Chapter 4: pn Junctions 
4.4.1. General discussion 
The current in a pn diode is due to carrier recombination or generation somewhere within the pn diode structure. Under forward bias, the diode current is due to recombination. This recombination can occur within the quasineutral regions, within the depletion region or at the metalsemiconductor Ohmic contacts. Under reverse bias, the current is due to generation. Carrier generation due to light will further increase the current under forward as well as reverse bias. 
In this section, we first derive the ideal diode current. We will also distinguish between the "long" diode and "short" diode case. The "long" diode expression applies to pn diodes in which recombination/generation occurs in the quasineutral region only. This is the case if the quasineutral region is much larger than the carrier diffusion length. The "short" diode expression applies to pn diodes in which recombination/generation occurs at the contacts only. In a short diode, the quasineutral region is much smaller than the diffusion length. In addition to the "long" and "short" diode expressions, we also present the general result for pn diodes with arbitrary widths. 
Next, we derive expressions for the recombination/generation in the depletion region. Here we have to distinguish between the different recombination mechanisms  bandtoband recombination and ShockleyHallRead recombination  as they lead to different currentvoltage characteristics. 
4.4.2. The ideal diode current 
When calculating the current in a pn diode one needs to know the carrier density and the electric field throughout the pn diode which can then be used to obtain the drift and diffusion current. Unfortunately, this requires the knowledge of the quasiFermi energies, which is only known if the currents are known. The straightforward approach is to simply solve the driftdiffusion equations listed in section 2.10 simultaneously. This approach however does not yield an analytic solution. 
To avoid this problem we will assume that the electron and hole quasiFermi energies in the depletion region equal those in the adjacent ntype and ptype quasineutral regions. We will derive an expression for "long" and "short" diodes as well as a general expression, which is to be used if the quasineutral region is comparable in size to the diffusion length. 
The electric field and potential are obtained by using the full depletion approximation. Assuming that the quasiFermi energies are constant throughout the depletion region, one obtains the minority carrier densities at the edges of the depletion region, yielding: 
(4.4.1) 
and 
(4.4.2) 
These equations can be verified to yield the thermalequilibrium carrier density for zero applied voltage. In addition, an increase of the applied voltage will increase the separation between the two quasiFermi energies by the applied voltage multiplied with the electronic charge. 
The carrier density at the metal contacts is assumed to equal the thermalequilibrium carrier density. This assumption implies that excess carriers immediately recombine when reaching either of the two metalsemiconductor contacts. As recombination is typically higher at a semiconductor surface and is further enhanced by the presence of the metal, this is found to be a reasonable assumption. This results in the following set of boundary conditions: 
(4.4.3) 
and 
(4.4.4) 
The general expression for the ideal diode current is obtained by applying the boundary conditions to the general solution of the diffusion equation for each of the quasineutral regions, as described by equation (2.9.13) and (2.9.14): 
(2.9.13) 
(2.9.14) 
The boundary conditions at the edge of the depletion regions are described by (4.4.1), (4.4.2), (4.4.3) and (4.4.4). 
