
Quantum Confinement Effects 



Lets consider the peculiar quantum effects in
modulation doped structures by looking at some typical dimensions. 


The width of the various space charge layers must still be
given by formulas not too different from the ones we had for Si;
details are found in an other
advanced module. For a GaAlAs/GaAs system with a high doping around
10^{18} cm^{–3} in the wide band gap GaAlAs
side, the width of the dips with the high electron density on the GaAs
side is about 5 nm  10 nm while the lateral extension is large by
comparison. 


The mean free path
length of the (highly mobile) electrons is larger than the thickness of the
potential dip (better called potential well for the multijunction
configuration shown above) and this means that we now have essentially a
twodimensional electron gas. 

What does that mean? Especially if we make the
thickness of the layers extremely thin? 


It means that we have a periodic crystal in two dimensions
(x and y) and a onedimensional potential well in
the zdirection which is always the direction used in the
pictures above. 


The relevant Schrödinger equation is easy to write down,
especially in the free electron approximation with a constant potential (=
0) in x and y, and a potential
V(z) in zdirection: 


– 
2 
æ
ç
è 
1^{ }_{ }
m_{x}^{*} 
· 
¶^{2}
¶x^{2} 
+ 
1^{ }_{ }
m_{y}^{*} 
· 
¶^{2}
¶y^{2} 
+ 
1^{ }_{ }
m_{z}^{*} 
· 
¶^{2}
¶z^{2} 
ö
÷
ø 
y(r) 
– e · V(z) ·
y(r) = E ·
y(r) 




This equation is solved by 


y(r) 
= 
y_{vert}(z)
· y_{lateral}(x,y) 




The two functions y_{lateral}(x,y) and y_{vert}(z) are decoupled, the
solutions can be obtained separately. For the lateral part we simple have



y_{vert}(z) 
= 
Solutions of the twodimensional
free electron gas problem 




The vertical part of the solution comes frome solving the
remaining onedimensional Schrödinger equation 


– 
2 
æ
ç
è 
1^{ }_{ }
m_{z}^{*} 
· 
¶^{2}
¶z^{2} 
– e · V(z) 
ö
÷
ø 
y_{vert}(z) 
= E_{vert} ·
y_{vert}(z) 



It is rather clear that the structure of the
twodimensional problem will not be much different from that of the common
three dimenional problem if we introduce a periodic potential in (x,y).
We simply obtain Bloch waves in two dimensions instead of plane waves for
y_{lateral}(x,y). The energy
eigenvalues are unchanged, too, they
were for the free
electron gas (using the
effective masses
by now). 


E_{lateral} 
= 
k^{2}_{lat}
2m^{*}_{lat} 




The solution of the onedimensional problem in
zdirection depends of course on the precise shape of
V(z), but as a general feature of potential wells we must
expect a sequence of discrete energy
levels. For the most simple case of a rectangular well (with
infinite height), standard calculations show that 


E_{vert} 
= 
(p)^{2}
2m^{*}_{lat} 
· 
j ^{2}
d_{z}^{2} 




With j = 1, 2, 3, ... = quantum number, and
d_{z} = thickness of the potential well. 

The total energy of an electron is now given by



E_{total} 
= 
E_{lat} 
+ E_{vert} 

E_{total} 
= 
(k_{lat})^{2}
2m^{*}_{lat} 
+ 
(p)^{2}
2m^{*}_{lat} 
· 
j ^{2}
d_{z}^{2} 




This "simply" means that the states in the
conduction bands are now a discrete series
given by the quantum number j with a density of states per level
of 


D_{lat} 
= 
·
m^{*}_{lat}
2p 
= constant 




If you like to try your hand at a little math: The formula for
D_{lat} is rather easy to obtain if you follow the recipe
for the threedimensional
DOS for this case. 

What do we get from this? Well, a lot of special
effects for enthusiastic solid state physicists, but not necessarily big
advantages for devices. However..... 


Each quantum well layer is now something like a onedimensional atom (in contrast to the three
dimensional real atoms were the wave functions of the electrons were confined
in all three directions. If we move these "atoms" close together in
the zdirection, there must be a point where the wavefunctions in
zdirection (the y_{vert}(z)) start to overlap and do
all the things real atoms do at close
distance. 


The energy levels change and split, and  in analogy to a
crystal formed by real atoms  a onedimensional energy band may start to
develop with an energy range that is given by the
geometry of the system, i.e. the thickness of the layers, for a
multi quantum well structure. 


This is a momentous statement! Think about
it!



It means that we can make materials with energetic
properties that we can tailor at will
(within bonds and limits, of course). We no longer must just live with bandgaps
and other properties that mother nature provides, we
now can make our own systems. At least in principle. 


Something like that we call a metamaterial. 


Well, nobody has made a really hot device with periodic
quantum wells so far, but the time is near. Multiple and single quantum wells
are already part of recent devices as shown in the
backbone II subchapter. We
therefore will devote an own chapter to this issue (in time to
come). 