p-n Junction

4.1. Introduction

P-n junctions consist of two semiconductor regions of opposite type. Such junctions show a pronounced rectifying behavior. They are also called p-n diodes in analogy with vacuum diodes.

The p-n junction is a versatile element, which can be used as a rectifier, as an isolation structure and as a voltage-dependent capacitor. In addition, they can be used as solar cells, photodiodes, light emitting diodes and even laser diodes. They are also an essential part of Metal-Oxide-Silicon Field-Effects-Transistors (MOSFETs) and Bipolar Junction Transistors (BJTs).

4.2. Structure and principle of operation

4.2.1. Structure
4.2.2. Thermal equilibrium
4.2.3. The built-in potential
4.2.4. Forward and reverse bias

A p-n junction consists of two semiconductor regions with opposite doping type as shown in Figure 4.2.1. The region on the left is p-type with an acceptor density Na, while the region on the right is n-type with a donor density Nd. The dopants are assumed to be shallow, so that the electron (hole) density in the n-type (p-type) region is approximately equal to the donor (acceptor) density.

Figure 4.2.1 :

Cross-section of a p-n junction

We will assume, unless stated otherwise, that the doped regions are uniformly doped and that the transition between the two regions is abrupt. We will refer to this structure as an abrupt p-n junction.

Frequently we will deal with p-n junctions in which one side is distinctly higher-doped than the other. We will find that in such a case only the low-doped region needs to be considered, since it primarily determines the device characteristics. We will refer to such a structure as a one-sided abrupt p-n junction.

The junction is biased with a voltage Va as shown in Figure 4.2.1. We will call the junction forward-biased if a positive voltage is applied to the p-doped region and reversed-biased if a negative voltage is applied to the p-doped region. The contact to the p-type region is also called the anode, while the contact to the n-type region is called the cathode, in reference to the anions or positive carriers and cations or negative carriers in each of these regions.

4.2.1. Flatband diagram

The principle of operation will be explained using a gedanken experiment, an experiment, which is in principle possible but not necessarily executable in practice. We imagine that one can bring both semiconductor regions together, aligning both the conduction and valence band energies of each region. This yields the so-called flatband diagram shown in Figure 4.2.2.

Figure 4.2.2 :

Energy band diagram of a p-n junction (a) before and (b) after merging the n-type and p-type regions

Note that this does not automatically align the Fermi energies, EF,n and EF,p. Also, note that this flatband diagram is not an equilibrium diagram since both electrons and holes can lower their energy by crossing the junction. A motion of electrons and holes is therefore expected before thermal equilibrium is obtained. The diagram shown in Figure 4.2.2 (b) is called a flatband diagram. This name refers to the horizontal band edges. It also implies that there is no field and no net charge in the semiconductor.

4.2.2. Thermal equilibrium

To reach thermal equilibrium, electrons/holes close to the metallurgical junction diffuse across the junction into the p-type/n-type region where hardly any electrons/holes are present. This process leaves the ionized donors (acceptors) behind, creating a region around the junction, which is depleted of mobile carriers. We call this region the depletion region, extending from x = -xp to x = xn. The charge due to the ionized donors and acceptors causes an electric field, which in turn causes a drift of carriers in the opposite direction. The diffusion of carriers continues until the drift current balances the diffusion current, thereby reaching thermal equilibrium as indicated by a constant Fermi energy. This situation is shown in Figure 4.2.3:

Figure 4.2.3 :

Energy band diagram of a p-n junction in thermal equilibrium

While in thermal equilibrium no external voltage is applied between the n-type and p-type material, there is an internal potential, fi, which is caused by the workfunction difference between the n-type and p-type semiconductors. This potential equals the built-in potential, which will be further discussed in the next section.

4.2.3. The built-in potential

The built-in potential in a semiconductor equals the potential across the depletion region in thermal equilibrium. Since thermal equilibrium implies that the Fermi energy is constant throughout the p-n diode, the built-in potential equals the difference between the Fermi energies, EFn and EFp, divided by the electronic charge. It also equals the sum of the bulk potentials of each region, fn and fp, since the bulk potential quantifies the distance between the Fermi energy and the intrinsic energy. This yields the following expression for the built-in potential.

(4.2.1)

Example 4.1

An abrupt silicon p-n junction consists of a p-type region containing 2 x 1016 cm-3 acceptors and an n-type region containing also 1016 cm-3 acceptors in addition to 1017 cm-3 donors. Calculate the thermal equilibrium density of electrons and holes in the p-type region as well as both densities in the n-type region. Calculate the built-in potential of the p-n junction Calculate the built-in potential of the p-n junction at 400 K.

Solution

The thermal equilibrium densities are: In the p-type region:p = Na = 2 x 1016 cm-3 n = ni2/p = 1020/2 x 1016 = 5 x 103 cm-3In the n-type regionn = Nd - Na = 9 x 1016 cm-3 p = ni2/n = 1020/(1 x 1016) = 1.11 x 103 cm-3 The built-in potential is obtained from: Similarly, the built-in potential at 400 K equals: where the instrinsic carrier density at 400 K was obtained from example 2.4.b

4.2.4. Forward and reverse bias

We now consider a p-n diode with an applied bias voltage, Va. A forward bias corresponds to applying a positive voltage to the anode (the p-type region) relative to the cathode (the n-type region). A reverse bias corresponds to a negative voltage applied to the cathode. Both bias modes are illustrated with Figure 4.2.4. The applied voltage is proportional to the difference between the Fermi energy in the n-type and p-type quasi-neutral regions.

As a negative voltage is applied, the potential across the semiconductor increases and so does the depletion layer width. As a positive voltage is applied, the potential across the semiconductor decreases and with it the depletion layer width. The total potential across the semiconductor equals the built-in potential minus the applied voltage, or:

(4.2.1)

Figure 4.2.4:

Energy band diagram of a p-n junction under reverse and forward bias

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 Poisson’s equation for an abrupt p-n junction
4.3.8. The hetero p-n junction
4.3.9. Solution to Poisson’s 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

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

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

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 4.2

An 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. Calculate the built-in potential of this p-n junction. Calculate the total width of the depletion region if the applied voltage Va equals 0, 0.5 and -2.5 V. Calculate maximum electric field in the depletion region at 0, 0.5 and -2.5 V. 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

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 4.3

Consider 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

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

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 Poisson’s equation for an abrupt p-n junction

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

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 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 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

4.4. The p-n diode current

4.4.1. General discussion
4.4.2. The ideal diode current
4.4.3. Recombination-Generation current
4.4.4. I-V characteristics of real p-n diodes
4.4.5. The diffusion capacitance
4.4.6. High Injection Effects
4.4.7. Heterojunction Diode Current

4.4.1. General discussion

The current in a p-n diode is due to carrier recombination or generation somewhere within the p-n diode structure. Under forward bias, the diode current is due to recombination. This recombination can occur within the quasi-neutral regions, within the depletion region or at the metal-semiconductor 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 p-n diodes in which recombination/generation occurs in the quasi-neutral region only. This is the case if the quasi-neutral region is much larger than the carrier diffusion length. The "short" diode expression applies to p-n diodes in which recombination/generation occurs at the contacts only. In a short diode, the quasi-neutral region is much smaller than the diffusion length. In addition to the "long" and "short" diode expressions, we also present the general result for p-n 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 - band-to-band recombination and Shockley-Hall-Read recombination - as they lead to different current-voltage characteristics.

4.4.2. The ideal diode current

4.4.2.1. General discussion and overview

When calculating the current in a p-n diode one needs to know the carrier density and the electric field throughout the p-n diode which can then be used to obtain the drift and diffusion current. Unfortunately, this requires the knowledge of the quasi-Fermi energies, which is only known if the currents are known. The straightforward approach is to simply solve the drift-diffusion 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 quasi-Fermi energies in the depletion region equal those in the adjacent n-type and p-type quasi-neutral regions. We will derive an expression for "long" and "short" diodes as well as a general expression, which is to be used if the quasi-neutral region is comparable in size to the diffusion length.

4.4.2.2. Assumptions and boundary conditions

The electric field and potential are obtained by using the full depletion approximation. Assuming that the quasi-Fermi 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 thermal-equilibrium carrier density for zero applied voltage. In addition, an increase of the applied voltage will increase the separation between the two quasi-Fermi energies by the applied voltage multiplied with the electronic charge.

The carrier density at the metal contacts is assumed to equal the thermal-equilibrium carrier density. This assumption implies that excess carriers immediately recombine when reaching either of the two metal-semiconductor 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)

4.4.2.3. General current expression

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 quasi-neutral 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 quasi-neutral region widths, wn' and wp', 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 p-type region, the maximum hole current in the n-type regions and the current due to recombination within the depletion region. The maximum currents in the quasi-neutral 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 Is can be written in the following form:

(4.4.14)

4.4.2.4. The p-n diode with a "long" quasi-neutral region

A diode with a "long" quasi-neutral region has a quasi-neutral region, which is much larger than the minority-carrier diffusion length in that region, or wn' > Lp and wp' > Ln. 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 quasi-neutral 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 n-type region. The current due to recombination in the depletion region is then given by:

(4.4.21)

so that Ir 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 p-n diodes usually satisfy this requirement, while gallium arsenide p-n diodes rarely do because of the short carrier lifetime and diffusion length.

As an example we now consider a silicon p-n diode with Na = 1.5 x 1014 cm-3 and Nd = 1014 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 p-n diode, shown in Figure 4.4.1:

Figure 4.4.1 :

Electron and hole density throughout a forward biased p-n diode.

The majority carrier densities in the quasi-neutral region simply equal the doping density. The minority carrier densities in the quasi-neutral 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 quasi-Fermi energy in the depletion region equals the electron/hole quasi-Fermi energy in the quasi-neutral n-type/p-type region. The corresponding band diagram is shown in Figure 4.4.2:

Figure 4.4.2 :

Energy band diagram of a p-n diode. Shown are the conduction band edge, Ec, and the valence band edge, Ev, the intrinsic energy, Ei, the electron quasi-Fermi energy, Fn, and the hole quasi-Fermi energy, Fp.

The quasi-Fermi energies were obtained by combining (4.4.16) and (4.4.17) with (2.10.5) and (2.10.6). Note that the quasi-Fermi energies vary linearly within the quasi-neutral 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.

