Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

178 CHAPTER 4. PN JUNCTIONS

and a hole is accompanied by the transfer of energy (and momentum) to another free electron (or hole), the parametric dependence of such a process, commonly known as the Auger recombination, involves a cubiclaw dependence on carrier concentrations. A simplified representation is given in [4.12] as

(4.116)

where Cn and cp are Auger recombination coefficients for electrons and holes, respectively. Their default values set in SEDAN are Cn = 2.8 X 10-31 cm6 /sec and cp = 9.9 X 10-32 cm6 /sec. The second term in the above expression looks very much like the numerator of Eq. (4.35) while the first parenthesized term represents the dependence on the needed extra carrier momentum shift. This first term also replaces the dependence on Nt, the trapping site/recombination center density.

The above discussion has given three major causes for the increased recombination in the n+ region. They are the increase of the density of recombination centers, the bandgap narrowing, and the onset of Auger SRR recombination. Specifically, both the lifetime and intrinsic carrier densities change due to increased doping. In addition, the increased carrier densities make direct band-to-band recombination possible by means of energy/momentum transfer to the third electron (or hole).

It is useful to become familiar with the modeling features of SEDAN which can be used to explore the physical effects. Referring to the model card input statement for SEDAN, there are three sets of variables that are of special interest:

1.SRR recombination/generation parameters

2.Auger (AUG) recombination/generation parameters

3.Bandgap narrowing (BGN) coefficients

and in addition,

4 Series Resistance (SRE) parameters

in the device card. We will now briefly discuss the features associated with each set of coefficients. The SRR recombination/generation formula follows that given by Eqs. (4.34) and (4.113). The SRR statement requires a logical (-srh if you don't want it be considered) input

whereas the other parameters require numerical values as listed below:

The final angle-bracketed values indicate the defaults used by SEDAN if none are given and only the keyword srh is specified. For the Auger recombination term, Eq. (4.116) is used. The other two parameters are the coefficients in Eq. (4.116) so that

where the bracketed values again denote the default values used if none is specified. Similarly, the bandgap narrowing is activated by the "bgnw" statement as a logical input. The final parameter of special interest for diode calculations is the series resistance parameter which is set by the logical "sres" statement. The following parameters may be specified:

Since SEDAN works in terms of current densities, the "ear" factor is used to determine total current so that external (also called extrinsic) voltage drops can be computed. Having discussed these model parameter inputs we can now consider real device structures and see the results of both the technology and the physical models.

The role of the above physical models (excluding for the moment the series resistance effects) is strongly dependent on doping level. While the example used in this section utilizes a highly doped n+ region, there was no specific correlation of the physical structure to a fabrication sequence. The profile used for the calculation is shown in Figure 4.12(a). Figures 4.21-4.23 compare plots of u( x) vs. x for several different assumptions about the lifetime, bandgap narrowing, and Auger recombination terms using the physical profile shown in Figure 4.12 (a) (the uniform substrate). Bandgap narrowing and the change in lifetime with doping playa major role in controlling the shape of the curves for u. Of lesser importance is the Auger recombination. In part this is due to the

-. - SRH+BGN

-- SRH+Auger+BGN

10'6 L-~~~~~~~~~~~-L~~~~~~~~

Depth (urn)

Figure 4.23: Adding band gap narrowing (also the effect of deleting Auger).

fact that the doping level is below the solid solubility. Both the BGN and SRH terms have a strong dependence on doping and the No terms. In Figures 4.21-4.23 we have simply varied the model coefficients to see these effects.

4.7.2Analysis of High-Level Injection

So far, we have discussed the detailed mechanisms involving current continuity and recombination process in various regions ofthe n+p junction. We now extend our discussion to higher bias levels and consider such secondary effects as high-level injection and ohmic drops in the neutral regions. Figure 4.24 shows a semi-logarithmic plot of current versus voltage for the same n+p diode considered in the above example. From the plot we can see several very distinct regions of device behavior. The point labeled C corresponds to the bias level used for the discussion given above with reference to Figure 4.15. The two dashed lines above and below the solid line correspond to slope values of q/2kT and q/kT, respectively. As discussed in Section 4.6, these values, respectively, correspond to the total domination of the simplified space-charge

5

4

3

2

.. 0

~

~ -I

~

'E- -2 ~ -3

:l

U -4

~-5

-6 -7

-8

-9

-10

Applied Volloge

Figure 4.24: Semi-logarithmic plot of I vs. V for the n+p diode. Points indicated are "A" low-level injection where space charge recombination dominates, "B" medium-level injection ("ideal" slope), "C" knee of high-level injection, "D" high-level injection.

recombination model and to the ideal diffusion-dominated flow of minority carriers in the neutral regions. The slope around the point A is somewhere between these limits. The discussion and analysis associated with Figure 4.15 should help to clarify the meaning of the voltage dependence observed.

