Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

16

:>~ 12

'b

4

in comparison to the p-side, using Eq. (4.73) for conceptual purposes we can see that In/lp ~ N A/N D ~ 1. In addition to this complication of In ~ Ip, we have a nonuniformly doped n-region. In this case, even without the n+p junction, it is necessary for the n-side to set up an electric field to hold the nonuniform doping in place. We can use Eq. (4.58) to estimate the magnitude of this field at each point. As we shall see shortly, its magnitude is generally much smaller than that of the field in the space-charge region. Figure 4.13 shows the plot of £(x) which corresponds to the doping and carrier distributions plotted in Figure 4.12. Certainly in the uniformly doped p-region £(x) changes linearly. However, near the peak and into the n-region the results are very much non-ideal. Near the peak we can observe a super-linear behavior which should be expected for non-constant doping. Finally, on the n-side we see the £(x) tail deep into the surface region which lies beyond the position where in might be expected, owing to the nonuniform doping of the n+ region.

Quantitative understanding of these results is somewhat complex. Possibly the clearest explanation comes from applying Eq. (4.59) and following the result in Eq. (4.60). Using the plot shown in Figure 4.13 we can integrate the field to find potential as shown in Figure 4.14.

~0.2 iiij

~ 0

-0.2

-0.4 L-~~--,-~_.L.---,--~--'"_'---'--~--'-~~~-'

Figure 4.14: Potential versus distance (V = 0) for distribution in Figure 4.12.

Taking the p-side as the reference, one can somewhat arbitrarily pick4 the electric field point marked £(~n) = 1.3 X 104 V/ em and the corresponding change in potential fj.:IP(~n) = 0.76 V as an estimate of the depletion edge on the n-side. Using the total change in potential to that point and the given value of Ppo = 2.0 X 1015 cm-3 , we find that PnO(~n) = 3.45 x 102 cm-3 , which corresponds to nnO = nUPno(~n) = 4.46 x 1017 cm-3 . These values are consistent with the results depicted in Figure 4.12. Exact extraction of the depletion edge on the n-side and understanding the physical meaning of "edge" in light of the nonuniform doping in that region are not possible. Nonetheless, the self-consistent application of Eqs. (4.58) and (4.60) provide the essential physical basis to interpret the results shown in Figures 4.12-4.14. One further application of the results presented in Figure 4.12 is the calculation of the actual depletion layer width_ Given the highly asymmetric nature of the junction, we know that Ip > In. Thus to a good approximation the device is a "one-sided" junction where I(V) ~ Ip(V). Using Eq. (4.76) and the value of 1/Jo calculated using the results discussed above, we see

4The criterion used here is to extrapolate two lines to an intersection. The difference in slopes result from the n+ doping gradient compared to the np junction space charge.

the value calculated using Eq. (4.76). Unfortunately, this analytic result can only be applied for the uniform case. As we shall see, for more practical junction profiles, the p-region also becomes nonuniformly doped. Nonetheless, the results presented to this point are relatively easy to understand in light of the analytic results presented in Section 4.5.

Despite the very positive aspect of the physical results presented above, one can observe a somewhat anomalous result in Figure 4.12. Look closely at the electron distribution within the space charge region. Right at the junction, its value on the p-side exceeds the p-type dopant level. Hence, in considering the Poisson solution based on complete depletion at that point, the observed carrier distribution is actually contrary to the assumption that mobile carriers can be neglected compared to the ionized dopant atoms. If one integrates this charge, it constitutes 10-15% of the total and hence it is difficult to see a quantitative effect at this point, in light of the other difficulties related to In and 'l/Jo discussed above. With increased forward bias this charge plays an important role in device performance by adding to the device capacitance. Namely it is the so-called "neutral capacitance" discussed in [4.13].

Now, having considered primarily the equilibrium conditions in the pn junction, we will turn to the nonequilibrium bias conditions. Figure 4.15 shows the log plot of electron and hole carrier distributions versus distance for a forward bias of 0.6V. In contrast to the idealized plots shown in Figure 4.8, Figure 4.15 reveals continuous plots of the variables with no clear delineation of space charge and neutral regions. Moreover, the fall-off of carriers with distance looks quite different when plotted on a semi-logarithmic scale. One unique feature of simulation is that it allows us to look carefully at all important variables on a pointwise basis. Since the simulator has solved for 'I/J, n, and p everywhere in the device, this information can in fact be used to our benefit. Moreover, in the process of solving the Poisson's and continuity equations, key variables such as the net generation/recombination rate are also available. It is the careful inspection of these results which gives us the clearest insight into features of device operation. Figures 4.16-4.20 give the detailed listings of the point-by-point values of variables as solved for by SEDAN. In the discussion that follows we will give an accounting of the key variables and how their use for simple calculations will deepen our understanding of the device physics. The underlined values at a depth of 0.324 {lm provide a useful point for later calculations - this

Figure 4.19: Electrostatic and quasi-Fermi potentials versus x (li = 0.6V).