4.4.2.5. The p-n diode with a "short" quasi-neutral region

A "short" diode is a diode with quasi-neutral regions, which are much shorter than the minority-carrier diffusion lengths. As the quasi-neutral region is much smaller than the diffusion length one finds that recombination in the quasi-neutral region is negligible so that the diffusion equations are reduced to:

(4.4.23)

The resulting carrier density varies linearly throughout the quasi-neutral 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 n-type and p-type quasi-neutral region.

The current in a "short" diode is again obtained by adding the maximum diffusion currents in each of the quasi-neutral regions and ignoring the current due to recombination in the depletion region, yielding:

(4.4.27)

where the saturation current, Is 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 quasi-neutral 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 one-sided n+-p diode.The p-type semiconductor has a width, wp, 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 p-type region is ideal so that the excess density is zero at x = wp'. 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/wp' = 1) = 0 . The ratio of the diffusion length to the width of the quasi-neutral 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 quasi-neutral region. For the case where the diffusion length is much smaller than the width (Ln << wp'), the electron density decays exponentially and the "long" diode expression can be used. If the diffusion length is much longer than the width (Ln >> wp'), 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 quasi-neutral 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 Ln > 2 wp' and when using the long diode expression with Ln < wp'/2. Note that the best approximation is not necessarily the same in each region of the same p-n diode.

Example 4.4

An abrupt silicon p-n junction (Na = 1016 cm-3 and Nd = 4 x 1016 cm-3) is biased with Va = 0.6 V. Calculate the ideal diode current assuming that the n-type region is much smaller than the diffusion length with wn' = 1 mm and assuming a "long" p-type region. Use mn = 1000 cm2/V-s and mp = 300 cm2/V-s. The minority carrier lifetime is 10 ms and the diode area is 100 mm by 100 mm.

Solution

The current is calculated from: with Dn = mn Vt = 1000 x 0.0258 = 25.8 cm2/V-s Dp = mp Vt = 300 x 0.0258 = 7.75 cm2/V-s np0 = ni2/Na = 1020/1016 = 104 cm-3 pn0 = ni2/Nd = 1020/4 x 1016 = 2.5 x 103 cm-3 yielding I = 40.7 mA Note that the hole diffusion current occurs in the "short" n-type region and therefore depends on the quasi-neutral width in that region. The electron diffusion current occurs in the "long" p-type region and therefore depends on the electron diffusion length in that region.

4.4.3. Recombination-Generation current

We now calculate the recombination-generation current in the depletion region of a p-n junction. We distinguish between two different possible recombination mechanisms: band-to-band recombination and Shockley-Hall-Read recombination.

4.4.3.1. Band-to-band Recombination-Generation current

The recombination/generation current due to band-to-band recombination/generation is obtained by integrating the net recombination rate, Ub-b, 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 quasi-Fermi 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 band-to-band 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.

4.4.3.2. Shockley-Hall-Read Recombination-Generation current

The current due to trap-assisted recombination in the depletion region is also obtained by integrating the trap-assisted 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 quasi-Fermi 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 trap-assisted 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, xd, as it is always larger than the effective width.

4.4.4. I-V characteristics of real p-n diodes

The forward biased I-V characteristics of real p-n 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 I-V characteristics of a silicon p+-n diode with Nd = 4 x 1014 cm-3, tp = 10 ms, and mp = 450 cm2/V-s. The I-V characteristics are plotted on a semi-logarithmic 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 semi-logarithmic 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 trap-assisted 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 p-n 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 Va = 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, Va*, relative to the internal voltage, Va, considered so far.

(4.4.42)

Where I is the diode current and Rs is the series resistance.

These four regions can be observed in most p-n diodes although the high-injection region rarely occurs, as the series resistance tends to limit the current first.

Figure 4.4.5:

Current-Voltage characteristics of a silicon diode under forward bias.

4.4.5. The diffusion capacitance

As a p-n diode is forward biased, the minority carrier distribution in the quasi-neutral region increases dramatically. In addition, to preserve quasi-neutrality, 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 minority-carrier charge is obtained by integrating the charge density over the quasi-neutral 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, pn(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 minority-carrier charge, DQp, in the quasi-neutral region, is proportional to the area defined by the solid and dotted lines.

Using equation (4.4.18), the excess charge, DQp, becomes:

(4.4.45)

where Is,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 quasi-neutral region, the current equals the excess charge divided by the average time needed to recombine with the majority carriers, i.e. the carrier lifetime, tp. 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 tr,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 quasi-neutral 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 quasi-neutral region, i.e. the transit time, tr,p.

The total diffusion capacitance is obtained by adding the diffusion capacitance of the n-type quasi-neutral region to that of the p-type quasi-neutral 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

Calculate the diffusion capacitance of the diode described in Example 4.4 at zero bias. Use mn= 1000 cm2/V-s, mp = 300 cm2/V-s, wp' = 1 mm and wn' = 1 mm. The minority carrier lifetime equals 0.1 ms. For the same diode, find the voltage for which the junction capacitance equals the diffusion capacitance.

Solution

The diffusion capacitance at zero volts equals using and where the "short" diode expression was used for the capacitance associated with the excess charge due to electrons in the p-type region. The "long" diode expression was used for the capacitance associated with the excess charge due to holes in the n-type region.The diffusion constants and diffusion lengths equal Dn = mn x Vt = 25.8 cm2/s Dp = mp x Vt = 7.75 cm2/s And the electron transit time in the p-type region equals The voltage at which the junction capacitance equals the diffusion capacitance is obtained by solving yielding Va = 0.442 V

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 = xn and those with subscript p at x = -xp. Solving the resulting quadratic equation yields:

(4.4.52)

(4.4.53)

where the second terms are approximations for large Va. From these expressions one can calculate the minority carrier diffusion current assuming a "long" diode in both quasi-neutral regions. We also ignore carrier recombination in the depletion region.

(4.4.54)

This means that high injection in a p-n diode will reduce the slope on the current-voltage characteristic on a semi-logarithmic scale to 119mV/decade.

High injection also causes a voltage drop across the quasi-neutral region. This voltage can be calculated from the carrier densities. Let's assume that high injection only occurs in the (lower doped) p-type region. The hole density at the edge of the depletion region (x = xp) equals:

(4.4.55)

where V1 is the voltage drop across the p-type quasi-neutral region. This equation can then be solved for V1 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 Va, 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 n-type 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 non-degenerate and that the difference between the electron and hole quasi Fermi energies in electron volt equals the applied voltage in volt. Quasi-neutrality implies that the excess densities are the same, which yields a quadratic equation for the minority carrier density pn:

(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 p-type 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 "quasi-neutral" region. This voltage in the n-type region is given by:

(4.4.66)

and similarly for the p-type region:

(4.4.67)

The potentials across the "quasi-neutral" 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 = fi – Va + Vn + Vp does not become zero.

As an example we now consider an abrupt one-sided p-n diode. The current is shown as function of the voltage in Figure 4.4.7. It is calculated for a rather low n-type doping of 1013 cm-3 to further highlight the high injection effects.

Figure 4.4.7:

Current-Voltage 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 n-type 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 quasi-neutral 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 p-n 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 quasi-neutral regions (including recombination at the contacts) whereas recombination and generation in the depletion region yield enhanced leakage or photocurrents.

4.4.7.1. Ideal diode equation

For the derivation of the ideal diode equation we will again assume that the quasi-Fermi 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 ni,n and ni,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 LpLn are the hole respectively the electron diffusion lengths in the n-type and p-type 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 Jideal:

(4.4.74)

(4.4.75)

This expression is valid only for a p-n diode with infinitely long quasi-neutral regions. For diodes with a quasi-neutral 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 quasi-neutral 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 n-type and p-type material can cause a significant difference between the electron and hole current and that independent of the doping concentrations.

4.4.7.2. Recombination/generation in the depletion region

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: band-to-band recombination and Shockley-Hall-Read recombination.

4.4.7.2.1. Band-to-band recombination

The recombination/generation rate is due to band-to-band 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 cm3s-1. For (or under forward bias conditions) recombination dominates, whereas for (under reverse bias conditions) thermal generation of electron-hole pairs occurs. Assuming constant quasi-Fermi levels in the depletion region, this rate can be expressed as a function of the applied voltage by using the "modified" mass-action 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 p-n heterojunction consisting of two uniformly doped regions with different bandgap, the integral becomes:

(4.4.80)

4.4.7.2.2. Schockley-Hall-Read recombination

Provided bias conditions are "close" to thermal equilibrium the recombination rate due to a density Nt of traps with energy Et and a recombination/generation cross-section s is given by

(4.4.81)

where nI is the intrinsic carrier concentration, vth is the thermal velocity of the carriers and EI is the intrinsic energy level. For EI = Et and t0 = 1/Ntsvth 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 USHR,max is larger than or equal to USHR anywhere within the depletion layer one finds that x' has to be smaller than xd = xn + xp. (Note that for a p-i-N or p-qw-N 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'.

4.4.7.3. Recombination/generation in a quantum well

4.4.7.3.1. Band-to-band recombination

Recognizing that band-to-band 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 En,e and En,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)

4.4.7.3.2. SHR recombination

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.

4.4.7.4. The graded p-n diode

4.4.7.4.1. General discussion of a graded region

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 non-linear 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 single-quantum-well graded-index separate-confinement heterostructure (GRINSCH) as used in edge-emitting laser diodes.

Figure 4.4.10:

Flat band diagram of a graded AlGaAs p-n 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 p-n diode shown above under forward bias. Va = 1.5 V, Na = 4 x 1017 cm-3, Nd = 4 x 1017 cm-3. Shown are the conduction and valence band edges (solid lines) as well as the quasi-Fermi energies (dotted lines).

The first difference between the two diagrams is that the conduction band edge in the n-type graded region as well as the valence band edge in the p-type graded region are almost constant. This assumption is correct if the majority carrier quasi-Fermi 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 quasi-Fermi 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 Ec0(x) and Ev0(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 p-n 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 quasi-Fermi level:

(4.4.100)

(4.4.101)

where it was assumed that the electron density is non-degenerate. 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 Ec0(x), yielding:

(4.4.102)

This expression will be used in the next section to calculate the ideal diode current in a graded p-n 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, Jp.

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 quasi-neutral region. The derivation below can be applied to a p-n 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 ion-implanted 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 quasi-Fermi 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 quasi-neutral 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 quasi-neutral region with the correct boundary condition and including the minority carrier bandedge gradient. For electrons in a p-type quasi-neutral 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 p-type 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 np0(xp) at the edge of the depletion region (at x = xp) is strongly voltage dependent since it is exponentially dependent on the actual bandgap at x = xp.

