Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

L J

(a)

p'

(b)

Figure 3.11: Two views of pn junctions that occur in integrated circuits,

(a) shows the parasitic pn junctions associated with an n-channel MOS device, (b) shows both the intrinsic junctions (solid lines) and parasitic junctions associated with a bipolar transistor.

and/or compare with the points discussed above.

3.4.5Analysis of a pn Junction Diode

The pn junction diode is a critical component for both MOS and bipolar integrated circuits. For MOS technology the junction diode represents an unavoidable parasitic associated with source and drain regions (see Figure 3.11 (a)). For bipolar technology, the diode appears both as an essential control variable for device operation and as a parasitic device, as shown in Figure 3.11 (b) for a junction-isolated bipolar technology. In the following discussion and example, we will consider the source n+p diode shown in Figure 3.11 (a). In order to have the most realistic values and data with which to work, we will choose the CMOS technology discussed in Chapter 1 and in particular we will use the technology

cross section shown in Figure 1.17.

Figure 3.12 (a) shows the appropriate SUPREM input deck (a minimum specification) needed to create the profile shown in Figure 3.12

(b). We have modified the substrate profile somewhat compared to Figure 1.17 in that the extra boron implant used to adjust the n-channel threshold is omitted for simplicity. Hence the boron profile is flat in Figure 3.12 (b), whereas in the real process it is increasing toward the junction as observed in Figure 1.17.

As discussed and considered in Chapter 2, the high concentration arsenic profile shows a flat region at high concentrations due to the enhanced diffusivity in that region. In one sense this profile shows the most ideal profile from the perspective of a step junction as is frequently assumed in the literature. Based on this profile we will now use PISCES to understand in greater detail many of the points discussed in Section 3.4.1.

Figure 3.13 (a) shows a PISCES input deck which demonstrates the principles of junction operation discussed in Section 3.4.2. Several output plots are specified in order to demonstrate PISCES capabilities and to further explore the diode effects discussed above. Figures 3.13 (b) and (c) show electric field and potential plots across the device both for V = 0 and V = O.5V. Comparing these numerical results with the analytic results presented in Figure 3.8 (c) and (d) we find that they look qualitatively similar, which supports the assumptions made in Section 3.4.3 for device analysis. In particular, the electric field is indeed contained in a narrow "space-charge" region. Also, both for V = 0 and V = O.5V virtually all change in the potential plot occurs across the space charge as was suggested in the discussion of Figure 3.9 (c).

Yet in addition to these qualitative features which show good agreement, there are several features apparent in Figure 3.13 which cannot be explained so easily based only on the discussion given in Section 3.4.2. Two specific points are as follows. First, the electric field plot, Figure 3.13 (b), shows two distinct regions - one extending a significant distance into the n+ region. The electric field peak results from the nonuniform doping profile of arsenic. That is, there is an electric field needed to hold the electron distribution in the shape determined by the arsenic profile. The more gradual tail region in fact corresponds to the constant doping result discussed in Section 3.4.2. With applied bias one can note that while the £(x) distribution changes significantly in the neighborhood of the space-charge region, the portion of £(x) vs. x

(b)

Figure 3.12: SUPREM simulation input and output for the n+ source/drain region of an nMOS device: (a) shows a simplified SUPREM input file and (b) the resulting n+ junction profile into the p- type substrate.

(b)

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- eJ .4 13.121121 121.2121 121.'1121 121.6121 121.8121 1.11Jrll 1.20 1."t12l 1.6

Distance (um)

(c)

Figure 3.13: (Cont'd) (b) shows the electric field versus distance for two bias conditions, and (c) shows the potential for the same bias conditions.

in the n+ region changes negligibly. As we will see in later discussion, this can be represented by a bias-independent expression for [o(x). If we next look at Figure 3.13 (c) we can see that indeed the majority of t::..'ljJ does occur across the space-charge region. Looking at the effect of [o(x) on potential, we see that indeed it does not really affect where the applied voltage appears. The second discrepancy observed in the plots shown in Figures 3.13 relates to the definition of Xn and xp. Looking specifically at the space-charge edge on the p-side, we see a significant "smearing" of [(x) as it approaches zero. In fact, from these plots it is difficult to unambiguously define a space-charge edge. Our above discussion suggests that the use of a "complete depletion" approximation would be appropriate. We will find that this approximation does indeed help to define a depletion edge on the p-side. However, if the p region is nonuniformly doped, the problem quickly becomes intractable using the simplest notion of a depletion edge since an [o( x) exists for some distance beyond the extrapolated x n .

