 |
We have encountered the Einstein
relation before. It is of such fundamental importance that we give two derivations:
one in this paragraph, another one in an advanced module. |
|
First, we consider the internal current (density)
in a material with a gradient
of the carrier density (ne or nh). |
|
 |
Fick's first law then tells us
that the diffusion-driven particle current jp,diff
is given by |
| |
|
jp,diff = – De,h · Ñ
ne,h |
|
|
|
|
If the particles are carrying a charge q, this particle
current is also an electrical
current (which obviously is a diffusion current, then), given by |
| |
| je,h | = |
q · jp,diff | = |
– q · De,h · Ñ
ne,h |
|
|
|
|
Considering only the one-dimensional case for electrons (i.e. q = –e;
holes behave in exactly the same way with q = +e), we have |
| |
| je (x) | = |
e · De · |
dne(x)
dx |
|
|
|
Since there can be no net
current in a piece of material just lying around (which nevertheless might still have a density gradient in the carrier
density, e.g. due to a gradient in the doping density), the carriers displaced by diffusion always generate an electrical
field that will drive the other carriers back. |
|
|
Any field E(x) ( written
in mauve
to avoid confusion with energies) now will cause a (so far one-dimensional) current given by
|
| |
| j | = |
s · E(x) = q
· n(x ) · µ · E(x) |
|
|
|
|
With s = conductivity, µ = mobility.
|
|
|
Note that the result is always the technical current density, which is positive for positive
charge carriers. Yet this equation also works for electrons because for them, effectively, two minus signs cancel: one from
their negative charge and the other from their direction of movement opposite to the electric field. This means that in
a strict sense, their mobility should be negative. However, in this equation one only considers positive charges and positive
mobilities – also for electrons. Therefore, to use ths equation in full generality, we write it as |
| |
|
|
|
The
total (one-dimensional) current in full generality is then |
| |
| jtotal(x) = e · n(x) · µ · E(x ) – q · D · |
dn( x )
dx |
|
|
|
|
We will need this equation later. |
|
For our case of no net current
and only fields caused by the diffusion current, both currents have to be equal in magnitude: |
| |
| e · n(x) · µ · E(x)
| = | q · D · |
dn(x)
dx |
|
|
|
|
This is an equation that comes up repeatedly; we will encounter it again later when we derive
the Debye length. |
|
|
Note that in this equation, the sign on the right-hand side depends on the type of charge
carriers (since q = ±e). This is balanced on the left-hand side by the direction of the electric field. |
|
Now we are stuck. We need some additional equation in
order to find a correlation between D and µ. |
|
|
This equation is the Boltzmann distribution (here used
as an approximation to the Fermi distribution), because we have equilibrium
in our material. |
|
|
However, we also know that, in this equilibrium situation, we have spatially varying charge
carrier densities and electric fields. We know such a situation from the p–n junction in equilibrium. There, this
was only possible due to the band bending, i.e. that the band edges were functions of the lateral position. This we also
consider here. |
|
|
Just to derive the relation to the electric field in the above equation, for the moment we
just consider the case of electrons as majority carriers. For their local density it holds that |
| |
| n(x ) | = |
Neff · exp – |
EC(x) – EF
kT |
|
|
|
|
Differentiation of the Boltzmann distribution gives us |
| |
dn
dx | = |
– Neff · 1/(kT) · |
dEC(x)
dx |
· exp – |
EC(x) – EF
kT |
| | | |
| | |
dn
dx | = |
– n(x) · 1/(kT) · |
dEC(x )
dx |
| |
|
|
|
|
The slope of the conduction band comes directly from the spatially varying electric potential
V (x); to convert the electric potential to the absolute energy of a charge carrier, the elementary
charge e is needed as an additional factor. From the p–n junction we know that the sign (i.e., the direction)
of the electric field is identical to that of the slope of the conduction band. Thus, altogether we have |
| |
| E(x) |
= – |
dV(x)
dx | = |
1/e | · |
dEC(x)
dx |
|
|
|
|
Using this relation, the current balance from above becomes |
| |
| e · n(x) · µ · E(x) |
= | q · D · |
dn(x )
dx |
| | | |
| |
| e · n(x) · µ ·
E(x) | = |
– q · n(x) · D · e/(kT) · |
E(x) | | |
| | |
| | D |
= | µkT/e |
|
|
|
|
In words: Equilibrium between diffusion currents and electrical currents for charged particles
demands a simple, but far reaching relation between the diffusion constant D and the mobility µ.
