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Lets look at the
band diagrams of an isotype heterojunction in equilibrium; we chose a somewhat
extreme case: |
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Electrons were transferred form the (heavily
n-doped) wide-gap material 1 to the (lightly n-doped;
almost intrinsic) small gap material 2. Space charge regions form; on
the left hand side by positively charged ionized dopant atoms; on the right
hand side by the increased electron concentration. |
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In other words: In the low gap semiconductor we
have carrier accumulation
like in
Si MOS devices |
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What happens if we apply a bias
U? |
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Well, draw the band diagrams. Move one side up by
e · U, and
make sure that the band
discontinuities do not change. What you get for the structure from above is
shown below. |
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The current is always carried by majority carriers
- i.e. electrons in our case. Inspecting the (exaggerated) drawings, it is
clear that this is relatively easy from left to right, but not from right to left. Without going into the
details of the characteristics, there are several novel features emerging with
possible uses for devices. The first one of these effects is clear; the
following ones need consideration: |
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1. We may use an isotype heterojunction to
inject majority carriers from the wide band gap material into the small band
gap material as in the case
of diode-type junction, and |
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2. We may use an isotype
heterojunction to spatially separate the carriers generated by doping in the
wide band gap material from the doped region. |
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3. We might have
peculiar new quantum effects. |
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While the first point is relatively clear,
including its usability for light emitting devices (again, try to figure this out yourself),
the second and third point needs some explaining. |
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Lets look at the isotype junction in equilibrium
again to understand the second point. |
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What happened is essentially that the electrons missing in
the space charge region on the left hand side where transferred to the
potential dip on the right hand side. Of course, electrons are also running
down the slope from the right, but the essential contribution is from the wide
band gap material on the left (which, after all, is the cause for the
dip). |
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Since we picked a highly doped material 1 and a lightly
doped material 2, we now have a lot of electrons (their density is
essentially given by the doping concentration in material 1) in a
crystal with few ionized dopant atoms. |
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And what this means is that we now have a high density of highly mobile electrons, because the
mobility at high doping concentration is always severely decreased by
scattering at the ionized dopants - cf. the
paragraph to
this topic. This effect is most pronounced at low temperature and can lead
to a mobility enhancement of an order of magnitude or even more. |
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This is not only generally useful, but can be
carried to extremes. All we have to do is to make sandwiches as shown below. |
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With properly chosen dimensions, deep potential wells will
form in the low band gap material that contain most of the electrons from the
highly doped wide band gap material. This amounts to a novel way of doping
material 2 called modulation
doping. |
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If the potential wells are small enough -
which is usually the case - the confinement of the electrons in the wells leads
to pronounced quantum effects, we therefore call these potential wells "quantum wells" (QW) and distinguish
single quantum wells (SQW) and multiple quantum wells (MQW). A SQW is obtained by
sandwiching just on small gap semiconductor, a MQW as shown above. The
introductory picture of the
heterojunction subchapter showed examples of both types. |
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We may even improve on that by inserting an extremely thin
layer (say 10 nm) of intrinsic material of a suitable band gap between
the two basic materials. If properly done, this layer, while not impeding
carrier flow into the potential wells for equilibration, keeps the carriers
from being scattered at the interface and thus increases the mobility even
more. |
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But with that we have not yet exhausted the
possibilities of heterojunctions - we will now turn to special quantum effects.
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Lets consider the peculiar quantum effects in
modulation doped structures by looking at some typical dimensions. |
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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
1018 cm3 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. |
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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 multi-junction
configuration shown above) and this means that we now have essentially a
two-dimensional electron gas. |
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What does that mean? Especially if we make the
thickness of the layers extremely thin? |
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It means that we have a periodic crystal in two dimensions
(x and y) and a one-dimensional potential well in
the z-direction which is always the direction used in the
pictures above. |
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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 z-direction: |
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2 |
æ
ç
è |
1
mx* |
· |
¶2
¶x2 |
+ |
1
my* |
· |
¶2
¶y2 |
+ |
1
mz* |
· |
¶2
¶z2 |
ö
÷
ø |
y(r) |
e · V(z) · y(r) = E ·
y(r) |
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This equation is solved by |
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y(r) |
= |
yvert(z) ·
ylateral(x,y) |
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The two functions ylateral(x,y) and yvert(z) are decoupled, the
solutions can be obtained separately. For the lateral part we simple have
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yvert(z) |
= |
Solutions of the two-dimensional
free electron gas problem |
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The vertical part of the solution comes frome solving the
remaining one-dimensional Schrödinger equation |
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2 |
æ
ç
è |
1
mz* |
· |
¶2
¶z2 |
e · V(z) |
ö
÷
ø |
yvert(z) |
= Evert · yvert(z) |
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It is rather clear that the structure of the
two-dimensional 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
ylateral(x,y). The energy
eigenvalues are unchanged, too, they
were for the free
electron gas (using the
effective masses
by now). |
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Elateral |
= |
k2lat
2m*lat |
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The solution of the one-dimensional problem in
z-direction 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 |
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Evert |
= |
( p)2
2m*lat |
· |
j 2
dz2 |
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With j = 1, 2, 3, ... = quantum number, and
dz = thickness of the potential well. |
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The total energy of an electron is now given by
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Etotal |
= |
( klat)2
2m*lat |
+ |
( p)2
2m*lat |
· |
j 2
dz2 |
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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 |
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Dlat |
= |
· m*lat
2p |
= constant |
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If you like to try your hand at a little math: The formula for
Dlat is rather easy to obtain if you follow the recipe
for the three-dimensional DOS for this
case. |
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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..... |
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Each quantum well layer is now something like a one-dimensional 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 z-direction, there must be a point where the wavefunctions in
z-direction (the yvert(z)) start to overlap and do
all the things real atoms do at close
distance. |
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The energy levels change and split, and - in analogy to a
crystal formed by real atoms - a one-dimensional 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. |
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This is a
momentous statement! Think about it! |
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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. |
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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). |
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