 |
First, we look at
the junction in equilibrium, i.e. there is no net current and the Fermi energy
is the same everywhere (we have the same situation as
shown before; but for diversity's
sake, the p- and n-side reversed). |
|
|
|
 |
In what follows, there will be a lot
of shuffling formulas around – and somehow, like by magic, the
I–V characteristics of a p–n junction will
emerge. So let's be clear about what we want to do in the major steps
(highlighted by the cyan background). |
 |
The first basic goal is to
find an expression for the carrier density at the edge
of the space charge region. |
|
 |
We know in a qualitative way
from the consideration of pure
diffusion currents that the minority carrier density around the edge of the
space charge region is somewhat larger than in the bulk under equilibrium
conditions – there is a Dnmin given by Dnmin = nmin(x)
– nmin(bulk), and Dnmin will increase for forward
bias, i.e., for non-equilibrium conditions. |
|
 |
We
also know that Dnmin induces a diffusion current
and that we therefore need a "real" current to maintain a constant
Dnmin. Finding Dnmin thus will automatically give us the necessary currents
belonging to the non-equilibrium as defined by the voltage. |
 |
We first look at the
various energies involved: |
|
 |
The energy difference between the left side (=
p-side, raised index
"p") and the right side (= n-side,
raised index "n")
of the junction
is given by |
|
|
ECp –
ECn |
= |
EVp –
EVn |
= |
e · DV |
|
|
|
 |
Here, DV is the difference of the
(yet-to-be-determined) electrostatic potential between the n- and
p-side, taken far away from the junction; the details (especially about
its sign) will be given below. |
 |
The band energy levels
ECn,p(x) and the potential
V(x) are functions of x, which makes all
densities functions of x, too. From now
on, however, we will omit the "(x)" when it's not considered
necessary. |
|
 |
As long as we discuss
equilibrium, the Fermi energy is constant and the carrier densities are given
by their usual expression. We consider
them separately for the left- and right hand side of the junction, i.e., for
the n- and p-part. (Note that here the usual minus sign in the
exponent was used to change the order of the terms in the differences.) |
|
|
nep |
= |
Neffe · exp
|
EF –
ECp
kT |
|
density of
electrons on the p-side |
|
|
|
|
|
|
nhp |
= |
Neffh · exp
|
EVp –
EF
kT |
|
density of
holes on the p-side |
|
|
|
|
|
|
nen |
= |
Neffe · exp
|
EF –
ECn
kT |
|
density of
electrons on the n-side |
|
|
|
|
|
|
nhn |
= |
Neffh · exp
|
EVn –
EF
kT |
|
density of
holes on the n-side |
|
|
|
 |
We also have the
mass action law, here
applicable everywhere since we are in full equilibrium: |
|
|
nep ·
nhp |
= |
nhn ·
nen |
= |
nmin ·
nmaj = |
|
|
ni2 |
|
|
|
|
= |
Neffe ·
Neffh · exp |
æ
è |
– |
Eg
kT |
ö
ø |
|
|
 |
What we need to know to get on is the x-dependence of the energies or of the
potential V(x) – this simply means we need the
quantitative band diagram that so far we always just drew "by
feeling". (This is one of the essential points why we reconsider
the p–n junction here; the other one will be the usage of the quasi-Fermi
energies.) |
|
 |
For this we need to solve the
Poisson
equation and this demands to specify the total charge
r(x) so that we can write down the
charge as a function of x. This is easy
in principle: |
 |
The total (space) charge r(x) at any point along the junction is the sum
of all charges: Electrons (ne(x)), holes
(nh(x)), ionized donors
(ND+(x)), and ionized aceptors
(NA–(x)). We have
as before |
|
|
r(x)
|
= e · |
æ
è |
nh(x) –
ne(x) +
ND+(x) –
NA–(x) |
ö
ø |
|
|
|
 |
Inserting r(x) into the Poisson equation gives |
|
|
– |
e e0
e |
· |
d2V(x)
dx2 |
= |
nh(x) –
ne(x) +
ND+(x) –
NA–(x) |
|
|
|
 |
Solving this equation with the proper boundary
conditions will yield V(x) and everything else –
but not so easily because the situation is
complicated: Since nh(x) and
ne(x) depend on V(x) via
the Fermi distribution, this is also an implicit equation for V(x).
|
|
 |
It is, however, not too difficult to find
good approximations for
"normal", i.e. highly idealized junctions; this is shown in an
advanced module accessible through the link. |
|
|
|
 |
For our final goal, which is to
describe the current–voltage characteristic of a p–n
junction, we use the same approximations
and conventions, namely: |
 |
1. The zero point of the electrostatic potential is
identical to the valence band edge in the p-side of the junction, i.e.
eV p = EVp = 0 as shown
in the complete
illustration to the situation shown in the picture above. This is a simple
convention without any physical
meaning. |
 |
2.