To explore this point further we can plot the hole and electron distributions along with the doping profiles. Figure 3.14 (a) shows this plot for V = 0 and Figure 3.14 (b) shows the results for V = O.5V. In contrast to Figure 3.13 with electric fields, which has a rather simple set of interpretations, these plots are rather formidable. Specifically, the vertical axis is logarithmic, which is not the easiest to understand. Be this as it is, let us briefly review what we have assumed based on simple first-order heuristics. Looking at Figure 3.14 (a) we see that the hole concentration has a simple-looking dependence with respect to the dopants. Namely, it falls off rapidly from NA (x), which is consistent with our earlier discussion of complete depletion. From the data analysis point of view it is not so clear how to define "complete depletion." One fairly reasonable assumption is to pick the point xp such that

where 7] is a number such as 2 or 4. That is, select an xp as the point at which the hole concentration is 1/2 or 1/4 of the doping concentration. Because of the nature ofthe data shown in Figure 3.14, the values of xp chosen in this way will most often be a unique nearest-neighbor point in the PISCES data (that is, the error due to the choice of 7] is not large). By contrast, looking at the comparison of n(x) and ND(X) we see a very confusing picture. First, we observe that any estimated Xn is more

Distance (uml

(a)

21 net doping

221

19

(Y)18

17

D1stance (uml

(b)

Figure 3.14: Semi-logarithmic plots of concentration versus distance for the n+p junction under (a) equilibrium (V = O.OV) and (b) a forward bias of V = O.5V.

than an order-of-magnitude smaller than xp. The integral f~xn ND( X )dx should equal Ie? NA (x )dx to conserve charge, where the junction position is chosen as the origin. But clearly, due to the logarithmic vertical scale, the n+ side requires only a tiny distance for Xn owing to the much higher values of ND (Le. ND » NA). Thus, trying to resolve Xn from this plot is probably a futile exercise. Moreover, if we try to define a condition similar to that used for the p-side in Eq. (3.34), we come to a contradictory point of view. Specifically, looking at Figure 3.14 (a) in the n+ region we see that n actually exceeds ND near the junction. As shown more clearly in Figure 3.13 (b), this is the local electric field [o(x) which is needed to compensate for the effects of nonuniform arsenic doping. Thus if we simply look for a depletion in n with respect to ND we can indeed be confused by the [(x) plot. Fortunately from a device performance point of view these local features of the [(x) plot have a minor impact on the current-voltage behavior. That is, since the n+ region injects substantially more electrons into the p-region than the p-region injects holes into the n+ region, we will be able to correctly determine current flow and charge distributions without having significant errors due to the local t:o(x) effects.

Figure 3.14 (b) shows the same plot of hole and electron distributions but for a forward bias of V = O.5V. Again we can see features similar to those discussed above. However, we can now also see the significant effect of electron injection into the p-region. First, if we qualitatively compare the excess electrons and their slope in the p-region with the excess holes and their slope in the n+ -region, we can see that the component of JT (the total current) due to electrons is significantly larger than that due to holes. Specifically, we can confirm the following

where Xl and X2 are chosen somewhat arbitrarily and we have ignored the local effects of [o(xI) for the holes at x = Xl, although this could easily be included so that we can show that

(3.36)

At this point it is useful to point out and expand on the observation made in Section 3.4.4. As we look at the plots of p( x) and for

126 CHAPTER 3. DEVICE CAD

the minority carriers, analytically we can think of Eqs. (3.26-3.27) (ignoring for the moment the effect of to( x)). As we choose x values different from the Xl and X2 points indicated in Figure 3.14 (b), the slopes change. This indicates that recombination is occurring, and the change in slope reflects this reduction in population of minority carriers. However, aside from this qualitative observation, it is difficult to be too specific about how much recombination is occurring. Moreover, across the space-charge region there is a dramatic change in concentration but here it is virtually impossible to distinguish the rapid slope changes observed in Figures 3.14 (a) and (b). This brings us back to the further discussion of general use of Eqs. (3.32-3.33) in numerical simulation.

Figures 3.15 illustrates the use of Eqs. (3.32-3.33) and shows how the numerical simulation results can be most helpful. Figure 3.15 (a)

shows the point values of u(x) while Figure 3.15 (b) shows the values for Jp(x) and In(x) over the same range for a forward bias of V = O.5V. A very prominent feature of the u(x) curve is the sharp peak in the space-charge region. As we will see in Chapter 4, this result is not at all surprising given the nature of recombination statistics. However, there is no way we could have directly determined this result from Figure 3.14.

As indicated in Eqs. (3.32-3.33), we can consider the change in current across any region Llx to be the integral of u( x) across this region. Looking specifically at the space-charge region roughly defined between points Xl and X2 in Figure 3.14 (b), we can say that the decrease in

Readers with an advanced background in pn junction theory may recall that in an ideal theory, this change in I n and Jp due to "space-charge recombination" is usually assumed to be small compared to the integral of u from x = Xl to the p-side contact. This assumption does not seem totally justified for the results shown in Figures 3.15. By choosing a larger forward bias, we could drive the results closer to the assumed ideal case. Again, we will come back to discuss these points further in Chapter 4. However, to summarize the key points of Figure 3.15 the following observations are useful. First, the plot of u(x) is indeed helpful to understand where generation/recombination occurs and to gain a quantitative picture of the deviation from equilibrium. Such a picture cannot be easily seen just from the differential form of the equations used for device analysis. The second key point of Figure 3.15 is that the conservation of current across a diode can most easily be observed

(b)

Figure 3.15: Semi-logarithmic plots of recombination and current components versus distance, (a) shows the net u while (b) shows the hole

and electron (In) current densities.