|
|
|
Distinguishing again between electrons and holes gives as the final result the
famous Einstein–Smoluchowski relations: |
| |
| De | = |
µe · kT
e |
| | | |
| Dh | = |
µh · kT
e |
|
|
|
You may want to have a look at a different
derivation in an advanced module. |
|
|
|
Non-Equilibrium Currents |
| | |
|
In the consideration above we postulated that there is no net current flow; in other words, we postulated total equilibrium.
Now let's consider that there is some net current flow
and see what we have to change to arrive at the relevant equations. |
|
In order to be close to applications, we treat the extrinsic
case and, since we do not assume equilibrium per se, we automatically do not assume that the carrier densities have
their equilibrium values n e(equ) and nh(equ),
but arbitrary
values that we can express by some Delta to the equilibrium value. We thus start with |
| |
|
|
|
Since carriers above the equilibrium density are often created in
pairs we have for this special, but rather common case |
| |
| Dne | =
| D nh |
= | Dn |
| | |
| |
|
| D n | = |
ne – ne(equ) |
= |
nh – nh(equ) |
|
|
|
|
This is a crucial assumption! |
|
This allows us to concentrate on one kind of carrier,
let's say we look at n-type Si with electrons as the majority carriers. We now focus
on holes as the minority carriers since we always can compute the electron density ne by
|
| |
| ne | = |
ne(equ) + Dne | =
| ne(equ) + nh – nh (equ) |
|
|
|
|
We now must consider Fick's
second law or the continuity
equation (it is the same thing for special cases, but the continuity equation is more general). |
|
For the net (mobile) charge density r (which
is the
difference of the electron and hole density, r = e · (nh
– ne), in contrast to the total particle density,
which is the sum !) we have |
|
|
|
|
|
With jtotal = je + jh =
sum of the electron and hole currents. |
|
In the simplest form we have for the holes |
| |
¶n
¶t | = |
– (1/e) · div (jh) |
|
|
|
The factor 1/e is needed to convert an electrical current j
to a particle current jpart via j = q · jpart,
with q = ±e. Here, as always, we have to pick the right sign for the elementary charge e (negative
for electrons, positive for holes). |
|
|
This is simply the statement that the charge is conserved.
It would be sufficient that no holes disappear or are created in any differential volume dV considered, i.e.
div jh = 0, to satisfy that condition. |
| |
But this is, of course, a condition that we know not to be true.
|
|
|
In all semiconductors, we have constant generation and recombination of holes (and electrons)
as discussed before. In in equilibrium, of course,
the generation rate G and the recombination rate R are equal, so they cancel each other in a
balance equation and need not be considered – since div jh = 0 is correct on
average. |
|
|
We are, however, considering non-equilibrium
, so we must primarily consider the recombination of the surplus
minority carriers given by |
|
|
|
|
|
Why? Because, as stated before, the generation
essentially does not change, so it still balances against the recombination rate of
the equilibrium density, and only the recombination rate of the surplus minorities, RD
= [nh – nh(equ)]/t needs to be considered (t is the minority carrier life time). |
|
|
R
D = [nh – nh(equ)]/t
is the rate with which carriers disappear by recombination, we thus must subtract
it from the carrier balance as expressed in the continuity equation, and obtain |
| |
dn
dt |
= – |
nh – nh(equ)
t |
– (1/e) · div (jh) |
|
|
|
The current j can always be expressed as the sum of a
field current and a diffusion current as we did
above by |
| |
| jh,total(x) | = |
e · n(x) · µ · Ex(
x) – e · Dh · |
dnh(x)
dx |
|
|
|
If we let Ex = 0 and consequently
