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Essentially, the semiconducting
properties of Silicon stem from the sp3 hybrid bonds formed
between electrically neutral atoms. |
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Two Si atoms donate one
electron each to all four sp3 hybrid bonds, forming the
familiar diamond type lattice. |
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Substituting a Si atom by a group
III or group V element produces a mobile hole or electron
and an immobile ion in the familiar
way. |
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All III-V compounds
essentially keep this structure. They form sp3 hybrid
orbitals, but there is now a big difference
to Si (or Ge, or diamond-C): |
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The atoms must become
ionized, at least to some extent. The group V elements
N, P, As, or Sb donate an electron to the group
III elements Ga or In. |
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This allows both partners to form the
sp3 hybrid orbitals needed for forming a diamond type
lattice. |
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The binding, which was totally
covalent for the elemental semiconductors, now has an ionic
component. The percentage p of the ionic binding
energy varies for the various compounds |
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The percentage p of the ionic
binding energy is closely related to the so-called
electronegativity of the
elements and varies for the various compounds. The electronegativity describes
the affinity to electrons of the element; in a binding situation the more
electronegative atoms will more strongly bind the electrons then its partner an
therefore carry a net negative charge. The difference in electronegativity of
the atoms in a compound semiconductor therefore gives a first measure for
p. |
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To give some examples: For Si we have
p = 0, for GaAs we find p = 0,08. |
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Doping is still achieved by
introducing specific atoms as substitutional
impurities. But in contrast to elemental semiconductors, we now have
more possibilities as can be seen by looking at the relevant part of the
periodic table. |
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| II |
III |
IV |
V |
VI |
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B |
C (2,5) |
N (3,1) |
O (3,5) |
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Al (1,5) |
Si (1,7) |
P (2,1) |
S (2,4) |
| Zn (1,7) |
Ga (1,8) |
Ge (2,0) |
As (2,2) |
Se |
| Cd (1,5) |
In (1,5) |
Sn (1,7) |
Sb (1,8) |
Te (2,0) |
| Hg |
Tl |
Pb (1,6) |
Bi (1,7) |
Po |
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The numbers in brackets give the
electronegativity, the
elements in yellow cells are never used for doping of compound
semiconductors. |
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A more electronegative element
replacing a certain lattice atom will attract the electrons from the partner
more strongly, become more negatively charged, and thus increase the ionic part
of the binding. |
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That has nothing to do with its ability to donate
electrons to the conduction band or to accept electrons from the valence band,
however. If a foreign atom acts as a donor or acceptor depends only on the
energy levels it introduces in the band gap. |
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We now can look a the basic
possibilities open for doping: |
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We can replace the group III elements by
group II elements to produce acceptors and the group V elements by group
VI elements to produce donors - in
principle. |
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But we can also replace both the group III
and group V elements by group IV elements - which may generate
donors or acceptor levels in the band gap
of some compounds - we have amphoteric
doping. |
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We could also replace the atoms of the compound
by an isoelectronic atom - group
III elements by some other group III elements and the same thing
for the group V partner. In Si, this would mean replacing a
Si atom by e.g. a Ge or C atom - which is not very
exciting. In compounds, however, doping with
isoeletronic atoms produces differences in the ionic part of the
binding and therefore local potential
differences with noteworthy effects as we shall see below. |
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We could even replace an atom of the lattice by a
small molecule - isoelectronic or not - and achieve a doping effect. |
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Well, in Si we could also use
molecules and all group III or group V elements - but in reality
only B, As, P, and sometimes Sb is used. |
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We do not use the group III elements
Ga, Al, or In; neither the group IV elements
N and Bi. The reasons are "technical": The may be
difficult to incorporate in a crystal, their solubility may be too small, their
diffusivity too high (or too low?), or their energy levels in the band gap not
suitable. |
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The same situation occurs with
compound semiconductors. |
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There are optimum
solutions to doping, depending on the type of semiconductor, the
technology available or mandated by other criteria, and so on and so
forth. |
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There are therefore no general rules for optimal
doping, and here we will only discuss amphoteric doping and isoelectronic doping in somewhat more detail. |
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The
energy levels of some
dopants and other impurities in some III-V semiconductors are shown
in an illustration module |
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A prime case of amphoteric doping is
the incorporation of Si into GaAs. If the Si atoms replace
Ga atoms, they act as donors, and as
acceptors if they occupy As lattice
sites. How does this work? |
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A Si atom has four electrons disposable
for binding. If it replaces a Ga atom that had only three atoms, the
As partner does not have to supply an electron any more to make up for
its "deficient" partner Ga, and the surplus electron will only
be weakly bound - it will easily escape into the conduction band |
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Si on
a Ga site thus causes the release of
an electron from the As - it acts as donor even so the electron is actually supplied by
the As. |
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Contrariwise, if an As atom is replaced by a Si atom, the new twosome is now short one
