Chapter 4: p-n Junctions

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4.3. Electrostatic analysis of a p-n diode

4.3.1. General discussion - Poisson's equation
4.3.2. The full-depletion approximation
4.3.3. Full depletion analysis
4.3.4. Junction capacitance
4.3.5. The linearly graded p-n junction
4.3.6. The abrupt p-i-n junction
4.3.7. Solution to Poissonís equation for an abrupt p-n junction
4.3.8. The hetero p-n junction
4.3.9. Solution to Poissonís equation for an abrupt p-n junction

The electrostatic analysis of a p-n 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 capacitance-voltage characteristics of the diode. The analysis is very similar to that of a metal-semiconductor junction (section 3.3). A key difference is that a p-n diode contains two depletion regions of opposite type.

4.3.1. General discussion - Poisson's equation

Next Subsection

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 n-type region, far away from the p-n interface.

This second-order non-linear 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 full-depletion approximation

Next Subsection

The full-depletion approximation assumes that the depletion region around the metallurgical junction has well-defined edges. It also assumes that the transition between the depleted and the quasi-neutral region is abrupt. We define the quasi-neutral 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 full-depletion 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 p-type region, xp, and the depletion region width in the n-type region, xn. The sum of the two depletion layer widths in each region is the total depletion layer width xd, 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 quasi-neutral 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 xp and xn, from which all other parameters can be obtained.

4.3.3. Full depletion analysis

Next Subsection

Once the full-depletion 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 p-n diode with doping densities, Na and Nd, 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 p-n 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 full-depletion approximation. The total charge per unit area in each region is also indicated on the figure. The charge in the n-type region, Qn, and the charge in the p-type region, Qp, 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 = -xp and x = xn. 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 p-n 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, xp and xn, namely:

(4.3.12)

This equation expresses the fact that the total positive charge in the n-type depletion region, Qn, exactly balances the total negative charge in the p-type depletion region, Qp. We can then combine equation (4.3.4) with expression (4.3.12) for the total depletion-layer width, xd, 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 piece-wise parabolic potential versus position as shown in Figure 4.3.1 (c)

The total potential across the semiconductor must equal the difference between the built-in potential and the applied voltage, which provides a second relation between xp and xn, namely:

(4.3.16)

The depletion layer width is obtained by substituting the expressions for xp and xn, (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, xp and xn are obtained:

(4.3.18)
(4.3.19)
Example 4.2An abrupt silicon (nI = 1010 cm-3) p-n junction consists of a p-type region containing 1016 cm-3 acceptors and an n-type region containing 5 x 1016 cm-3 donors.
  1. Calculate the built-in potential of this p-n junction.
  2. Calculate the total width of the depletion region if the applied voltage Va equals 0, 0.5 and -2.5 V.
  3. Calculate maximum electric field in the depletion region at 0, 0.5 and -2.5 V.
  4. Calculate the potential across the depletion region in the n-type semiconductor at 0, 0.5 and -2.5 V.
Solution

The built-in potential is calculated from:

The depletion layer width is obtained from:

the electric field from

and the potential across the n-type region equals

where

one can also show that:

This yields the following numeric values:

4.3.4. Junction capacitance

Next Subsection

Any variation of the charge within a p-n diode with an applied voltage variation yields a capacitance, which must be added to the circuit model of a p-n diode. This capacitance related to the depletion layer charge in a p-n 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, xd, as a function of voltage, reveals that the expression for the junction capacitance, Cj, 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, xd.

The capacitance of a p-n diode is frequently expressed as a function of the zero bias capacitance, Cj0:

(4.3.23)

Where

(4.3.24)

A capacitance versus voltage measurement can be used to obtain the built-in voltage and the doping density of a one-sided p-n diode. When plotting the inverse of the capacitance squared, one expects a linear dependence as expressed by:

(4.3.25)

The capacitance-voltage characteristic and the corresponding 1/C2 curve are shown in Figure 4.3.2.

Figure 4.3.2 :Capacitance and 1/C2 versus voltage of a p-n diode with Na = 1016 cm-3, Nd = 1017 cm-3 and an area of 10-4 cm2.

The built-in voltage is obtained at the intersection of the 1/C2 curve and the horizontal axis, while the doping density is obtained from the slope of the curve.

(4.3.26)
Example 4.3Consider an abrupt p-n diode with Na = 1018 cm-3 and Nd = 1016 cm-3. Calculate the junction capacitance at zero bias. The diode area equals 10-4 cm2. Repeat the problem while treating the diode as a one-sided 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 one-sided 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 one-sided approximation.

A capacitance-voltage measurement also provides the doping density profile of one-sided p-n 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 xd = esA/Cj. 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 capacitance-voltage data obtained on a 6H-SiC p-n diode. The diode consists of a highly doped p-type region on a lightly doped n-type region on top of a highly doped n-type substrate. The measured capacitance as well as 1/C2is 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 p-n 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/C2 versus voltage of a 6H-SiC p-n 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 n-type layer. No information is obtained at the interface (x = 0) as is typical for doping profiles obtained from C-V 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 p-n junction

Next Subsection

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 xn in the n-type region and xp in the p-type 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 = - xp and x = xn = xp twice resulting in:

(4.3.29)

Where the built-in 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 built-in 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 built-in potential and then solves for the depletion layer width. A possible initial value for the built-in 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 built-in potential and repeats the calculation of the depletion layer width. As one repeats this process, one finds that the values for the built-in 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 zero-bias capacitance or:

(4.3.35)

Where Cj0 is the capacitance at zero bias, which is given by:

(4.3.36)

4.3.6. The abrupt p-i-n junction

Next Subsection

A p-i-n junction is similar to a p-n junction, but contains in addition an intrinsic or un-intentionally doped region with thickness, d, between the n-type and p-type 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 p-i-n junctions.

