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Before you start
this subchapter, you should consult the advanced module "Isotype Junctions, Modulation Doping, and
Quantum Effects" - it leads right up to the topics of this
subchapter. |
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Lets first look at an ideal
single quantum well (SQW), rectangular
and with an extension dz and infinite depth (the index
"z" serves to remind us, that we always have a three-dimensional
system with the one-dimensional quantum structures along the z-axis). |
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We have already solved the Schrödinger
equation for this problem: It is nothing else but the one-dimensional free
electron gas with dz instead of the length L used before. |
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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 |
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with kz = +/- nz·
2p/dz, and n =
+/-(0,1,2,3,...) |
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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 |
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kz =
jz· p/dz, 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 |
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The absolute value of the energy
levels and the spacing in between increases with decreasing width of the SQW,
i.e. with decreasing thickness of the small band gap semiconductor sandwiched
between the two large band gap semiconductors. |
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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. |
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The SQW has a depth of 0,4 eV; if it disappears
for dz = 0, we simply have a constant energy of 0,4 eV
for the ground state; all excites states stop at that level. |
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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. |
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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. |
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Since single atoms may also be
described as SQWs (for one electron you just have the hydrogen atom type with a
Coulomb potential), we must expect that the wave function of the electrons
start to overlap as soon as the single SQWs in the MQW structure are close
enough. |
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The situation is completely analogous to the
qualitative formation of a crystal with a periodic potential from atoms. The
discrete energy levels must split into many level, organized in bands. |
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This is exactly what happened; it was
Leo Esaki who first did
the calculations (and more) for this case for which he was awarded the Nobel
prize. |
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Her is the result for the same system shown
above. |
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Energy bands with respectable band gaps develop
indeed, and we now should redraw the band diagram from before to include this
fact: |
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We have "mini-bands" in the quantum
wells (for symmetry reasons also for the holes in the quantum well along the
valence band); green denotes occupied levels; blue empty levels. |
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What doe we gain by this (besides a
Noble prize)? |
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A structure that is used
in commercially sold LASER diodes!. In other words, quite crucial
parts of the information technology rely on MQWs. |
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We will come back to this issue in the context of
semiconductor lasers. |