Home Mathematics

# THE IDEAL PN JUNCTION AT EQUILIBRIUM

We imagine two pieces of doped semiconductor materials, one P-type and one N-type coming close together and forming a PN junction. Of course this is not the way PN junctions are manufactured, but it is a useful thought process in order to understand the real

FIGURE 3.11 Schematic carrier and dopant concentrations of a P and an N type semiconductor piece before they are brought into contact to make a PN junction.

processes occurring at the interface of the P- and N-type materials. Figure 3.11 portrays the situation when the two pieces have not touched each other and figure 3.12 gives the situation when a PN junction has been formed and an equilibrium has been achieved, as we show below.

When the two neutral semiconducting pieces make contact with each other, electrons from the N-type semiconductor will diffuse to the P-type (because the concentration of electrons is much lower there) and likewise holes from the P-type semiconductor will diffuse towards the N-type (because the concentration of holes is much lower there). On their way, the two types of carriers will annihilate themselves by recombination, leaving near the surface the charged impurities of each side (portrayed as circles in figure 3.11) without a corresponding neutralizing charge, i.e. a space charge region will be established, also called depletion layer, as shown in figure 3.12. The layer is called the depletion layer because the concentration of both electrons and holes is much reduced there.

The positively charged donors in the N-type semiconductor and the negatively charged acceptors in the P-type semiconductor near the surface create an electric field £ whose direction opposes the further diffusion of electrons and holes. Note that the concentration gradients of both electrons and holes does not become zero anywhere inside the depletion region but a dynamic steady state is reached: the “built in” electric field £ is creating a current density in the opposite direction of the diffusion current that keeps the concentration gradients

constant. Note also that this situation is an equilibrium situation since no energy enters or leaves the system. The above description pertains to the internal electronic density structure before any voltage is applied to the junction.

Before we give a microscopic view of the charge flow in the PN junction, we deal first with the electrostatics of the junction. Certain simplifying assumptions, which are reasonably accurate, are needed in order to get closed expressions for the potential and the electric field built in the diode. We assume that complete annihilation of the mobile charge carriers n and p has occurred inside the space charge-depletion layer, and the charge density, there, is given by the respective doping concentrations (with the right sign) instead of the more general expression p = e{^N^ + p-N~A —я). Furthermore, we assume that this annihilation has abrupt limits. Further assumptions will be required for the PN junction under bias. For the time, we assume that the depletion layer extends from -xp to 0 in the P side and from 0 to x„ in the N side of the juction. The charge density p in the limit of an abrupt junction is shown in figure 3.13a. Therefore, Poisson’s equation for this 1-D problem inside the depletion layer becomes

where, as noted, xp and x„ are the lengths of the P and N part of the PN junction respectively. We have also assumed that the dielectric constant £( of the semiconductor has not been affected by the formation of the depletion layer. We note that the semiconductor junction was initially neutral and the space charge layer has appeared as a result of mutual annihilation of electrons and holes. Therefore, the total charge has been preserved and we have

By the definition of x„ and xp> we know the length of the depletion region (also called width sometimes) W is

The boundary conditions for the simple differential equation 3.39 are that the electric field

£ = - —j— is zero outside the depletion region. Therefore £(xp) = £(x„) = 0. We furthermore

denote by Vbi = V(x„)- V(-Xp) the built-in potential difference inside the space charge layer or depletion layer. Integrating 3.39 once and using the boundary conditions we get

FIGURE 3.13 Change density (a), electric field (b), and potential (c) along the length of a PN junction, and

Note that at x = 0, equations 3.42a and 3.42b must give the same value. This value is easily seen to be the maximum of the absolute value of the electric field which we denote as £max. The graph of £(x) is shown in figure 3.13b.

The built-in potential, defined above, is the integral of the electric field, and by simple geometry From equations 3.42a, 3.42b we get

Hence the length W of the depletion region is

and by the use of 3.43 and the fact that at room temperature all impurities are ionized Furthermore, using 3.44a and 3.44b in 3.42a and 3.42b respectively for the electric field we get

Note that both the above expressions give £(0) = —£max and each gives zero at the respective boundaries -xp, x,„ see figure 3.13b. Integrating once more and choosing V (-xp j = 0 we get

and using Vbi = V(x„)~ v(xp) = V(x„)

The graph of V(x) is shown in figure 3.13c.

FIGURE 3.14a Position of Fermi levels before contact of the P and N semiconductor pieces.

We now come to a microscopic description of the PN junction, meaning a description of the energy bands and charge flow in the junction. Figure 3.14a reproduces the standard band model of an isolated P-type and an isolated N-type semiconductor. This figure is a partial reproduction of figure 2.14a-b that we have discussed in chapter 2 with a slight modification of notation. The Fermi level in the N-type semiconductor is denoted Ep„ and the Fermi level in the P-type £^. When the two pieces are brought together to form the junction (a thought experiment as we have emphasized), the two Fermi levels must be equalized since any system under equilibrium has a unique Fermi level. It does not matter if Efp will come up in energy or Ef„ will go down. What matters is that a unique £/., constant in space must exist. The only way that this can happen is if the bands bend near the interface as figure 3.14b shows.

What will cause the bands to bend? The electrostatic potential energy present in the space charge region or depletion layer constitutes an additional energy that is added to that

of the crystalline potential energy so that the bending of the bands is a direct result of the built-in potential we have just calculated. It may surprise the reader that the electrostatic potential we have just calculated (see equation 3.47) is monotonically increasing, whereas the bending of the bands is monotonically decreasing. This is only due to the fact that bands denote energies of electrons, whereas the Poisson’s equation refers to the potential of a positive test charge. Hence we must multiply the solution of the Poisson equation by (-e) to get the band bending. This relation between band bending and the solution of the Poisson equation derives from the argument we presented in the first section of this chapter regarding voltage and potential energy of electrons.

Under the absence of an external field, the current is zero. This zero current however is the sum of two non-zero currents which add up to zero as we have already mentioned. We will consider each of them in detail using our previous notation for the concentrations of electrons and holes by adding a subscript denoting the region in which the concentration is calculated. Thus nn denotes the concentration of electrons in the N-type semiconductor side of a PN junction and np denotes the concentration of electrons in the P-side side of a PN junction and so on.

We first observe that n„ » np in the PN junction and hence there will be a diffusion current of electrons from the N-side to the P-side. However this diffusion current is much smaller than what would have been calculated by Fick s law because there is a barrier to the motion of electrons from the N-side to the P-side due to the built-in potential Vbi. On the contrary the potential difference Vbi acts as a down-hill potential for the electrons coming from the P-side to the N-side. This current is a drift current because electrons move under the action of the electric field producing the Vbi. In an obvious notation we therefore write

Likewise we observe a concentration difference between pp and p„. Therefore, a diffusion current of holes will appear from the P-side to N-side of the PN junction. Again this diffusion current is opposed by the barrier presented by the band bending. At first sight this seems to be a down-hill potential instead of an up-hill one, but we have to remember that we are dealing with holes where as the energies shown are those of electrons, so again the energy gradient shown by the band bending has to be inverted. Also, in analogy with the electrons in the P-side, the holes in the N-type semiconductor will drift by the existing built-in field to the P-type semiconductor side of the junction, as shown by the arrows in figure 3.14b. The two currents just described will cancel each other, so we will have

 Related topics