Materials in Electronics/Electrons in Conductors

Current as a Flow of Electrons

Current is defined as the flow of (positive) charge past a point per unit time:

${\displaystyle I={\frac {Q}{t}}}$

where

• I is current, in amperes
• Q is charge, in coulombs
• t is time, in seconds

Remember that since electrons hold a negative charge, a flow of electrons is in the opposite direction to the conventional current, I. In the diagram below, the conventional current flows from right to left in the conductive material (grey). The electron flow produces a current in the same direction, but the particles themselves are moving the other way, from left to right.

Electrons in a Vacuum

Consider an electron gun that produces a stream of electrons which are accelerated using a potential difference as below:

The kinetic energy of an electron from this electron gun is given by

${\displaystyle T={\frac {1}{2}}m_{e}v^{2}}$

where

• T is the kinetic energy in J
• me is the mass of an electron in kg
• v is the velocity of the electron in m s-1

Now, this is equal to the energy provided to the electron by the electric field, E, which is given by

${\displaystyle T=QV\,}$

where

• t is the energy in J
• Q is the charge in C
• V is the voltage between the plates

Since Q=-e, we can write

${\displaystyle {\frac {1}{2}}m_{e}v^{2}=eV}$
${\displaystyle v={\sqrt {{2eV} \over {m_{e}}}}}$

Electrons in a Resistive Medium

This subsection will derive the drift velocity of an electron in a resistive medium, the conductivity (and therefore also resistivity) and provide a demonstration of the validity of Ohm's Law. It will also introduce mobility.

When electrons flow in a resistive medium, they collide with and scatter off the atoms that make up the medium:

These constant collisions means that, averaged over a length of time, the electrons do not accelerate because every time they collide the velocity is "reset" to, on average, zero. They travel instead at an average velocity, the drift velocity , vd. We shall now derive the value of the drift velocity.

An electric field, E is set up across the medium by the applied voltage, V. This field is given by:

${\displaystyle E={\frac {V}{L}}}$

From Newton's Second Law, we can say that the acceleration, af, of the electron (when not in collision) due to the field is given by

${\displaystyle a_{f}={\frac {F}{m_{e}}},}$

where F is the force exerted by the electric field. This is given by ${\displaystyle F=QE}$. This gives:

${\displaystyle a_{f}=-{\frac {eE}{m_{e}}}}$

The negative sign comes about because of the negative charge on the electron.

Every time the electron collides, it velocity returns to, on average, zero. Call the time between collisions τ. This means that the acceleration due to the collisions, ac is given by the change in velocity over τ. Since the electron can be said to be always travelling at the drift velocity (the average velocity), we have

${\displaystyle a_{c}={\frac {-v_{d}}{\tau }}}$

Since the electron travels at a net constant velocity, the acceleration due to the field and the acceleration due the collisions sum to zero:

${\displaystyle a_{f}+a_{c}=0\,}$

So,

${\displaystyle -{\frac {eE}{m_{e}}}={\frac {v_{d}}{\tau }}}$
${\displaystyle v_{d}=-{\frac {e\tau }{m_{e}}}E}$

We now introduce a new term, electron mobility, which is the velocity of the charge carrier per unit field. This has the symbol μe and is given by

${\displaystyle \mu _{e}={\frac {v_{d}}{E}}=-{\frac {e\tau }{m_{e}}}}$

Now, consider the equation

${\displaystyle I=nAv_{d}q\,}$

where

• n is the charge carrier density in m-3
• a is the cross sectional area in m2

Substituting for vd and q, we get

${\displaystyle I=-enA{\frac {-e\tau E}{m_{e}}}}$

Substituting E=V/l,

${\displaystyle I={\frac {nAe^{2}\tau }{m_{e}}}{\frac {V}{l}}}$

Rearranging for convenience,

${\displaystyle I={\frac {ne^{2}\tau }{m_{e}}}{\frac {A}{l}}V}$

We now have current, I as a function of the voltage, V and constants. Now, consider the definition of conductivity, σ:

${\displaystyle \sigma ={\frac {Il}{VA}}}$

Substituting for I, and simplifying

${\displaystyle \sigma ={\frac {ne^{2}\tau }{m_{e}}}}$
${\displaystyle \sigma =\mu _{e}ne\,}$

Also, we can demonstrate Ohm's Law from here. Note that

${\displaystyle I={\frac {ne^{2}\tau }{m_{e}}}{\frac {A}{l}}V}$

can be written as

${\displaystyle V=IR\,}$

where

${\displaystyle {\frac {1}{R}}={\frac {ne^{2}\tau }{m_{e}}}{\frac {A}{l}}}$

This serves as a proof that in a conductor, so long as none of the other parameters change, voltage is proportional to current.