Chapter 4: pn Junctions 
The electrostatic analysis of a pn diode is of interest since it provides knowledge about the charge density and the electric field in the depletion region. It is also required to obtain the capacitancevoltage characteristics of the diode. The analysis is very similar to that of a metalsemiconductor junction (section 3.3). A key difference is that a pn diode contains two depletion regions of opposite type. 
4.3.1. General discussion  Poisson's equation 
The general analysis starts by setting up Poisson's equation: 
(4.3.1) 
where the charge density, r, is written as a function of the electron density, the hole density and the donor and acceptor densities. To solve the equation, we have to express the electron and hole density, n and p, as a function of the potential, f, yielding: 
(4.3.2) 
with 
(4.3.3) 
where the potential is chosen to be zero in the ntype region, far away from the pn interface. 
This secondorder nonlinear differential equation (4.3.2) cannot be solved analytically. Instead we will make the simplifying assumption that the depletion region is fully depleted and that the adjacent neutral regions contain no charge. This full depletion approximation is the topic of the next section. 
4.3.2. The fulldepletion approximation 
The fulldepletion approximation assumes that the depletion region around the metallurgical junction has welldefined edges. It also assumes that the transition between the depleted and the quasineutral region is abrupt. We define the quasineutral region as the region adjacent to the depletion region where the electric field is small and the free carrier density is close to the net doping density. 
The fulldepletion approximation is justified by the fact that the carrier densities change exponentially with the position of the Fermi energy relative to the band edges. For example, as the distance between the Fermi energy and the conduction band edge is increased by 59 meV, the electron concentration at room temperature decreases to one tenth of its original value. The charge in the depletion layer is then quickly dominated by the remaining ionized impurities, yielding a constant charge density for uniformly doped regions. 
We will therefore start the electrostatic analysis using an abrupt charge density profile, while introducing two unknowns, namely the depletion layer width in the ptype region, x_{p}, and the depletion region width in the ntype region, x_{n}. The sum of the two depletion layer widths in each region is the total depletion layer width x_{d}, or: 
(4.3.4) 
From the charge density, we then calculate the electric field and the potential across the depletion region. A first relationship between the two unknowns is obtained by setting the positive charge in the depletion layer equal to the negative charge. This is required since the electric field in both quasineutral regions must be zero. A second relationship between the two unknowns is obtained by relating the potential across the depletion layer width to the applied voltage. The combination of both relations yields a solution for x_{p} and x_{n}, from which all other parameters can be obtained. 
4.3.3. Full depletion analysis 
Once the fulldepletion approximation is made, it is easy to find the charge density profile: It equals the sum of the charges due to the holes, electrons, ionized acceptors and ionized holes: 
(4.3.5) 
where it is assumed that no free carriers are present within the depletion region. For an abrupt pn diode with doping densities, N_{a} and N_{d}, the charge density is then given by: 
(4.3.6) 
This charge density, r, is shown in Figure 4.3.1 (a). 
Figure 4.3.1:  (a) Charge density in a pn junction, (b) Electric field, (c) Potential and (d) Energy band diagram 
As can be seen from Figure 4.3.1 (a), the charge density is constant in each region, as dictated by the fulldepletion approximation. The total charge per unit area in each region is also indicated on the figure. The charge in the ntype region, Q_{n}, and the charge in the ptype region, Q_{p}, are given by: 
(4.3.7) 
(4.3.8) 
The electric field is obtained from the charge density using Gauss's law, which states that the field gradient equals the charge density divided by the dielectric constant or: 
(4.3.9) 
The electric field is obtained by integrating equation (4.3.9). The boundary conditions, consistent with the full depletion approximation, are that the electric field is zero at both edges of the depletion region, namely at x = x_{p} and x = x_{n}. The electric field has to be zero outside the depletion region since any field would cause the free carriers to move thereby eliminating the electric field. Integration of the charge density in an abrupt pn diode as shown in Figure 4.3.1 (a) is given by: 
(4.3.10) 
The electric field varies linearly in the depletion region and reaches a maximum value at x = 0 as can be seen on Figure 4.3.1(b). This maximum field can be calculated on either side of the depletion region, yielding: 
(4.3.11) 
This provides the first relationship between the two unknowns, x_{p} and x_{n}, namely: 
(4.3.12) 
This equation expresses the fact that the total positive charge in the ntype depletion region, Q_{n}, exactly balances the total negative charge in the ptype depletion region, Q_{p}. We can then combine equation (4.3.4) with expression (4.3.12) for the total depletionlayer width, x_{d}, yielding: 
(4.3.13) 
and 
(4.3.14) 
The potential in the semiconductor is obtained from the electric field using: 
(4.3.15) 
We therefore integrate the electric field yielding a piecewise parabolic potential versus position as shown in Figure 4.3.1 (c) 
The total potential across the semiconductor must equal the difference between the builtin potential and the applied voltage, which provides a second relation between x_{p} and x_{n}, namely: 
(4.3.16) 
The depletion layer width is obtained by substituting the expressions for x_{p} and x_{n}, (4.3.13) and (4.3.14), into the expression for the potential across the depletion region, yielding: 
(4.3.17) 
from which the solutions for the individual depletion layer widths, x_{p} and x_{n} are obtained: 
(4.3.18) 
(4.3.19) 
Example 4.2  An abrupt silicon (n_{I} = 10^{10} cm^{3}) pn junction consists of a ptype region containing 10^{16} cm^{3} acceptors and an ntype region containing 5 x 10^{16} cm^{3} donors.