The electron current at x = xp 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 non-graded junction:

(4.4.107)

while for strongly graded diodes (aLn >> 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.

4.5. Reverse bias breakdown

4.5.1. General breakdown characteristics
4.5.2. Edge effects
4.5.3. Avalanche breakdown
4.5.4. Zener breakdown

4.5.1. General breakdown characteristics

The maximum reverse bias voltage that can be applied to a p-n diode is limited by breakdown. Breakdown is characterized by the rapid increase of the current under reverse bias. The corresponding applied voltage is referred to as the breakdown voltage.

The breakdown voltage is a key parameter of power devices. The breakdown of logic devices is equally important as one typically reduces the device dimensions without reducing the applied voltages, thereby increasing the internal electric field.

Two mechanisms can cause breakdown, namely avalanche multiplication and quantum mechanical tunneling of carriers through the bandgap. Neither of the two breakdown mechanisms is destructive. However heating caused by the large breakdown current and high breakdown voltage causes the diode to be destroyed unless sufficient heat sinking is provided.

Breakdown in silicon at room temperature can be predicted using the following empirical expression for the electric field at breakdown.

(4.5.1)

Assuming a one-sided abrupt p-n diode, the corresponding breakdown voltage can then be calculated, yielding:

(4.5.2)

The resulting breakdown voltage is inversely proportional to the doping density if one ignores the weak doping dependence of the electric field at breakdown. The corresponding depletion layer width equals:

(4.5.3)

4.5.2. Edge effects

Few p-n diodes are truly planar and typically have higher electric fields at the edges. Since the diodes will break down in the regions where the breakdown field is reached first, one has to take into account the radius of curvature of the metallurgical junction at the edges. Most doping processes including diffusion and ion implantation yield a radius of curvature on the order of the junction depth, xj. The p-n diode interface can then be approximated as having a cylindrical shape along a straight edge and a spherical at a corner of a rectangular pattern. Both structures can be solved analytically as a function of the doping density, N, and the radius of curvature, xj.

The resulting breakdown voltages and depletion layer widths are plotted below as a function of the doping density of an abrupt one-sided junction.

Figure 4.5.1 :

Breakdown voltage and depletion layer width at breakdown versus doping density of an abrupt one-sided p-n diode. Shown are the voltage and width for a planar (top curves), cylindrical (middle curves) and spherical (bottom curves) junction with 1 mm radius of curvature.

4.5.3. Avalanche breakdown

Avalanche breakdown is caused by impact ionization of electron-hole pairs. This process was described previously in section 2.8.When applying a high electric field, carriers gain kinetic energy and generate additional electron-hole pairs through impact ionization. The ionization rate is quantified by the ionization constants of electrons and holes, an and ap. These ionization constants are defined as the change of the carrier density with position divided by the carrier density or:

(4.5.4)

The ionization causes a generation of additional electrons and holes. Assuming that the ionization coefficients of electrons and holes are the same, the multiplication factor M, can be calculated from:

(4.5.5)

The integral is taken between x1 and x2, the region within the depletion layer where the electric field is assumed constant and large enough to cause impact ionization. Outside this range, the electric field is assumed to be too low to cause impact ionization. The equation for the multiplication factor reaches infinity if the integral equals one. This condition can be interpreted as follows: For each electron coming to the high field at point x1 one additional electron-hole pair is generated arriving at point x2. This hole drifts in the opposite direction and generates an additional electron-hole pair at the starting point x1. One initial electron therefore yields an infinite number of electrons arriving at x2, hence an infinite multiplication factor.

The multiplication factor is commonly expressed as a function of the applied voltage and the breakdown voltage using the following empirical relation:

(4.5.6)

4.5.4. Zener breakdown

Quantum mechanical tunneling of carriers through the bandgap is the dominant breakdown mechanism for highly doped p-n junctions. The analysis is identical to that of tunneling in a metal-semiconductor junction (section 3.4.4.3) where the barrier height is replaced by the energy bandgap of the material.

The tunneling probability equals:

(4.5.7)

where the electric field equals = Eg/(qL).

