Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

where CA is the substitutional (or otherwise immobile) impurity concentration, CAY is the concentration of impurity atoms paired with vacancies and CAl is the concentration of impurity atoms paired with interstitials. Since the diffusion takes place by means of the point defect mechanisms, only those impurity atoms paired with point defects will diffuse through the crystal. With this point in mind, the flux of dopants due to diffusion can be written as [2.36]

where the negative sign has been neglected.

Assuming that the relative contributions of vacancies and interstitials do not change significantly over the region of interest, using the chain rule, Eq. (2.29) can be written as

(2.30)

where CA. represents an effective dopant concentration, assumed to be mobile, and the bracketed term represents an effective diffusivity D A •• The use of an effective diffusivity is a major convenience. To simplify further, we define the fraction of dopant atoms diffusing by means of interstitials as:

where CAl is the ratio of dopant atoms diffusing with interstitials to the total dopant concentration, CAr/CAo, and CAV is the ratio of dopant atoms diffusing with vacancies to the total dopant concentration, CAV / CA•. Thus, if one wants just the interstitial component of dopant flux, it can be approximated as

68 CHAPTER 2. INTRODUCTION TO SUPREM

This equation is certainly the easiest to implement in SUPREM since there are relatively few coefficients. However, shortcomings result from the lack of kinetics to dynamically account for the coupled effects involved in diffusion. In particular, the interstitials and vacancies interact strongly and directly via recombination mechanisms, and the impurities "feel" the effect of the impurity concentrations by enhanced diffusivity. In advanced 2D versions of SUPREM more exact approximations have been included (2.37].

In SUPREM III, Eq. (2.34) is used to model the role of interstitials in the diffusion process. However, the values used for fA! are much lower than that expected based on the definition given in Eq. (2.31) and the experimental evidence [2.35]. The reason is one of convenience.

The vast majority of available data and extracted coefficients have been interpreted only in terms of the vacancy model D:4.v' As we will see

later, the fAl term is indeed important in modeling oxidation enhancement effects. Hence, its value must be non-zero. However, by keeping the value small compared to unity the existing body diffusion models can be directly used in SUPREM III.

2.4.2Concentration Dependent Diffusion

The models used in SUPREM for the concentration dependence of diffusivity are based on vacancy diffusion mechanisms (Figure 2.14 (a)) under non-oxidizing conditions. This is somewhat of an expedience because such models have been developed quantitatively to the point where they can be implemented in a general-purpose simulator. This is not true at the present time of interstitial-based diffusion models. However, as shown above, the definition of fA! allows us to attribute a portion of the diffusivity to the interstitialcy component. The vacancy model, coupled with numerical solution of the diffusion equation, does provide satisfactory agreement with experiment. Using the vacancy model, we can consider the diffusivity to be altered by the presence of charged vacancies which interact with the dopant. Such vacancies exist in neutral, positively charged, negatively charged and double-negatively charged states. In general, the effective impurity diffusion coefficient can be expressed as [2.38J

(2.35)

Figure 2.15: Semi-logarithmic plot oftotal arsenic concentration profiles diffused under two conditions. The upper curve shows concentration enhanced diffusion. The lower curve shows results from intrinsic diffusion.

Several features of Eq. (2.42) are worth discussing. Obviously, as n increases, arsenic diffusivity increases. This effect is observed frequently since arsenic is most often used for high concentration contacts and emitter regions. Not so obvious is the fact that arsenic diffusivity decreases in p-type silicon in which case n/ni is less than 1. In other words, the D As component of diffusivity is suppressed in p-type material, which "retards" diffusion as compared to the intrinsic case. The SUPREM program allows the user to modify the Dis and DAs terms directly. Figure 2.15 shows a comparison of calculated diffusion profiles for arsenic under intrinsic and extrinsic conditions. Both calculations were made at the same processing time and temperature. It is clear from the figure that diffusivity is enhanced substantially for increased

Figure 2.16: Log-log plot of arsenic diffusivity versus concentration. The straight line dependence at high concentrations agrees well with Eq. (2.43).

doping.

It should be noted that SUP REM allows for the use of both negatively charged and double-negatively charged vacancy's components in the diffusion expression. This corresponds to the general form of the model as given by Eq. (2.35). For arsenic, the doubly negative term is not used, whereas for phosphorus both the D- and the D= terms are used. This difference can be regarded from the user's point of view simply as a matter of data fitting. The best numerical fits to experimentally obtained profiles end up using both D- and D= for phosphorus and only D- for arsenic. Figure 2.16 shows the agreement of the model to measured data for the default arsenic model coefficients used in SUPREM

72 CHAPTER 2. INTRODUCTION TO SUPREM

/3=0.5

fv

Figure 2.17: Log-log plot of normalized diffusivity vs. concentration (normalized to ni) using the simplified Eq. (2.44) [2.40].