The analysis is also similar to that of a p-n diode, although the potential across the undoped region, fu, must be included in the analysis. Equation (4.3.16) then becomes:

(4.3.37)
(4.3.38)

while the charge in the n-type region still equals that in the p-type region, so that (4.3.12) still holds:

(4.3.39)

Equations (4.3.37) through (4.3.39) can be solved for xn yielding:

(4.3.40)

From xn and xp, all other parameters of the p-i-n junction can be obtained. The total depletion layer width, xd, 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 = -xn was assumed to be zero.

4.3.6.1. Capacitance of the p-i-n junction

The capacitance of a p-i-n 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 Poissonís equation for an abrupt p-n junction

Next Subsection

Applying Gauss's law one finds that the total charge in the n-type depletion region equals minus the charge in the p-type depletion region:

(4.3.46)

Poisson's equation can be solved separately in the n-type and p-type 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 fn and fp are assumed negative if the semiconductor is depleted. Their relation to the applied voltage is given by:

(4.3.48)

One obtains fn and fp as a function of the applied voltage by solving the transcendental equations.

For the special case of a symmetric doping profile, or Nd = Na, 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 p-n junction

Next Subsection

Heterojunction p-n 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 electro-static analysis of heterojunction p-n diodes.

The heterojunction p-n 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 p-n diode concepts need almost no changing. An excellent detailed treatment can be found in Wolfe et al.

4.3.8.1. Band diagram of a heterojunction p-n diode under Flatband conditions

The flatband energy band diagram of a heterojunction p-n diode is shown in the figure below. As a convention we will assume DEc to be positive if Ec,n > Ec,p and DEv to be positive if Ev,n < Ev,p.

Figure 4.3.5 :Flat-band energy band diagram of a p-n heterojunction

4.3.8.2. Calculation of the contact potential (built-in voltage)

The built-in potential is defined as the difference between the Fermi levels in both the n-type and the p-type 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 built-in 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 built-in voltage independent of the free carrier concentrations:

(4.3.56)

where ni,n and ni,p are the intrinsic carrier concentrations of the n-type and p-type region, respectively. DEc and DEv are positive quantities if the bandgap of the n-type region is smaller than that of the p-type 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 built-in junction of a homojunction if the material parameters in the n-type region equal those in the p-type region. If the effective densities of states are the same, the expression for the heterojunction reduces to:

(4.3.57)

4.3.8.3. Abrupt p-n heterojunction

For the calculation of the charge, field and potential distribution in an abrupt p-n 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 built-in 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 xn and xp for a homojunction can still be used if one replaces Na by Na es,p/es , Nd by Nd es,n/es, xp by xp es/es,p , and xn by xn es/es,n. Adding xn and xp yields the total depletion layer width xd:

(4.3.61)

The capacitance per unit area can be obtained from the series connection of the capacitance of each layer:

(4.3.62)

4.3.8.4. Abrupt P-i-N heterojunction

For a P-i-N heterojunction the above expressions take the following modified form:

(4.3.63)
(4.3.64)
(4.3.65)

Where fu 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 xn, yielding:

(4.3.68)

A solution for xp can be obtained from (4.3.68) by replacing Nd by Na, Na by Nd, es,n by es,p, and es,p by es,n. Once xn and xp are determined all other parameters of the P-i-N 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 = -xn was assumed to be zero.

An example of the charge distribution, electric field, potentials and energy band diagram throughout the P-i-N 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 p-n heterojunction with Va = 0.5 V, x = 0.4 on the left and x = 0 on the right. Nd = Na = 1017cm-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 n-type and p-type 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.

4.3.8.5. A P-M-N junction with interface charges

Real P-i-N 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 Nm = Ndm - Nam and a dielectric constant es,m. A charge Q1 is assumed between the N and M layer, and a charge Q2 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 xn and xp 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 p-i-n heterojunction with Va = 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 Nd = Na = 1017cm-3

4.3.8.6. Quantum well in a p-n junction

Next, we consider a p-n junction with a quantum well located between the n and p region as shown in Figure 4.3.8.

Figure 4.3.8:Flat-band energy band diagram of a p-n 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 p-type and nĖtype 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 DEn,e and DEn,h given by:

(4.3.81)
(4.3.82)

where En,e and En,h are the nth 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 xn equals xp and fn equals fp. Assuming that only one energy level namely the n = 1 level is populated in the quantum well one finds:

(4.3.83)

where Eg 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 p-n 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 Na and Nd are 4 x 1017 cm-3 and Va = 1.4 V.

From the numeric simulation of a GaAs n-qw-p 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 quasi-Fermi level one can rewrite the hole density as:

(4.3.85)

Since the 2-D 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