Solution  The builtin potential is calculated from: The depletion layer width is obtained from: the electric field from and the potential across the ntype region equals where one can also show that: This yields the following numeric values:

4.3.4. Junction capacitance 
Any variation of the charge within a pn diode with an applied voltage variation yields a capacitance, which must be added to the circuit model of a pn diode. This capacitance related to the depletion layer charge in a pn diode is called the junction capacitance. 
The capacitance versus applied voltage is by definition the change in charge for a change in applied voltage, or: 
(4.3.20) 
The absolute value sign is added in the definition so that either the positive or the negative charge can be used in the calculation, as they are equal in magnitude. Using equation (4.3.7) and (4.3.18) one obtains: 
(4.3.21) 
A comparison with equation (4.3.17), which provides the depletion layer width, x_{d}, as a function of voltage, reveals that the expression for the junction capacitance, C_{j}, seems to be identical to that of a parallel plate capacitor, namely: 
(4.3.22) 
The difference, however, is that the depletion layer width and hence the capacitance is voltage dependent. The parallel plate expression still applies since charge is only added at the edge of the depletion regions. The distance between the added negative and positive charge equals the depletion layer width, x_{d}. 
The capacitance of a pn diode is frequently expressed as a function of the zero bias capacitance, C_{j0}: 
(4.3.23) 
Where 
(4.3.24) 
A capacitance versus voltage measurement can be used to obtain the builtin voltage and the doping density of a onesided pn diode. When plotting the inverse of the capacitance squared, one expects a linear dependence as expressed by: 
(4.3.25) 
The capacitancevoltage characteristic and the corresponding 1/C^{2} curve are shown in Figure 4.3.2. 
Figure 4.3.2 :  Capacitance and 1/C^{2} versus voltage of a pn diode with N_{a} = 10^{16} cm^{3}, N_{d} = 10^{17} cm^{3} and an area of 10^{4} cm^{2}. 
The builtin voltage is obtained at the intersection of the 1/C^{2} curve and the horizontal axis, while the doping density is obtained from the slope of the curve. 
(4.3.26) 
Example 4.3  Consider an abrupt pn diode with N_{a} = 10^{18} cm^{3} and N_{d} = 10^{16} cm^{3}. Calculate the junction capacitance at zero bias. The diode area equals 10^{4} cm^{2}. Repeat the problem while treating the diode as a onesided diode and calculate the relative error. 
Solution  The built in potential of the diode equals: The depletion layer width at zero bias equals: And the junction capacitance at zero bias equals: Repeating the analysis while treating the diode as a onesided diode, one only has to consider the region with the lower doping density so that And the junction capacitance at zero bias equals The relative error equals 0.5 %, which justifies the use of the onesided approximation.

A capacitancevoltage measurement also provides the doping density profile of onesided pn diodes. For a p^{+,/sup>n diode, one obtains the doping density from: } 
(4.3.27) 
while the depth equals the depletion layer width, obtained from x_{d} = e_{s}A/C_{j}. Both the doping density and the corresponding depth can be obtained at each voltage, yielding a doping density profile. Note that the capacitance in equations (4.3.21), (4.3.22), (4.3.25), and (4.3.27) is a capacitance per unit area. 