Before applying the boundary conditions, it is convenient to rewrite the general solution in terms of hyperbolic functions: 
(4.4.5) 
(4.4.6) 
where A^{*}, B^{*}, C^{*} and D^{*} are constants whose value remains to be determined. Applying the boundary conditions then yields: 
(4.4.7) 
(4.4.8) 
Where the quasineutral region widths, w_{n}^{'} and w_{p}^{'}, are defined as: 
(4.4.9) 
and 
(4.4.10) 
The current density in each region is obtained by calculating the diffusion current density using equations (2.7.24) and (2.7.25): 
(4.4.11) 
(4.4.12) 
The total current must be constant throughout the structure since a steady state case is assumed. No charge can accumulate or disappear somewhere in the structure so that the charge flow must be constant throughout the diode. The total current then equals the sum of the maximum electron current in the ptype region, the maximum hole current in the ntype regions and the current due to recombination within the depletion region. The maximum currents in the quasineutral regions occur at either side of the depletion region and can therefore be calculated from equations (4.4.11) and (4.4.12). Since we do not know the current due to recombination in the depletion region we will simply assume that it can be ignored. Later, we will more closely examine this assumption. The total current is then given by: 
(4.4.13) 
where I_{s} can be written in the following form: 
(4.4.14) 
A diode with a "long" quasineutral region has a quasineutral region, which is much larger than the minoritycarrier diffusion length in that region, or w_{n}^{'} > L_{p} and w_{p}^{'} > L_{n}. The general solution can be simplified under those conditions using: 
(4.4.15) 
yielding the following carrier densities, current densities and saturation currents: 
(4.4.16) 
(4.4.17) 
(4.4.18) 
(4.4.19) 
(4.4.20) 
We now come back to our assumption that the current due to recombination in the depletion region can be simply ignored. Given that there is recombination in the quasineutral region, it would be unreasonable to suggest that the recombination rate would simply drop to zero in the depletion region. Instead, we assume that the recombination rate is constant in the depletion region. To further simplify the analysis we will consider a p^{+}n junction so that we only need to consider the recombination in the ntype region. The current due to recombination in the depletion region is then given by: 
(4.4.21) 
so that I_{r} can be ignored if: 
(4.4.22) 
A necessary, but not sufficient requirement is therefore that the depletion region width is much smaller than the diffusion length for the ideal diode assumption to be valid. Silicon and germanium pn diodes usually satisfy this requirement, while gallium arsenide pn diodes rarely do because of the short carrier lifetime and diffusion length. 
As an example we now consider a silicon pn diode with N_{a} = 1.5 x 10^{14} cm^{3} and N_{d} = 10^{14} cm^{3}. The minority carrier lifetime was chosen to be very short, namely 400 ps, so that most features of interest can easily be observed. We start by examining the electron and hole density throughout the pn diode, shown in Figure 4.4.1: 
Figure 4.4.1 :  Electron and hole density throughout a forward biased pn diode. 
The majority carrier densities in the quasineutral region simply equal the doping density. The minority carrier densities in the quasineutral regions are obtained from equations (4.4.16) and (4.4.17). The electron and hole densities in the depletion region are calculating using the assumption that the electron/hole quasiFermi energy in the depletion region equals the electron/hole quasiFermi energy in the quasineutral ntype/ptype region. The corresponding band diagram is shown in Figure 4.4.2: 
The quasiFermi energies were obtained by combining (4.4.16) and (4.4.17) with (2.10.5) and (2.10.6). Note that the quasiFermi energies vary linearly within the quasineutral regions. 
Next, we discuss the current density. Shown in Figure 4.4.3 is the electron and hole current density as calculated using (4.4.18) and (4.4.19). The current due to recombination in the depletion region was assumed to be constant. 
Figure 4.4.3 :  Electron and hole current density versus position. The vertical lines indicate the edges of the depletion region. 
A "short" diode is a diode with quasineutral regions, which are much shorter than the minoritycarrier diffusion lengths. As the quasineutral region is much smaller than the diffusion length one finds that recombination in the quasineutral region is negligible so that the diffusion equations are reduced to: 
(4.4.23) 
The resulting carrier density varies linearly throughout the quasineutral region and in general is given by: 
(4.4.24) 
where A, B, C and D are constants obtained by satisfying the boundary conditions. Applying the same boundary conditions at the edge of the depletion region as above (equations (4.4.3) and (4.4.4)) and requiring thermal equilibrium at the contacts yields: 
(4.4.25) 
(4.4.26) 
for the hole and electron density in the ntype and ptype quasineutral region. 
The current in a "short" diode is again obtained by adding the maximum diffusion currents in each of the quasineutral regions and ignoring the current due to recombination in the depletion region, yielding: 
(4.4.27) 
where the saturation current, I_{s} is given by: 
(4.4.28) 
A comparison of the "short" diode expression with the "long" diode expression reveals that they are the same except for the use of either the diffusion length or the quasineutral region width in the denominator, whichever is smaller. 
