 |
In this subchapter we will give the
pn-junction a new look using a somewhat more advanced point of view.
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 |
A full treatment of pn-junctions in any
kind of three-dimensional semiconductor,
allowing for arbitrary doping profiles,
finite size, and effects of the interfaces and surfaces, is one of the more difficult things to do
in semiconductor theory; we will not attempt it here. |
| |
|
In all standard treatments of junctions, we
always look at special (unrealistic) cases and use (lots of) approximations.
This, admittedly, can become somewhat confusing. |
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|
It is thus very advisable, to become really well
acquainted with the simple treatments given in subchapter 2.2, this will clear the mind for the
essentials. |
|
Here we look at an advanced treatment
of a pn-junction, but we still will have to make some assumptions. |
|
|
We consider a "narrow" one-dimensional junction in an
infinitely long crystal, which is formed by a sudden change of doping, i.e.
there are no gradients of the doping
concentrations ND and NA to
the left or right of the junction. |
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|
"Narrow" then means that the width of
the space charge layer is much smaller than the diffusion length of minority
carriers, but still much larger than the
mean free path length
needed for thermalization of carriers |
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In consequence, we do not consider recombination
in the SCR, and we can assume thermal equilibrium in the bands, i.e. we
can use the Quasi-Fermi energies. |
|
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 diversities
sake, the p- and n-side reversed). |
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In what follows, there will be a lot
of shuffling formulas around - and somehow, like by magic, the
I-V-characteristics of a pn-junction will emerge. So let's
be clear about what we want to do in the major steps (highlighted in blue) |
|
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 concentration around the edge
of the space charge region is somewhat larger then 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 and, if necessary, Dnmax, thus will automatically give us the necessary currents
belonging to the non-equilibrium as defined by the voltage. |
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We first look at the various energies
involved: |
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The energy difference between the left side (=
n-side; index "n") and
the right side (= p-side; index
"p"; these indizes will
appear "up" or "down" as dictated by the limitations of
HTML) of the junction is given by |
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| E Cn – ECp
|
= |
EVn –
EVp |
= |
e · (V n – V p) |
= |
e · DV |
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With V n, V
p = constant potentials at
the n or p-side and far away from the junction, respectively.
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The band energy levels
ECn, p = ECn,
p(x) and the potential V(x) are functions
of x, which makes all concentrations functions of
x, too. We will, however, write all
these quantities without the "(x)" from now
on. |
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As long as we discuss
equilibrium, the Fermi energy is constant and the carrier concentrations 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. |
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| npe |
= |
N eeff · exp |
EF – ECp
kT |
= |
|
concentration of
electrons on the p-side |
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| nph |
= |
N peff · exp |
EVp – EF
kT |
= |
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concentration of
holes on the p-side |
| |
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| nne |
= |
N eeff · exp |
EF – ECn
kT |
= |
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concentration of
electrons on the n-side |
| |
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| nnh |
= |
N peff · exp |
EVn – EF
kT |
= |
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concentration of
holes on the n-side |
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We also have the
mass action law
applicable everywhere in equilibrium |
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| npe · nph
|
= |
nnh ·
nne |
= |
ni2 |
= |
N eeff · N
peff · exp – |
Eg
kT |
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Algebraic manipulation gives us any
number of relations, some of which may be useful later on. Here some
examples: |
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npe
nne |
= |
nph
nnh |
= |
exp |
e · (V n – V p)
kT |
| ECp – EVn
|
= |
– kT · ln |
nne
N eeff |
– kT · ln |
nph
N heff |
| e · (V n – V p) |
= |
ECp –
ECn |
= |
Eg + kT · |
nne · nph
N eeff · N
heff |
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This is about all
we can do in full generality. |
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What we need to know to get on is the x-dependence of the energies or of the
potential V(x) - this simply measn we need the
(quantitative) banddiagram that so far we always just drew "by
feeling". |
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For this we need to solve the
Poisson equation and
this demands to specify the total charge r(x) so we can write dow the charge as a
function of x. This is easy in
principle: |
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The total (space) charge r(x) at any point along the junction is the sum
of all charges: Electrons, holes, ionized donors
(N+D(x)), and ionized aceptors
(N–A(x)). We have
as before (with the necessary
"minus" sign being part of the elementary charge e) |
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| r(x) |
= e · |
æ
è |
nh(x) –
ne(x) +
ND+(x ) +
NA–(x) |
ö
ø |
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Inserting r in
the Poisson equation gives |
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|
e e0
e |
· |
d2V(x)
dx2 |
= |
nh(x) –
ne(x) +
ND+(x) +
NA–(x) |
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Solving this equation with the proper boundary
conditions will yield V(x) and everything else - but not so easily because
nh(x) and ne(x)
are complicated functions of V(x) via the Fermi
distribution. |
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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. |
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For our final goal, which is to
describe the current-voltage characteristics of a pn-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.
e · V 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. |
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2.