As an example, we consider the measured capacitancevoltage data obtained on a 6HSiC pn diode. The diode consists of a highly doped ptype region on a lightly doped ntype region on top of a highly doped ntype substrate. The measured capacitance as well as 1/C^{2}is plotted as a function of the applied voltage. The dotted line forms a reasonable fit at voltages close to zero from which one can conclude that the doping density is almost constant close to the pn interface. The capacitance becomes almost constant at large negative voltages, which corresponds according to equation (4.3.27) to a high doping density. 
Figure 4.3.3 :  Capacitance and 1/C^{2} versus voltage of a 6HSiC pn diode. 
The doping profile calculated from the date presented in Figure 4.3.3 is shown in Figure 4.3.4. The figure confirms the presence of the highly doped substrate and yields the thickness of the ntype layer. No information is obtained at the interface (x = 0) as is typical for doping profiles obtained from CV measurements. This is because the capacitance measurement is limited to small forward bias voltages since the forward bias current and the diffusion capacitance affect the accuracy of the capacitance measurement. 
Figure 4.3.4 :  Doping profile corresponding to the measured data, shown in Figure 4.3.3. 
4.3.5. The linearly graded pn junction 
A linearly graded junction has a doping profile, which depends linearly on the distance from the interface. 
(4.3.28) 
To analyze such junction we again use the full depletion approximation, namely we assume a depletion region with width x_{n} in the ntype region and x_{p} in the ptype region. Because of the symmetry, we can immediately conclude that both depletion regions must be the same. The potential across the junction is obtained by integrating the charge density between x =  x_{p} and x = x_{n} = x_{p} twice resulting in: 
(4.3.29) 
Where the builtin potential is linked to the doping density at the edge of the depletion region such that: 
(4.3.30) 
The depletion layer with is then obtained by solving for the following equation: 
(4.3.31) 
Since the depletion layer width depends on the builtin potential, which in turn depends on the depletion layer width, this transcendental equation cannot be solved analytically. Instead it is solved numerically through iteration. One starts with an initial value for the builtin potential and then solves for the depletion layer width. A possible initial value for the builtin potential is the bandgap energy divided by the electronic charge, or 1.12 V in the case of silicon. From the depletion layer width, one calculates a more accurate value for the builtin potential and repeats the calculation of the depletion layer width. As one repeats this process, one finds that the values for the builtin potential and depletion layer width converge. 
The capacitance of a linearly graded junction is calculated like before as: 
(4.3.32) 
Where the charge per unit area must be recalculated for the linear junction, namely: 
(4.3.33) 
The capacitance then becomes: 
(4.3.34) 
The capacitance of a linearly graded junction can also be expressed as a function of the zerobias capacitance or: 
(4.3.35) 
Where C_{j0} is the capacitance at zero bias, which is given by: 
(4.3.36) 
4.3.6. The abrupt pin junction 
A pin junction is similar to a pn junction, but contains in addition an intrinsic or unintentionally doped region with thickness, d, between the ntype and ptype layer. Such structure is typically used if one wants to increase the width of the depletion region, for instance to increase the optical absorption in the depletion region. Photodiodes and solar cells are therefore likely to be pin junctions. 
The analysis is also similar to that of a pn diode, although the potential across the undoped region, f_{u}, must be included in the analysis. Equation (4.3.16) then becomes: 
(4.3.37) 
(4.3.38) 
while the charge in the ntype region still equals that in the ptype region, so that (4.3.12) still holds: 
(4.3.39) 
Equations (4.3.37) through (4.3.39) can be solved for x_{n} yielding: 
(4.3.40) 
From x_{n} and x_{p}, all other parameters of the pin junction can be obtained. The total depletion layer width, x_{d}, is obtained from: 
(4.3.41) 
The potential throughout the structure is given by: 
(4.3.42) 
(4.3.43) 
(4.3.44) 
where the potential at x = x_{n} was assumed to be zero. 
The capacitance of a pin diode equals the series connection of the capacitances of each region, simply by adding both depletion layer widths and the width of the undoped region: 
(4.3.45) 
4.3.7. Solution to Poisson’s equation for an abrupt pn junction 
Applying Gauss's law one finds that the total charge in the ntype depletion region equals minus the charge in the ptype depletion region: 
(4.3.46) 
Poisson's equation can be solved separately in the ntype and ptype region as was done in section 3.3.7 yielding an expression for (x = 0) which is almost identical to equation (3.3.22): 
(4.3.47) 
where f_{n} and f_{p} are assumed negative if the semiconductor is depleted. Their relation to the applied voltage is given by: 
(4.3.48) 
One obtains f_{n} and f_{p} as a function of the applied voltage by solving the transcendental equations. 
