Resistivity [112,114] is a material’s resistance to the flow of electrical current. A better understanding can be formulated through exploring a model of electrical conduction such as the Drude model. Electrical conduction occurs from the net
movement of the charge carriers in a material that has been placed under the influence of an applied electric field in the direction of the field. This movement of the carriers gives rise to an electric current density, J, which is defined as the net amount of charge
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that flows across a unit area per unit time, 𝐽 = ∆𝑞
𝐴∆𝑡
If electrons are the carriers in material and become mobile from the presence of an electric field, they will move opposite the field and conventional current. These
electrons undergo a Coulombic force, eEx, in the x-dir due to their negative charge. The electrons in a material move about randomly, but when an external electric field is present, it cause them to acquire a net velocity in the direction of the field called a drift velocity. This drift velocity is calculated by averaging the velocities of the electrons in the x-direction at a time t as follows
𝑣𝑑𝑥 = 1
𝑁[𝑣𝑥1+ 𝑣𝑥2+ 𝑣𝑥3+ ⋯ ]
If n, the concentration of conduction electrons, is defined as the number of electrons per unit volume (N/V), and at a Δt the displacement of the electrons is given as Δx = vdxΔt , then the amount of charge passing through an area A can be written as
∆𝑞
𝐴 = 𝑒 (𝑛𝐴∆𝑥) = 𝑒𝑛𝐴𝑣𝑑𝑥Δ𝑡
which is the number of electrons crossing this area for a certain amount of time Δt. The
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electron current density is then rewritten Jx= ∆𝑞
𝐴Δ𝑡 =
𝑒𝑛𝐴𝑣𝑑𝑥Δ𝑡
𝐴Δ𝑡 = 𝑒𝑛𝑣𝑑𝑥
The drift velocity can be rewritten to be 𝑣𝑑𝑥 = 𝑒𝜏
𝑚𝑒𝐸𝑥 = 𝜇𝑑𝐸𝑥
where e and me is the charge and mass of the electron respectively, and τ is the mean free time or time between collisions. The formulation of these variables as shown is given as the drift mobility, μd. The drift mobility is a measure of how fast the electrons will drift when driven by an electric field if the electrons are not highly scattered. So if the time between collisions, τ, is large μd is also large. The mean frequency of the collisions is defined as τ -1. Using the drift mobility, the current density can again be
rewritten as
Jx = 𝑒𝑛𝜇𝑑𝐸𝑥
The first part of this equation
σ = 𝑒𝑛𝜇𝑑
is called the electrical conductivity, σ. The electrical resistivity is the inverse of this
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𝜌 = 1 σ =
1 𝑒𝑛𝜇𝑑
The energy corresponding to the highest filled state at 0 K is called the Fermi energy, εf. The conduction band is the lowest lying electron energy band that is empty of
electrons at εf. The valence band is the energy band that contains the valence electrons.
Metals have conduction bands and valence bands that overlap. Therefore, metals have loosely bound electrons that are able to move freely. Very little energy is required to move the valence electrons into the low lying empty states in the conduction band.
Because of the freedom of these electrons, electrical conductivity is high in metals.
In semiconductors there is a gap between the conduction band and valence band.
Valence electrons must be promoted across an energy gap, Eg, into the conduction band.
The larger the band gap, the more energy that is needed to promote the electrons into the conduction band, and the lower the electrical conductivity. A large band gap makes the material an insulator. In all materials, an increase in temperature increases the thermal excitation and therefore gives rise to a higher electrical conductivity, so the electrical conductivity is temperature dependent. In metals this gives rise to a linear dependence.
The electrical properties of semiconductors are very sensitive to even the smallest
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amount of impurities. Intrinsic semiconductors have electrical properties that are based on the electronic structure inherent to the pure material. An extrinsic semiconductor’s electrical behavior is determined by the impurities which introduce excess carriers.
Introduction of excess carriers can be done by doping, which can changes the properties of the semiconductor. When the number of electrons far exceeds the number of holes in the valence band, the semiconductor is said to be n-type (donor impurities). When holes are in higher concentration, the semiconductor is p-type (acceptor impurities). Generally there are both holes and electrons present in any material. The one with the higher concentration is considered the majority carriers and the other is the minority carrier.
The type of carrier in control of the properties can change over the temperature range.
If the electrons in a metal are only being scattered by their own thermal vibrations
the resistivity can be written as
ρT = 𝑚𝑒𝑇 𝑒2𝑛𝑐
giving it a linear temperature dependence. Impurities in a material hinder the electrons movements causing the electrical resistivity to be larger. These impurities distort the crystal structure. As the electron is traveling through the lattice, it feels a force due to the change in the potential energy as it reaches the impurity. The electron has one
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scattering time from the thermal vibrations, τT, and one due to the scattering from the impurities, τI. Matthiessen’s rule states that these can be thought of as independent of each other. To find the electron’s probability of being scattering, the terms can be added as
where the final scattering time, τ, is smaller than both contributions. Because of their
relationships with τ,
The mobility and electrical resistivity can both be rewritten using Matthiessen’s rule as follows
1 𝜇𝑑 = 1
𝜇𝑇+ 1 𝜇𝐼
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and
ρ = ρT+ ρR
where the contribution due to thermal vibrations has a linear temperature dependence and the residual contribution which includes impurities, dislocations, defects, grain boundaries, and vacancies, is constant.