All dopants are ionized, their
density is constant up to the junction, and there is only one kind on each side
of the junction, i.e. |
|
|
nhp(bulk) |
= |
NA |
= |
NA– |
|
|
|
|
|
nen(bulk) |
= |
ND |
= |
ND+ |
|
|
|
 |
This is a crucial
assumption. Note that while
nh,ep,n(bulk) are constant, this is not
required for nh,ep,n(x) around the
junction. |
 |
3. We also
assume that away from the junction, the Si extends into infinity (or at
least to a distance much larger than several diffusion lengths) to both sides
of the junction – in total we use the "abrupt"
"large" junction approach |
|
 |
This gives us for the carrier densities in
equilibrium anywhere in the junction: |
|
|
nh(x) |
= |
NA · exp |
æ
è |
– |
e · V(x)
kT |
ö
ø |
|
|
|
|
|
ne(x) |
= |
ND · exp |
æ
è |
– |
e · [V n –
V(x)]
kT |
ö
ø |
|
|
|
 |
Here, V n is the constant
value of the potential deep in the n-type region. Note that, having
chosen the zero point for V(x) at the p-side of the
junction where there is the negative pole of the electric field, it holds
inside the SCR that 0 £
V(x) £ V n.
|
|
 |
These equations mean that the carrier density is whatever you have in the
undisturbed p- or n-part (i.e., the dopant density) times the
Boltzmann factor of the energy shifts relative to this situation. |
 |
V n is the
built-in potential for equilibrium conditions, it is
thus determined by the difference
in the Fermi energies of the n- and the p-side before contact
(relative to the band edges) – our
simple view of a junction is totally
correct on this point. |
|
 |
With and without an external voltage
Uext we have |
|
|
V n(Uext=0) |
= |
1
e |
· |
(EFn
– EFp) |
|
|
|
|
|
V n(Uext)
|
= |
1
e |
· |
(EFn
– EFp + e ·
Uext) |
|
|
|
 |
Here, the sign of Uext
is such that a positive external voltage increases the built-in potential difference. Note
that this is just an interim choice; later on we will replace it by the usual
standard. |
|
 |
In the general case, the maximum potential at
the n-side, V n(bulk), becomes |
|
|
V n(bulk) |
= |
V n(Uext=0) +
Uext |
= |
1
e |
· |
DEF +
Uext |
= V n + Uext
|
= DV |
|
|
|
 |
Looking at the
proper solution of the
Poisson equation for our case, we realize that the space charge region was
defined as the part of the Si where the potential was not yet constant.
This means that V n(bulk) = V n|SCR
edge on the n-side, and V p|SCR
edge = 0. This is an essential point, even so it is matter-of-course.
|
|
|
|
 |
We now can move
towards our primary goal and find an
expression for the carrier density at the edge of the SCR by
considering the ratio of a
carrier species on both sides of the junction. From the
equations above, we obtain for the edge of the SCR: |
|
|
nep
nen |
÷
÷
÷ |
SCR
edge |
= |
nhn
nhp |
÷
÷
÷ |
SCR
edge |
= exp |
æ
è |
– |
e · DV
kT |
ö
ø |
= exp |
æ
è |
– |
e · (V n +
Uext)
kT |
ö
ø |
|
|
|
 |
The minority carrier densities (always at the edge of the SCR without indicating it
anymore) can now be written as |
|
|
nep(Uext)
|
= |
nen(Uext)
· exp |
æ
è |
– |
e · (V n + Uext)
kT |
ö
ø |
|
electrons on
the p-side |
|
|
|
|
|
|
nhn(Uext)
|
= |
nhp(Uext)
· exp |
æ
è |
– |
e · (V n + Uext)
kT |
ö
ø |
|
holes on
the n-side |
|
|
|
 |
These equations are nothing but the Boltzmann
distribution giving the number of particles (nmin)
that make it to the energy e(V n +
Uext) out of a total number nmaj
– in thermal equilibrium. We
used essentially the same equation before, but now we know the kind of
approximations that were necessary and that means we also know what we would
have to do for "better" solutions of the problem. |
 |
Since this is important, let's review
the approximations we made: |
|
 |
Besides the
"abrupt" "large" junction, we used the approximations
from the simple solution
to the Poisson equation which implies that the potential stays constant right
up to the edge of the SCR and then changes monotonously. |
|
 |
This means that for equilibrium we must obtain
the same equations by computing the minority carrier density from the
mass action law, i.e.
