4.6.2 The ideal p-n diode current

1. General discussion and overview
2. Basic assumptions
3. The p-n diode with a "long" quasi-neutral region
4. The p-n diode with a "short" quasi-neutral region
5. General current expression
6. Example

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" as well as "short" diodes. A so called "long" diode has an infinite quasi-neutral region, while a "short" diode has a quasi neutral region which is much smaller than the diffusion length. In addition we derive a general expression which is to be used if the quasi-neutral region is comparable in size to the diffusion length. A final expression is derived for diodes with a finite recombination velocity at the metal-semiconductor contacts.

(pnc1)

(pnc2)

The general solution of the diffusion equation in the quasi-neutral regions is given by:

(pnc3)

(pnc4)

(pnc5)

and

(pnc6)

were the constants A and D were chosen so that the boundary conditions at x = x_n and x = -x_p are satisfied.

The diffusion current densities in the quasi-neutral regions due to minority carriers is then obtained from the derivative of the minority carrier density, yielding:

(pnc7)

(pnc8)

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. Also since we do not know the current due to recombination in the depletion region we will simply assume that it can be ignored. Later on we will more closely examine this assumption. The total current is then given by:

(pnc9)

where I_s can be written in the following forms:

(pnc12)

using the definition of the diffusion length, namely

(pnc14)

and

(pnc15)

while the thermal equilibrium minority carrier densities, n_p0 and p_n0, are given by:

(pnc2a)

and

(pnc1a)

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. In the next section we will show that this assumption is actually correct if band-to-band recombination dominates in the depletion region and that it underestimates the total recombination current if Shockley-Hall-Read recombination dominates. 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 therefore given by:

(pnc10)

so that I_r can be ignored if:

(pnc11)

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 almost always satisfy this requirement, while gallium arsenide p-n diodes rarely do because of the short carrier life time and diffusion length.

As an example we now consider a silicon p-n diode with N_a = 1.5 x 10¹⁴ cm^-3 and N_d = 10¹⁴ cm^-3. The minority carrier life time was chosen to be very short, namely 400 ps, so that most features of interest can be easily observed. We start by examining the electron and hole density throughout the p-n diode, shown in the figure below:

pncurr.xls - pndens.gif

Fig.4.6.1 Electron (red curve) and hole (blue curve) 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 the equations derived in section 4.6.2.3. 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 the figure below:

pncurr.xls - pncband.gif

Fig.4.6.2 Energy band diagram of a p-n diode. Shown are the conduction band edge (top black curve) and the valence band edge (bottom black curve), the intrinsic energy (yellow curve), the electron quasi-fermi energy (red curve) and the hole quasi-fermi energy (blue curve)

Next we discuss the current density. Shown in the figure below are the electron and hole current density

pncurr.xls - pncurr.gif

Fig.4.6.3 Electron (red curve) and hole (blue curve) current density versus position.

(pnc13a)

so that the carrier density varies linearly throughout the quasi-neutral region and in general is given by:

(pnc13b)

where A and B are constants are obtained by satisfying the boundary conditions. Applying the same boundary conditions at the edge of the depletion region as above and requiring thermal equlibrium at the contacts yields:

(pnc13c)

for the hole density in the n-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:

(pnc9)

where the saturation current, I_s is given by:

(pnc13)

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 denominater, whichever is smaller.

(pnc16)

finite recombination velocity v_s

(pnc17)

length.xls - length1.gif

Fig.4.6.4 Excess electron density versus position as obtained by solving the diffusion equation with dn(x = 0) = 1 and dn(x = w) = 0 and w = 10 micron. 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 normalized current as calculated using the "long" diode, the "short" diode and general expressions are shown in the figure below as well as the relative error of the "long and "short" diode expressions:

length.xls - length.gif

Fig.4.6.5 Normalized current as a function of the ratio of the diffusion length to the quasi-neutral region width as calculated using the "long" diode expression (blue curves), the "short" diode expressions (red curves) and the general expression (black curve).