
Lets first look at an ideal
single quantum well (SQW),
rectangular and with an extension d_{z} and infinite
depth (the index "z" serves to remind us, that we always have
a threedimensional system with the onedimensional quantum structures along
the zaxis). 


We have already solved the Schrödinger
equation for this problem: It is nothing else but the onedimensional free
electron gas with d_{z} instead of the length
L of the crystal
used before. 


We thus can take over the solutions for the
energy levels; but being much wiser now, we use the
effective mass
instead of the real mass for the electrons and obtain 


E 
= 
( ·
k_{z})^{2}
2m_{e}^{*} 




With k_{z} = ±
n_{z} · 2p/d_{z}, and n = ±
(0,1,2,3,...). 


We have used periodic
boundary conditions for this case, which is physically sensible for
large crystals. The wave functions are propagating plane waves in this case. It
is, however, more common and sensible to use fixed boundary conditions,
especially for small dimensions. The wave functions then are standing waves.
Both boundary conditions produce identical results for energies, density of
states and so on, but the set of wave vectors and quantum numbers are
different; we have 


k_{z} =
j_{z}· p/d_{z}, and j = 1,2,3,..
(we use j as quantum number to indicate a change in the system).
For the energy levels in a single quantum well we now have the somewhat
modified formula 


E 
= 
^{2} ·
p^{2}
2m_{e}^{*} 
· 
j ^{2}
d_{z}^{2} 



The absolute value of the energy
levels and the spacing in between increases with decreasing width of the
SQW, i.e. with decreasing thickness d_{z} of the
small band gap semiconductor sandwiched
between the two large band gap semiconductors. 


Large differences in energy levels might be
useful for producing light with interesting wave lengths. In infinitely deep
ideal SQWs this is not a problem, but what do we get for real
SQWs with a depth below 1 eV? This needs more involved
calculations, the result is shown in the following figure. 





The SQW has a depth of 0,4 eV; if
it disappears for d_{z} = 0, we simply have a constant
energy of 0,4 eV for the ground state; all excited states stop at that
level. 

For layer thicknesses in the
nm region (which is technically accessible) energy differences of 0,2
eV – 0,3 eV are possible, which are certainly interesting, but not so
much for direct technical use. 


While SQWs are relatively easy to produce
and provide a wealth of properties for research (and applications), we will now
turn to multiple quantum wells obtainable
by periodic staggering of different semiconductors as
shown before. 