electron. It therefore will fill its hole by an electron from the valence band
- Si now acts as an acceptor. |
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While this seems to offer an elegant
way for doping, the tough question now is: How do we control which lattice sites are occupied by
Si. |
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In other words: On what kind of lattice sites
will we find the Si atoms after it was ion- implanted, diffused into the
crystal from the outside world, or incorporated directly during crystal growth
procedures or thin film growth? |
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This question cannot be easily
answered from first principles. Two guidelines are: |
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At low
temperatures (T £ 700
oC), Si will prefer to sit on As sites - it acts
as acceptor with an energy level about
0,35 eV above the valence band edge. |
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At high
temperatures (T ³ 900
oC), it tends to sit at Ga sites and acts as donor with an energy level about 0,006 eV
below the conduction band edge. |
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If Si is incorporated into a GaAs
melt, large Si concentrations tend
to produce donors, small concentrations acceptors. |
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More about amphoteric doping and its
practical aspects in the various chapters about specific compound
semiconductors. |
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If we introduce isoelectronic
elements in the lattice - e.g. a group V atom like N or B
substituting a P atom, or even a molecule with 5 electrons to
share like ZnO substituting a GaP pair in the GaP lattice
- we do not "dope" in the conventional sense of the word. We rather
change the ionic component of the local binding. |
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Since the introduction of isoelectronic elements
is deliberate with a specific purpose in mind, we deal with it under the
general heading "doping", keeping in mind that we do not change the
carrier concentration in this way, but
their recombination behavior. |
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Doping with isoelectronic elements may not do
much in most compound semiconductors, but it can have pronounced effects in
others by providing new radiative
recombination channels by an interaction of the isoelectronic dopant and
electron-hole pairs called excitons. |
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The paradigmatic semiconductor for isoelectronic
doping is GaP, an indirect
semiconductor. It is used as a strong emitter of green light, however, by
doping it with isoelectronic elements and using the
radiative decay of excitons bound to the doping elements. |
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How does this work? There are several
crucial ingredients, all from rather involved solid state physics: |
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First, we need excitons. |
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Second, we need to have the excitons bound to isoelectronic dopants. |
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Third, we need to have a
radiative recombination of the electron and hole constituting the
exciton - despite the nominal violation of the crystal
momentum. |
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A detailed treatment of these points
(explaining, among other things, why this "works" in GaP, but
not in many other compound semiconductors) is not possible in the context of
this lectures course. We will only superficially look at the basics; somewhat
more information is contained in an
advanced module. |
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What is an exciton? Imagine the generation of an electron - hole
pair, e.g. by irradiating a semiconductor with light. |
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If the photon energy is large enough to lift an
electron all the way from the valence band to the conduction band in a direct
transition, you now have a free electron
and a free hole which move about the
crystal in a random way. Their coordinates in real space
(r-space) are arbitrary or undetermined, while their coordinates
in k-space are well-defined: they are identical and do not
change. |
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Now imagine that the hole and the electron stay
so close to each other (i.e. less than a
Debye
length) in real space that they feel
the Coulomb attraction. They are then bound
to each other with a certain binding energy Eex and
their coordinates in r-space are (nearly) identical, but still
undetermined. |
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We already know a system where
one negative elementary charge is bound to
one positive one - it is called
"Hydrogen atom". The only
difference between an exciton and an Hydrogen atom is that the mass of the hole
is much smaller that the mass of the proton (and, of course, that our exciton
can only exist in a solid). |
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From a more advanced treatment of the Hydrogen
atom, where the electron mass is not
neglected with respect to the proton mass, we immediately can carry over the
solution of the relevant Schrödinger equation to an exciton. Of interest
are especially the allowed energy states, we obtain for
Eex |
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| Eex (k) =
Egap – |
mred ·
q4
8 · (n · e0
er · h)2 |
+ |
h2 k2
8p2 · (me +
mh) |
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With q = elementary charge,
n = quantum number = 1,2,3,..., mred
= reduced mass (from
1/mred = 1/me +
1/mh). |
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The first term simply accounts for the crystal
energy, the second one is straight from the hydrogen atom, and the third term
is a correction if the two particles are not at the same place in
k-space (it is zero for ke = –
kh or ke = kh = 0 as
it will be for most direct semiconductors). |
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In total we have a system of energy
levels right below the conduction band with
the "deepest" level defined by the energy difference |
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| DEex = |
mred ·
q4
2e0 er · h |
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This formula would give us a constant value of about 10 meV for all semiconductors - except that they all have
different masses because we have to take the
effective masses,
of course, and different er. |
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Properly calculated values, e.g. for GaAs
give: DEex(n = 1) = 4,4
meV which is close to the experimental one.
