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 a 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.  
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 (n_{e} or n_{h}).  
Fick's first law then tells us that the diffusiondriven particle current j_{p,diff} is given by  
 
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  
 
Considering only the onedimensional case for electrons (i.e. q = –e; holes behave in exactly the same way with q = +e), we have  
 
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 onedimensional) current given by  
 
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 (onedimensional) current in full generality is then  
 
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:  
 
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 righthand side depends on the type of charge carriers (since q = ±e). This is balanced on the lefthand 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  
 
Differentiation of the Boltzmann distribution gives us  
 
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  
 
Using this relation, the current balance from above becomes  
 
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:  
 
You may want to have a look at a different derivation in an advanced module.  
NonEquilibrium 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 n_{h}(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  
 
This is a crucial assumption!  
This allows us to concentrate on one kind of carrier, let's say we look at ntype Si with electrons as the majority carriers. We now focus on holes as the minority carriers since we always can compute the electron density n_{e} by  
 
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 · (n_{h} – n_{e}), in contrast to the total particle density, which is the sum !) we have  
 
With j_{total} = j_{e} + j_{h} = sum of the electron and hole currents.  
In the simplest form we have for the holes  
 
The factor 1/e is needed to convert an electrical current j to a particle current j_{part} via j = q · j_{part}, 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 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 – since div j_{h} = 0 is correct on average.  
We are, however, considering nonequilibrium , 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, R^{D} = [n_{h} – n_{h}(equ)]/t needs to be considered (t is the minority carrier life time).  
R^{ D} = [n_{h} – n_{h}(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  
 
The current j can always be expressed as the sum of a field current and a diffusion current as we did above by  

Inserting this equation in our continuity equation yields  
 
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 n_{h}(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 threedimensional 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 p–n 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 p–n junction, but the realistic minority carrier situation is now included.  
 
The region outside the space charge region, while now showing a density gradient of the minority carrier density, is essentially field free or at least has only a small electrical field strength. 
If we let E_{x} = 0 and consequently ¶E_{x} (x)/¶x = 0, too, the current equation from above reduces to  
 
Since ¶n_{h}/¶t = ¶[n_{h}(equ) + Dn_{h}]/¶t = ¶Dn_{h}/ ¶t, and correspondingly ¶^{2}n _{h}(x)/¶x^{2} = ¶ ^{2}Dn_{h}(x)/¶ x^{2}, we have  
 
If we consider steady state , we have ¶Dn_{h}/¶ 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 , nonchanging 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:  
 
The solution (for a onedimensional 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  
 
By simple differentiation of our density equation from above we obtain  
 
Insertion into the current equation yields the final result  
 
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 BandBending 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 onedimensional 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(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.  
 
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  
 
Thereby, the Poisson equation becomes  
 
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 surfacenear region and thus to induce some bandbending .  
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 E_{F} still constant (currents are not possible). The bands are still bent, however, this means that E_{C} – E_{F} incrases in the direction toards the surface.  
If the majority carrier density then is becoming very small in surfacenear regions (it scales with exp [– (E_{C} – E_{F} )/(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 surfacenear 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 ntype materials. The surface charge is symbolized by the green spheres or blue squares on the left.  
 
Between depletion and accumulation must be the flatband 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 – ntype 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 E_{D} – E_{F} >> kT.  
We then have only two charged entities:  
 
This means in what follows we only consider the majority carriers.  
The Poisson equation then reduces to  
 
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 fieldfree situation, i.e. from E_{C}^{0} = E_{C}(x = ¥ ). We thus write  
 
The exponential term of the Poisson equation can now be rewritten, we obtain  
 
The first part of the right hand side gives just the electron density in a fieldfree part of the semiconductor, which – in our approximations – is identical to the density N_{D} of donor atoms. This leaves us with a usable form of the Poisson equation for the case of accumulation :  
 
DE_{C} 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 E_{C} .  
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 quasineutrality which will give us the allimportant Debye length.  
The other cases can be found in advanced modules:  
Quasineutrality is the mathematically most simple case; it treats only small deviations from equilibrium and thus from charge neutrality.  
The condition for quasineutrality is simple: We assume DE_{C} << kT.  
We then can approximate the exponential function by its Taylor series and stop after the second term. This yields  
 
That is easy now, the solution is  
 
The solution defines L_{Db} = Debye length for ntype semiconductors = Debye length for electrons, we have  
 
Obviously the Debye length L_{Db} for holes in ptype semiconductors is given by  
 
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 ptype material (as it is conventionally done), the electrostatic potential is related to the conduction band edge by E_{C} (x) = E_{g} – 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  
 
Note that in the case of accumulation at the surface of an ntype semiconductor, D E_{C}(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.  
L_{Db} 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 fieldinduced 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 t_{d} .  
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 t_{0}; i.e. we still assume quasineutrality.  
The Poisson equation now is extremely simple, we write it directly for the electrical field strength and have  
 
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 quasineutrality, 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  
 
Inserting dE/dx from the Poisson equation gives  
 
r is the total carrier density, we can write it as r = r_{0} + 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  
 
With t_{d} = 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 (L_{Db} = {(e · e_{0} · kT)/(e · r )}^{1/2}, pluck it into the definition of the dielectric relaxation time (again replacing e · N_{D} by r) and obtain  
 
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, t_{d} is in the order of picoseconds and L_{Db} 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.  
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 density is very small; t_{d} and L_{D} consequently are no longer small.  
Moreover, whatever disturbance occurs in the density of minorities, there are plenty of majorities that can react very quickly (with their t_{d}) to the electrical field always tied to a Dr_{min}.  
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.  
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 density gradient (which is automatically introduced, too) is possible.  
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!).  
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. 
© H. Föll (Semiconductors  Script)