The tunneling current is obtained from the product of the carrier charge, velocity and carrier density. The velocity equals the Richardson velocity, the velocity with which on average the carriers approach the barrier while the carrier density equals the density of available electrons multiplied with the tunneling probability, yielding: 4.6. Optoelectronic devices 4.6.1. Light absorption and emission 4.6.2. Photodiodes 4.6.3. Solar cells 4.6.4. LEDs 4.6.5. Laser diodes P-n junctions are an integral part of several optoelectronic devices. These include photodiodes, solar cells light emitting diodes (LEDs) and semiconductor lasers. In this section, we discuss the principle of operation of these devices and derive equations for key parameters. 4.6.1. Light absorption and emission A large number of optoelectronic devices consist of a p-type and n-type region, just like a regular p-n diode. The key difference is that there is an additional interaction between the electrons and holes in the semiconductor and light. This interaction is not restricted to optoelectronic devices. Regular diodes are also known to be light sensitive and in some cases also emit light. The key difference is that optoelectronic devices such as photodiodes, solar cells, LEDs and laser diodes are specifically designed to optimize the light absorption and emission, resulting in a high conversion efficiency. Light absorption and emission in a semiconductor is known to be heavily dependent on the detailed band structure of the semiconductor. Direct bandgap semiconductors, i.e semiconductors for which the minimum of the conduction band occurs at the same wavevector, k, as the maximum of the valence band, have a stronger absorption of light as characterized by a larger absorption coefficient. They are also the favored semiconductors when fabricating light emitting devices. Indirect bandgap semiconductors, i.e. semiconductors for which the minimum of the conduction band does not occur at the same wavevector as the maximum of the valence band, are known to have a smaller absorption coefficient and are rarely used in light emitting devices. Figure 4.6.1: E-k diagram illustrating a) Photon absorption in a direct bandgap semiconductor b) Photon absorption in an indirect bandgap semiconductor assisted by phonon absorption and c) Photon absorption in an indirect bandgap semiconductor assisted by phonon emission. This striking difference is further illustrated with Figure 4.6.1 and can be explained based on the energy and momentum conservation required in the electron-photon interaction. The direct bandgap semiconductor, which has a vertically aligned conduction and valence band, is shown in Figure 4.6.1(a). Absorption of a photon is obtained if an empty state in the conduction band is available for which the energy and momentum equals that of an electron in the valence band plus that of the incident photon. Photons have little momentum relative of their energy since they travel at the speed of light. The electron therefore makes an almost vertical transition on the E-k diagram. For an indirect bandgap semiconductor, the conduction band is not vertically aligned to the valence band as shown in Figure 4.6.1(b). Therefore a simply interaction of an incident photon with an electron in the valence band will not provide the correct energy and momentum corresponding to that of an empty state in the conduction band. As a result absorption of light requires the help of another particle, namely a photon. Since a phonon, i.e a particle associated with lattice vibrations, has a relatively low velocity close to the speed of sound in the material, it has a small energy and large momentum compared to that of a photon. Conservation of both energy and momentum can therefore be obtained in the absorption process if a phonon is created or an existing phonon participates. The phonon assisted absorption processes are illustrated with Figure 4.6.1(b) and (c). Figure 4.6.1(b) illustrates the absorption of a photon aided by the simultaneous absorption of a phonon, while Figure 4.6.1(c) depicts the absorption of a photon, which results in the emission of a phonon. The minimum photon energy that can be absorbed is slightly below the bandgap energy in the case of phonon absorption and has to be slightly above the bandgap energy in the case of phonon emission. Since the absorption process in an indirect bandgap semiconductor involves a phonon in addition to the electron and photon, the probability of having an interaction take place involving all three particles will be lower than a simple electron-photon interaction in a direct bandgap semiconductor. As a result one finds that absorption is much stronger in a direct bandgap material. Similarly, in the case of light emission, a direct bandgap material is also more likely to emit a photon than an indirect bandgap material. While indirect bandgap materials are occasionally used for some LEDs, they result in a low conversion efficiency. Direct bandgap materials are used exclusively for semiconductor laser diodes. 4.6.2. Photodiodes Photodiodes and crystalline solar cells are essentially the same as the p-n diodes, which have been described in this chapter. However, the diode is exposed to light, which yields a photocurrent in addition to the diode current so that the total diode current is given by: (4.6.1) where the additional photocurrent, Iph,is due to photogeneration of electrons and holes shown in Figure 4.6.2. These electrons and holes are pulled into the region where they are majority carriers by the electric field in the depletion region. Figure 4.6.2 : Motion of photo-generated carriers in a p-n photodiode. The photo-generated carriers cause a photocurrent, which opposes the diode current under forward bias. Therefore, the diode can be used as a photodetector - using a reverse or even zero bias voltage - as the measured photocurrent is proportional to the incident light intensity. The diode can also be used as a solar cell - using a forward bias – to generate electrical power. The primary characteristics of a photodiode are the responsivity, the dark current and the bandwidth. The responsivity is the photocurrent divided by the incident optical power. The maximum photocurrent in a photodiode equals (4.6.2) Where Pin is the incident optical power. This maximum photocurrent occurs when each incoming photon creates one electron-hole pair, which contributes to the photocurrent. The maximum photocurrent in the presence of a reflection, R at the surface of the photodiode and an absorption over a thickness d, in a material with an absorption coefficient, a, is given by: (4.6.3) This photocurrent is obtained by integrating the generation rate (2.8.12) over the absorption region with thickness, d. The photocurrent is further reduced if photo-generated electron-hole pairs recombine within the photodiode instead of being swept into the regions where they are majority carriers. The dark current is the current through the diode in the absence of light. This current is due to the ideal diode current, the generation/recombination of carriers in the depletion region and any surface leakage, which occurs in the diode. The dark current obviously limits the minimum power detected by the photodiode, since a photocurrent much smaller than the dark current would be hard to measure. However, the true limitation is the shot noise generated by the current through the diode. The shot noise as quantified by the average of the square of the noise current is given by: (4.6.4) Where I is the diode current and Df is the bandwidth of the detector. The bandwidth of the diode is affected by the transit time of the photo-generated carriers through the diode and by the capacitance of the diode. The carrier transit time yields the intrinsic bandwidth of the diode while the capacitance together with the impedance of the amplifier or the transmission line connected to the diode yields a the parasitic RC delay. 4.6.3. Solar cells Solar cells are typically illuminated with sunlight and are intended to convert the solar energy into electrical energy. The solar energy is in the form of electromagnetic radiation, more specifically "black-body" radiation as described in section 1.2.3. The sun’s spectrum is consistent with that of a black body at a temperature of 5800 K. The radiation spectrum has a peak at 0.8 eV. A significant part of the spectrum is in the visible range of the spectrum (400 - 700 nm). The power density is approximately 100 mW/cm2. Only part of the solar spectrum actually makes it to the earth's surface. Scattering and absorption in the earth's atmosphere, and the incident angle affect the incident power density. Therefore, the available power density depends on the time of the day, the season and the latitude of a specific location. Of the solar light, which does reach a solar cell, only photons with energy larger than the energy bandgap of the semiconductor generate electron-hole pairs. In addition, one finds that the voltage across the solar cell at the point where it delivers its maximum power is less than the bandgap energy in electron volt. The overall power-conversion efficiency of single-crystalline solar cells ranges from 10 to 30 % yielding 10 to 30 mW/cm2. The calculation of the maximum power of a solar cell is illustrated by Figure 4.6.3 and Figure 4.6.4. The sign convention of the current and voltage is shown as well. It considers a current coming out of the cell to be positive as it leads to electrical power generation. The power generated depends on the solar cell itself and the load connected to it. As an example, a resistive load is shown in the diagram below. Figure 4.6.3 : Circuit diagram and sign convention of a p-n diode solar cell connected to a resistive load. The current and the power as function of the forward bias voltage across the diode are shown in Figure 4.6.4 for a photocurrent of 1 mA: Figure 4.6.4: Current-Voltage (I-V) and Power-Voltage (P-V) characteristics of a p-n diode solar cell with Iph = 1 mA and Is = 10-10 A. The crosshatched area indicates the power generated by the solar cell. The markers indicate the voltage and current, Vm and Im, for which the maximum power, Pm is generated. We identify the open-circuit voltage, Voc, as the voltage across the illuminated cell at zero current. The short-circuit current, Isc, is the current through the illuminated cell if the voltage across the cell is zero. The short-circuit current is close to the photocurrent while the open-circuit voltage is close to the turn-on voltage of the diode as measured on a current scale similar to that of the photocurrent. The power equals the product of the diode voltage and current and at first increases linearly with the diode voltage but then rapidly goes to zero as the voltage approaches the turn-on voltage of the diode. The maximum power is obtained at a voltage labeled as Vm with Im being the current at that voltage. The fill factor of the solar cell is defined as the ratio of the maximum power of the cell to the product of the open-circuit voltage, Voc, and the short-circuit current, Isc, or: (4.6.5) Example 4.6 A 1 cm2 silicon solar cell has a saturation current of 10-12 A and is illuminated with sunlight yielding a short-circuit photocurrent of 25 mA. Calculate the solar cell efficiency and fill factor. Solution The maximum power is generated for: where the voltage, Vm, is the voltage corresponding to the maximum power point. This voltage is obtained by solving the following transcendental equation: Using iteration and a starting value of 0.5 V one obtains the following successive values for Vm: Vm = 0.5, 0.542, 0.540 V and the efficiency equals: The current, Im, corresponding to the voltage, Vm, was calculated using equation (4.6.1) and the power of the sun was assumed 100 mW/cm2. The fill factor equals: where the open circuit voltage is calculated using equation (4.6.1) and I = 0. The short circuit current equals the photocurrent. 4.6.4. LEDs Light emitting diodes are p-n diodes in which the recombination of electrons and holes yields a photon. This radiative recombination process occurs primarily in direct bandgap semiconductors where the lowest conduction band minimum and the highest valence band maximum occur at k = 0, where k is the wavenumber. Examples of direct bandgap semiconductors are GaAs, InP, and GaN while most group IV semiconductors including Si, Ge and SiC are indirect bandgap semiconductors. The radiative recombination process is in competition with non-radiative recombination processes such as trap-assisted recombination. Radiative recombination dominates at high minority-carrier densities. Using a quantum well, a thin region with a lower bandgap, positioned at the metallurgical junction, one can obtain high carrier densities at low current densities. These quantum well LEDs have high internal quantum efficiency as almost every electron injected in the quantum well recombines with a hole and yields a photon. The external quantum efficiency of planar LEDs is much lower than unity due to total internal reflection. As the photons are generated in the semiconductor, which has a high refractive index, only photons traveling normal to the semiconductor-air interface can exit the semiconductor. For GaAs with a refractive index of 3.5, the angle for total internal reflection equals 17o so that only a few percent of the generated photons can escape the semiconductor. This effect can be avoided by having a spherical semiconductor shape, which ensures that most photons travel normal to the interface. The external quantum efficiency can thereby be increased to values larger than 50%. 4.6.5. Laser diodes Laser diodes are very similar to LEDs since they also consist of a p-n diode with an active region where electrons and holes recombine resulting in light emission. However, a laser diode also contains an optical cavity where stimulated emission takes place. The laser cavity consists of a waveguide terminated on each end by a mirror. As an example, the structure of an edge-emitting laser diode is shown in Figure 4.6.5. Photons, which are emitted into the waveguide, can travel back and forth in this waveguide provided they are reflected at the mirrors. Figure 4.6.5: Structure of an edge-emitting laser diode. The light in the waveguide is amplified by stimulated emission. Stimulated emission is a process where a photon triggers the radiative recombination of an electron and hole thereby creating an additional photon with the same energy and phase as the incident photon. This process is illustrated with Figure 4.6.6. This "cloning" of photons results in a coherent beam. Figure 4.6.6: Stimulated emission of a photon. The stimulated emission process yields an increase in photons as they travel along the waveguide. Combined with the waveguide losses, stimulated emission yields a net gain per unit length, g. The number of photons can therefore be maintained if the roundtrip amplification in a cavity of length, L, including the partial reflection at the mirrors with reflectivity R1 and R2 equals unity. This yields the following lasing condition: (4.6.6) If the roundtrip amplification is less than one, then the number of photons steadily decreases. If the roundtrip amplification is larger than one, the number of photons increases as the photons travel back and forth in the cavity. The gain required for lasing therefore equals: (4.6.7) Initially, the gain is negative if no current is applied to the laser diode as absorption dominates in the waveguide. As the laser current is increased, the absorption first decreases and the gain increases. The current for which the gain satisfies the lasing condition is the threshold current of the laser, Ith. Below the threshold current very little light is emitted by the laser structure. For an applied current larger than the threshold current, the output power, Pout, increases linearly with the applied current, as each additional incoming electron-hole pair is converted into an additional photon. The output power therefore equals: (4.6.8) where hn is the energy per photon. The factor, h, indicates that only a fraction of the generated photons contribute to the output power of the laser as photons are partially lost through the other mirror and throughout the waveguide. Figure 4.6.7: Output power from a laser diode versus the applied current.

(4.5.8)

The tunneling current therefore depends exponentially on the bandgap energy to the 3/2 power.

4.7. Photodiodes

4.7.1. P-i-N photodiodes
4.7.2. Photoconductors
4.7.3. Metal-Semiconductor-Metal (MSM) Photodetectors

4.7.1. P-i-N photodiodes

4.7.1.1 Responsivity of a P-i-N photodiode
4.7.1.2 Noise in a photodiode
4.7.1.3 Switching of a P-i-n photodiode

P-i-N photodiodes are commonly used in a variety of applications. A typical P-i-N photodiode is shown in Figure 4.7.1. It consists of a highly-doped transparent p-type contact layer on top of an undoped absorbing layer and an n-type highly doped contact layer on the bottom. Discrete photodiodes are fabricated on a conductive substrate as shown in the figure, which facilitates the formation of the n-type contact and reduces the number of process steps. The top contact is typically a metal ring contact, which has a low contact resistance and still allows the light to be absorbed in the semiconductor. An alternative approach uses a transparent conductor such as Indium Tin Oxide (ITO). The active device area is formed by mesa etching or by proton implantation of the adjacent area, which makes it isolating. A dielectric layer is added around the active area to reduce leakage currents and to ensure a low parasitic capacitance of the contact pad.

Figure 4.7.1 :

Top view and vertical structure through section A-A' of a P-i-N heterostructure photodiode.

Grading of the material composition between the transparent contact layer and the absorbing layer is commonly used to reduce the n-n+ or p-p+ barrier formed at the interface. The above structure evolved mainly from one basic requirement: light should be absorbed in the depletion region of the diode to ensure that the electrons and holes are separated in the electric field and contribute to the photocurrent, while the transit time must be minimal. This implies that a depletion region larger than the absorption length must exist in the detector. This is easily assured by making the absorbing layer undoped. Only a very small voltage is required to deplete the undoped region. If a minimum electric field is required throughout the absorbing layer, to ensure a short transit time, it is also the undoped structure, which satisfies this condition with a minimal voltage across the region, because the electric field is constant. An added advantage is that the recombination/generation time constant is longest for undoped material, which provides a minimal thermal generation current. It also implies that the top contact layer should be transparent to the incoming light. In silicon photodiodes one uses a thin highly doped contact layer to minimize the absorption. By using a contact layer with a wider band gap (also called the window layer) absorption in the contact layer can be eliminated (except for a small fraction due to free carrier absorption) which improves the responsivity. Electron-hole pairs, which are absorbed in the quasi-neutral regions, can still contribute to the photocurrent provided they are generated within one diffusion length of the depletion region.

However, the collection of carriers due to diffusion is relatively inefficient and leads to long tails in the transient response. It therefore should be avoided. Because of the large difference in refractive index between air and most semiconductors, there is a substantial reflection at the surface. The reflection at normal incidence between two materials with refractive index n1 and n2 is given by:

0

For instance, the reflection between air and GaAs (n = 3.5) is 31 %. By coating the semiconductor surface with a dielectric material (anti-reflection coating) of appropriate thickness this reflection can largely be eliminated.

The reflectivity for an arbitrary incident angle is:

(4.7.2)

(4.7.3)

with

where qi is the incident angle, and qt the transmitted angle. RTE is the reflectivity if the electric field is parallel to the surface while RTM is the reflectivity if the magnetic field is parallel to the surface. The reflectivity as a function of qi, for an air-GaAs interface is shown in Figure 4.7.2:

Figure 4.7.2 :

Reflectivity versus incident angle for a transverse electric, RTE, and transverse magnetic, RTM, incident field.