Specifica.liy, both space charge recombination and neutral region diffusion currents are responsible for the observed curve. The point labeled B corresponds to an "ideal" I - V dependence. For biases above point A there is a rapid increase in injected carriers outside the space charge region and within the space-charge region itself the recombination increases only at the rate of eQV/2kT, which can soon be neglected with respect to the more rapidly increasing eqV/ kT term of carrier injection. Hence, at point B, and for several hundred millivolts around that point, the slope follows the nearly ideal q/ kT dependence. At the point labeled D in Figure 4.24, we observe a dramatic level-off from the ideal q/kT

-doping

_. - ole

--hoi"

10' ~~~~L.....~~~,l......,~~~...1..-~~~

Figure 4.25: Doping and carrier distribution for n+p diode at V = 0.9V.

slope as observed for point B. At this point we have encountered highlevel conditions. These consist primarily of several physical effects, all related to non-negligible field-effects in the neutral regions. To understand these conditions more completely, Figure 4.25 shows the carrier distributions at this bias point and Figures 4.26-4.30 give the tabulated values of essential variables. From Figure 4.25 one fact is immediately apparent - the injected electron density now exceeds the acceptor doping concentration. Generally speaking, this is a necessary condition for high-level injection. At this point, the usual assumption concerning drift and diffusion for minority carriers become invalid. Namely, one can no longer neglect the drift component for minority carriers. Moreover, the combined electron and hole distributions are very much affected by the electric field. In fact, at these current levels, the voltage drop in the neutral regions can be significant. One can see this by using the electric field and carrier density information given in Figures 4.26-4.30 to compute the actual drift component of the electron current. For this particular bias condition, the drift component is 50% of the total electron current at x = 1.0/-Lm. Computing the total drop of electrostatic potential in the p-region from x = 0.33/-Lm to x = 2.0 J-lm we see that it is 140 mV. It is important to note that this voltage is the total voltage necessary to satisfy the Poisson's equation - both in modulating the space charge and sustaining the mobile charge distributions. In Figure 4.24, the voltage is depicted as well as the extrapolated ideal

Figure 4.27: SEDAN output of electric field versus depth (n+p diode, V = O.9V).

which now has the factor of two in the denominator of the exponent and VHL marks the bias at the HL condition. By comparison, for the low-level (LL) condition we normally assume

as determined in Eq. (4.65).

By equating n(lp) with p(lp) in the above equation we find that Eq. (4.118) reduces to exactly Eq. (4.117). The fact that the minority carriers now reach the level of the majority carrier density and that both increase with further applied voltage gives the result that the high-level injection requires greater applied voltages across the space-charge region (in this case, actually the metallurgical junction, for the SCR drastically diminishes under high-level injection) to maintain the same current if

Figure 4.28: SEDAN output of mobility values versus depth (n+ p diode example).

the edge of HL injection. Using the injected electron concentration of 3.5 x 1017 cm-3 at the physical junction (x = 0.33 JLm) in Eq. (4.117) one finds that V = 0.888 V. In turn, this would require that only 12 mV be dropped elsewhere in the device. Based on this improved estimation of the boundary condition for carriers, we are now faced with only 12 mV compared to the earlier estimate of 135 m V drop. In the following we will consider both the electrostatic potential and then return to the evaluation of how the applied voltage appears across the device.

The discussion begins with consideration of the hole and electron concentrations for 0.33 JLm < x < 2.0 J.Lm (Le. from the metallurgical pn junction to the contact to the p-region). From Figure 4.25 we see that n(x) ~ p(x) as we might expect for a charge-neutral region. To get a first-order constraint on carrier distributions and fields, let us assume a linear fall-off for electrons. Hence, at xo = 0.33 JLm, no = n(xo) = 3.5 x 1017 cm-3 (from SEDAN) and at the p-region contact,

Figure 4.29: SEDAN output of net recombination and electron and hole current densities versus depth (n+p diode, V = 0.9V).

where ~x = X-Xo is in units of pm, and npo is the electron concentration in the p-substrate at the equilibrium. Up to the point where n(~x) = Ppo, we may also use the above expression for the hole concentration.

In the region where n ~ p, we can conclude that the electron component of the current should be much larger than that of its hole counterpart. The reason is clear: the supremacy of the majority carrier in dominating the total current in the QNR due to drift current is no longer valid as the populations of both minority and majority carriers become roughly equal. On the contrary, in this particular example, the drift component of the minority carriers (electrons) even exceeds that of majority carriers (holes) because of the fact that the electron mobility is larger than hole mobility. Furthermore, the direction of drift and diffusion components for electrons are the same in the p-type quasi-neutral