitude of the vibrations When the conduction electrons are only scattered by thermal vibrations of the metal ion, then in the mobility expression refers to the mean time between scattering events by this process.

2.2 Temperature dependence of resistivity Lattice-scattering-limited conductivity : the resistivity of a pure metal wire increase linearly with the temperature, due to the scattering of conduction electrons by thermal vibrations of the atoms.  feature of a metal (cf. semiconductors) The thermal vibrations of the atom can be considered to be simple harmonic motion, much the same way as that of a mass M attached to a spring. From the kinetic theory of matter, This makes sense because raising the T increases atomic vibrations. Thus Since the mean time between scattering events τ is inversely proportional to the area that scatters the ‘e’, results in (to show a relation with T) So, the resistivity of a metal

2.3 MATTHIESSEN’s and NORDHEIM’s Rules. 2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity : The theory of conduction that considers scattering from lattice vibrations only works well with pure metal and it fails for metallic alloys. Their resistivities are weakly T-dependent, and so, different type of scattering mechanism is required for metallic alloys. Let’s consider a metal alloy that has randomly distributed impurity atoms. We have two mean free times between collision. In unit time, a net probability of scattering, is given by Then, since drift mobility depends on effective scattering time, effective drift mobility is given by

2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity is the lattice-scattering-limited drift mobility, is the impurity-scattering-limited drift mobility. Since effective resistivity of the material is simply This summation rule of resistivities from different scattering mechanisms is called Matthiessen’s rule. Furthermore, in a general from, effective resistivity can be given by Since residual resistivity shows very little T-dependence whereas ρT = AT .

2.3.1 Matthiessen’s rule and the temperature coefficient of resistivity Temperature coefficient of resistivity (TCR) Eqn. 2.17 indicates that the resistivity of a metal varies with T, with A and B depending on the material. Instead of listing A and B in resistivity tables, we prefer a temperature coefficient that refers to small, normalized changes around a reference temperature. - temp sensitivity of the resistivity of metals If the resistivity follows the behavior like in Eqn. 2.17, then Eqn. 2.18 leads to where a0 is constant over a temperature range T0 to T, &

Resistivity of various metals vs. T However is only an approximation for some metals and not true for all metals. This is because the origin of the scattering may be different depending on the temperature. Scattering from vibration Scattering from impurity

2.3.2 Solid solution and Nordheim’s rule How does the resistivity of solid solutions change with alloy composition ? In an isomorphous alloy of two metals, that is, a binary alloy that forms a solid solution (Ni-Cr alloy), we would expect Eqn 2.15 to apply, with the temperature-independent impurity contribution increasing with the concentration of solute atoms. This means that as alloy concentration increases, resistivity increases and becomes less temperature dependent as ρI, overwhelms ρT, leading to αo << 1/273. This (temperature independency) is the advantage of alloys in resistive components.

2.3.2 Solid solution and nordheim’s rule How does the concentration of solute atoms affect on ρI ? Nordheim’s rule for solid solutions: an important semiempirical Eqn. that can be used to predict the resistivity of an alloy, which relates the impurity resistivity to the atomic fraction X of solute atoms in a solid solution, as follows: C (Nordheim’s coefficient): represents effectiveness of the solute atom in increasing the resistivity. consistent Nordheim rule is useful for predicting the resistivities of dilute alloys, particularly in the low-concentration region. %Nordheim’s rule assumes that the solid solution has the solute atoms randomly distributed in the lattice, and these random distributions of impurities cause the ‘e’ to become scattered as they whiz around the crystal.

2.3.2 Solid solution and Nordheim’s rule Combination of Matthiessen and Nordheim rules leads to a general expression for ρ of the solid solution: Exception: at some concentrations of certain binary alloys, Cu and Au atoms are not randomly mixed but occupy regular sites, which decrease the resistivity. ------------