Transient and AC conditions

As the probability of ionization P approaches unity, the carrier multiplication (and therefore the reverse current through the junction) increase without limit. Actually, the limit on the current will be dictated by the external circuit.

The relation between multiplication and P was easy to write in Eq. (5-43); however, the relation of P to parameters of the junction is much more complicated. Physically, we expect the ionization probability to increase with increasing electric field, and therefore to depend on the reverse bias. Measurements of carrier multiplication M in junctions near breakdown lead to an empirical relation.

Where the exponent n varies from about 3 to 6, depending on the type of material used for the junction.

In general, the critical reverse voltage for breakdown increases with the band gap of the material, since more energy is required for an ionizing collision. Also, the peak electric field within w increases with increased doping on the more lightly doped side of the junction (prob. 5.15) therefore, V decreases as doping increases, as Fig. 5-19 indicates.

5.5. Transient and A-C conditions

We have considered the properties of p-n junctions under equilibrium conditions and with steady state current flow. Most of the basic concepts of junction devices can be obtained from these properties, except for the important behavior of junctions under transient of a-c conditions. Since most solid state devices are used for switching processing a-c signals, we cannot claim to understand p-n junctions without knowing at least the basics of time - dependent processes. Unfortunately, a complete analysis of these effects involver more mathematical manipulation than is appropriate for an introductory discussion. Basically, the problem involves solving the various current flow equations in two simultaneous variables, space and time. We can, however, obtain the basic results for several special cases which represent typical time-dependent applications of junction devices.

In the section we investigate the important influence of excess carriers in transient and a-c problem. The switching of a diode from its forward state to its reverse state is analyzed to illustrate a typical transient problem. Finally, these concepts are applied to the case of small a-c signals to determine the equivalent capacitance of a p-n junction.

5.5.1. Time variation of stored charge

Another look at the excess carrier distributions of a p-n junction under bias (e.g. Fig 5-12) tells us that any change in current must lead to change of charge stored in the carrier distributions. Since time is required in building up or depleting a charge distributions. Since time is required in building up of depleting a charge distribution, however, the stored charge must inevitably lag behind the current in a time-dependent problem. This is inherently a capacitive effect, as we shall see in section 5.5.3.

For a proper solution of s transient problem, we must use the time-dependent continuity equations, eqs. (4-31). We can obtain each component of the current at positions x and time t from these equations; for example, from Eq. (4-31a) we can write

To obtain the instantaneous current density, we integrate both sides at time t to obtain

For injection into a long n region from a p+ region, we can take the current at xn = 0 to be all hole current, and jp at xn = to be zero (fig 5-15). Then the total injected current, including time variations, is

This result indicates that the hole current injected across the p+ -n junction (and therefore approximately the total diode current) is determined by two charge storage effects: (1) the usual recombination term Qp / Tp in the wich the excess carrier distribution is replaced every Tp seconds, and (2) a charge buildup (or depletion) term dQp / dt, which allows for the fact that the distribution of excess carriers can be increasing or decreasing in the time-dependent problem. For steady state the dQp / dt term is zero, and Eq. (5-47) reduces to Eq (5-40), as expected. In fact, we could have written Eq. (5-47) intuitively rather than having obtained it from the continuity equation, since it is reasonable that the hole current injected at any given time must supply minority carriers for recombination and for whatever variations occur in the total stored charge.

We can solve for the stored charge as a function of time for a given current transient. For example, the step turn-off transient (Fig. 5-20a), in which a current “I” is suddenly removed at t = 0, leaves the diode with stored charge. Since the excess holes in the n region must die out by recombination with the matching excess electro population, some time is required fo Qp (t) reach zero. Solving Eq. (5-47) with Laplace transforms, with i(t>0) = 0 and Qp (0) = ITp, we obtain

As expected, the stored charge dies out exponentially from initial value ITp with a time constant equal to the hole lifetime in the n material.

An important implication of Fig. 5-20 is that even though the current is suddenly terminated, the voltage across the junction persists until Qp disappears. Since the excess hole concentration can be related to junction voltage by formulas derive in Section 5.3.2., we can presumably solve for v(T). we already know that at any time during the transient, the excess hole concentration at Xn = 0 is

So the finding Pm (T) will easily give us the transient voltage. Unfortunately, it is not simple to obtain Pm(T) exactly from our expression for Qp (t). the problem is that the hole distribution does not remain in the convenient exponential from is has steady state. As Fig. 5-20 suggests, the quantity P(Xn,T) becomes markedly nonexponential as the transient proceeds. For example, since the injected hole current is proportional to the gradient of the hole distribution at Xn = 0 (Fig, 5-13a) zero current implies zero gradient, thus the slope of the distribution must be exactly zero at Xn = 0 throughout the transient. This zero slope at the point of injection distorts the exponential distribution, particularly in the region near the junction. As time professes in Fig. 5-20c, P (and therefore N) decreases as the excess electrons and holes recombine. To find the exact expression for nP (Xn, T) during the transient would require a rather difficult solution of the time-dependent continuity equation.

An approximate solution for v(t) can be obtained by assuming and exponential distribution for P at every instant during the decay. This type of quasi-steady state approximation neglects distortion due to the slope requirement Xn = 0 and the effects of diffusion during the transient. Thus we would expect the calculation to give rather crude results.

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