Now that we have two approximate expressions, it is of interest to know when to use one or the other. To this end, we now consider a onesided n^{+}p diode.The ptype semiconductor has a width, w_{p}, and we normalize the excess electron density relative to its value at the edge of the depletion region (x = 0). The Ohmic contact to the ptype region is ideal so that the excess density is zero at x = w_{p}^{'}. The normalized excess carrier density is shown in Figure 4.4.4 for different values of the diffusion length. 
Figure 4.4.4 :  Excess electron density versus position as obtained by solving the diffusion equation with dn(x = 0) = 1 and dn(x/w_{p}^{'} = 1) = 0 . The ratio of the diffusion length to the width of the quasineutral region is varied from 0.1 (Bottom curve), 0.3, 0.5, 1 and ¥ (top curve) 
The figure illustrates how the excess electron density changes as the diffusion length is varied relative to the width of the quasineutral region. For the case where the diffusion length is much smaller than the width (L_{n} << w_{p}^{'}), the electron density decays exponentially and the "long" diode expression can be used. If the diffusion length is much longer than the width (L_{n} >> w_{p}^{'}), the electron density reduces linearly with position and the "short" diode expression can be used. If the diffusion length is comparable to the width of the quasineutral region width one must use the general expression. A numeric analysis reveals that the error is less than 10 % when using the short diode expression with L_{n} > 2 w_{p}^{'} and when using the long diode expression with L_{n} < w_{p}^{'}/2. Note that the best approximation is not necessarily the same in each region of the same pn diode. 
Example 4.4  An abrupt silicon pn junction (N_{a} = 10^{16} cm^{3} and N_{d} = 4 x 10^{16} cm^{3}) is biased with V_{a} = 0.6 V. Calculate the ideal diode current assuming that the ntype region is much smaller than the diffusion length with w_{n}^{'} = 1 mm and assuming a "long" ptype region. Use m_{n} = 1000 cm^{2}/Vs and m_{p} = 300 cm^{2}/Vs. The minority carrier lifetime is 10 ms and the diode area is 100 mm by 100 mm. 
Solution  The current is calculated from:
with
yielding I = 40.7 mA Note that the hole diffusion current occurs in the "short" ntype region and therefore depends on the quasineutral width in that region. The electron diffusion current occurs in the "long" ptype region and therefore depends on the electron diffusion length in that region. 
4.4.3. RecombinationGeneration current 
We now calculate the recombinationgeneration current in the depletion region of a pn junction. We distinguish between two different possible recombination mechanisms: bandtoband recombination and ShockleyHallRead recombination. 
The recombination/generation current due to bandtoband recombination/generation is obtained by integrating the net recombination rate, U_{bb}, over the depletion region: 
(4.4.29) 
where the net recombination rate is given by (2.8.3): 
(4.4.30) 
The carrier densities can be related to the constant quasiFermi energies and the product is independent of position: 
(4.4.31) 
This allows the integral to be solved analytically yielding: 
(4.4.32) 
The current due to bandtoband recombination has therefore the same voltage dependence as the ideal diode current and simply adds an additional term to the expression for the saturation current. 
The current due to trapassisted recombination in the depletion region is also obtained by integrating the trapassisted recombination rate over the depletion region width: 
(4.4.33) 
Substituting the expression (2.8.4) for the recombination rate yields: 
(4.4.34) 
where the product of the electron and hole densities was obtained by assuming that the quasiFermi energies are constant throughout the depletion region, which leads to: 
(4.4.35) 
The maximum recombination rate is obtained when the electron and hole densities are equal and therefore equals the square root of the product yielding: 
(4.4.36) 
From which an effective width can be defined which, when multiplied with the maximum recombination rate, equals the integral of the recombination rate over the depletion region. This effective width, x^{'}, is then defined by: 
(4.4.37) 
and the associated current due to trapassisted recombination in the depletion region is given by: 
(4.4.38) 
This does not provide an actual solution since the effective width, x^{'}, still must be determined by performing a numeric integration. Nevertheless, the above expression provides a way to obtain an upper estimate by substituting the depletion layer width, x_{d}, as it is always larger than the effective width. 
