 | 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 Dn
min 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 |
| |
EC
p – 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. We will, however,
write all these quantities without the "(x)" from now on. |
|  | 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 – EC
p kT | |
density of electrons on the p-side |
| | |
| | |
nhp | = |
Neffh · exp |
EVp – EF
kT | |
density of holes on the p-side | |
| | | | |
nen | = |
N effe · exp | EF
– ECn kT
| | density of
electrons on the n-side | | |
| | |
| nh
n | = | 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: |
| |
ne p · nhp
| = | nhn
· nen | = | nmin
· n maj = | ni2 |
| | |
= | Neffe
· Neffh · exp – | Eg
kT | | |
 |
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)
+ N D+(x) – NA–
(x) | ö ø |
| |
|  |
Inserting r(x) into
the Poisson equation gives |
| | – | e e 0 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. |
| | nh p(bulk)
| = | NA |
= | NA–
| | |
| | | n en(bulk) | = |
N D | = | 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) | = | N
A · exp – | e · V(x) kT | |
| | |
| ne(x
) | = | ND ·
exp – | e · [Vn
– V(x)] kT | |
|
|  | Here,
Vn 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) £
Vn. |
|  | 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. |
| Vn is the difference
of 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 |
| |
Vn(Uext=0) | = | 1 e |
· | (EFn
– EF p) | |
| | |
| V
n(Uext) | = | 1 e | · |
(EF n – 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,
Vn(bulk), becomes |
|
| Vn(bulk) | = | V n(Uext=0)
+ Uext | = | 1 e | · | DEF + Uext | =
Vn + 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 Vn
(bulk) = Vn|SCR edge on the n-side, and Vp
|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 nh p |
÷ ÷ ÷ | SCR
edge | = exp – | e
· DV kT |
= exp – | e ·
(Vn + Uext) kT |
| |
|  |
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 – | e · (Vn
+ Uext) kT | |
electrons on the p-side | |
| | |
| | nhn(Uext) | =
| nhp(Uext) ·
exp – | e · (Vn + Uext)
kT | | 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 ne
n |SCR edge and n hp |SCR edge
are given by |
| |
nhp | =
| Neffp · exp – |
EF kT |
| |
| | |
ne n | =
| Neffe · exp + | EF – ECn
kT | |
|
|  |
Do you remember them? These are two of our first equations from
above, but given here for the choice of EV p = 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. |
| |
ne,h
n,p(Uext) | ÷
÷ | SCR
edge | = | ne,h
n,p(equ) | ÷ ÷ |
SCR edge |
= | ne,hn,p(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 EF
maj 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 |
| |
Dne,hp,n | ÷ ÷ | SCR edge | = | ne,h
p,n(Uext) – ne,hp,n(U
ext=0) | | | |
|  | It
comes out as |
| |
Dne,hp,n |
= | ne,hn,p
· | æ ç è |
exp – | e · (Vn
+ Uext ) kT | –
| exp – | eV
n kT | ö
÷ ø | | |
| | |
| |
| = | n
e,hn,p | · | exp
– | eVn kT |
· | æ
ç è | exp – |
eUext kT | – |
1 | ö
÷ ø | | |
 |
Inserting the general expressions
for the minority carrier density from above for the case
U ext = 0 yields the final formula for our first goal: |
| |
Dne,hp,n | ÷ ÷ | SCR edge | = | n
e,h p,n(equ) · | æ
ç è | exp – | e
Uext kT | – 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 | · D
ne | ÷ ÷ |
SCR edge |
| or | |
| je(Uext) | | =
| e · De Le |
· nep(equ) · |
æ ç è | exp – | eUext
kT | – 1 |
ö ÷ ø | | jh(Uext)
| | = | e · Dh Lh | · nh
n(equ) · | æ ç
è | exp – | eUext
kT | – 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 := –U
ext, 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
L h | 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 |
| |
ne,hp,n |
= | ni2 nh,e p,n | |
| | n
ep | = | n
i2 NA | |
| | n
h n | = | n
i2 ND | |
|
| Shuffling
everything around with these identities gives us – among many other equivalent formulations
– . . . |
| |
jD(U D) | =
| æ ç è |
e · Le· ni2 te· NA |
+ | e · Lh·
ni2 th
· ND | ö ÷
ø | · |
æ ç è | exp | e
UD 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 means that a deep level first emits a hole to the valence band, and then an electron
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 | E
DL – 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, Dn e,
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 – 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 D
ne, 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) |
| |
jF
(SCR) | = | e · ni
· d 2tG |
· exp | e UD 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 (Semiconductors - Script)