(p.447) APPENDIX E THE DIODE EQUATION
(p.447) APPENDIX E THE DIODE EQUATION
Before entering a quantitative analysis of the diode characteristics, it is useful to review the relationships that determine carrier concentrations. To avoid confusion with exponential functions the symbol q _{e} will be used for the electronic charge instead of e.
E.1 Carrier concentrations in pure semiconductors
In thermal equilibrium the probability that an electron state in the conduction band is filled is given by the Fermi–Dirac distribution
In silicon the bandgap is 1.12 eV. If the Fermi level is at midgap, the bandedges will be 0.56 eV above and below E _{F}. As is apparent from Figure E.1, relatively large deviations from the Fermi level, i.e. extremely small occupancies, will still yield significant carrier densities.
The number of occupied electron states N _{e} is determined by summing over all available states multiplied by the occupation probability for each individual state
The second obstacle to a simple analytical solution of the integral is the intractability of integrating over the Fermi distribution. Fortunately, if E − E _{F} is at least several times kT, the Fermi distribution can be approximated by a Boltzmann distribution, as shown in Figure E.2. (p.449)
At energies 2.3 kT beyond the Fermi level the difference between the Boltzmann approximation and the Fermi distribution is < 10%, for energies > 4.5 kT it is less than 1%.
Applying the approximation to the occupancy of hole states, the probability of a hole state being occupied, i.e. a valence state being empty, is
With these simplifications the number of electrons in the conduction band in thermal equilibrium
Using the above results
A remarkable result is that the product of the electron and hole concentrations
This result, the law of mass action, is very useful in semiconductor device analysis. It requires only that the Boltzmann approximation holds. Qualitatively, it says that if one carrier type exceeds this equilibrium concentration, recombination will decrease the concentrations of both electrons and holes to maintain $np={n}_{i}^{2}$, a relationship that also holds in doped crystals.
E.2 Carrier concentrations in doped crystals
The equality n = p only holds for pure crystals, where all of the electrons in the conduction band have been thermally excited from the valence band. In practical semiconductors the presence of impurities tips the balance towards either electrons or holes.
Impurities are an unavoidable byproduct of the crystal growth process, although special techniques can achieve astounding results. For example, as noted above, in the purest semiconductor crystals – “ultrapure” Ge – the net impurity concentration is about 3 · 10^{10} cm^{−3}.
In semiconductor device technology impurities are introduced intentionally to control the conductivity of the semiconductor. Let ${N}_{d}^{+}$ be the concentration of ionized donors and ${N}_{a}^{}$ the concentration of ionized acceptors. Overall charge neutrality is preserved, as each ionized dopant introduces a charged carrier and an oppositely charged atom, but the net carrier concentration is now
If the conductivity is dominated by only one type of carrier, the Fermi level is easy to determine. If, for example, n ≫ p then eqn E.16 can be written as
In reality the impurity levels of common dopants are not close enough to the band edge for the Boltzmann approximation to hold, so the calculation must use the Fermi distribution and solve numerically for E _{F}. Nevertheless, the qualitative conclusions derived here still apply.
It is often convenient to refer all of these quantities to the intrinsic level E _{i}, as it accounts for both E _{c} and E _{v}. Then
E.3 pnjunctions
A pnjunction is formed at the interface of a p and an ntype region. Since the electron concentration in the nregion is greater than in the pregion, electrons will diffuse into the pregion. Correspondingly, holes will diffuse into the nregion. As electrons and holes diffuse across the junction, a space charge due to the ionized donor and acceptor atoms builds up. The field due to this space charge is directed to impede the flow of electrons and holes.
The situation is dynamic. The concentration gradient causes a continuous diffusion current to flow, whereas the field due to the space charge drives a drift current in the opposite direction. Equilibrium is attained when the two currents are equal, i.e. the sum of the diffusion and drift currents is zero. The net hole current density is
(p.452) To solve this equation we make use of the following relationships: The hole concentration is
The remaining ingredient is the Einstein relationship, which relates the mobility to the diffusion constant
For the Fermi level to be flat, the band structure must adapt, since on the pside the Fermi level is near the valence band, whereas on the nside it is near the conduction band (Figure 2.19). If we assume that the dopants are exclusively donors on the nside and acceptors on the pside, the difference in the respective Fermi levels is
(p.453) As either N _{a} or N _{d} increases relative to n _{i}, the respective Fermi level moves closer to the band edge, increasing the builtin voltage. With increasing doping levels the builtin voltage approaches the equivalent potential of the bandgap E _{g}/q _{e}.
E.4 The forwardbiased pnjunction
Applying an external bias leads to a condition that deviates from thermal equilibrium, i.e. the Fermi level is no longer constant throughout the junction. If a positive voltage is applied to the pelectrode relative to the nelectrode, the total variation of the electric potential across the junction will decrease (Figure 2.20). Since this reduces the electric field across the junction, the drift component of the junction current will decrease. Since the concentration gradient is unchanged, the diffusion current will exceed the drift current and a net current will flow.
This net current leads to an excess of electrons in the pregion and an excess of holes in the nregion. This “injection” condition leads to a local deviation from equilibrium, i.e. $pn>{n}_{i}^{2}$. Equilibrium will be restored by recombination.
Note that a depletion region exists even under forward bias, although its width is decreased. The electric field due to the space charge opposes the flow of charge, but the large concentration gradient overrides the field.
Consider holes flowing into the nregion. They will flow through the depletion region with small losses due to recombination, as the electron concentration is small compared with the bulk. When holes reach the nside boundary of the depletion region the concentration of electrons available for recombination increases and the concentration of holes will decrease with distance, depending on the crosssection for recombination, expressed as a diffusion length. Ultimately, all holes will have recombined with electrons. The required electrons are furnished through the external contact from the power supply.
On the pside, electrons undergo a similar process. The holes required to sustain recombination are formed at the external contact to the pregion by electron flow toward the power supply, equal to the electron flow toward the ncontact. The following derivation follows the discussions by Shockley (1949, 1950) and Grove (1967).
The steadystate distribution of charge is determined by solving the diffusion equation,
The first boundary condition required for the solution of the diffusion equation is that the excess concentration of electrons vanish at large distances x,
To analyze the regions with nonequilibrium carrier concentrations Shockley introduced a simplifying assumption by postulating that the product pn is constant. In this specific quasiequilibrium state this constant will be larger than ${n}_{i}^{2}$, the pnproduct in thermal equilibrium. In analogy to thermal equilibrium, this quasiequilibrium state is expressed in terms of a “quasiFermi level”, which is the quantity used in place of E _{F} that gives the carrier concentration under nonequilibrium conditions.
The postulate pn = const is equivalent to stating that the nonequilibrium carrier concentrations are given by a Boltzmann distribution, so the concentration of electrons is
Using the quasiFermi level and the Einstein relationship, the electron current entering the pregion becomes
These relationships describe the behavior of the quasiFermi level in the depletion region. How does this connect to the neutral region?
(p.455) In the neutral regions the majority carrier motion is dominated by drift (in contrast to the injected minority carrier current, which is determined by diffusion). Consider the ntype region. Here the bulk electron current that provides the junction current
In the space charge region, pn is constant, so the quasiFermi levels for holes and electrons must be parallel, i.e. both will remain constant at their respective majority carrier equilibrium levels in the neutral regions.
If an external bias V is applied, the equilibrium Fermi levels are offset by V, so it follows that the quasiFermi levels are also offset by V,
If the majority carrier concentration is much greater than the concentration due to minority carrier injection (“lowlevel injection”), the hole concentration at the edge of the pregion remains essentially at the equilibrium value. Consequently, the enhanced pnproduct increases the electron concentration.
This is the diode equation (or Shockley equation), which describes the current–voltage characteristic both under forward and reverse bias. Under forward bias the current indreases exponentially. Under reverse bias (negative V), the exponential term vanishes when the bias exceeds several kT/q _{e} and the current becomes the reverse saturation current J = −J _{0}. For a uniform junction crosssection the current densities J _{n}, J _{p}, and J _{0} can be replaced by their respective currents.
Note that in the diode equation:

