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A Microscopic View of Conduction (Chapter 2: Theory)

Updated: Sep 2, 2021

In this article we will cover:

- A microscopic view of copper

- The free-electron model and quantum model

- Work function and the 4 special processes to eject electrons

- Drift velocity and the application of voltage

- Drift velocity equation

A Microscopic View of Copper

Microscopically, copper resembles a lattice of copper balls (positive lattice ions) which form a face-centered cubic lattice. A cloud of free electrons fill the gaps between the positive lattice ions, and "glue" them together. This is known as metallic bonding. The free electrons collide randomly with the positive lattice ions and other impurities within the metal. They have random speeds and directions.

A face-centred cubic lattice structure.

Free-Electron and Quantum Models

In the free-electron model, the free electrons travel freely throughout the crystal as a gas of non-interacting charges. In other words, when using this model, the electrical interactions between the electrons and the positive lattice ions are neglected, as are the interactions between the electrons.

In this model, during normal conditions (room temperature with no voltage applied), every free electron has a thermal velocity (the typical velocity of the thermal motion). This value, as the name suggests, is dependent on temperature. The average distance a free electron travels before colliding with something is called the mean free path (denoted using λ) and the average time between collisions is denoted using the letter tau. Therefore, the thermal velocity is calculated using the equation:

thermal velocity = mean free path/average time between collisions

Electron motion due to heat is fast but random

This a model based of Newtonian mechanics. Models involving quantum mechanics, whilst the free-electron model is a useful tool, are more accurate. These models dictate that the movement of electrons depend on quantum ideas. Unlike thermal velocity, Fermi velocity, is not dependent on temperature. It is the velocity of electrons in the highest energy state (where they are in the conduction band) at zero temperature. In other words, it is the velocity that electrons travel purely due to quantum effects instead of due to thermal effects.

Due to the random nature of electron movement caused by both thermal and quantum effects, the average Fermi velocity and thermal velocity of an electron in the conductor is zero. However, in one particular direction, thermal velocity and Fermi velocity are both high since thermal energy and Fermi energy (the maximum energy state occupied by an electron) are very high and the mass of an electron is very low (E = 1/2*m*v^2 applies here because energy is kinetic).

Work Function and the 4 Special Processes to Eject Electrons

Normally, at room temperature, it is impossible for electrons to escape the surface of the metal due to the electrostatic attraction of the positive lattice ions in the metal. The energy binding the electrons to the surface of the metal is known as the work function.

However, if an electron possesses more energy than the work function it can leave the metal. There are 4 special processes by which an electron can do this:

1) Thermionic emission - Electrons are provides with enough energy through heating to overcome the work function.

2) Field emission - A strong enough positive field does enough work on the electrons to overcome the work function.

3) Secondary emission - Energy is transferred from high-speed electrons bombarding the surface to the surface electrons, leading to the surface electrons having enough energy to leave the surface.

4) Photoelectric emission - If a photon hitting the surface has an energy greater than or equal to the work function, the electrons gain all of that energy in a 1-on-1 interaction and therefore have enough energy to be emitted from the surface.

Albert Einstein won the 1921 Nobel Prize in Physics for his discovery of the photoelectric effect.

Drift Velocity and the Application of a Voltage

When a voltage is applied across a conductor, an electric field is set up through that conductor (due to the excess of electrons being pumped into the negative end of the wire). This means that the electrons experience a force pulling them towards the positive end of the wire. According to Newton's 2nd Law, this should cause an acceleration in that direction. However, the constant collisions the electrons experience in the conductor, creates a kind of "drag" force that, on average, balances out the force caused by the field giving the electrons a constant velocity towards the positive end of the conductor.

The electrons experience this constant velocity towards the positive end of the conductor (a drift velocity) because the field causes the electrons' path to deviate, creating a more parabolic path and the electrons are shifted towards the positive end of the conductor compared to electrons that are not under an electric field. However, this "shift" is very small because the thermal velocity of the electrons is so great, it requires a large force to change their momentum even slightly. Since there is a small displacement towards the positive end per second, the drift velocity is low. All electrons are moving in the same direction towards the positive end and so a current flows.

Current can still flow despite the tiny drift velocity because, when a voltage is applied, the electric field created means that the electron has a repulsive force on its neighbour, and that one has a repulsive force on its neighbour and so on. This then creates a chain reaction that propagates through the conductor at almost the speed of light. For instance, water flows out of a hosepipe immediately because the hosepipe is already filled with water, so the water particles near the nozzle get a slight push from those behind and flow out. This is the same thing that happens with electrons in a filament bulb, so the bulb turns on immediately.

This is one of the diagrams in this section of the book comparing the magnitudes of the thermal velocity, drift velocity and signal velocity.

Drift Velocity Equation

The book then gives an equation for drift velocity:

drift velocity = current density/(charge of an electron x free-electron density)

v = J/eP

current density = current/cross-sectional area (measured in A/m^2)

free-electron density = elects/volume (measured in electrons/m^3)

This can be derived by doing the following:

To find the total charge in a conductor -

Total charge = (charge of an individual electron) x (number of electrons per unit volume) x (volume)

Q = ePAl

I = dQ/dt

I = d(ePAl)/dt

Since dl/dt = velocity and l is the length of the conductor, velocity is in that direction, therefore it can represent drift velocity.

I = ePAv

Rearranged, this gives v = J/eP

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