4.4.4. IV characteristics of real pn diodes 
The forward biased IV characteristics of real pn diodes are further affected by high injection and the series resistance of the diode. To illustrate these effects while summarizing the current mechanisms discussed previously we consider the IV characteristics of a silicon p^{+}n diode with N_{d} = 4 x 10^{14} cm^{3}, t_{p} = 10 ms, and m_{p} = 450 cm^{2}/Vs. The IV characteristics are plotted on a semilogarithmic scale and four different regions can be distinguished as indicated on Figure 4.4.5. First, there is the ideal diode region where the current increases by one order of magnitude as the voltage is increased by 60 mV. This region is referred to as having an ideality factor, n, of one. This ideality factor is obtained by fitting a section of the curve to the following expression for the current: 
(4.4.39) 
The ideality factor can also be obtained from the slope of the curve on a semilogarithmic scale using: 
(4.4.40) 
where the slope is in units of V/decade. To the left of the ideal diode region there is the region where the current is dominated by the trapassisted recombination in the depletion region described in section 4.4.3.2. This part of the curve has an ideality factor of two. To the right of the ideal diode region, the current becomes limited by high injection effects and by the series resistance. 
High injection occurs in a forward biased pn diode when the injected minority carrier density exceeds the doping density. High injection will therefore occur first in the lowest doped region of the diode since that region has the highest minority carrier density. 
Using equations (4.4.1) and (4.4.2), one finds that high injection occurs in a p^{+}n diode for the following applied voltage: 
(4.4.41) 
or at V_{a} = 0.55 V for the diode of Figure 4.4.5 as can be verified on the figure as the voltage where the ideality factor changes from one to two. For higher forward bias voltages, the current no longer increases exponentially with voltage. Instead, it increases linearly due to the series resistance of the diode. This series resistance can be due to the contact resistance between the metal and the semiconductor, due to the resistivity of the semiconductor or due to the series resistance of the connecting wires. This series resistance increases the external voltage, V_{a}^{*}, relative to the internal voltage, V_{a}, considered so far. 
(4.4.42) 
Where I is the diode current and R_{s} is the series resistance. 
These four regions can be observed in most pn diodes although the highinjection region rarely occurs, as the series resistance tends to limit the current first. 
Figure 4.4.5:  CurrentVoltage characteristics of a silicon diode under forward bias. 
4.4.5. The diffusion capacitance 
As a pn diode is forward biased, the minority carrier distribution in the quasineutral region increases dramatically. In addition, to preserve quasineutrality, the majority carrier density increases by the same amount. This effect leads to an additional capacitance called the diffusion capacitance. 
The diffusion capacitance is calculated from the change in charge with voltage: 
(4.4.43) 
Where the charge, DQ, is due to the excess carriers. Unlike a parallel plate capacitor, the positive and negative charge is not spatially separated. Instead, the electrons and holes are separated by the energy bandgap. Nevertheless, these voltage dependent charges yield a capacitance just as the one associated with a parallel plate capacitor. The excess minoritycarrier charge is obtained by integrating the charge density over the quasineutral region: 
(4.4.44) 
We now distinguish between the two limiting cases as discussed when calculating the ideal diode current, namely the "long" diode and a "short" diode. The carrier distribution, p_{n}(x), in a "long" diode is illustrated by Figure 4.4.6 (a). 
Figure 4.4.6:  Minority carrier distribution in (a) a "long" diode, and (b) a "short" diode. The excess minoritycarrier charge, DQ_{p}, in the quasineutral region, is proportional to the area defined by the solid and dotted lines. 
Using equation (4.4.18), the excess charge, DQ_{p}, becomes: 
(4.4.45) 
where I_{s,p} is the saturation current for holes, given by: 
(4.4.46) 
Equation (4.4.45) directly links the excess charge to the diffusion current. Since all injected minority carriers recombine in the quasineutral region, the current equals the excess charge divided by the average time needed to recombine with the majority carriers, i.e. the carrier lifetime, t_{p}. This relation will be the corner stone of the charge control model of bipolar junction transistors (section 5.6.2). 
