4.2. P-NJUNCTION IN EQUILIBRIUM 147
Let us start withp-andn-type semiconductors without forming a junction as shown in fig-
ure 4.1(a). The electron affinityeχ, defined as the energy difference between the conduction
band and vacuum level, is shown along with the work function (eφsporeφsn). The work func-
tion represents the energy required to remove an electron from the semiconductor to the “free”
vacuum level and is the difference between the vacuum level and the Fermi level.
Let us now examine what happens when thep-andn-type materials are made to form a
junction. In the absence of any applied bias, there is no net current in the system and the Fermi
level is uniform throughout the structure. In figure 4.1(b) we show a schematic of the band
diagram of ap-njunction.
Majority carriers near the interface on both sides diffuse across the junction (holes frompside
and electrons fromnside), as a result of the difference in electron and hole densities across the
junction. Most of the electrons which diffuse to thep-side recombine with holes, and most of the
holes which diffuse to then-side recombine with electrons. As a result, a region is formed near
the junction that has been depleted of mobile carriers. An electric field exists in this depletion
region that sweeps out any electrons and holes that enter the region.
Three regions can be identified as seen in figure 4.1(b):
i) Thep-type region where the material is neutral and the bands are flat. The density of
acceptors exactly balances the density of holes (assuming that all of the acceptors are ionized);
ii) Then-type region away from the junction where again the material is neutral and the
density of immobile donors exactly balances the free electron density. Again we assume that all
of the donors are ionized. In general the majority carrier (holes in thep-region and electrons
in then-region) densities are equal to the density of ionized dopants as long as minority carrier
densities are negligible.
iii) Around the junction there is a depletion region where the bands are bent and a field exists
that sweeps the mobile carriers, leaving behind negatively charged acceptors in thep-region and
positively charged donors in then-region. The depletion region extends a distanceWpin the
p-region and a distanceWnin then-region.
Due to the field in the depletion region electrons or holes which enter the depletion region are
swept away. Thus, once equilibrium is established, a drift current exists that counterbalances the
diffusion current. Let us calculate the width of the depletion region, and the electric field. To
obtain analytical results we make some simplifying assumptions:
i) The junction is uniformly doped.
ii) The mobile charge density in the depletion region is not zero, but it is much smaller than
the background dopant density. TosolvethePoissonequationwewillassumethatthemobile
carrierdensityisessentiallyzero, the depletion approximation.
The various current flow terms in the diode are as follows: the electron drift current and
electron diffusion current as well as the hole drift and hole diffusion current, as shown in fig-
ure 4.2(b). When there is no applied bias, these currents cancel each otherindividually. Let us
consider these current components. The hole current density is
Jp(x)=e⎡
⎢⎢
⎣︸μpp(x︷︷)E(x)︸
drift−Dpdp(x)
︸ ︷︷dx︸
diffusion