2.3.4 Useful Relations

There is no way to cover all relevant semiconductor physics within the scope of this course. This subchapter provides some important or useful relations needed for the understanding of the topics.
It also serves as the "gate" to an number of modules providing additional information.
This subchapter therefore is more open than the other ones; it will fill out and sprout a network in connection with the lecture course that cannot be predicted by now.

Einstein Relation

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 an homogeneous material with a gradient of the carrier concentration (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 · Ñne,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
j  =  q · jp,diff  =  q · De,h · Ñne,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
j(x)  =  e · De · dne(x)
Since there can be no net current in a piece of material just lying around (which nevertheless might still have a concentration gradient in the carrier density, e.g. due to a gradient in the doping concentration), 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.
The total (one-dimensional) current in full generality is then
jtotal(x)  =  q · n(x) · µ · E(x)  –  q · D · dn(x)
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:
q · n(x) · µ · E(x)  =    q · D  ·   dn(x)
This is an equation that comes up repeatedly, we had it, e.g., at the simple derivation of the Debye length.
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.
Additionally, we denote the electrostatic potential energy correlated to the electrical field by V(x). Thus, altogether we have
E(x)  =  –  dV(x)
  n(x)  =  n0 · exp –  eV(x)
Differentiation of the Boltzmann distribution gives us
 =   n0 · e/(kT) · dV(x)
  · exp –  eV(x)
 =   n(x) · e/(kT) · dV(x)
Using this equation, the current balance from above becomes
– µ · n(x) · dV(x)
 =  D · e/(kT) · n(x) · dV(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
Dh  =  µh · kT
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 lets 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 concentrations have their equilibrium values ne(equ) and nh(equ), but arbitrary values that we can express by some Delta to the equilibrium value. We thus start with
ne  =  ne(equ) + Dne