¶Ex (x)/¶x
= 0, too, the current equation from above reduces to |
|
|
¶nh
¶t | = – |
nh – nh (equ)
t | + D · |
¶2nh ( x)
¶x2 |
|
|
|
|
Since ¶nh/¶t
= ¶[nh(equ) + Dnh]/¶t = ¶Dnh/ ¶t, and correspondingly ¶2n
h(x)/¶x2 = ¶
2Dnh(x)/¶
x2, we have |
|
|
¶D nh
¶t |
= – |
D nh
t |
+ D · |
¶2D
nh(x)
¶ x2 |
|
|
|
If we consider steady state , we have ¶Dnh/¶ t = 0,
and the solution of the differential equation is now mathematically easy. |
|
But how can steady state be achieved in practice? How can we provide for a constant , non-changing
density of minority carriers above equilibrium? |
|
|
For example by having a defined source of (surplus) holes at x = 0. In the illustration
this is the (constant) hole current that makes it over the potential barrier of the p–n junction. |
|
|
But we could equally well imagine holes generated by light a x = 0 at a constant
rate. The surplus hole density then will assume some distribution in space which will be constant after a short initiation
time - i.e. we have steady state and a simple differential equation: |
| |
| D · |
¶2[D
nh(x)]
¶x2 |
– |
D nh(x)
t | = | 0 |
|
|
|
The solution (for a one-dimensional bar extending from x
= 0 to x = ¥) is |
| |
|
|
|
The length L is given by |
| |
|
|
|
L is simply the diffusion length of the minority carriers (= holes in
the example) as defined in the "simple" (but in this case accurate)
introduction of life times and diffusion length. |
|
|
This solution is already shown in the drawing above which also shows the direct geometrical
interpretation of L. |
|
The important point to realize is that the steady
state tied to this solution can only be maintained if the hole current at x = 0 has a constant,
time independent value resulting from Fick's 1st law since
we have no electrical fields that could drive a current. |
|
|
This gives us |
|
|
| jh(x = 0) |
= – e · D |
¶ Dnh(x)
¶ x | ÷
÷ | x = 0 |
|
|
|
|
By simple differentiation of our density equation from above we obtain |
| |
¶
Dnh(x)
¶
x | ÷
÷ |
x = 0 | = – |
Dn0
L |
|
|
|
|
Insertion into the current equation yields the final result |
| |
| jh (x = 0) | = |
e · Dh
Lh |
· Dnh(x = 0) |
|
|
|
|
The physical meaning is that the hole part of the current
will decrease from this value as x increases, while the total current
stays constant – the remainder is taken up by the electron current. |
| | |
|
General Band-Bending and Debye
Length |
| |
|
|
The Debye length and the dielectric
relaxation time are important quantities for majority carriers (corresponding
to the diffusion length and the minority carrier life time
for minority carriers). Let's see why this is so in this paragraph. |
|
|
Both quantities are rather general and come up whenever density gradients cause currents that
are counteracted by the developing electrical field. |
|
|
An alternative simple treatment of the
Debye length can be found in a basic module. |
|
Let us start with the Poisson equation for an arbitrary one-dimensional semiconductor with a
varying electrostatic potential V(x) caused by charges with a density r(x)
distributed somehow in the material. We then have |
|
|
| – e · e0 ·
| d2 V(x)
dx2 |
= |
e · e0 · |
d E(x)
dx |
= | r(x) |
|
|
|
|
E(x) is the electrical field strength; always minus the derivative of the potential V. |
|
|
The charge r(x) at any one point
can only result from our usual charged entities, which are electrons, holes, and ionized doping atoms. r(x)
is always the net sum of this charges, i.e. |
| |
| r(x) |
= e · | æ
è
|
nh(x) + ND+(x) – [ne (x) + NA–(
x)] | ö
ø |
|
|
|
|
The electrostatic potential V needed for the Poisson equation
is now a function of x and shifts the conduction and valence band by the potential energy
qV relative to some reference point for which one has V = 0.