The diffusion capacitance then equals: 
(4.4.47) 
Similarly, for a "short" diode, as illustrated by Figure 4.4.6 (b), one obtains: 
(4.4.48) 
Where t_{r,p} is the hole transit time given by: 
(4.4.49) 
Again, the excess charge can be related to the current. However, in the case of a "short" diode all minority carriers flow through the quasineutral region and do not recombine with the majority carriers. The current therefore equals the excess charge divided by the average time needed to traverse the quasineutral region, i.e. the transit time, t_{r,p}. 
The total diffusion capacitance is obtained by adding the diffusion capacitance of the ntype quasineutral region to that of the ptype quasineutral region. 
The total capacitance of the junction equals the sum of the junction capacitance, discussed in section 4.3.4, and the diffusion capacitance. For reverse biased voltages and small forward bias voltages, one finds that the junction capacitance is dominant. As the forward bias voltage is further increased the diffusion capacitance increases exponentially and eventually becomes larger than the junction capacitance. 
Example 4.5 

Solution 

4.4.6. High Injection Effects 
High injection of carriers causes to invalidate one of the approximations made in the derivation of the ideal diode characteristics, namely that the majority carrier density equals the thermal equilibrium value. Excess carriers will dominate the electron and hole concentration and can be expressed in the following way: 
(4.4.50) 
(4.4.51) 
where all carrier densities with subscript n are taken at x = x_{n} and those with subscript p at x = x_{p}. Solving the resulting quadratic equation yields: 
(4.4.52) 
(4.4.53) 
where the second terms are approximations for large V_{a}. From these expressions one can calculate the minority carrier diffusion current assuming a "long" diode in both quasineutral regions. We also ignore carrier recombination in the depletion region. 
(4.4.54) 
This means that high injection in a pn diode will reduce the slope on the currentvoltage characteristic on a semilogarithmic scale to 119mV/decade. 
High injection also causes a voltage drop across the quasineutral region. This voltage can be calculated from the carrier densities. Let's assume that high injection only occurs in the (lower doped) ptype region. The hole density at the edge of the depletion region (x = x_{p}) equals: 
(4.4.55) 
where V_{1} is the voltage drop across the ptype quasineutral region. This equation can then be solved for V_{1} yielding 
(4.4.56) 
High injection occurs (by definition) when the excess minority carrier density exceeds the doping density in the material. It is under such conditions that also the majority carrier density increases since, for charge neutrality to exist, the excess electron density has to equal the excess hole density: If there exists a net charge, the resulting electric field causes the carriers to move so that charge neutrality is restored. 
We repeat the high injection analysis without assuming a “large” applied voltage V_{a}, while providing more detail and deriving equations (4.4.52), (4.4.53) and (4.4.56). 
The analysis starts by assuming a certain excess carrier density so that the total density can be written as the sum of the thermal equilibrium density plus the excess carrier density. For an ntype region this yields the following equations: 
(4.4.57) 
(4.4.58) 
The product of the carrier densities can be expressed as a function of the intrinsic density in the following way: 
(4.4.59) 
where it was assumed that the semiconductor is nondegenerate and that the difference between the electron and hole quasi Fermi energies in electron volt equals the applied voltage in volt. Quasineutrality implies that the excess densities are the same, which yields a quadratic equation for the minority carrier density p_{n}: 
(4.4.60) 
which yields a value for the minority carrier density at the edge of the depletion region. 
(4.4.61) 
The associated current is a diffusion current and using a procedure similar to that for calculating the ideal diode current in a "long" diode one obtains the following hole current. 
(4.4.62) 
The electron current due to diffusion of electrons in the ptype region is given by a similar expression: 
(4.4.63) 
These expressions can be reduced to the ideal diode expressions provided that the excess minority density is much smaller than one quarter of the doping density, or: 
(4.4.64) 
while if the excess minority carrier density is much larger than one quarter of the doping density and an expression is obtained which is only valid under high injection conditions: 
(4.4.65) 
A closer examination of the problem prompts the question whether the full depletion approximation is still valid since the sign of potential across the semiconductor reverses. However the increase of the majority carrier density beyond the doping density causes a potential variation across the "quasineutral" region. This voltage in the ntype region is given by: 
(4.4.66) 
and similarly for the ptype region: 
(4.4.67) 
The potentials across the "quasineutral" region causes a larger potential across the depletion layer, since they have opposite sign, so that the depletion layer width as calculated using the modified potential f = f_{i} – V_{a} + V_{n} + V_{p} does not become zero. 