nh  =  nh(equ) + Dnh
Since carriers above the equilibrium concentration are often created in pairs we have for this special, but rather common case
Dne  =  Dnh  =  Dn
Dne  =  ne –  ne(equ)  =   nh –  nh(equ)
This is a crucial assumption!
This allows us to concentrate on one kind of carrier, lets 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) + nhnh(equ)
We now must consider Ficks 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 = nenh,  in contrast to the particle density, which is the sum!) we have
 =  – div (jtotal)
With jtotal = j e + j h = sum of the electron and hole current.
In the simplest form we have for the holes
 =  – (1/e) · div (j h)
The factor 1/e is needed to convert a particle current jpart to an electrical current j  via  j = e · jpart. 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 j h = 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 - div j h = 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
Dnh  =  nhnh(equ)
Why? Because, as stated before, the generation essentially does not change, so it still balances against the recombination rate of the equilibrium concentration, and only the recombination rate of the surplus minorities, RD = [nhnh(equ)]/t needs to be considered (t is the minority carrier life time).
RD = [nhnh(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
 =  –   nh  –  nh(equ)
 –  (1/e) · div (j h)
The current j can always be expressed as the sum of a field current and a diffusion current as we did above by
j htotal(x)  =  e · n(x) · µ · Ex(x)  –  e · Dh · dnh(x)
Inserting this equation in our continuity equation yields
 =  –  nh(x) – nh(equ)
 –  nh(x) · µ · E(x)
  –  E(x) · µ · nh(x)
 +  D · 2nh(x)
This is an important, if not so simple equation. It is not so simple, because the electrical field strength E(x) at x is a function of the carrier density nh(x) at x, which is what we want to calculate! We have used the symbols for partial derivatives ("") to emphasize that it is in reality a three-dimensional equation.
We will now look at some applications of this equation.
Pure Diffusion Currents
Consider the minority carrier situation just outside of the space charge region of a biased pn-junction.
If it is forwardly biased, a lot of majority carriers are flowing to the respective other side where they become minority carriers. They will eventually disappear by recombination, but the minority carrier density right at the edge of the space charge region will be larger than in equilibrium and will decrease as we go away from the junction.
This is now shown in the illustration used before in the simple model of the pn-junction, but the realistic minority carrier situation is now included.
Realistic minOrity cONCentratiOn in p-n-junction
The region outside the space charge region, while now showing a concentration gradient of the minority carrier concentration, is essentially field free or at least has only a small electrical field strength.
If we let Ex = 0 and consequently Ex(x)/x = 0, too, the current equation from above reduces to
 =  –   nh  –  nh(equ)
 +  D · 2nh(x)
Since nh/t = [nh(equ) + Dnh]/t = Dnh/t, and correspondingly 2nh(x)/x2 = 2Dnh(x)/x2, we have
 =  –   Dnh
 +  D · 2Dnh(x)
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 concentration 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 pn-junction.
But we could equally well imagine holes generated by light a x = 0 at a constant rate. The surplus hole concentration 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[Dnh(x)]
 –   Dnh(x)
 =  0
The solution (for a one-dimensional bar extending from x = 0 to x = ¥) is
Dn(x)    =  Dn0 · exp –  x 
The length L is given by
L  =  æ
Dh · t ö
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 Ficks 1st law since we have no electrical fields that could drive a current.
This gives us
j h(x = 0)  =  –  e · D Dnh(x)
x = 0
By simple differentiation of our concentration equation from above we obtain
x = 0  =  –   Dn0
Insertion into the current equation yields the final result
j h(x = 0)  =   e · Dh
  · 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 concentration 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 · d2V(x)
 =  e · e0· dE(x)
 =   r(x)
E(x) is the electrical field strength; always 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
nh(x) + ND+(x) – [ne(x) + NA(x)] ö
(The sign in this formulation is negative if more negative charges are present than positive ones - which takes care of the minus sign usually attached to Poissons equation).
The electrostatic potential V needed for the Poisson equation is now a function of x and shifts the conduction and valence band up or down by the amount eV relative to some zero point at x = 0. We thus may write
EC(x)  =  EC(V = 0) + e · V(x)
EV(x)  =  EV(V = 0) + e · V(x)
The Poisson equation becomes
e · e0 · d2V(x)
 =  e · e0
 ·  d2EC(x)
 =  e æ
nh(x) + ND+(x) – [ne(x) + NA(x)] ö
If we now insert the proper equations for the four concentrations, we obtain a formidable differential equation that is not easy to solve, but of prime importance for semiconductor physics and devices.
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 (imagine a fictive 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 concentration 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 for the surface charge; the field can penetrate for a considerable distance.
However: In contrast to what we learned about SCRs in pn-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 ECEF incrases in the direction toards the surface.
If the majority carrier concentration then is becoming very small in surface near regions (it scales with exp – (ECEF) after all), the minority carrier concentration 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.
Band diagrams fOr fields
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 EDEF >> kT.
We then have only two charged entities:
ND+  =  ND  
ne   =  Neeff  · exp – ECEF
This means in what follows we only consider the majority carriers.
The Poisson equation then reduces to
e · e0
  ·  d2EC(x)
  =   – e æ
ND –  Neeff  · exp –   EC(x) – EF
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
EC(x)   =  EC0 + DEC(x)
The exponential term of the Poisson equation can now be rewritten, we obtain
e · Neeff  · exp – EC(x) – EF
 =  e · Neeff  · exp – EC0EF
· exp – DEC(x)
The first part of the right hand side gives just the electron (charge) 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:
 =  d2DEC
 = –   e · ND
e · e0
 ·  æ
1  –  exp – DEC
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 DEC.
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 develop the exponential function in a Taylor series and stop after the second term. This yields
 =  e2 ·ND
e · e0
 ·  DEC
That is easy now, the solution is
DEC(x)  =  DEC(x = 0) · exp – x 
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.
Since the field strength E(x) is the derivative of the electrostatic potential, with the latter, in turn, following simply from e · V(x) = EC(x), we have
E(x)  =  –  dV(x)
 =  –  1 
 e · LDb
 · DEC(x)
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 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 lenght 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
Lets start from the same assumption 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)
 =  –   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 dr/dt. The Poisson equation won't help because it does not explicitely contain the time dependence.
But simply using the continuity equation from above, and replacing nh by Dr (because this is the relevant density of charged carries now), provides a dDr/dt term. Moreover, we can hugely simplify this equation, because we
  • Look only at majority carriers; i.e. we may neglect the first term [nhnh(equ)]/ t.
  • We are treating quasi neutrality, ie. we neglect all terms with gradients in the carrier concentration
This leaves us with the following continuity equation
  =  –  e · r · µ · E(x)
Inserting dE/dx from the Poisson equation gives
 =  –   e · r · µ
 · Dr
r is the total carrier concentration, we can write it as r = r0 + Dr » r0 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   =   ee0
e · µ ·r0
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 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 e · µ · r = s, the conductivity of the material (for the carriers in question), we have the simple and fundamental relation
td  =  ee0
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 = {(ee0 · kT)/(e2 · r)}1/2, pluck it into the definition of the dielectric relaxation time (again replacing ND by r) and obtain
td  =  LDb2
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 concentration is high, td is in the order of pico seconds and LDb extends over nano meters. Any deviation from equilibrium is thus almost instantaneously wiped out, or, if that is not possible, contained within a very small scale.
And this is the regular situation for majority carriers. The few minority carriers always present in the semiconductor, too, can be safely neglected.
For minority carriers, however, the situation is entirely different.
Their concentration is very small; td and LD consequently are no longer small.
Moreover, whatever disturbance occurs in the concentration of minorities, there are plenty of majorities that can react very quickly (with their td) to the electrical field always tied to a Dr.
The majority carriers are always attracted to the minorities and thus will quickly surround any excess minority charge with a "cloud" of majority carriers, essentially compensating the electrical field of the excess minorities to zero.
They will, of course, eventually remove the excess charge by recombination, but that takes far longer than the time needed to do the screening.
Since the electrical field is now zero, the excess charge cannot disappear or spread out by field currents - only spreading by diffusion in the concentration gradient (which is automatically introduced, too), is possible.
But this is exactly the process that we have neglected in this discussion (we had all concentration gradients in the continuity equation set to zero!).
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!
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.

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© H. Föll (Semiconductors - Script)