More values can be found in
the advanced module. |
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In k-space for
GaP this looks like this: |
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It simply takes DEex less energy to kick an electron
from the top of the valence band to the lowest exciton state - because you safe
the binding energy. |
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Since DEex for a free exciton - that moves
about the crystal, transporting energy, but not charge - is just a few
meV, comparable to typical donor levels, it will not live very long at
room temperature. The thermal energy then is enough to ionize the exciton, i.e.
to remove the electron (or the hole; your choice; everything is rather
symmetrical), and we are left with a free hole and electron. |
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Even so the electron and the hole are almost at
the same place, recombination is not
possible without the help of phonons, so it is rather unlikely -
as stated before. Excitons in
most semiconductors therefore only make their presence known at low
temperatures - and in the absorption of light, because you will already find
some absorption for light with an energy slightly below the band gap
energy! |
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Now imagine an isoelectronic dopant in GaP, e.g. N
instead of P. It distorts the potential for electrons a little bit
and strictly locally; and in GaP
this will lower the energy of the electrons
locally - inside a radius comparable to the lattice constant. |
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An electron thus may become
bound to the isoelectronic dopant - i.e. it "revolves" around the
isoelectronic atom. It may now attract a hole by Coulomb interaction over
comparatively large distances and thus form a bound
exciton. This is an easy, albeit oversimplified way, to conceive bound
excitons. |
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The net effect of a positive
("binding") interaction of an isoelectronic dopant and an exciton is
twofold: |
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The exciton energy levels move "down"
from the free exciton level by an amount equal to the binding energy of the
electron to the isoelectronic dopant; i.e. DEex increases. In some cases - naturally for GaP
- the binding energy may be in the order of 10 meV, and this pushes the
exciton levels so far below the conduction band that the bound exciton is now
relatively stable at room temperature. |
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The exciton is now localized in space. This demands that its
coordinates in k-space must be somewhat undetermined thanks to
the uncertainty principle which requires
that |
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D( k) ·
D(r) |
> |
h |
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With
k =
momentum. Since
Dr is in the order of the length scale
of the attractive potential, i.e. a lattice
constant, Dk will be in the
order of a/2p, i.e. a Brillouin zone
width. |
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In other words, the bound exciton can have any wave vector in the 1st Brillouin zone
with a certain, not too small probability. |
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In sum total we conclude: Recombination for bound excitons is easy! It is
still an indirect recombination that needs a phonon as a third partner. But in
contrast to indirect recombination between free electrons and holes, which
needs a photon with a precisely matched k-vector, any phonon will do in this case because it always
matches one of the k-vectors from the spectrum accessible to the
bound exciton. |
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If bound excitons exist (at room temperature),
their recombination provides a very efficient channel for establishing
equilibrium and thus a possibility to generate light with an energy given by
the bandgap minus a small exciton energy. |
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This is essentially the mechanism to extract
light out of the indirect semiconductor
GaP! |
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You should now have
a lot of questions: |
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Why GaP? How about other
III-V compound semiconductors? |
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How about more exotic semiconductors? The
II-VI system, organic semiconductors? |
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Anything similar for elemental semiconductors?
After all, putting Ge into Si also changes the potential
locally. |
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How about other defects, not necessarily
isoelectronic? For example, ionized donors and acceptors also attract and
possibly "bind" free electrons or holes, respectively? |
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Well, this is not an advanced solid
state lecture course. And even there, you may not find all the answers easily.
Some answers I would have to look up, too, some, however, you can work out for
yourself - at least sort of. Otherwise, turn to the advanced module where some
answers will be given (in due time). |
© H. Föll (Semiconductor - Script)