As an example we now consider an abrupt onesided pn diode. The current is shown as function of the voltage in Figure 4.4.7. It is calculated for a rather low ntype doping of 10^{13} cm^{3} to further highlight the high injection effects. 
Figure 4.4.7:  CurrentVoltage characteristics of a p^{+}n diode including the effects of high injection and a linear series resistance. 
The dotted line on Figure 4.4.7 fits the current at high forward bias. The slope is 1 decade/120 mV, which corresponds to an ideality factor of 2. Figure 4.4.8 and Figure 4.4.9 provide a more detailed look at the high injection condition. Figure 4.4.8 presents the carrier densities in the ntype region. Once can observed that the majority carrier (electron) density increases beyond the doping density and tracks the minority carrier (hole) density in the region up to 50 mm away from the junction. 
Figure 4.4.8:  Electron and hole density under high injection conditions. 
Figure 4.4.9 provides the energy band diagram at a 0.6 V bias. Band bending can be observed in the quasineutral region. 
Figure 4.4.9:  Energy band diagram under high injection conditions. 
4.4.7. Heterojunction Diode Current 
This section is very similar to the one discussing currents across a homojunction. Just as for the homojunction we find that current in a pn junction can only exist if there is recombination or generation of electron and holes somewhere throughout the structure. The ideal diode equation is a result of the recombination and generation in the quasineutral regions (including recombination at the contacts) whereas recombination and generation in the depletion region yield enhanced leakage or photocurrents. 
For the derivation of the ideal diode equation we will again assume that the quasiFermi levels are constant throughout the depletion region so that the minority carrier densities at the edges of the depletion region and assuming "low" injection are still given by: 
(4.4.68) 
(4.4.69) 
Where n_{i,n} and n_{i,p} refer to the intrinsic concentrations of the n and p region. Solving the diffusion equations with these minority carrier densities as boundary condition and assuming a "long" diode we obtain the same expressions for the carrier and current distributions: 
(4.4.70) 
(4.4.71) 
(4.4.72) 
(4.4.73) 
Where L_{p}L_{n} are the hole respectively the electron diffusion lengths in the ntype and ptype material, respectively. The difference compared to the homojunction case is contained in the difference of the material parameters, the thermal equilibrium carrier densities and the width of the depletion layers. Ignoring recombination of carriers in the base yields the total ideal diode current density J_{ideal}: 
(4.4.74) 
(4.4.75) 
This expression is valid only for a pn diode with infinitely long quasineutral regions. For diodes with a quasineutral region shorter than the diffusion length, and assuming an infinite recombination velocity at the contacts, the diffusion length can simply be replaced by the width of the quasineutral region. For more general boundary conditions, we refer to section 4.4.2.3. 
Since the intrinsic concentrations depend exponentially on the energy bandgap, a small difference in bandgap between the ntype and ptype material can cause a significant difference between the electron and hole current and that independent of the doping concentrations. 
Recombination/generation currents in a heterojunction can be much more important than in a homojunction because most recombination/generation mechanisms depend on the intrinsic carrier concentration which depends strongly on the energy bandgap. We will consider only two major mechanisms: bandtoband recombination and ShockleyHallRead recombination. 