For the special case of a symmetric doping profile, or N_{d} = N_{a}, these equations can easily be solved yielding: 
(4.3.49) 
The depletion layer widths also equal each other and are given by: 
(4.3.50) 
Using the above expression for the electric field at the origin, we find: 
(4.3.51) 
where is the extrinsic Debye length. The relative error of the depletion layer width as obtained using the full depletion approximation equals: 
(4.3.52) 
So that for = 1, 2, 5, 10, 20 and 40, one finds the relative error to be 45, 23, 10, 5.1, 2.5 and 1.26 %. 
4.3.8. The hetero pn junction 
Heterojunction pn diodes can be found in a wide range of heterojunction devices including laser diodes, high electron mobility transistors (HEMTs) and heterojunction bipolar transistors (HBTs). Such devices take advantage of the choice of different materials, and the corresponding material properties, for each layer of the heterostructure. We present in this section the electrostatic analysis of heterojunction pn diodes. 
The heterojunction pn diode is in principle very similar to a homojunction. The main problem that needs to be tackled is the effect of the bandgap discontinuities and the different material parameters, which make the actual calculations more complex even though the pn diode concepts need almost no changing. An excellent detailed treatment can be found in Wolfe et al. 
The flatband energy band diagram of a heterojunction pn diode is shown in the figure below. As a convention we will assume DE_{c} to be positive if E_{c,n} > E_{c,p} and DE_{v} to be positive if E_{v,n} < E_{v,p}. 
Figure 4.3.5 :  Flatband energy band diagram of a pn heterojunction 
The builtin potential is defined as the difference between the Fermi levels in both the ntype and the ptype semiconductor. From the energy diagram we find: 
(4.3.53) 
which can be expressed as a function of the electron concentrations and the effective densities of states in the conduction band: 
(4.3.54) 
The builtin voltage can also be related to the hole concentrations and the effective density of states of the valence band: 
(4.3.55) 
Combining both expressions yields the builtin voltage independent of the free carrier concentrations: 
(4.3.56) 
where n_{i,n} and n_{i,p} are the intrinsic carrier concentrations of the ntype and ptype region, respectively. DE_{c} and DE_{v} are positive quantities if the bandgap of the ntype region is smaller than that of the ptype region and the sum of both equals the bandgap difference. The band alignment must also be as shown in Figure 4.3.5. The above expression reduces to that of the builtin junction of a homojunction if the material parameters in the ntype region equal those in the ptype region. If the effective densities of states are the same, the expression for the heterojunction reduces to: 
(4.3.57) 
For the calculation of the charge, field and potential distribution in an abrupt pn junction we follow the same approach as for the homojunction. First of all we use the full depletion approximation and solve Poisson's equation. The expressions derived in section 4.3.3 then still apply. 
(4.3.58) 
(4.3.59) 
(4.3.60) 
The main differences are the different expression for the builtin voltage and the discontinuities in the field distribution (because of the different dielectric constants of the two regions) and in the energy band diagram. However the expressions for x_{n} and x_{p} for a homojunction can still be used if one replaces N_{a} by N_{a} e_{s,p}/e_{s} , N_{d} by N_{d} e_{s,n}/e_{s}, x_{p} by x_{p} e_{s}/e_{s,p} , and x_{n} by x_{n} e_{s}/e_{s,n}. Adding x_{n} and x_{p} yields the total depletion layer width x_{d}: 
(4.3.61) 
The capacitance per unit area can be obtained from the series connection of the capacitance of each layer: 
(4.3.62) 
For a PiN heterojunction the above expressions take the following modified form: 
(4.3.63) 
(4.3.64) 
(4.3.65) 
Where f_{u} is the potential across the middle undoped region of the diode, having a thickness d. The depletion layer width and the capacitance are given by: 
(4.3.66) 
(4.3.67) 
Equations (4.3.63) through (4.3.65) can be solved for x_{n}, yielding: 
(4.3.68) 
A solution for x_{p} can be obtained from (4.3.68) by replacing N_{d} by N_{a}, N_{a} by N_{d}, e_{s,n} by e_{s,p}, and e_{s,p} by e_{s,n}. Once x_{n} and x_{p} are determined all other parameters of the PiN junction can be obtained. The potential throughout the structure is given by: 
(4.3.69) 
(4.3.70) 
(4.3.71) 
where the potential at x = x_{n} was assumed to be zero. 