Since the band structure refers to the energy of electrons, we have that q = –e and thus may write |
| |
| EC (x) | = |
EC( V = 0) – e · V(x) |
| | | |
| E V(x) | = |
EV(V = 0) – e · V(x) |
|
|
|
|
Thereby, the Poisson equation becomes |
| |
| – e · e0 ·
| d2V (x )
dx2 |
= |
e · e0
e | · | d2
EC (x)
dx2 |
= e | æ
è |
n
h(x) + ND+(x) – [ne( x) + NA–(
x)] | ö
ø |
|
|
|
|
If we now insert the proper equations for the four
densities, we obtain a formidable differential equation that is of prime importance for semiconductor physics and devices,
but not easy to solve. |
|
|
However, even if we could solve the differential equation (which we most certainly cannot),
it would not be of much help, because we also a need a "gut feeling" of what is going on. |
|
The best way to visualize the basic situation is to imagine a
homogeneously doped semiconductor with a fixed charge density at its surface and no net currents (think of a fictional
insulating layer with infinitesimal thickness that contains some charge on its outer surface). |
|
|
Carriers of the semiconducor thus can not
neutralize the charge, and the surface charge will cause an electrical field which will penetrate into the semiconductor
to a certain depth. |
|
|
This is the most general case for disturbing the carrier density in a surface-near region
and thus to induce some band-bending . |
| |
There are two distinct major situations: |
|
1. The surface charge has the same polarity as the majority carriers in the semiconductor,
thus pushing them into the interior of the material. |
|
|
This exposes the ionized dopant atoms with opposite charge and a large space charge layer (SCR) will built up. This is also called the depletion
case. |
|
|
The SCR is large because the dopant density is low and the dopant atoms cannot
move to the interface. Many dopant atoms have to be "exposed" to be able to compensate the surface
charge; the field can penetrate for a considerable distance. |
|
|
However: In contrast to what we learned
about SCRs in p–n junctions, even for large fields (corresponding to large reverse voltages at a junction),
the Fermi energy is EF
still constant (currents are not possible). The bands are still bent, however, this
means that EC – EF incrases in the direction toards the surface. |
|
|
If the majority carrier density then is becoming very small in surface-near regions (it scales
with exp [– (EC – EF )/(kT)] after all), the minority carrier
density increases due to the mass action law until minority carriers become the majority – we have the case of inversion . |
|
2. The surface charge has the opposite polarity
as the majority carriers in the semiconductor, thus accumulating them at the surface-near region of the material. |
|
|
Then majority carriers can move to the surface near region and compensate the external charge.
The field cannot penerrate deeply into the material. |
|
|
This case is called accumulation. |
|
The situation is best visualized by simple band diagrams, we chose the case for
n-type materials. The surface charge is symbolized by the green spheres or blue squares on the left. |
| |
|
|
|
Between depletion and accumulation must be the flat-band
case as another prominent special case. This is not necessarily tied to a surface charge of zero (as shown in the drawing
where a blue square symbolizes some positive surface charge), but for the external charge that compensates the charge due to intrinsic surface states. |
|
We have some idea
about the width of the space charge region that comes with the depletion case. But how
wide is the region of appreciable band bending in the case of accumulation? |
|
|
Qualitatively, we know that it can be small - at least in comparison to a SCR - because
the charges in the semiconductor compensating the surface charges are mobile and can, in principle, pile up at the interface |
|
For the quantitative answer for all cases, we have to solve the Poisson equation from above. However, because we cannot do that in full generality, we look at some special
cases. |
|
First we restrict ourselves to the usual case of one
kind of doping – n-type for the following example – and temperatures where the donors are fully ionized,
which means that the Fermi energy is well below the donor level or ED – EF
>> kT. |
|
|
We then have only two charged entities: |
| |
| ND + | = |
ND |
| | |
|
| ne | = |
Neffe · exp – |
EC – EF
kT |
|
|
|
|
This means in what follows we only consider the majority
carriers. |
|
The Poisson equation then reduces to |
| |
e · e
0
e | · |
d2EC (x)
dx2 |
= e | æ
è
|
ND – Neffe · exp – |
EC( x) – EF
kT | ö
ø
|
|
|
|
|
And this, while special but still fairly general, is still not easy to solve . |
|
We will have to specialize even more. But before we do this, we will rewrite the
equation somewhat. |
|
|
For what follows, it is convenient to express the band bending of the conduction band in
terms of its deviation from the field-free situation, i.e. from EC0 = EC(x