All dopants are ionized, their
concentration is constant up to the junction, and there is only one kind on
both sides of the junction, i.e. |
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| nhp(bulk) |
= |
NA |
= |
N –A |
| |
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| nen(bulk) |
= |
ND |
= |
N +D |
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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. |
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3. We also
assume that the Si extends into infinity (or at least to a distance much
larger than a diffusion length) to both sides of the junction - in total we use
the "abrupt" "large" junction approach |
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This gives us for the carrier concentrations in
equilibrium anywhere in the junction: |
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| nh(x) |
= |
NA · exp – |
e · V(x)
kT |
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| |
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| ne(x) |
= |
ND · exp – |
e · [V(x) – V
n]
kT |
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With V n = constant
value of the potential deep in the n-type region. |
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These equations are already a special case of the more general equations above.
What they mean is that the carrier concentration is whatever you have in the
undisturbed p- or n-part (i.e the dopant concentration) times the
Boltzmann factor of the energy shifts relative to this situation. |
|
V n is the
difference of the built-in potentials for equilibrium conditions, it is
thus determined by the difference
in the Fermi energies of the two semiconductors before contact - our
simple view of a junction is totally
correct on this point |
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With and without out an external voltage
U we have |
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| V n(U = 0) |
= |
1
e |
· |
(E nF – E
pF) |
| |
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| V n(U) |
= |
1
e |
· |
(E nF – E
pF + e · U) |
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The maximum potential at the n-side,
V n(bulk) becomes |
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| V n(bulk) |
= |
V n(U = 0) + U |
= |
1
e |
· |
DEF + U |
:= V n |
for brevity |
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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 =
V n|edge SCR on the n-side, and
V p|edge SCR = 0. This is an essential
point, even so it is matter-of-course. |
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We know can move
towards our primary goal and find an
expression for the carrier density at the edge of the SCR by inserting
the proper potential in the n pe and
n ne equation from above. We obtain for the edge of
the SCR: |
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npe
nne |
÷
÷
÷ |
edge
SCR |
= |
nnh
nph |
÷
÷
÷ |
edge
SCR |
= |
exp – |
e · (V n + U)
kT |
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The minority carrier concentrations (always at the edge of the SCR without indicating it
anymore) can now be written as |
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| npe(U) |
= |
nne(U) · exp – |
e · (V n + U)
kT |
|
electrons on
the p-side |
| |
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| nnh(U) |
= |
n ph(U) · exp – |
e · (V n + U)
kT |
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holes on
the n-side |
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These equations are nothing but the Boltzmann
distribution giving the number of particles (nmin)
that make it to the energy e(Vn + U) 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. |
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Since this is important, let's
review the approximations we made: |
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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 than changes monotously. |
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This means that for equilibrium we must obtain
the same equations by computing the minority carrier concentration from the
mass action law, i.e.
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| npe(U = 0) |
= |
ni2
n ph(U = 0) |
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We will see if this is true in a little
exercise |
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| Exercise 2.3.5-1 |
| Show the equality of the two
equations for the minority carrier density |
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| |
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Now comes a
crucial point: We are looking at non-equilibrium. We first review the starting point
again: |
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At equilibrium (U =
0), the majority carrier concentrations
nne |edge SCR and
nph |edge SCR are given by
. |
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| nph |
= |
Npeff · exp – |
EF
kT |
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| |
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| nne |
= |
Neeff · exp + |
(EF – e · V
n)
kT |
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- our first equations
from above, but expressed now with the
normalized potential V. This is fully correct as implicitely
shown in the exercise. However, we are now going to look at non-equilibrium. |
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The essential point
for the majority carrier concentration 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
U, i.e |
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| nn, he, h(U) |
÷
÷ |
edge
SCR |
= |
nn, pe, h(equ) |
÷
÷ |
edge
SCR |
= |
nn, pe, h |
÷
÷ |
bulk |
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This can be seen from the equations above since
neither EF nor [EF –
e · (V n + U)]|edge SCR
depend on the voltage since the Fermi energy in the bulk moves up or down with
the band edges! |
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The minority carrier
concentrations npe |edge SCR and
nnh |edge SCR, however, depend
very much on the applied voltage as
expressed in the formula above if we replace Vn by
Vn + U. |
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In other words: While the
quasi-Fermi energy
EmajF for majority carriers remains at the
equilibrium value EF near (and even for some distance
in) the SCR, the quasi-Fermi energy
for the minority carriers, EminF, branches
off early (the illustration below shows this,
albeit in a slightly different connection). |
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We now ask for the
difference of the minority carrier concentration
relative to equilibrium, i.e. for |
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| D np, ne, h |
÷
÷ |
edge
SCR |
= |
n p, ne, h(U) –
n p, ne, h(U = 0) |
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It comes out as |
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| Dn pe |
= |
n ne · |
æ
ç
è |
exp – |
e · (V n + U)
kT |
– exp – |
eVn
kT |
ö
÷
ø |
| |
