Page 330 - Electrical Properties of Materials

312 Lasers

12.7.2 Wells, wires, and dots
On the whole, is it an advantage to have multiple quantum wells? One might
be permitted to see only disadvantages. Surely, the more quantum wells, the
more complicated must be the production process.
In order to see the advantages, we need to investigate what happens as we
further reduce the thickness of the active region. The main effect is that the
discrete nature of the energy levels will be more manifest. Let us remember
(eqn 6.1) the energy levels in a potential well:

2 2
h n
E = . (12.39)
8mL 2
If the lateral dimension of the well is 10 nm, then the lowest energy level comes
to 0.056 eV (where we have taken the effective mass of the electron at m m –1 =
∗
0.067). In terms of the energies we talk about this is not negligible. It comprises
about 4% of the energy gap of GaAs. If this is the lowest energy available above
the bottom of the conduction band, and similarly, there is a highest discrete
level for holes in the valence band, then the wavelength of emitted radiation is
determined by the energy difference between these levels. Thus, one advantage
should now be clear. Our laser can be tuned by choosing the thickness of the
active layer in a Multiple Quantum Well (MQW) device. The tuning range
might be as much as 20%.
Are there any other advantages? To answer this question, we need to make a
digression and look again at the density of states function which we worked out
in Chapter 6. Let us start with the energy levels of a three-dimensional well, as
given by eqn (6.2) but permitting well dimensions to be different:

2
h 2 n 2 x n y n 2 z
E = + + . (12.40)
8m L 2 L 2 L 2
x y z
In MQW lasers the dimensions L y and L z are much larger than L x = d,the
thickness of the active region. We may just as well take L y = L z = l, with which
eqn (12.40) modifies to

!

d 2 2 2
2
E = E 0 n + n + n , (12.41)
x y z
l
where
h 2
E 0 = . (12.42)
8md 2
It is clear from eqn (12.41) that n x has a much higher influence on the al-
lowed energies then n y and n z . There will be big steps at n x = 1, 2, 3,etc.Our
primary interest is in the density of states because that will tell us how many
electrons within an energy range dE can make the plunge downwards.
Next, let us determine the density of states in the region between n x = 1 and
n x = 2. This is then determined by n y and n z . Within a radius of n( 1) the