|
|
|
nep(Uext=0)
|
= |
ni2
nhp(Uext=0) |
|
|
|
 |
We will see if this is true in a little exercise:
|
|
|
Exercise 2.3.5-1 |
Show the equivalence of the two equations
for the minority carrier density! |
|
 |
Now comes a
crucial point: We are looking at stationary
non-equilibrium. We first review the starting point again: |
|
 |
At equilibrium
(Uext = 0), the majority carrier densities
nen |SCR edge and
nhp |SCR edge are given by |
|
|
nhp |
= |
Neffp · exp
|
æ
è |
– |
EF
kT |
ö
ø |
|
|
|
|
|
nen |
= |
Neffe · exp |
æ
è |
+ |
EF –
ECn
kT |
ö
ø |
|
|
|
 |
Do you remember them? These are two of our first
equations from above, but given here for the
choice of EVp = 0.
|
 |
The essential point
for the majority carrier density at the edge of the space charge region for
non-equilibrium is that it
remains practically unchanged
(approximately at its bulk value) if we now apply a voltage
Uext, i.e. |
|
|
nen(Uext)
|
÷
÷ |
SCR
edge |
= |
nen(equ) |
÷
÷ |
SCR
edge |
= |
nen(bulk) |
|
nhp(Uext)
|
÷
÷ |
SCR
edge |
= |
nhp(equ) |
÷
÷ |
SCR
edge |
= |
nhp(bulk) |
|
|
|
 |
The trick here is that we consider the majority
carrier density at the SCR edge
– and the position of the latter may vary with the applied voltage! |
|
 |
Nevertheless, beyond that point we have the bulk
behaviour of the majorities – because that's how we have defined the
SCR edge: The bulk potential stays constant right up to the edge, and
this is only possible for a constant density of majority carriers.
|
 |
The minority
carrier densities nep |SCR edge
and nhn |SCR edge, however,
depend very much on the applied voltage as
expressed in the formulae above. |
|
 |
Thus, we have to adjust the
minority carrier density independent of the majority density, which means we
have to use the quasi-Fermi energies. |
|
 |
In other words: While the
quasi-Fermi energy
EFmaj for majority carriers remains at the
equilibrium value EF near the SCR, the quasi-Fermi energy for the minority carriers,
EFmin, branches off early; the details will
be shown below. |
 |
We now ask about
the difference of the minority carrier density
relative to equilibrium, i.e. we look at |
|
|
Dnep |
÷
÷ |
SCR
edge |
= |
nep(Uext)
– nep(Uext=0)
|
|
Dnhn |
÷
÷ |
SCR
edge |
= |
nhn(Uext)
– nhn(Uext=0)
|
|
|
|
 |
For this, we can only use the relationship to
the corresponding majority carrier density at the other side of the junction,
because only then we include the effect of the applied voltage. |
|
 |
For both types of carriers it comes out as |
|
|
Dnmin |
= |
nmaj · |
æ
ç
è |
exp |
æ
è |
– |
e · (V n+
Uext)
kT |
ö
ø |
– exp |
æ
è |
– |
eV n
kT |
ö
ø |
ö
÷
ø |
|
|
|
|
|
|
|
|
|
= |
nmaj |
· |
exp |
æ
è |
– |
eV n
kT |
ö
ø |
· |
æ
ç
è |
exp |
æ
è |
– |
eUext
kT |
ö
ø |
– |
1 |
ö
÷
ø |
|
|
 |
Inserting the
general expressions for the minority carrier density
from above for the case
Uext = 0 yields the final formula for our first goal:
|
|
|
Dnmin |
÷
÷ |
SCR
edge |
= |
nmin(equ) · |
æ
ç
è |
exp |
æ
è |
– |
eUext
kT |
ö
ø |
– 1 |
ö
÷
ø |
|
|
|
 |
In other words: The density of
minority carriers at the edge of the SCR will be changed by an external
voltage. |
 |
In steady
state conditions (which does not
imply equilibrium, just that nothing changes) this density must remain constant
as a function of time. |
|
 |
Since deep in the material the minority carrier
density is unchanged and has its equilibrium value, we now must have a current,
driven by the density gradient alone, and this current
must be maintained by the voltage/current source if we want steady
state. |
|
 |
Physically speaking, the excess density of
minority carriers will diffuse around and disappear after some diffusion
lengths – deep in the material they are not noticeable any more. |
 |
This is exactly the
situation treated under "useful
relations" for
pure diffusion
currents. |
|
 |
We can take the formula derived
there with Dnp,ne,h(x=0)
given by the equation from above and obtain immediately for the current–voltage
relationship of a p–n junction (just considering the
absolute magnitudes): |
|
|
| je(Uext) | |
= |
e · De
Le |
· Dne |
÷
÷ |
SCR
edge |
|
or |
|
| je(Uext) | |
= |
e · De
Le |
· nep(equ) ·
|
æ
ç
è |
exp |
æ
è |
– |
eUext
kT |
ö
ø |
– 1 |
ö
÷
ø |
| jh(Uext) | |
= |
e · Dh
Lh |
· nhn(equ) ·
|
æ
ç
è |
exp |
æ
è |
– |
eUext
kT |
ö
ø |
– 1 |
ö
÷
ø |
|
|
 |
We now see that the external voltage,
as we have introduced it, raises the potential barrier and therefore decreases
the minority carrier density – and, thus, also the current flow.