An example of the charge distribution, electric field, potentials and energy band diagram throughout the PiN heterostructure is presented in Figure 4.3.6: 
Figure 4.3.6 :  Charge distribution, electric field, potential and energy band diagram of an AlGaAs/GaAs pn heterojunction with V_{a} = 0.5 V, x = 0.4 on the left and x = 0 on the right. N_{d} = N_{a} = 10^{17}cm^{3} 
The above derivation ignores the fact that  because of the energy band discontinuities  the carrier densities in the intrinsic region could be substantially larger than in the depletion regions in the ntype and ptype semiconductor. Large amounts of free carriers imply that the full depletion approximation is not valid and that the derivation has to be repeated while including a possible charge in the intrinsic region. 
Real PiN junctions often differ from their ideal model, which was described in section section 4.3.8.4. The intrinsic region could be lightly doped, while a fixed interface charge could be present between the individual layers. We now consider the middle layer to have a doping concentration N_{m} = N_{dm}  N_{am} and a dielectric constant e_{s,m}. A charge Q_{1} is assumed between the N and M layer, and a charge Q_{2} between the M and P layer. Equations (4.3.63) through (4.3.65) then take the following form: 
(4.3.72) 
(4.3.73) 
(4.3.74) 
These equations can be solved for x_{n} and x_{p} yielding a general solution for this structure. Again it should be noted that this solution is only valid if the middle region is indeed fully depleted. 
Solving the above equation allows to draw the charge density, the electric field distribution, the potential and the energy band diagram. An example is provided in Figure 4.3.7. 
Figure 4.3.7:  Charge distribution, electric field, potential and energy band diagram of an AlGaAs/GaAs pin heterojunction with V_{a} = 1.4 V, x = 0.4 on the left, x = 0 in the middle and x = 0.2 on the right. d = 10 nm and N_{d} = N_{a} = 10^{17}cm^{3} 
Next, we consider a pn junction with a quantum well located between the n and p region as shown in Figure 4.3.8. 
Figure 4.3.8:  Flatband energy band diagram of a pn heterojunction with a quantum well at the interface. 
Under forward bias, charge can accumulate within the quantum well. In this section, we will outline the procedure to solve this structure. The actual solution can only be obtained by solving a transcendental equation. Approximations will be made to obtain useful analytic expressions. 
The potentials within the structure can be related to the applied voltage by: 
(4.3.75) 
where the potentials across the ptype and n–type regions are obtained using the full depletion approximation: 
(4.3.76) 
The potential across the quantum well is to first order given by: 
(4.3.77) 
where P and N are the hole and electron density per unit area in the quantum well. This equation assumes that the charge in the quantum well Q = q (P  N) is located in the middle of the well. Applying Gauss's law yields the following balance between the charges: 
(4.3.78) 
where the electron and hole densities can be expressed as a function of the effective densities of states in the quantum well: 
(4.3.79) 
(4.3.80) 
with DE_{n,e} and DE_{n,h} given by: 
(4.3.81) 
(4.3.82) 
where E_{n,e} and E_{n,h} are the n^{th} energies of the electrons respectively holes relative to the conduction respectively valence band edge. These nine equations can be used to solve for the nine unknowns by applying numerical methods. A quick solution can be obtained for a symmetric diode, for which all the parameters (including material parameters) of the n and p region are the same. For this diode N equals P because of the symmetry. Also x_{n} equals x_{p} and f_{n} equals f_{p}. Assuming that only one energy level namely the n = 1 level is populated in the quantum well one finds: 
(4.3.83) 
where E_{g} is the bandgap of the quantum well material. 
Numeric simulations of the general case reveal that, especially under large forward bias conditions, the electron and hole density in the quantum well are the same to within a few percent. An example is presented in Figure 4.3.9. 
Figure 4.3.9:  Energy band diagram of a GaAs/AlGaAs pn junction with a quantum well in between. The aluminum concentration is 40 % for both the p and n region, and zero in the well. The doping concentrations N_{a} and N_{d} are 4 x 10^{17} cm^{3} and V_{a} = 1.4 V. 
From the numeric simulation of a GaAs nqwp structure we find that typically only one electron level is filled with electrons, while several hole levels are filled with holes or 
(4.3.84) 
If all the quantized hole levels are more than 3kT below the hole quasiFermi level one can rewrite the hole density as: 
(4.3.85) 
Since the 2D densities of states are identical for each quantized level. The applied voltage is given by: 
(4.3.86) 
with 
(4.3.87) 
or 
(4.3.88) 
Boulder, December 2004 