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| Dnnh |
= |
n ph |
æ
ç
è |
exp + |
e · U
kT |
|
– 1 |
ö
÷
ø |
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Inserting the
general expressions for the minority carrier concentration
from above yields the final formula for our
first goal: |
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| Dnpe |
÷
÷ |
edge
SCR |
= |
npe(equ) · |
æ
ç
è |
exp |
eU
kT |
– 1 |
ö
÷
ø |
| |
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| Dnnh |
÷
÷ |
edge
SCR |
= |
nnh(equ) · |
æ
ç
è |
exp |
eU
kT |
– 1 |
ö
÷
ø |
|
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In other words: The
concentration 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 concentration must remain constant
as a function of time. |
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Since deep in the material the minority carrier
concentration is unchanged and has its equilibrium value, we now must have a
current, driven by the concentration gradient alone, and this current must be maintained by the voltage/current
source if we want steady state. |
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Physically speaking, the excess concentration of
minority carriers will diffuse around and disappear after some diffusion
lengths - deep in the material they are not noticeable any more. |
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This is exactly the
situation treated under "useful
relations" for
pure diffusion
currents. |
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We can take the formula derived
there with Dn e, hp,
n(x = 0) given by the equation from above and obtain
immediately for the current-voltage
relationship of a pn-junction |
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| j e(U) |
= |
e · De
Le |
· Dne |
÷
÷ |
edge
SCR |
|
or |
|
| j e(U) |
= |
e · De
Le |
· nep(equ) · |
æ
ç
è |
exp |
e · U
kT |
– 1 |
ö
÷
ø |
| j h(U) |
= |
e · Dh
Lh |
· nhn(equ) · |
æ
ç
è |
exp |
e · U
kT |
– 1 |
ö
÷
ø |
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In terms of sign conventions we assumed that
positive voltages lower the barrier. |
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For the final result we add the electron and hole currents,
drop suffixes and functional arguments now unnecessary, and obtain the
diode equation: |
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|
|
| jtotal(U) |
= |
æ
ç
è |
e · n ep · De
Le |
+ |
e · n hn · Dh
Lh |
ö
÷
ø |
· |
æ
ç
è |
exp |
e · U
kT |
– 1 |
ö
÷
ø |
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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: |
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From the
Einstein relation we have |
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| L e, h |
= |
æ
è |
De, h · te, h
) |
ö
ø |
1/2 |
| |
D e, h |
= |
(L e, h ) 2 |
· te, h |
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| |
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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) anyway) we get |
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| ne,hp,n |
= |
ni2
nh, ep, n |
| |
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| n ep |
= |
ni2
NA |
| |
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| n hn |
= |
ni2
ND |
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Shuffling
everything around with these identities gives us - among many other equivalent
formulations |
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|
| jtotal(U) |
= |
æ
ç
è |
e · L e · ni2
te · NA |
+ |
e · L h · ni2
th · ND |
ö
÷
ø |
· |
æ
ç
è |
exp |
e · U
kT |
– 1 |
ö
÷
ø |
|
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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! |
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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. |
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Here we looked at the surplus of minorities accounting for the forward
current and which has to be moved away from the junction. |
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Think about, why this is the same thing (start
with U = 0)! |
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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. |
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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 shown
in the picture. |
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The Quasi-Fermi levels must behave in the way
shown (the details do not matter), because otherwise the concentration of
carriers in the junction would be too high (it must be close to zero!). The
decisive point is that in any given thin slice of the SCR, we may
consider to be in local equilibrium, and
that the Quasi-Fermi energy of the electrons is lower
than that of the holes. |
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|
This the exact opposite of the situation that we
have considered so far in the recombination business, where we looked at an
increased concentration of carriers, e.g.
produced by irradiation with light. Then recombination outweighs generation and
UDL, the difference between recombination and generation was
positive. |
|
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. |
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Pair production means that a deep level first
emits a hole to the valence band, and than an electron to the conduction
band. |
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|
Lets look at this using the formula for
UDL: |
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|
|
| UDL
= |
v · se · NDL
· (ne · nh –
ni2) |
| ne + nh +
2ni · |
cosh |
EDL – EMB
kT |
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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 |
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|
| UDL |
= – |
v · se ·
NDL · ni
2 |
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For these assumptions
we have seen that 1/v
·se · NDL =
te, or since we do have holes
and electrons now on an equal footing, we drop the suffixes and write
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). |
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This leaves us with a
net generation of one kind of carrier of |
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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, yielding |
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With d = width of the
SCR. |
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|
and 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 to the forward current from
the SCR? |
|
|
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 concentration of surplus minority carriers,
Dne, hp, n 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 - in our running
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 Dne, hp, n 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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|
|
|
|
| jR |
= |
e · ni · d
2tG |
· exp – |
eU
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! |
© H. Föll (Semiconductor - Script)
