2.2 Electrical resistivity of metals and its application
2.2.1 Drude model
In 1897, Thomson discovered the electron and it was quickly realized that electrons contribute to the electric currents. Three years later, Drude proposed a model based on the classic kinetic theory of gases to calculate resistivity. In his model, he treated the free electrons as a classical ideal gas; although the Drude model has many shortcomings, it is still used today as a quick practical way to form simple pictures and rough estimates of the electrical properties of metals.
He made four basic assumptions in the model [152, 153]:
1) Free electron approximation. Drude assumed that when atoms of a metallic element are brought together to form a metal, the valence electrons become detached and wander
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freely through the metal. They are the so-called free electrons and they are the charge carriers. The positively charged ion cores remain intact and are immobile. The free electrons can collide with the ion cores while they are moving. These collisions instantaneously change their velocity. Besides the collisions, the electrons do not interact with the ion cores.
2) Independent electron approximation. The free electrons do not interact with each other at all: There is no coulomb interaction, and as opposed to a classical gas model, they do not collide with each other either. Thus in the absence of externally electromagnetic fields, each electron moves in a straight line until it collides with the ion cores. When an external field is applied, each electron is taken to move as determined by Newton’s laws of motion. Fig. 2-2 illustrates the free electron approximation and independent electron approximation of the Drude Model.
Fig. 2-2 Schematic diagrams of the Drude Model.
a) without external electric field, b) with external electric field E.
3) Relaxation time and mean free path. Drude assumed that the probability that an electron experiences a collision with the ion cores per unit time is 1/τ. It means that an electron picked at random will, on average, travel for a time τ since its last collision. The time τ is variously known as the relaxation time. It is assumed independent of the electron position and is independent of time. In between collisions, the electrons move freely.
The mean length of this free movement is called the mean free path.
4) Electrons will achieve thermal equilibrium with their surroundings through collisions with the ion cores. Drude presumed the electrons maintain local thermodynamic equilibrium in a particularly simple way: immediately after each collision, an electron obtains a velocity uncorrelated to its velocity before the collision, this velocity is
15 randomly directed and its speed is related to the temperature where the collision occurred.
With the basic assumptions of the Drude Model, we can now explain the resistivity of metals.
Consider the movement of an electron when an electric field E is applied. The equation of motion, according to Newton’s law, is [152-154]:
𝑚𝑒𝑑𝒗
𝑑𝑡 = − 𝑬 (2-3)
𝑚𝑒 is the mass of the electron, 𝒗 is the average electronic velocity and is the proton charge.
According to the fourth assumption of Drude Model, the electrons are at thermal equilibrium with their surroundings before the electric field is applied. Since there is no transfer of the electrical current in the absence of externally applied electromagnetic fields, the average electronic velocity 𝒗 equals zero. When an electric field E is applied, the electrons will achieve a new thermal equilibrium with their surroundings in time τ. Therefore the average electronic velocity in an external electric field E is:
𝒗𝐸 =− 𝑬
𝑚𝑒 (2-4)
With the average electronic velocity, we can calculate the current density J. Consider an area A perpendicular to the electric field. The amount of charge passing through the area per unit time is:
− 𝑛𝒗𝐸𝐴
n is the conduction electron density, that is, the number of conduction electrons per unit volume.
Therefore, the current density J is:
𝐽 = − 𝑛𝒗𝐸 (2-5)
Now with the Eq.(2-4) we can get:
𝐽 =𝑛 2
𝑚𝑒 𝑬 (2-6)
Consider the Eq.(2-2) we know that:
𝜌 = 𝑚𝑒
𝑛 2 (2-7)
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The Drude model thus explains the electrical resistivity quantitatively. In Eq.(2-7), 𝑚𝑒 and are physical constants, so the electrical resistivity is determined by the conduction electron density n and relaxation time . n is calculated by assuming that every atom contributes Z conduction electrons (the valence electrons) [153] when atoms are brought together to form metal. Then n is calculated:
𝑛 = 𝑁𝐴𝑍𝜌𝑚
𝑀 (2-8)
𝑁𝐴 is the Avogadro’s number, 𝜌𝑚 is the density of the solid in kg/m3, M is the atomic mass in kilograms per atom. The relaxation time is calculated by applying the experimental data into the Eq.(2-7). Table 2-2 gives the conduction electron density n, relaxation time and resistivity 𝝆 of selected elements. The Drude model does not seem to make any real predictions, because it determines resistivity only at the expense of introducing another unknown parameter, the relaxation time . It does, however, frame electrical resistivity in the terms that will be used later for more detailed calculation, as a balance between the force -eE causing electrons to accelerate, with the scattering events encoded in that causes them to decelerate [155].
Table 2-2 The conduction electron density, relaxation time and resistivity of selected elements at 273 K [152].
Element n (1022/cm3) (10-14 s) 𝜌 (mΩ∙cm)
Li 4.70 0.88 8.55
Na 2.65 3.20 4.20
K 1.40 4.10 6.10
Rb 1.15 2.80 11.00
Cs 0.91 2.10 18.8
Cu 8.47 2.70 1.56
Ag 5.86 4.00 1.51
Au 5.90 3.00 2.04
Mg 8.61 1.10 3.90
Fe 17.00 0.24 8.90
Zn 13.20 0.49 5.50
Al 18.10 0.80 2.45
17 The Drude Model was a great success at that time; it could explain Ohm’s law and the Hall effect. Drude also “successfully” explained the Wiedemann–Franz law quantitatively using his model. Despite its great success, it fails to explain the disparity between the expected heat capacities of metals compared to insulators. In addition, the Drude model fails to explain the existence of apparently positive charge carriers, as demonstrated by the positive Hall effect of certain materials [153]. Sommerfeld put forth the free electron model by combining the Drude model and quantum mechanical Fermi–Dirac statistics. It resolves the major problems of the classical Drude model [156, 157]. Nevertheless, the successes of the Drude model were considerable and it is still used today as a quick practical way to form simple pictures and rough estimates of properties.
2.2.2 Matthiessen’s rule