|
|
 |
This means that, in order to enhance the current flow over the p–n
junction, we have to apply the external voltage in a way that it lowers
the barrier. |
|
 |
Therefore, the forward voltage
is UD := –Uext, and since
it is the forward voltage, it is also the one which is taken as positive; the
subscript "D" refers to the p–n junction functioning as a
diode. |
 |
For the final result we add the electron and hole currents,
drop suffixes and functional arguments now unnecessary, and obtain the
diode equation (giving the total current
density, including the reverse current, counted in the standard way): |
|
|
jD(UD) |
= |
æ
ç
è |
e · nep ·
De
Le |
+ |
e · nhn ·
Dh
Lh |
ö
÷
ø |
· |
æ
ç
è |
exp |
eUD
kT |
– 1 |
ö
÷
ø |
|
|
 |
This is the same
equation as before if we take into account that the
pre-exponential factor can be written in many ways. To see that, we
use the following identies: |
|
 |
For the
diffusion length we
have |
|
|
Le,h |
= |
æ
è |
De,h · te,h |
ö
ø |
1/2 |
|
|
|
 |
From the
mass action law, which
is still valid for the bulk, and the general
approximation for the majority carrier density (that is already contained
in our equations) we get |
|
|
nep |
= |
ni2
NA |
|
|
|
nhn |
= |
ni2
ND |
|
|
 |
Shuffling
everything around with these identities gives us – among many other
equivalent formulations – . . . |
|
|
jD(UD) |
= |
æ
ç
è |
e · Le·
ni2
te· NA |
+ |
e · Lh·
ni2
th · ND |
ö
÷
ø |
· |
æ
ç
è |
exp |
eUD
kT |
– 1 |
ö
÷
ø |
|
|
|
 |
. . . and that is
exactly the equation we got before!
However, we did not have to "cut corners" this time and we did not
have to assume that some
proportionality constant equals 1! |
 |
More important, however: The
interpretation of what happens may now be different. Different in the sense of looking at one and the
same situation from a different point of
view, not different in the sense that it is something else. The two
points of view are complementary and not mutually exclusive; neither one is
wrong! |
|
 |
In the simple picture we looked at the minority
carriers that had to be generated to
account for the loss of carriers accounting for the
reverse current and running down the energy slope. |
|
 |
Here we looked at the surplus of minorities accounting for the forward
current and which has to be moved away from the junction. |
|
 |
Think about why this is the same thing! (Hint:
Start from UD = 0.) |
 |
What is left is just to consider the
quasi-Fermi energies relevant for the
forward direction; not only was the relevant drawing promised already above, it
will also show explicitly what is meant by "surplus of minorities, having
to be moved away from the junction" – because it will show us where
those minorities end up. |
|
 |
To cut a long story short, here it is: |
|
|
|
|
 |
That the quasi-Fermi energies of the majorities
remain constant throughout the SCR corresponds to the expressions giving
the ratio of each carrier type on both sides of the junction. |
 |
Note that in the dawing, deliberately
there are more minority carriers close to the SCR edges than deeper in
the bulk. Yes, that's where the surplus minorities go. But that's not the end
of the story: |
|
 |
That the quasi-Fermi energies of the minorities
outside the SCR linearly merge towards the majorities' ones corresponds
to the exponential decay of the surplus minority density away from the
SCR, with the decay constant given by the diffusion length – as
already discussed for the case of
pure diffusion
currents. |
|
 |
Think for yourself about why all this is the
case! And think about the possible consequences of the surplus minorities'
presence in the case of a direct semiconductor. |
|
|
|
|
Contributions from the Space Charge
Region |
|
|
 |
We now should include the
generation
currents from the space charge region,
as we
did (in a somewhat fishy way) in our simple consideration of a junction.