The recombination/generation rate is due to bandtoband transitions is given by: 
(4.4.76) 
where b is the bimolecular recombination rate. For bulk GaAs this value is 1.1 x 10^{10} cm^{3}s^{1}. For (or under forward bias conditions) recombination dominates, whereas for (under reverse bias conditions) thermal generation of electronhole pairs occurs. Assuming constant quasiFermi levels in the depletion region, this rate can be expressed as a function of the applied voltage by using the "modified" massaction law ,yielding: 
(4.4.77) 
The current is then obtained by integrating the recombination rate throughout the depletion region: 
(4.4.78) 
For uniform material (homojunction) this integration yielded: 
(4.4.79) 
Whereas for a pn heterojunction consisting of two uniformly doped regions with different bandgap, the integral becomes: 
(4.4.80) 
Provided bias conditions are "close" to thermal equilibrium the recombination rate due to a density N_{t} of traps with energy E_{t} and a recombination/generation crosssection s is given by 
(4.4.81) 
where n_{I} is the intrinsic carrier concentration, v_{th} is the thermal velocity of the carriers and E_{I} is the intrinsic energy level. For E_{I} = E_{t} and t_{0} = 1/N_{t}sv_{th} this expression simplifies to: 
(4.4.82) 
Throughout the depletion region, the product of electron and hole density is given by the "modified" mass action law: 
(4.4.83) 
This enables to find the maximum recombination rate which occurs for 
(4.4.84) 
The total recombination current is obtained by integrating the recombination rate over the depletion layer width: 
(4.4.85) 
which can be written as a function of the maximum recombination rate and an "effective" width x^{'}: 
(4.4.86) 
where 
(4.4.87) 
Since U_{SHR,max} is larger than or equal to U_{SHR} anywhere within the depletion layer one finds that x^{'} has to be smaller than x_{d} = x_{n} + x_{p}. (Note that for a piN or pqwN structure the width of the intrinsic/qw layer has to be included). 
The calculation of x^{'} requires a numerical integration. The carrier concentrations n and p in the depletion region are given by: 
(4.4.88) 
(4.4.89) 
Substituting these equations into ( 4.4.82) then yields x^{'}. 
Recognizing that bandtoband recombination between different states in the quantum well has a different coefficient, the total recombination including all possible transitions can be written as: 
(4.4.90) 
with 
(4.4.91) 
and 
(4.4.92) 
where E_{n,e} and E_{n,h} are calculated in the absence of an electric field. To keep this derivation simple, we will only consider radiative transitions between the n = 1 states for which: 
(4.4.93) 
(4.4.94) 
both expressions can be combined yielding 
(4.4.95) 
A straightforward extension of the expression for bulk material to two dimensions yields 
(4.4.96) 
and the recombination current equals: 
(4.4.97) 
This expression implies that carriers from any quantum state are equally likely to recombine with a midgap trap. 
Graded regions can often be found in heterojunction devices. Typically they are used to avoid abrupt heterostructures, which limit the current flow. In addition they are used in laser diodes where they provide a graded index region, which guides the lasing mode. An accurate solution for a graded region requires the solution of a set of nonlinear differential equations. 
Numeric simulation programs provide such solutions and can be used to gain the understanding needed to obtain approximate analytical solutions. A common misconception regarding such structures is that the flatband diagram is close to the actual energy band diagram under forward bias. Both are shown in the figure below for a singlequantumwell gradedindex separateconfinement heterostructure (GRINSCH) as used in edgeemitting laser diodes. 
Figure 4.4.10:  Flat band diagram of a graded AlGaAs pn diode with x = 40 % in the cladding regions, x varying linearly from 40 % to 20 % in the graded regions and x = 0% in the quantum well. 
Figure 4.4.11:  Energy band diagram of the graded pn diode shown above under forward bias. V_{a} = 1.5 V, N_{a} = 4 x 10^{17} cm^{3}, N_{d} = 4 x 10^{17} cm^{3}. Shown are the conduction and valence band edges (solid lines) as well as the quasiFermi energies (dotted lines). 