4.7.1.1. Responsivity of a P-i-N photodiode

4.7.1.1.1 Generation of electron hole pairs
4.7.1.1.2 Photocurrent due to absorption in the depletion region
4.7.1.1.3 Photocurrent due to absorption in the quasi-neutral region
4.7.1.1.4 Absorption in the p-contact region
4.7.1.1.5 Total responsivity:
4.7.1.1.6 Dark current of the Photodiode:

4.7.1.1.1. Generation of electron hole pairs

The generation of electron-hole pairs in a semiconductor is directly related to the absorption of light since every absorbed photon generates one electron-hole pair. The optical generation rate gop is given by:

(4.7.4)

where A is the illuminated area of the photodiode, Popt is the incident optical power, a is the absorption coefficient and hn is the photon energy. Note that the optical power is position dependent and obtained by solving:

(4.7.5)

The resulting generation rate must be added to the continuity equation and solved throughout the photodiode, which results in the photocurrent.

4.7.1.1.2. Photocurrent due to absorption in the depletion region

Assuming that all the generated electron-hole pairs contribute to the photocurrent, the photocurrent is simply the integral of the generation rate over the depletion region:

(4.7.6)

where d is the thickness of the undoped region. The minus sign is due to the sign convention indicated on Figure 4.7.1. For a P-i-N diode with heavily doped n-type and p-type regions and a transparent top contact layer, this integral reduces to:

(4.7.7)

where Pin is the incident optical power and R is the reflection at the surface.

4.7.1.1.3. Photocurrent due to absorption in the quasi-neutral region

To find the photocurrent due to absorption in the quasi-neutral region, we first have to solve the diffusion equation in the presence of light. For holes in the n-type contact layer this means solving the continuity equation:

(4.7.8)

Where the electron-hole pair generation gop depends on position. For an the n-type contact layer with the same energy bandgap as the absorption layer, the optical generation rate equals:

(4.7.9)

and the photocurrent due to holes originating in the n-type contact layer equals:

(4.7.10)

The first term is due to light whereas the second term is the due to thermal generation of electron-hole pairs. This derivation assumes that the thickness of the n-type contact layer is much larger than the diffusion length.

4.7.1.1.4. Absorption in the p-contact region

Even though the contact layer was designed so that no light absorbs in this layer, it will become absorbing at shorter wavelengths. Consider a worst-case scenario where all the electron-hole pairs, which are generated in the p-type contact layer, recombine without contributing to the photocurrent. The optical power incident on the undoped region is reduced by exp(-a*wp’) where wp’ is the width of the quasi-neutral p region and a* is the absorption coefficient in that region.

4.7.1.1.5. Total responsivity:

Combining all the above effects the total responsivity of the detector - ignoring the dark current - equals:

(4.7.11)

Note that a*, a and hn are wavelength dependent. For a direct bandgap semiconductor these are calculated from:

(4.7.12)

The quantum efficiency then equals:

(4.7.13)

4.7.1.1.6. Dark current of the Photodiode:

The dark current of a p-n diode including the ideal diode current, as well as recombination/generation in the depletion region is given by:

(4.7.14)

Under reverse bias conditions this expression reduces to:

(4.7.15)

The ideal diode current due to recombination of electrons has been ignored since np0 = ni,p2/Na is much smaller than pn0 because the p-layer has a larger band gap. In the undoped region, one expects the trap-assisted generation to be much larger than bimolecular generation. Which further reduces the current to:

(4.7.16)

The trap-assisted recombination tends to dominate for most practical diodes.

4.7.1.2. Noise in a photodiode

4.7.1.2.1 Shot noise sensitivity
4.7.1.2.2 Equivalence of shot noise and Johnson noise
4.7.1.2.3 Examples.
4.7.1.2.4 Noise equivalent Power and ac noise analysis

4.7.1.2.1. Shot noise sensitivity

Noise in a p-i-n photodiode is primarily due to shot noise; the random nature of the generation of carriers in the photodiode yields also a random current fluctuation. The square of the current fluctuations equals:

(4.7.17)

where Ij are the currents due to different recombination/generation mechanisms and Df is the frequency range. Including the ideal diode current, Shockley-Hall-Read and band-to-band recombination as well as generation due to light one obtains:

(4.7.18)

The minimum detectable input power depends on the actual signal and the required signal to noise ration. As a first approximation, we now calculate the minimum detectable power as the power, which generates a current equal to the RMS noise current. A more detailed model for sinusoidal modulated signals is described in section 4.7.1.2.4.

(4.7.19)

The minimal noise current is obtained at Va = 0 for which the noise current and minimal power equal:

(4.7.20)

(4.7.21)

4.7.1.2.2. Equivalence of shot noise and Johnson noise

The following derivation illustrates that shot noise and Johnson noise are not two independent noise mechanisms. In fact, we will show that both are the same for the special case of an ideal p-n diode under zero bias. At zero bias the photodetector can also be modeled as a resistor. Therefore the expression for Johnson noise should apply:

(4.7.22)

The resistance of a photodiode with is

(4.7.23)

or for zero bias, the Johnson noise current is given by:

(4.7.24)

whereas the shot noise current at Va = 0 is given by:

(4.7.25)

where we added the noise due to the diffusion current to the noise due to the (constant) drift current, since both noise mechanisms do not cancel each other. Equations (4.7.24) and (4.7.25) are identical, thereby proving the equivalence between shot noise and Johnson noise in a photodiode at zero voltage. Note that this relation does not apply if the current is dominated by trap-assisted recombination/generation in the depletion region because of the non-equilibrium nature of the recombination/generation process.

4.7.1.2.3. Examples.

For a diode current of 1mA, a bandwidth Df of 1 GHz and a responsivity, R, of 0.2A/W, the noise current equals 18 nA, corresponding to a minimum detectable power of 89 nW or -40.5 dBm. Johnson noise in a 50 W resistor, over a bandwidth Df of 1 GHz, yields a noise current of 0.58 mA and Pmin = 2.9 mW or -25.4 dBm.

If the diode current is only due to the optical power, or I = PminR, then

(4.7.26)

The sensitivity for a given bandwidth can also be expressed as a number of photons per bit:

(4.7.27)

For instance, for a minimal power of -30 dBm and a bandwidth of 1 GHz, this sensitivity corresponds to 4400 photons per bit.

4.7.1.2.4. Noise equivalent Power and ac noise analysis

Assume the optical power with average value P0 is amplitude modulated with modulation depth, m, as described by:

(4.7.28)

The ac current (RMS value) in the photodiode with responsivity, R, is then

(4.7.29)

which yields as an equivalent circuit of the photodiode a current source in parallel with a resistance, Req, where Req is the equivalent resistance across the diode and is the noise source, which is given by:

(4.7.30)

where the equivalent dark current also includes the Johnson noise of the resistor, Req:

(4.7.31)

The signal to noise ratio is then given by:

(4.7.32)

from the above equation one can find the required optical power P0 needed to obtain a given signal to noise ratio, S/N:

(4.7.33)

The noise equivalent power is now defined as the ac (RMS) optical power needed to obtain a signal-to-noise ratio of one for a bandwidth of 1 Hz or:

(4.7.34)

We now consider two limiting case in which the NEP is either limited by the optical power or by the dark current.

For it is the average optical power rather than the dark current which limits the NEP.

(4.7.35)

The noise equivalent power can also be used to calculate the ac optical power if the bandwidth differs from 1Hz from:

(4.7.36)

where the noise equivalent power has units of W/Hz. However, the optical power is mostly limited by the dark current for which the expressions are derived below.

For it is the dark current (including the Johnson noise of the resistor) which limits the NEP

(4.7.37)

Again one can use the noise equivalent power to calculate the minimum detectable power for a given bandwidth:

(4.7.38)

where the noise equivalent power has now units of

4.7.1.3. Switching of a P-i-n photodiode

4.7.1.3.1 Solution in the presence of drift, diffusion and recombination
4.7.1.3.2 Harmonic solution
4.7.1.3.3 Time response due to carriers generated in the Q.N. region
4.7.1.3.4 Dynamic range of a photodiode

A rigorous solution for the switching time of a P-i-n photodiode starts from the continuity equations for electrons and holes:

(4.7.39)

(4.7.40)

with

(4.7.41)

(4.7.42)

and the electric field is obtained from Gauss's law. For a P-i-n diode with generation only at t = 0 and neglecting recombination and diffusion these equations reduce to:

(4.7.43)

Where the electric field, , is assumed to be a constant equal to:

(4.7.44)

replacing n(x,t) by n*(x - vnt) and p(x,t) by p*(x - vpt) yields vn = -mn and vp = mp.

The carrier distributions therefore equal those at t = 0 but displaced by a distance mnt for holes and -mpt for electrons. The total current due to the moving charge is a displacement current which is given by:

(4.7.45)

(4.7.46)

(4.7.47)

for t < |d/vn| and t < |d/vp| . For a uniform carrier generation this reduces to:

(4.7.48)

(4.7.49)

In the special case where vn = vp or mn = mp, the full width half maximum (FWHM) of the impulse response is:

(4.7.50)

Note: Rule of thumb to convert a pulse response to –3 dB frequency: Assuming the photodiode response to be linear, the FWHM can be related to the half-power frequency by calculating the Fourier transform. For a gaussian pulse response (which also yields a gaussian frequency response) this relation becomes

(4.7.51)

Since the bandwidth depends on the transit time, which in turn depends on the depletion layer width, there is a tradeoff between the bandwidth and the quantum efficiency.

4.7.1.3.1. Solution in the presence of drift, diffusion and recombination

If we simplify the SHR recombination rate to n/t and p/t and assume a constant electric field and initial condition n(x,0) = n0 , the electron concentration can be obtained by solving the continuity equation, yielding:

(4.7.52)

where

(4.7.53)

with

(4.7.54)

For this analysis we solved the continuity equation with n(0,t) = n(L,t) = 0 implying infinite recombination at the edges of the depletion region. The initial carrier concentration n0 can also be related to the total energy which is absorbed in the diode at time t = 0:

(4.7.55)

and the photo current (calculated as described above) is

(4.7.56)

with Ck given by

(4.7.57)

The above equations can be used to calculate the impulse response of a photodiode. Each equation must be applied to electrons as well as holes since both are generated within the diode. Typically electrons and holes have a different mobility, which results in two regions with different slopes. This effect is clearly visible in GaAs diodes as illustrated with the figure below.