= ¥ ). We thus write |
|
|
|
|
|
The exponential term of the Poisson equation can now be rewritten, we obtain |
| |
| Neffe · exp – |
EC(x) – EF
kT | = |
N effe · exp – |
EC0 – EF
kT | · exp – |
DE
C(x )
kT |
|
|
|
|
The first part of the right hand side gives just the electron density in a field-free part
of the semiconductor, which – in our approximations – is identical to the density ND
of donor atoms. This leaves us with a usable form of the Poisson equation for the case
of accumulation : |
|
|
d2EC
dx2 |
= |
d2 ( DEC )
dx2
| = | e2 · ND
e · e0 |
· |
æ
ç
è |
1 – exp | æ
è
| – | D EC
kT | ö
ø |
ö
÷
ø |
|
|
| |
DEC characterizes the amount
of band bending. We can now proceed to simplify and solve the differential equation by considering different cases for the
sign and magnitude of D EC . |
|
|
Unfortunately, this is one of the more tedious (and boring) exercises in
fiddling around with the Poisson equation. The results, however, are of prime importance – they contain the very basics
of all semiconductor devices. |
|
We will do one
approximative solution here
for the most simple case of quasi-neutrality which will give us the all-important Debye
length. |
|
|
The other cases can be found in advanced modules: |
|
Quasi-neutrality
is the mathematically most simple case; it treats only small deviations from
equilibrium and thus from charge neutrality. |
|
|
The condition for quasi-neutrality is simple: We assume |DEC|
<< kT. |
|
|
We then can approximate the exponential function by its Taylor
series and stop after the second term. This yields |
|
|
d2(DEC)
dx2
| = |
e2 ·ND
e · e0 |
· | DEC
kT |
|
|
|
|
That is easy now, the solution is |
|
|
| DEC(x) |
= |
DEC (x = 0) · exp – |
x
LDb |
|
|
|
|
The solution defines LDb
= Debye length for n-type semiconductors
= Debye length for electrons, we have |
|
|
| LDb | = |
Ö |
e · e0 · kT
e2
· ND |
|
|
|
|
Obviously the Debye length LDb for holes
in p-type semiconductors is given by |
|
|
| LDb | = |
Ö |
e · e0 · kT
e2
· NA |
|
|
|
For added value, our solution also gives the field strength of the electrical
field extending from the surface charges into the depth of the sample. |
|
|
Setting it to zero at the top of the valence band in the p-type material (as it is conventionally done), the electrostatic potential is related to the conduction band
edge by EC (x) = Eg – e · V(x). As discussed
already above, the minus sign stems from the negative charge of an electron. |
|
|
Since the field strength E(x) is minus the
derivative of the electrostatic potential, we now have |
|
|
| E(x) |
= – |
dV(x)
dx | = |
1/e | · |
dEC(x)
dx |
= – | 1
e · LDb |
· DEC(x) |
|
|
|
|
Note that in the case of accumulation at the surface of an n-type semiconductor, D
EC(x) is negative, so the electric field comes out positive – in full agreement with
the surface (at x = 0) being positively charged in this case. |
|
The Debye length gives the typical length within which a small
deviation from equilibrium in the total charge density – which for doped semiconductors
is always dominated by the majority carriers – is relaxed or screened; in other
words, it is no longer felt. |
|
|
LDb
is a direct material parameter – its definition contains nothing but
prime material parameters (including the doping). |
|
|
For medium to high doping densities, it becomes rather small. The dependence
of the Debye length on material parameters is shown in an illustration. |
|
|
The Debye length is also a prime material quantity in materials other than semiconductors
- especially in ionic conductors and electrolytes (for which it was originally introduced). It also applies to metals, but
there it is so small that it rarely matters. |
|
The Debye length comes up in all kinds of equations. Some examples are given in
the advanced modules dealing with the other cases of field-induced band bending |
|
|
The Debye length is to majority carriers what the diffusion
length is to minorities. And just as the diffusion length is linked to the
minority carrier lifetime t
, the Debye length correlates to a specific time, too, called the dielectric relaxation
time td . |
|
|
This will be the subject of the next paragraph. |
| | |
|
Dielectric
Relaxation Time |
| |
|
|
Let's start from the same situation that lead to the Debye length: A doped semiconductor,
all dopants ionized, and some small disturbance in the charge equilibrium expressed as some small Dr(x, t) somewhere, starting at some time t0; i.e. we still
assume quasi-neutrality. |
|
|
The Poisson equation now is extremely simple, we write it directly for the electrical field
strength and have |
| |
dE(x, t)
dx
| = |
Dr(x, t)
e · e0 |
|
|
|
We now want to find out about how long it takes to establish a steady
state, so we need some expression for d(Dr)/dt. The Poisson equation won't help
because it does not explicitly contain the time dependence. |
|
|
But simply using the continuity equation
for the relevant charge density Dr
provides a d(Dr)/dt term.
