Page 15 - Photonics Essentials an introduction with experiments

Electrons and Photons

Electrons and Photons 9

Figure 2.1. A schematic picture of a collection of atoms in a gas. The arrows give the
magnitude and direction of the velocity of each atom. If the gas is contained in a bottle on
your lab bench, then the average velocity of the atoms relative to you is 0. However, the
average of the square of the velocity is a positive number.

So, what is this constant? Boltzmann’s constant, of course!
Pr(E = E 1 ) = A·e –(E/k B T)
k B T 0.026 eV @ 295 K = room temp (2.3)

If the total number of gas molecules in the bottle is N T , the number
of molecules having energy E 1 is given by the total number of mole-
cules times the probability that a molecule has energy E 1 :

n(E 1 ) = N T Pr(E = E 1 ) = N T · e –(E 1 /k B T) (2.4)

The number of molecules at energy E 2 relative to those at energy E 1
is readily expressed:
n(E 2 )
= e –(E 2 –E 1 )/k B T (2.5)
n(E 1 )
The Boltzmann relation given in Eq. 2.5 is a fundamental tool that
you use to determine how photonic devices operate. The Boltzmann
relation can be applied to electrons as well as to molecules, provided
that these electrons is are equilibrium. With suitable and simple mod-
ifications, it is possible to use this relationship under nonequilibrium
conditions. The current–voltage expression for a p-n diode is exactly
that adjustment. We will use this tool over and over throughout this
book. Its importance cannot be overestimated.

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