Figure 4.7.3 :

Photocurrent calculated using equation (5.1.50) for a GaAs diode with fI -Va = 0.3 V, Epulse = 10-13 J, Eph = 2 eV and d = 2mm.

4.7.1.3.2. Harmonic solution

Whereas section 4.7.1.3.1 provides a solution to the pulse response, one can also solve the frequency response when illuminating with a photon flux F1ejwt. If the photodiode has a linear response, both methods should be equivalent. To simplify the derivation, we assume that the total flux (in photons/s cm2) is absorbed at x = 0. This is for instance the case for a p-i-n diode with a quantum well at the interface between the p-type and intrinsic region and which is illuminated with long wavelength photons, which only absorb in the quantum well. The carriers moving through the depletion region cause a conduction current, Jcond(x):

(4.7.58)

where vn = mn is a constant velocity.

From Ampere's law applied to a homogeneous medium, we find:

(4.7.59)

And the total current is the sum of the conduction and the displacement current:

(4.7.60)

If we assume that the electric field is independent of time, the total photo current equals

(4.7.61)

with transit time . The corresponding –3 dB frequency is:

(4.7.62)

4.7.1.3.3. Time response due to carriers generated in the Q.N. region

For an infinitely long quasi-neutral (Q.N.) region and under stationary conditions, the generated carriers are only collected if they are generated within a diffusion length of the depletion region. The average time to diffuse over one diffusion length is the recombination time, t. Postulating a simple exponential time response we find that the current equals

(4.7.63)

Because of the relatively long carrier lifetime in fast photodiodes, carriers absorbed in the quasi-neutral region produce a long "tail" in the pulse response and should be avoided.

4.7.1.3.4. Dynamic range of a photodiode

The dynamic range is the ratio of the maximal optical power which can be detected to the minimal optical power. In most applications the dynamic range implies that the response is linear as well. The saturation current, defined as the maximum current which can flow through the external circuit, equals:

(4.7.64)

which yields an optimistic upper limit for the optical power:

(4.7.65)

and the dynamic range is defined as the ratio of the maximum to the minimum power:

(4.7.66)

Using (4.7.19) for the minimum power the dynamic range becomes independent of the responsivity and equals:

(4.7.67)

For example, if the equivalent noise current equals Ieq = 1mA, the bandwidth Df = 1 GHz, the impedance R = 50 W, and the applied voltage Va = 0, then the dynamic range equals 1.35 x 106 (for fi = 1.2 V) or 61.3 dB.

4.7.2. Photoconductors

Photoconductors consist of a piece of semiconductor with two Ohmic contacts. Under illumination, the conductance of the semiconductor changes with the intensity of the incident optical power. The current is mainly due to majority carriers since they are free to flow across the Ohmic contacts. However the majority carrier current depends on the presence of the minority carriers. The minority carriers pile up at one of the contacts, where they cause additional injection of majority carriers until the minority carriers recombine. This effect can cause large "photoconductive" gain, which depends primarily of the ratio of the minority carrier lifetime to the majority carrier transit time. Long carrier lifetimes therefore cause large gain, but also a slow response time. The gain-bandwidth product of the photoconductor is almost independent of the minority carrier lifetime and depends only on the majority carrier transit time.

Consider now a photoconductor with length, L, width W and thickness d, which is illuminated a total power, P. The optical power, P(x), in the material decreases with distance due to absorption and is described by:

(4.7.68)

The optical power causes a generation of electrons and holes in the material. Solving the diffusion equation (4.7.8) for the steady state case and in the absence of a current density gradient one obtains for the excess carrier densities:

(4.7.69)

Where it was assumed that the majority carriers, which primarily contribute to the photocurrent, are injected from the contacts as long as the minority carriers are present. The photo current due to the majority carriers (here assumed to be n-type) is:

(4.7.70)

where tr is the majority carrier transit time given by:

(4.7.71)

The equation above also includes the power reduction due to the reflection at the surface of the semiconductor. The normalized photocurrent is plotted in Figure 4.7.4 as a function of the normalized layer thickness for different ratio of lifetime to transit time.

Figure 4.7.4:

Normalized current versus normalized thickness ad as a function of the ratio of the minority carrier lifetime to the majority carrier transit time, t/tr, ranging from 0.01 (bottom curve) to 100 (top curve)

As an example, consider a silicon photoconductor with mn = 1400 cm2/V-s and t = 1 ms. The photoconductor has a length of 10 micron and width of 100 micron. For an applied voltage of 5 Volt, the transit time is 143 ps yielding a photoconductive gain of 7000. For a normalized distance ad = 1 and incident power of 1 mW the photocurrent equals 1.548 mA. A reflectivity of 30 % was assumed at the air/silicon interface.

High photoconductive gain is typically obtained for materials with a long minority carrier lifetime, t, high mobility, mn, and above all a photoconductor with a short distance, L, between the electrodes.

4.7.3. Metal-Semiconductor-Metal (MSM) Photodetectors

4.7.3.1 Responsivity of an MSM detector
4.7.3.2 Pulse response of an MSM detector
4.7.3.3 Equivalent circuit of an MSM detector.

Metal-semiconductor-metal photodetectors are the simplest type of photodetectors since they can be fabricated with a single mask. They typically consist of a set of interdigitated fingers, resulting in a large active area and short distance between the electrodes.

4.7.3.1. Responsivity of an MSM detector

The responsivity for a detector with thickness, d, surface reflectivity, R, finger spacing, L, and finger width, w, is given by:

(4.7.72)

Where a is the absorption length and the reflectivity, R, of the air-semiconductor interface as a function of the incident angle is given by:

(4.7.73)

(4.7.74)

with

with qi the incident angle, and qt the transmitted angle. RTE is the reflectivity if the electric field is parallel to the surface while RTM is the reflectivity if the magnetic field is parallel to the surface. The reflectivity as a function of qi, for an air-GaAs interface is shown in Figure 4.7.5:

Figure 4.7.5:

Angular dependencies of the reflectivity of an Air-to-GaAs interface

Including drift, diffusion and recombination the responsivity becomes:

(4.7.75)

with

(4.7.80)

The above expression can be used to calculate the current as a function of the applied voltage. An example is shown in Figure 4.7.6. Both the electron and the hole current are plotted as is the total current. The difference between the electron and hole current is due to the recombination of carriers. For large voltages all photo-generated carriers are swept out yielding a saturation of the photocurrent with applied voltage, whereas for small voltages around zero diffusion is found to be the dominant mechanism. The ratio of the transit time to the diffusion time determines the current around zero volt. In the absence of velocity saturation both transit times depend on the carrier mobility so that the ratio becomes independent of the carrier mobility. This causes the I-V curves to be identical for electrons and holes in the absence of recombination.

Figure 4.7.6:

Current - Voltage characteristic of an MSM photodiode.

4.7.3.2. Pulse response of an MSM detector

The pulse response can be calculated by solving the time dependent continuity equation, yielding:

(4.7.81)

with Ck given by:

(4.7.82)

where

(4.7.83)

and

(4.7.84)

This solution is plotted in Figure 4.7.7.

Figure 4.7.7:

Transient behavior (Pulse energy, Epulse = 0.1 pJ, Va = 0.3V)

4.7.3.3. Equivalent circuit of an MSM detector.

The equivalent circuit of the diode consists of the diode capacitance, Cp, a parallel resistance, Rp, obtained from the slope of the I-V characteristics at the operating voltage in parallel to the photocurrent, Iph, which is obtained by calculating the convolution of the impulse response and the optical input signal. A parasitic series inductance, LB, primarily due to the bond wire, and a series resistance, Rs, are added to complete the equivalent circuit shown in Figure 4.7.8.

Figure 4.7.8:

Equivalent circuit of an MSM detector

4.8. Solar cells

4.8.1. The solar spectrum
4.8.2. Calculation of maximum power
4.8.3. Conversion efficiency for monochromatic illumination
4.8.4. Effect of diffusion and recombination in a solar cell
4.8.5. Spectral response
4.8.6. Influence of the series resistance

Solar cells are p-i-n photodiodes, which are operated under forward bias. The intention is to convert the incoming optical power into electrical power with maximum efficiency

4.8.1. The solar spectrum

The solar spectrum is shown in Figure 4.8.1. The spectrum as seen from a satellite is referred to as the AM0 spectrum (where AM stands for air mass) and closely fits the spectrum of a black body at 5800 K. The total power density is 1353 W/m2.

Figure 4.8.1 :

The solar spectrum under of AM1 conditions

The solar spectrum as observed on earth is modified due to absorption in the atmosphere. For AM1 (normal incidence) the power density is reduced to 925 W/cm2 whereas for AM1.5 (45o above the horizon) the power density is 844 W/m2. The irregularities in the spectrum are due to absorption at specific photon energies. The corresponding cumulative photocurrent is presented in Figure 4.8.2 as a function of the photon energy.

Figure 4.8.2 :

Cumulative Photocurrent versus Photon Energy under AM1 conditions

4.8.2. Calculation of maximum power

The current through the solar cell can be obtained from:

(4.8.1)

where Is is the saturation current of the diode and Iph is the photo current (which is assumed to be independent of the applied voltage Va). This expression only includes the ideal diode current of the diode, thereby ignoring recombination in the depletion region. The short circuit current, Isc, is the current at zero voltage which equals Isc = -Iph. The open circuit voltage equals:

(4.8.2)

The total power dissipation is then:

(4.8.3)

The maximum power occurs at dP/dVa = 0. The voltage and current corresponding to the maximal power point are Vm and Im.

(4.8.4)

This equation can be rewritten as:

(4.8.5)

by using equation (4.8.2) for the open circuit voltage Voc. A more accurate solution is obtained by solving this transcendental equation and substituting into equations (4.8.1) and (4.8.3). The maximum power can be approximated by:

(4.8.6)

(4.8.7)

or

(4.8.8)

where

(4.8.9)

The energy Em is the energy of one photon, which is converted to electrical energy at the maximum power point. The total photo current is calculated as (for a given bandgap Eg)

(4.8.10)

and the efficiency equals:

(4.8.11)

4.8.3. Conversion efficiency for monochromatic illumination

This first order model provides an analytic approximation for the efficiency of a solar cell under monochromatic illumination. We start with the result of section 4.8.2:

(4.8.12)

and replace Voc by the largest possible open circuit voltage, Eg/q , yielding:

(4.8.13)

and

(4.8.14)

for a GaAs solar cell at 300K, Eg/kT = 55 so that the efficiency equals h = 85%

4.8.4. Effect of diffusion and recombination in a solar cell

4.8.4.1. Photo current versus voltage

(4.8.15)

as well as a similar equation for holes. The photo current is obtained from

(4.8.16)

Once this photocurrent is obtained the total current is obtained from:

(4.8.17)

To obtain the corresponding maximum power one has to repeat the derivation of section 5.3.2.

