SEMICONDUCTOR JUNCTIONS

The hole produced by an acceptor impurity as in Fig. 2.19 is likewise bound, and it also requires only a small amount of energy to be freed. Its Bohr energy levels, calledacceptor levels, lie just above the top of the valence band.

In summary, the doping of a semiconductor with a donor or acceptor does not by itself produce conduction-band electrons or valence-band holes, as assumed in our discussion based on Figs. 2.18 and 2.19. However, the energy required to “ionize” these donors or acceptors is so small that thermal excitation at moderate temperatures will do the job. † Another important semiconductor, germanium, is similar to silicon in that it is tetravalent. It may, therefore, be doped in the same ways to producen-type andp-type materials. In addition to such elemental semiconductors are “III – V” binary semi- conductors such as gallium arsenide or indium antimonide, in which trivalent and pentavalent atoms share in covalent bonding as a result of unfilled spsubshells (Problem 2.5).

opposes further diffusion of electrons and holes and thus keeps the depletion region confined to a narrow layer at the boundary. In other words, there is some voltage drop, which we callV0, in going from thenside to thepside (Fig. 2.23).

In thermal equilibrium there is a diffusion of individual electrons and holes across the junction, but no net flow of current. Holes on thenside, for instance, have no diffi- culty dropping down the potential-energy hill and going over to thepside. Holes on thepside, on the other hand, have to cross the potential-energy barriereV0to get to thenside (hereeis understood to be positive). According to statistical mechanics, the fraction of holes able to cross the barrier is given by the Boltzmann factor exp(2eV0/ kBT), wherekBis Boltzmann’s constant (kB1:3810²³ J=K₄₀¹ eV=300K) and T is the absolute temperature. If the current due to holes diffusing from the n side to thep side is to be exactly balanced by the hole current in the opposite direction, therefore, we must have

Np(nside)¼Np(pside)e^eV0=kBT, (2:8:1) where N_p denotes the number of holes per unit volume. This equation shows, as expected, that there is a greater density of holes on thepside than on thenside. The same reasoning leads to the relation

N_n(pside)¼N_n(nside)e^eV0=kBT, (2:8:2) for the number densityNnof electrons. According to these relations, the productNnNpis the same for the two sides of the junction.

p

–

n

Donor ion

Electron Hole

Acceptor ion

– –

– +

+ +

+

– +

Figure 2.22 At apnjunction, electrons from thenside are attracted to thepside, and holes from the pside are attracted to thenside. When electron – hole pairs meet they are “annihilated” as the electron becomes part of a covalent bond. This results in a depletion region at the boundary, which is a region in which there are very few electrons or holes.

– –

– +

p n

– +

+ +

V₀

Figure 2.23 The charge separation due to donor and acceptor ions in the depletion layer of apn junction results in a potential differenceV0.

Now suppose that thepandnsides are connected to the terminals of a battery. If thep side is connected to the positive terminal and thenside to the negative terminal, the junction is said to beforward biased(Fig. 2.24a). Since the depletion layer is much more resistive to current than the bulk regions, most of the applied voltageVis dropped across the depletion layer. In other words, the applied voltage has the effect of lowering the barrier voltageV0toV02V. We will assume for the present thatVis small compared toV0.

This lowering of the barrier potential by a forward-biased voltage does not affect very much the diffusion current of holes from thenside to thepside because with or without it they have no potential-energy barrier to cross. Their diffusion current, there- fore, is unaffected by the applied voltage. The hole current from thepside to thenside, however, will increase with the applied voltage because now a fractione^(V0V)=kBT of them are able to cross over. For the net current of holes diffusing from thepside to thenside we have, therefore,

I_p/N_p(pside)e^e(V0V)=kBT N_p(nside)

¼Np(nside)e^eV=kBTNp(nside)

¼N_p(nside)(e^eV=kBT1), (2:8:3) where in the second line we have used (2.8.1). In other words, we have the relation

I_p¼I_p0(e^eV=kBT1), (2:8:4) for the net hole current flowing from thepside to thenside under forward bias, where I_n0 is the hole diffusion current from thenside to the pside. A similar expression is obtained for the net electron current under forward biasing:

In¼In0(e^eV=kBT 1), (2:8:5) whereI_n0is the electron diffusion current flowing from thepside to thenside. The total current flowing from thepside to thenside of a forward-biasedpnjunction is, therefore, I¼IpþIn¼I0(e^eV=kBT1) (forward biasing), (2:8:6) whereI₀¼I_p0þI_n0is called thesaturation currentof the junction.

p

(a) (b)

+ –

n p

– + n

Figure 2.24 Forward-biased (a) and reverse-biased (b)pnjunctions.

2.8 SEMICONDUCTOR JUNCTIONS 47

Suppose instead that the leads are reversed, as in Fig. 2.24b. In this case the junction is said to bereverse biased. The barrier potential is now increased fromV0toV0þV, and the net current is obtained simply by changingVto2Vin (2.8.6):

I¼I₀(e^eV=kBT1) (reverse bias): (2:8:7) Equations (2.8.6) and (2.8.7) give the so-called current – voltage (IV) characteristics of an idealpnjunction (Fig. 2.25). Under forward biasing, the current increases rapidly (exponentially) with increasing voltage. Under reverse bias, the currentsaturateswith increasing voltage to the valueI0. Typical barrier voltagesV0in silicon and germanium diodes are roughly on the order of half a volt. Saturation current densities are extremely small, perhaps less than 10²¹⁰A/cm²; for a typical junction area of 10²² cm², this amounts to a saturation currentI0of less than 10²¹²A. Formulas can be derived to esti- mateV0andI0as a function of carrier concentrations and other parameters, but we will not take the time to do so.

The key property of a pn junction, therefore, is that it will conduct current in one direction but not the other. That is, it can act as a diode, a sort of automatic switch that closes a circuit when voltage is applied in a forward sense, but blocks the flow of current otherwise. This diode can be used as a rectifier, converting ac current to dc current.

† Real semiconductor diodes do not display exactly the same IV characteristics as the idealized diode we have considered. For one thing, we have ignored electron – hole recombination within the depletion layer. This and other effects may be taken into account by replacing (2.8.6) by

I¼I0(e^eV=bkBT1), (2:8:8) where the “ideality factor”bis a dimensionless parameter between 1 and 2, depending onT.

Furthermore, at large reverse-bias voltages a real diode no longer blocks the flow of current.

At a certain “breakdown” voltage there is a sudden jump in the reverse current. One reason for this is that high-energy charge carriers colliding with atoms in the crystal lattice can ionize them, producing more charge carriers, which lead to further ionization and therefore increasing the current. This is calledavalanche breakdown. Another mechanism for reverse-current gener- ation is theZener effectin which electrons undergo a quantum mechanical “tunneling” from thep

side to thenside. †

V I

I₀

Negative bias Forward bias

Figure 2.25 Current – voltage characteristics of an idealpnjunction.