|
|
 |
This, however, is not so easy to do in a correct
(albeit still very approximate) fashion. |
 |
For the reverse part of the generation current from the
SCR, we can obtain an equation directly from the
Shockley-Read-Hall theory. All we
have to do is to consider the quasi-Fermi
energies of a junction in reverse
bias. This is schematically shown in the following picture: |
|
|
|
|
 |
The quasi-Fermi energies must behave in the way
shown (the details do not matter), because otherwise the density of charge
carriers (especially minority carriers!) in the junction would be too
high. |
|
 |
Note that in the dawing there are no minority
carriers close to the SCR edges (deliberately!); only in those regions
away from the SCR, where there is a single Fermi energy (shown in red),
minority carriers are depicted. There, the standard full-equilibrium mass
action law holds. |
 |
The decisive point is that we may
consider any given thin slice of the SCR to be in local equilibrium, and that the quasi-Fermi energy of the electrons is lower than that
of the holes throughout the SCR. |
|
 |
The latter is a direct consequence of the applied
reverse bias, increasing and steepening the potential barrier in the
SCR, in combination with the diffusion length of the minorities being
larger than the width of the SCR (remember the narrow junction
approximation from above). |
|
 |
This ordering of the quasi-Fermi energies is the
exact opposite of the situation that we have considered so far in the
recombination business, where we looked at an increased density of carriers, e.g. produced by
irradiation with light. Then recombination outweighs generation and
UDL, the difference between recombination and generation, was
positive. |
 |
Hence, in the case we are considering
here, UDL is
negative, i.e. there is more generation than recombination. And this
means that the space charge region is busily producing carriers, always in
pairs because of neutrality, which will run down the energy barrier producing
an additional reverse current. |
|
 |
Pair production at a deep level means that,
while the hole is created in the valence band, the corresponding electron
at first just occupies the deep level; only after some time it gets emitted
to the conduction band.
|
|
 |
Let's look at this using the formula for
UDL: |
|
|
UDL
= |
v · s · NDL ·
(ne · nh –
ni2) |
ne+ nh +
2ni · |
cosh |
EDL – EMB
kT |
|
|
|
|
 |
For making estimates easier, we assume a mid-band
level (i.e., cosh[(EDL –
EMB)/(kT)] = 1) and
ne, nh <<
ni. This leaves us with |
|
|
UDL |
= – |
v · s ·
NDL · ni
2 |
|
|
|
 |
For these assumptions
we have seen that,
treating holes and electrons on equal footing, 1/(v · s · NDL) = t. |
|
 |
However, because we now have more generation than recombination, t is now called the
generation life
time tG for this case.
(More to that topic in the link.) |
|
 |
This leaves us with a
net generation of one kind of carrier of |
|
|
|
 |
The current density from the net
generation of carriers in the SCR is then given by the product of the
net generation rate with the width d of the SCR; adding up
the holes and the electrons yields |
|
|
|
|
 |
This is
exactly the same formula (give or
take a factor of 2) as in our "quick and dirty" estimate from
before. The physical reasoning wasn't so different either, if you think about
it. |
 |
How about the
contribution of the SCR to the forward
current? |
|
 |
The proper treatment is much more complicated and
physically different from our simple explanation. The physical reasoning is as
follows: |
|
 |
We have seen
that we need to sustain a certain density of surplus minority carriers,
Dnep
and
Dnhn, at
the edges of the SCR to maintain local equilibrium. The surplus carriers
needed were injected from the other side of the junction and crossed the
junction without losses – at least in
our present approximation. |
|
 |
In reality, however, some injected holes from the p-side will recombine with
the injected electrons from the n-side. Recombination in the
SCR thus reduces the current needed to maintain Dnmin, and an
additional current has to be produced which exactly compensates the losses.
|
 |
The
necessary calculations are
shown in an advanced module, suffice it to state here that the final result for
the forward current from the SCR is (in a rather crude approximation)
|
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|
jF(SCR) |
= |
e · ni · d
2tG |
· exp |
eUD
2kT |
|
|
|
 |
Again, besides the factor 2 (and the new
kind of life time), the same formula
as before. But this time it was a kind of lucky coincidence, not really
very well justified. |
|
 |
Or was it?
Think about it! |
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© H. Föll (Semiconductors - Script)