4.8.5. Spectral response

Because of the wavelength dependence of the absorption coefficient one expects the shorter wavelengths to be absorbed closer to the surface while the longer wavelengths are absorbed deep in the bulk. Surface recombination will therefore be more important for short wavelengths while recombination in the quasi-neutral region is more important for long wavelengths.

4.8.6. Influence of the series resistance

(4.8.18)

(4.8.19)

(4.8.20)

Repeating the derivation of section 4.8.2 one can show that the maximum power condition is given by the following set of transcendental equations:

(4.8.21)

(4.8.22)

while the maximum external power equals: Pm,ext = Im (Vm + Im Rs) 4.9. LEDs 4.9.1. Rate equations 4.9.2. DC solution to the rate equations 4.9.3. AC solution to the rate equations 4.9.4. Equivalent circuit of an LED A light emitting diode consists of a p-n diode, which is designed so that radiative recombination dominates. Homojunction p-n diodes, heterojunction p-i-n diodes where the intrinsic layer has a smaller bandgap (this structure is also referred to as a double-hetero-structure) and p-n diodes with a quantum well in the middle are all used for LEDs. We will only consider the p-n diode with a quantum well because the analytical analysis is more straightforward and also since this structure is used often in LEDs and even more frequently in laser diodes. 4.9.1. Rate equations The LED rate equations are derived from the continuity equations as applied to the p-n diode: (4.9.1) where G is the generation rate per unit volume and R is the recombination rate per unit volume. This equation is now simplified by integrating in the direction perpendicular to the plane of the junction. We separate the integral in two parts: one for the quantum well, one for the rest of the structure. (4.9.2) where k refers to the quantum number in the well. If we ignore the carriers everywhere except in the quantum well and assume that only the first quantum level is populated with electrons/holes and that the density of electrons equals the density of holes, we obtain: (4.9.3) where the last term is added to include reabsorption of photons. The rate equation for the photon density including loss of photons due to emission (as described with the photon lifetime tph) and absorption (as described with the photon absorption time tab) equals: (4.9.4) The corresponding voltage across the diode equals: (4.9.5) Where the modified effective hole density of states in the quantum well, Nv,qw*, accounts for the occupation of multiple hole levels as described in section 4.3.8.6. The optical output power is given by the number of photons, which leave the semiconductor per unit time, multiplied with the photon energy: (4.9.6) where A is the active area of the device, R is the reflectivity at the surface and Qc is the critical angle for total internal reflection (4.9.7) The reflectivity and critical angle for a GaAs Air interface are 30 % and 16o respectively. 4.9.2. DC solution to the rate equations The time independent solution (indicated with the subscript 0) in the absence of reabsorption is obtained from: (4.9.8) (4.9.9) where B is the bimolecular recombination constant. Solving these equations yields: (4.9.10) For small currents this reduces to: (J << q/16tnr2B) (4.9.11) which indicates that SHR recombination dominates, whereas for large currents one finds: (J >> q/16tnr2B) (4.9.12) The dc optical output power is: (4.9.13) This expression explains the poor efficiency of an LED. Even if no non-radiative recombination occurs in the active region of the LED, most photons are confined to the semiconductor because of the small critical angle. Typically, only a few percent of the photons generated escape the semiconductor. This problem is most severe for planar surface emitting LEDs. Better efficiencies have been obtained for edge emitting, "super luminescent" LEDs (where stimulated emission provides a larger fraction of photons which can escape the semiconductor) and LEDs with curved surfaces. 4.9.3. AC solution to the rate equations We now assume that all variables can be written as a sum of a time independent term and a time dependent term (note that n(t) is still a density per unit area): (4.9.14) (4.9.15) (4.9.16) The rate equations for the time-dependent terms is then given by: (4.9.17) (4.9.18) Next, we assuming the ac current of the form j1 = j1,0ejwt and ignore the higher order terms. This results in a harmonic solution of the form: (4.9.19) with: (4.9.20) where teff depends on N0 as: (4.9.21) and the ac responsivity is: (4.9.22) At w = 0 this also yields the differential quantum efficiency (D.Q.E) (4.9.23) 4.9.4. Equivalent circuit of an LED The equivalent circuit of an LED consists of the p-n diode current source parallel to the diode capacitance and in series with a linear series resistance, R. The capacitance, C, is obtained from: (4.9.24) (4.9.25) or (4.9.26) with (4.9.27) for N0 << Nc and/or Nv, m = 2 while for N0 >> Nc and/or Nv, . 4.10. Laser Diodes 4.10.1. Introduction 4.10.2. Laser cavities and laser cavity modes 4.10.3. Emission, Absorption and modal gain 4.10.4. The rate equations for a laser diode. 4.10.5. Large signal switching of a laser diode 4.10.1. Introduction 4.10.1.1. Laser structure and principle of operation Laser diodes consist of a p-n diode with an active region where electrons and holes recombine resulting in light emission. In addition, a laser diode contains an optical cavity where stimulated emission takes place. The laser cavity consists of a waveguide terminated on each end by a mirror. As an example, the structure of an edge-emitting laser diode is shown in Figure 4.10.1. Photons, which are emitted into the waveguide, can travel back and forth in this waveguide provided they are reflected at the mirrors. The distance between the two mirrors is the cavity length, labeled L. Figure 4.10.1 : Structure of an edge-emitting laser diode. 4.10.1.2. Stimulated emission and modal gain The light in the waveguide is amplified by stimulated emission. Stimulated emission is a process where a photon triggers the radiative recombination of an electron and hole thereby creating an additional photon with the same energy and phase as the incident photon. This process is illustrated with Figure 4.10.2. This "cloning" of photons results in a coherent beam. Figure 4.10.2 : Stimulated emission of a photon. The stimulated emission process yields an increase in photons as they travel along the waveguide. 4.10.1.3. Lasing condition Combined with the waveguide losses, stimulated emission yields a net gain per unit length, g. The number of photons can therefore be maintained if the roundtrip amplification in a cavity of length, L, including the partial reflection at the mirrors with reflectivity R1 and R2 equals unity. This yields the following lasing condition: (4.10.1) If the roundtrip amplification is less than one then the number of photons steadily decreases. If the roundtrip amplification is larger than one, the number of photons increases as the photons travel back and forth in the cavity and no steady state value would be obtained. The gain required for lasing therefore equals: (4.10.2) Initially, the gain is negative if no current is applied to the laser diode as absorption dominates in the waveguide. As the laser current is increased, the absorption first decreases and the gain increases. 4.10.1.4. Output power The current for which the gain satisfies the lasing condition is the threshold current of the laser, Ith. Below the threshold current very little light is emitted by the laser structure. For an applied current larger than the threshold current, the output power, Pout, increases linearly with the applied current as illustrated with Figure 4.10.4. The output power therefore equals: (4.10.3) where hn is the energy per photon. The factor, h, indicates that only a fraction of the generated photons contribute to the output power of the laser as photons are partially lost through the other mirror and throughout the waveguide. Figure 4.10.3 : Output power from a laser diode versus the applied current. 4.10.2. Laser cavities and laser cavity modes A laser diode consists of a cavity, defined as the region between two mirrors with reflectivity R1 and R2, and a gain medium, in our case a quantum well. The optical mode originates in spontaneous emission, which is confined to the cavity by the waveguide. This optical mode is amplified by the gain medium and partially reflected by the mirrors. The modal gain depends on the gain of the medium, multiplied with the overlap between the gain medium and the optical mode which we call the confinement factor, G, or: (4.10.4) This confinement factor will be calculated in section 4.10.2. Lasing occurs when the round trip optical gain equals the losses. For a laser with modal gain g(N)G and waveguide loss, a, this condition implies: (4.10.5) where L is the length of the cavity. The distributed loss of the mirrors is therefore: (4.10.6) 4.10.2.1. Longitudinal modes in the laser cavity. Longitudinal modes in the laser cavity correspond to standing waves between the mirrors. If we assume total reflection at the mirrors this wave contains N/2 periods where N is an integer. For a given wave length l and a corresponding effective index, neff, this yields: (4.10.7) Because of dispersion in the waveguide, a second order model should also include the wavelength dependence of the effective index. Ignoring such dispersion effects, we find the difference in wavelength between two adjacent longitudinal modes from: (4.10.8) (4.10.9) (4.10.10) Longer cavities therefore have closer spaced longitudinal modes. An edge emitting (long) cavity with length of 300 mm, neff = 3.3, and l = 0.8 mm has a wavelength spacing Dl of 0.32 nm while a surface emitting (short) cavity of 3 mm has a wavelength spacing of only 32 nm. These wavelength differences can be converted to energy differences using: (4.10.11) so that 0.32 nm corresponds to -6.2 meV and 32 nm to 620 meV. A typical width of the optical gain spectrum is 60 meV, so that an edge emitter biased below threshold can easily contain 10 longitudinal modes, while for a surface emitter the cavity must be carefully designed so that the longitudinal mode overlaps with the gain spectrum. A more detailed analysis of a Fabry-Perot etalon is described in Appendix 15, providing the reflectivity, absorption and transmission as a function of photon energy. 