[2.38]. Because of the physical uncertainties at present concerning the diffusion of phosphorus, the model for extrinsic phosphorus diffusion will not be discussed further here. For the examples related to the Stanford CMOS process, extrinsic doping effects can be disregarded since the phosphorus diffuses under intrinsic, or nearly intrinsic, conditions. One final observation concerning the dependence of diffusivity on doping is useful. If we rewrite Eq. (2.42), a rather handy empirical form emerges:

where DAs! = Dis + D As and represents the intrinsic value of diffusivity. This is the model previously used in SUPREM II. The coefficient {3 is chosen to properly represent the transition from the neutral to the negatively-charged defect as the major contributor to diffusivity. Figure 2.17 shows a plot of normalized D vs. n/ni for different values of {3. The enhancement effect for n > ni and the retardation effect for

n < ni are clearly visible. For SUP REM II, the default parameters are {3 = 100 for arsenic and {3 = 3 for boron. However, based on the present SUP REM III default numbers, a value for boron of {3 = 20 seems to give a better fit to the data.

It is important to remember that diffusivity is retarded in the case of counterdoping at levels above ni. For example, for diffusion of boron in a highly n-doped region, the n+ term will be substantially suppressed compared to the intrinsic condition. Similarly, for phosphorus diffusion in a heavily p-doped material one would expect reduced diffusivity. These effects will be demonstrated and discussed again later by means of examples. At this point we will move on to other diffusion-related effects.

2.4.3Dopant Clustering

Although diffusivity increases markedly as n becomes larger than ni, there is a limit to the electrical activation possible for a given dopant at processing temperatures. Above a certain concentration the dopant becomes inactive and ceases to move by the normal diffusion process. As yet there is some uncertainty as to the exact reasons for this inactivity. One line of thought is that the dopants become clustered, meaning that they choose to locally associate with other atoms in a tightly bonded configuration. Alternatively, it may be the case that precipitation occurs above the solid solubility limit of the dopant in silicon, giving rise to another phase of material. For example, an arsenic silicide may be formed within the silicon lattice. Whichever case occurs physically, the result can be represented by removing a fraction of the total dopant from the diffusing stream.

Using the clustering representation, the electrically active portion of the dopant can be separated from the total dopant concentration as follows:

where Ga is the concentration of clusters, m is the cluster size, i.e. the number of atoms in each cluster, GT is the total chemical concentration, and GTA (e.g. As+) is the concentration of active and therefore diffusible impurities.

The following reaction describes one formulation of the clustering-

kd

where ke and kd are the clustering and declustering coefficients, respectively, and represent here the clustering and declustering processes. The dynamic relationship involving the clustering process is given by the following continuity equation:

(2.46)

where n is the electron concentration, G is the term for the generation of clusters and is proportional to nCfA from the law of mass action, and R is the recombination term, corresponding to declustering and proportional to the cluster concentration Cc. Combined with Eq. (2.44) we can create an equation for evaluating Cc as a function of time, and consequently the diffusive flux of CTA.

While the physics of such a formulation appear to be correct, the implementation can result in prohibitively long computational times. Namely, the time constants (ke and kd) and the diffusivity give rise to orders-of-magnitude difference in the time step required to control the accuracy. Hence, as for the case of point-defect modeling, a simplified approach is used, which is more consistent with computational resources typically used for process modeling.

In the case of SUPREM III, an equilibrium relationship is applied to Eq. (2.46) so that the reaction is balanced and the clustering/declustering occurs instantaneously based on this relationship. The equilibrium coefficient, defined as J(eq = ke/kd' is determined by Eq. (2.46) as:

where the square brackets indicate concentrations and n is the free electron density. Using this equilibrium coefficient and substituting it into Eq. (2.44), the following expression results in:

(2.48)

where m in this case is 3. In SUPREM III the dependence on electron concentration is in fact not included, so the expression simply involves

DEPTH (f/-m)

Figure 2.18: Semi-logarithmic plot of arsenic concentration versus depth. Both total (CT) and active (CTA) are shown [2.40j.

two terms with CTA. The results of this clustering representation as it impacts a diffused profile are shown in Figure 2.18. In this figure, the total arsenic concentration, measured by Rutherford backscattering, is shown along with the electrically active dopant profile [2.40j. Substantially less arsenic is active than the total concentration for values near 1020 cm-3 , a number closely related to the solid solubility. There are still discussion and debate on the details and use of clustering models, given the possibility of precipitation above the solid-solubility limit. At present SUPREM has the capability to model clustering by the relationship given in Eq. (2.48). The user may alter the coefficients in this equation, use default values, or turn the model "off."

2.4.4Dopant Segregation

The diffusive effects for low dopant concentrations, while "intrinsic" in terms of carrier concentrations, are strongly influenced by extrinsic effects such as oxidation. The most well known of these effects is the segregation phenomena, whereby dopants have a physical preference for