Since we are treating quasi-neutrality, we neglect all terms with gradients
in the carrier density (this will turn out to be fully justified). |
|
|
Since the only relevant current is the drift curent j(x) = s
E(x) = r · µ · E(x),
this leaves us with the following continuity equation |
| |
¶ (Dr)
¶ t |
= – r · µ · |
¶E (x)
¶x |
|
|
|
Inserting dE/dx from the Poisson equation gives |
|
|
¶(Dr )
¶t | = – |
r · µ
e · e0 |
· Dr |
|
|
|
|
r is the total carrier density, we can write it as r = r0 + Dr
»
r 0 since we have quasi neutrality; µ, as always, is the mobility of the carrier in question.
|
|
|
This is a differential equation for Dr(x,
t) with the simple solution |
|
|
| Dr(x, t) |
= |
Dr (x, 0) ·
exp – | t
td |
|
|
|
|
With td = dielectric
relaxation time =
another basic material constant for the same reason as the Debye length. |
|
|
The dielectric relaxation time tells us exactly what we
wanted to know: How long does it take for the majority carriers to respond to a disturbance in the charge density. |
|
While this definition of some special time is of some interest, but not overwhelmingly
so, the situation gets more exciting when we consider relations between our basic material constants obtained so far: |
|
|
Since µ · r = s
, the conductivity of the material (for the carriers in question),
we have the simple and fundamental relation |
|
|
|
|
Now let's see if there is a correlation to the Debye length: |
|
|
We use the Einstein relation
D = µ(kT/e), the Debye length definition (LDb
= {(e · e0 · kT)/(e · r
)}1/2, pluck it into the definition of the dielectric relaxation time (again replacing e · ND
by r) and obtain |
|
|
| td | = |
LDb2
D | | | |
| | LDb |
= | Ö |
D · td |
|
|
|
|
This is exactly the same relation for the majority
carriers between a characteristic time constant and a length
as in the case of the minority carriers where we had the minority lifetime t and the correlated
diffusion length L. |
|
|
The physical meaning is the same, too. In both cases the times and lengths give the numbers
for how fast a deviation from the carrier equilibrium will be equalized and over which distances small deviations are felt. |
|
This merits a few more thoughts. |
|
|
If the carrier density is high, td
is in the order of picoseconds and LDb
extends over nanometers. Any deviation from equilibrium is thus almost instantaneously
wiped out, or, if that is not possible, contained within a very small scale. |
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And this is the regular situation for majority
carriers. The few minority carriers always present in the semiconductor, too, can be safely neglected. |
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For minority carriers, however, the situation
is entirely different. |
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Their density is very small; td and LD
consequently are no longer small. |
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Moreover, whatever disturbance occurs in the density of minorities,
there are plenty of majorities that can react very quickly (with their td)
to the electrical field always tied to a Drmin. |
|
The majority carriers are always attracted to the minorities and thus will quickly
surround any excess minority charge with a "cloud" of majority carriers (which is called screening),
essentially compensating the electrical field of the excess minorities to zero. |
|
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They will, of course, eventually remove the excess charge by recombination, but that takes
far longer than the time needed to do the screening. |
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Since the electrical field is now zero, the excess charge cannot disappear or spread out by
field currents – only spreading by diffusion in the density gradient (which is automatically introduced, too) is possible. |
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But this is exactly the process that we have neglected in this discussion (we
had all density gradients in the continuity equation set to zero!). |
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Dielectric relaxation (i.e. the disappearance of charge
surpluses driven by electrical fields) is thus not applicable to minority carriers. Charge equilibration there is driven
by diffusion - which is a much slower process! |
|
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This then justifies the simple approach we took before,
where we only considered the diffusion of minorities and did not take into account the majority carriers. |
© H. Föll (Semiconductors - Script)