4.10.2.2. Waveguide modes The optical modes in the waveguide determine the effective index needed to calculate the longitudinal modes as well as the confinement factor. Starting from Maxwell's equations in the absence of sources: (4.10.12) (4.10.13) and assuming a propagating wave in the z-direction and no variation in the y-direction, we obtain the following one-dimensional reduced wave equation for a time harmonic field, = x ejwt, of a TM mode: (4.10.14) Where, b, is the propagation constant given by , and . The equation then becomes: (4.10.15) this equation is very similar to the Schrödinger equation. In fact, previous solutions for quantum wells can be used to solve Maxwell's equation by setting the potential V(x) equal to -n2 (x) and replacing by . The energy eigenvalues, El, can then be interpreted as minus the effective indices of the modes: -n2eff,l. One particular waveguide of interest is a slab waveguide consisting of a piece of high refractive index material, n1, with thickness d, between two infinitely wide cladding layers consisting of lower refractive index material, n2. From Appendix 17 one finds that only one mode exists for: (4.10.16) or, (4.10.17) For l = 0.8 mm, n1 = 3.5 and n2 = 3.3 one finds d £ 0.34 mm. 4.10.2.3. The confinement factor The confinement factor is defined as the ratio of the modal gain to the gain in the active medium at the wavelength of interest: (4.10.18) for a quantum well with width Lx, the confinement factor reduces to (4.10.19) which is approximately @ 0.02...0.04 for a typical GaAs single quantum well laser 4.10.3. Emission, Absorption and modal gain The analysis of a semiconductor laser diode requires a detailed knowledge of the modal gain, which quantifies the amplification of light confined to the lasing mode. To find the modal gain, one starts from the requirement that the emission as well as absorption of photons, must conserve both energy and momentum of all particles involved in the process. The conservation of energy requires that the photon energy equals the difference between the electron and hole energy: (4.10.20) with (4.10.21) (4.10.22) The conservation of momentum requires that the electron momentum equals that of the empty state it occupies in the valence band plus the momentum of the photon (4.10.23) The photon momentum is much smaller than that of the electron and hole, so that the electron and hole momentum are approximately equal. As a result we can replace kn and kp by a single variable k. Equations (4.10.20), (4.10.21), (4.10.22) and (4.10.23) then result in: (4.10.24) where Eg,qw1 is the energy between the lowest electron energy in the conduction band and the lowest hole energy in the valence band. mr* is the reduced effective mass given by: (4.10.25) The electron and hole energies, En and Ep, can then be expressed as a function of the photon energy by: (4.10.26) (4.10.27) The emission and absorption spectra (b(Eph) and a(Eph)) of a quantum well depend on the density of states and the occupancy of the relevant states in the conduction and valence band. Since the density of states in the conduction and valence band is constant in a quantum well, the emission and absorption can be expressed as a product of a maximum emission and absorption rate and the probability of occupancy of the conduction and valence band states, namely: (4.10.28) (4.10.29) Stimulated emission occurs if an incoming photon triggers the emission of another photon. The net gain in the semiconductor is the stimulated emission minus the absorption. The maximum stimulated emission equals the maximum absorption since the initial and final states are simply reversed so that the transition rates as calculated based on the matrix elements are identical. The net gain is then given by: (4.10.30) where the maximum stimulated emission and the maximum absorption were replaced by the maximum gain, gmax. The normalized gain spectrum is shown in Figure 4.10.4 for different values of the carrier density. The two staircase curves indicate the maximum possible gain and the maximum possible absorption in the quantum well. Figure 4.10.4: Normalized gain versus photon energy of a 10nm GaAs quantum well for a carrier density of 1012 (lower curve), 3 x 1012, 5 x 1012, 7 x 1012 and 9 x 1012 (upper curve) cm-2. The theoretical gain curve of Figure 4.10.4 exhibits a sharp discontinuity at Eph = Eg,qw1. The gain can also be expressed as a function of the carrier densities, N and P, when assuming that only one electron and one hole level is occupied: (4.10.31) The peak value at Eph = Eg,qw1, assuming quasi-neutrality (N = P) is then: (4.10.32) The maximum gain can be obtained from the absorption of light in bulk material: One can verify that the wavefunction of a free electron in bulk material is the same as the wavefunction in an infinite stack of infinitely deep quantum wells, provided the barriers are infinitely thin and placed at the nodes of the wavefunction. This means that for such a set of quantum wells the absorption would be the same as in bulk provided that the density of states is also the same. This is the case for Eph = Eqw1, so that the maximum gain per unit length is given by: (4.10.33) where Lx is the width of the quantum well. This expression shows that the total gain of a single quantum well due to a single quantized level is independent of the width. The corresponding value for GaAs quantum wells is 0.006 or 0.6%. Experimental gain curves do not show the discontinuity at Eph = Eqw1 due to inter-carrier scattering which limits the lifetime of carriers in a specific state. The line width of a single set of electron and hole levels widens as a function of the scattering time, which disturbs the phase of the atomic oscillator. Therefore, an approximation to the actual gain curve can be obtained by convoluting (4.10.30) with a Lorenzian line shape function: (4.10.34) with , where t is the carrier collision time in the quantum well. The original and convoluted gain curves are shown in Figure 4.10.5. Figure 4.10.5: Original and convoluted gain spectrum of a 10 nm GaAs quantum well with a carrier density of 3 x 1012 cm-2 and a collision time of 0.09 ps. For lasers with long cavities such as edge-emitter lasers, one finds that the longitudinal modes are closely spaced so that lasing will occur at or close to the peak of the gain spectrum. It is therefore of interest to find an expression for the peak gain as a function of the carrier density. A numeric solution is shown in Figure 4.10.6 where the peak gain is normalized to the maximum value of the first quantized energy level. Initially, the gain peak is linear with carrier concentration but saturates because of the constant density of states, until the gain peak associated with the second quantized level takes over. Since the peak gain will be relevant for lasing, we will consider it more closely. As a first order approximation we will set the peak gain g(N) equal to: (4.10.35) where is the differential gain coefficient. This approximation is only valid close to N = Ntr, and even more so for quantum well lasers as opposed to double-hetero-structure lasers. An approximate value for the differential gain coefficient of a quantum well can be calculated from (4.10.32) yielding: (4.10.36) Figure 4.10.6: Calculated gain versus carrier density for a 10 nm GaAs quantum well (solid line) compared to equation (4.10.32) From Figure 4.10.6 one finds that the material becomes "transparent" when the gain equals zero or: (4.10.37) which can be solved yielding: (4.10.38) The transparency current density is defined as the minimal current density for which the material becomes transparent for any photon energy larger than or equal to Eg,qw1. This means that the transparency condition is fulfilled for . The corresponding carrier density is referred to as Ntr, the transparency carrier density. The transparency carrier density can be obtained from by setting gmax = 0, yielding (4.10.39) This expression can be solved by iteration for Nv > Nc. The solution is shown in Figure 4.10.7. Figure 4.10.7: Normalized transparency carrier density versus the ratio of the effective density of states in the valence and conduction band. To include multiple hole levels one simply replaces Nv by Nv,qw* as described in section 4.3.8.6. 4.10.4. The rate equations for a laser diode. Rate equations for each longitudinal mode, l, with photon density Sl and carrier density Nl which couple into this mode are: (4.10.40) (4.10.41) Rather than using this set of differential equations for all waveguide modes, we will only consider one mode with photon density S, whose photon energy is closest to the gain peak. The intensity of this mode will grow faster than all others and eventually dominate. This simplification avoids the problem of finding the parameters and coefficients for every single mode. On the other hand it does not enable to calculate the emission spectrum of the laser diode. For a single longitudinal mode the rate equations reduce to: (4.10.42) (4.10.43) (4.10.44) 4.10.4.1. DC solution to the rate equations The time independent rate equations, ignoring spontaneous emission, are: (4.10.45) (4.10.46) where the photon life time is given by: (4.10.47) from which we can solve the carrier concentration while lasing: (4.10.48) which is independent of the photon density. The threshold current density is obtained when S0 = 0 (4.10.49) The photon density above lasing threshold, and power emitted through mirror R1, are given by: (4.10.50) and the power emitted through mirror 1 is: (4.10.51) The differential efficiency of the laser diode is: (4.10.52) and the quantum efficiency is: (4.10.53) Efficient lasers are therefore obtained by reducing the waveguide losses, increasing the reflectivity of the back mirror, decreasing the reflectivity of the front mirror and decreasing the length of the cavity. Decreasing the reflectivity of the mirror also increases the threshold current and is therefore less desirable. Decreasing the cavity length at first decreases the threshold current but then rapidly increases the threshold current. 4.10.4.2. AC solution to the rate equations Assuming a time-harmonic solution and ignoring higher order terms (as we did for the LED) the rate equations become: (4.10.54) (4.10.55) where teff is the same as for an LED and given by equation (4.9.21). Using these equations can be solved yielding: (4.10.56) We now replace n1 by relating it to the small signal voltage v1: (4.10.57) The equation for the small signal current, I1, can be then written as (4.10.58) with , and , where A is the area of the laser diode. (4.10.59) and (4.10.60) 4.10.4.3. Small signal equivalent circuit Adding parasitic elements and the circuit described by the equation (4.10.58) we obtain the equivalent circuit of Figure 4.10.8, where LB is a series inductance, primarily due to the bond wire, Rs is the series resistance in the device and Cp is the parallel capacitance due to the laser contact and bonding pad. Figure 4.10.8: Small signal equivalent circuit of a laser diode The resistor, Rd, in series with the inductor, L, is due to gain saturation and can be obtained by adding a gain saturation term to equation (4.10.35). The optical output power is proportional to the current through inductor L, i1,L, which is given by: (4.10.61) and the corresponding power emitted from mirror R1 (4.10.62) When further ignoring the parasitic elements and the gain saturation resistance, Rd, one finds the ac responsivity p1/I1 as: (4.10.63) from which we find the relaxation frequency of the laser: (4.10.64) The relaxation frequency is therefore proportional to the square root of the DC output power. The amplitude at the relaxation frequency relative to that at zero frequency equals: (4.10.65) 4.10.5. Large signal switching of a laser diode Because of the non-linear terms in the rate equations the large signal switching of a laser diode exhibits some peculiar characteristics. The response to a current step is shown in the Figure 4.10.1. The carrier density initially increases linearly with time while the photon density remains very small since stimulated emission only kicks in for N > N0. Figure 4.10.9: Optical power and normalized carrier concentration versus time when applying a step current at t = 0 from I = 0.95 Ith to I = 1.3 Ith. Both the carrier density and the photon density oscillate around their final value. The oscillation peaks are spaced by roughly 2p/w0, where w0 is the small signal relaxation frequency at the final current. The photon and carrier densities are out of phase as carriers are converted into photons due to stimulated emission, while photons are converted back into electron-hole pairs due to absorption. One can use the non-linear behavior to generate short optical pulses. By applying a current pulse, which is long enough to initiate the first peak in the oscillation, but short enough to avoid the second peak, one obtains a single optical pulse, which is significantly shorter that the applied current pulse. This method is referred to as gain switching or current spiking.