1. The bandgap does not appear explicitly (only implicitly in J _{0} via n _{i}).

2. The total current has two distinct components, due to electrons and holes.

3. The electron and hole currents are generally not equal. The ratio
(E.53)$$\frac{{J}_{n}}{{J}_{p}}=\frac{{N}_{d}}{{N}_{a}}\text{if\hspace{0.17em}}\frac{{D}_{n}}{{L}_{n}}=\frac{{D}_{p}}{{L}_{p}}.$$ 
4. Current flows for all values of V. However, when plotted on a linear scale, the exponential appears to have a knee, often referred to as the “turnon” voltage.

5. The magnitude of the turnon voltage is determined by J _{0}. Diodes with different bandgaps will show the same behavior if J _{0} is the same.
Figure E.3 shows measured I–V curves for commercial Si and Ge junction diodes (1N4148 and 1N34A). On a linear scale the Ge diode “turns on” at 200 – 300 mV, whereas the Si diode has a threshold of 500 – 600 mV. However, on a logarithmic scale it becomes apparent that both diodes pass current at all voltages > 0.
The reverse current (Figure E.4) shows why the Ge diode shows greater sensitivity at low voltages. The smaller bandgap leads to increased n _{i}. The Si diode shows a “textbook” exponential forward characteristic at currents > 10 nA, whereas the Ge diode exhibits a more complex structure.
The discrepancies in the forward current between the measured results and the simple theory require the analysis of all processes in the depletion zone.

1. Generationrecombination in the depletion region (see Appendix F).

2. Diffusion current (as just calculated for the ideal diode).
(p.457)

3. Highinjection region where the injected carrier concentration affects the potentials in the neutral regions.

4. Voltage drop due to bulk series resistance.
The reverse current is increased due to generation and recombination currents in the depletion zone, as discussed in Appendix F. In optimized photodiodes reverse bias currents of about 100 pA/cm^{2} have been achieved, which is about 3 times the theoretical value (Holland 2004).