The first difference between the two diagrams is that the conduction band edge in the ntype graded region as well as the valence band edge in the ptype graded region are almost constant. This assumption is correct if the majority carrier quasiFermi energy, the majority carrier density and the effective density of states for the majority carriers don't vary within the graded region. Since the carrier recombination primarily occurs within the quantum well (as it should be in a good laser diode), the quasiFermi energy does not change in the graded regions, while the effective density of states varies as the 3/2 power of the effective mass, which varies only slowly within the graded region. The constant band edge for the minority carriers implies that the minority carrier band edge reflects the bandgap variation within the graded region. It also implies a constant electric field throughout the grade region which compensates for the majority carrier bandgap variation or: 
(4.4.98) 
(4.4.99) 
where E_{c0}(x) and E_{v0}(x) are the conduction and valence band edge as shown in the flatband diagram. The actual electric field is compared to these equations in Figure 4.4.12. The existence of an electric field requires a significant charge density at each end of the graded regions, which is obtained by a depletion of carriers. This also causes a small cusp in the band diagram. 
Figure 4.4.12:  Electric Field within a graded pn diode. Compared are a numeric simulation (solid line) and equations (4.4.108) and (4.4.109) (dotted line). The field in the depletion regions around the quantum well was calculated using the linearized Poisson equation. 
Another important issue is that the traditional current equation with a drift and diffusion term must be modified. We now derive the modified expression by starting from the relation between the current density and the gradient of the quasiFermi level: 
(4.4.100) 
(4.4.101) 
where it was assumed that the electron density is nondegenerate. At first sight it seems that only the last term is different from the usual expression. However the equation can be rewritten as a function of E_{c0}(x), yielding: 
(4.4.102) 
This expression will be used in the next section to calculate the ideal diode current in a graded pn diode. We will at that time ignore the gradient of the effective density of states. A similar expression can be derived for the hole current density, J_{p}. 
We now calculate the ideal diode current in a graded heterojunction. Such calculation poses a special challenge since a gradient of the bandedge exists within the quasineutral region. The derivation below can be applied to a pn diode with a graded doping density as well as one with a graded bandgap provided that the gradient is constant. For a diode with a graded doping concentration this implies an exponential doping profile as can be found in an ionimplanted base of a silicon bipolar junction transistor. For a diode with a graded bandgap the bandedge gradient is constant if the bandgap is linearly graded provided the majority carrier quasiFermi level is parallel to the majority carrier band edge. 
Focusing on a diode with a graded bandgap we first assume that the gradient is indeed constant in the quasineutral region and that the doping density is constant. Using the full depletion approximation one can then solve for the depletion layer width. This requires solving a transcendental equation since the dielectric constant changes with material composition (and therefore also with bandgap energy). A first order approximation can be obtained by choosing an average dielectric constant within the depletion region and using previously derived expressions for the depletion layer width. Under forward bias conditions one finds that the potential across the depletion regions becomes comparable to the thermal voltage. One can then use the linearized Poisson equation or solve Poisson's equation exactly. The former approach was taken to obtain the electric field in Figure 4.4.12. 
The next step requires solving the diffusion equation in the quasineutral region with the correct boundary condition and including the minority carrier bandedge gradient. For electrons in a ptype quasineutral region we have to solve the following modified diffusion equation 
(4.4.103) 
which can be normalized yielding: 
(4.4.104) 
If the junction interface is at x = 0 and the ptype material is on the right hand side, extending up to infinity, the carrier concentrations equals 
(4.4.105) 
where we ignored the minority carrier density under thermal equilibrium, which limits this solution to forward bias voltages. Note that the minority carrier concentration n_{p0}(x_{p}) at the edge of the depletion region (at x = x_{p}) is strongly voltage dependent since it is exponentially dependent on the actual bandgap at x = x_{p}. 
The electron current at x = x_{p} is calculated using the above carrier concentration but including the drift current since the bandedge gradient is not zero, yielding: 
(4.4.106) 
The minus sign occurs since the electrons move from left to right for a positive applied voltage. For a = 0, the current equals the ideal diode current in a nongraded junction: 
(4.4.107) 
while for strongly graded diodes (aL_{n} >> 1) the current becomes: 
(4.4.108) 
For a bandgap grading given by: 
(4.4.109) 
one finds 
(4.4.110) 
and the current density equals: 
(4.4.111) 
where J_{n}(a = 0) is the current density in the absence of any bandgap grading. 
Boulder, December 2004 