Figure 4.20: Net recombination and currents versus x (V = 0.6V).

point is in the middle of the space-charge region. First, consider the two key ways to express current flow - Eqs. (4.47) and (4.67) - considering only holes for the moment. Strictly speaking the expressions are equivalent, however the first is in terms of the diffusive and drift components while the second uses the quasi-Fermi potential as the driving force. By simply using the computed values shown in Figures 4.16A 4.20 we can gain some very specific insight concerning the device operation. Picking the point x = 0.324 J.lm which is in the middle of the space charge region, and considering only the first term in Eq. (4.47), we obtain

The observed terminal current should in fact be negligible compared to these terms in order to apply the equilibrium relationship used to derive the boundary conditions for injected carriers at the QNR edges. Thus the above rather simple discussion allows us to see the difference between the apparent flow of carriers, via the detailed drift and diffusion balance of the carrier distributions, and the true macroscopic driving force for real current flow - the quasi-Fermi potential.

Another rather simple but useful set of calculations relates to the recombination/generation rate and the change in current density over regions. Namely, by considering a bounded region, the change in current components - either hole and electron current - is determined by the integral of the net recombination over this region. Actually, the net recombination acts like a current exchanger - what loss one current component suffers is what the other gains, and the total current is preserved under the steady state. For example, if we consider the region from x = 0.32 J-Lm to x = 0.332 J-Lm we find that

This is in fact a direct consequence of Eqs. (4.50-4.51) for steady state. Although this result is easily seen from the mathematics, it poses two rather interesting consequences.

First, we find that the result obtained across a portion of the depletion region contradicts Figure 4.9 exactly in that we assumed!!.Q change in I n or Jp across the space charge. Second, as we look in the neutral regions, and particularly the p-region, we see a rather small change in current compared to either the space-charge or contact regions. The answer to both these apparent problems is that the neutral p-region does indeed have a comparatively low recombination rate. Moreover, the space-charge and contact regions do have a higher recombination

176 CHAPTER 4. PN JUNCTIONS

rate. Finally, the n+ region shows an unusually high recombination rate as well. The effect at the contacts can be easily seen and understood in terms of the boundary condition that returns the excess carrier concentrations to their equilibrium values. Most of the flux of carriers across the p-region is in fact needed to sustain the high recombination rate at the contact. For the large recombination rates in the n+ and space charge regions, two rather different physical effects are responsible as described below.

In Section 4.3 we derived the equation for net recombination rate and Eq. (4.38) represents a simplified form of that result. Subsequently in the last section we used this expression to estimate recombination current in the space-charge region. In so doing we made the crude approximation that of the total separation of the quasi-Fermi levels for holes and electrons, half of the bias went to increase each carrier population (Eq. (4.104)). Using this result along with Eq. (4.38) we could then integrate the net recombination and use Eq. (4.103) to estimate the total current at the low biases. For the results shown in Figures 4.16 through 4.17, we have all the necessary, detailed information to find the current component due to the recombination in the SeR exactly. That is, we have exact hole and electron concentrations at each point which can be used to compute the net recombination rate. In fact, that is what we have computed in the above summation of u for 0.32 fLm < x < 0.332 jlm. The primary reason that this change in current is much higher than that in the p-region is that both the hole and electron populations are enhanced across the space charge region. That is, although the electron densities continue to be high in the p-region, the relative excess of holes and electrons - the pn product - is not as strong a driving force as it is within the space charge region. One can see this from the outputted u value or compute it directly from the hole and electron concentrations.

4.7.1Heavy Doping Effects

The above discussion concerns only the space charge region. Yet there is still no immediately obvious reason that the recombination in the n+ region should increase. The answer comes from considering again Eq. (4.38) but this time focusing on the denominator. Namely, the lifetimes Tp and Tn depend on the density of trap/recombination centers, Nt, which is in turn related to the total impurity concentration [4.7]. In

fact it is the dependence of Nt on doping that accounts in part for the increased recombination. The increased doping directly increases the number of trapping/recombination sites for generation/recombination. In SEDAN and in PISCES the equation used to represent this dependence is [4.11]

where TO is the intrinsic lifetime, that is, the carrier lifetime in lightly doped material; NT is the total impurity density; and Nref is the onset concentration for the fall-off of T. The user of the simulators can adjust both TO and Nref, and the default values in SEDAN are 5 x 10-7 sec and 5 x 1016 cm-3, respectively. Hence, in the n+ region the lifetime is reduced by roughly 1,000 times compared to that in the space-charge and p-type regions. In addition to the lifetime dependence on doping, at very heavy doping levels the energy band structure is thought to change so that Eg is reduced as follows [4.9J:

(4.114)

where

where VI = 9 meV and the default value for user-accessible parameter No is 1017 cm-3. NT again is the total doping concentration. The result of this so-called "bandgap narrowing" is to reduce Eg and thereby increase ni, (as given in Eq. (4.1)) which in turn affects the net generation rate.

Finally, at very high carrier densities the recombination processes for silicon change. Direct band-to-band processes become significant in addition to the processes that occur through intermediate trapping/ recombination states. For materials such as silicon with an indirect bandgap, the recombination process requires a shift in carrier momentum to occur along with the energy transition. In the case of trapping/recombination levels, those sites provide the needed momentum shift. At high carrier densities, the large popUlation of electrons allows electron-electron interactions which can accommodate the needed momentum shift. Because this direct recombination between an electron