# Dopant-Dependent Oxidation Modeling in ATHENA

Introduction

The fabrication of integrated circuit microelectronic structures and devices vitally depends on the thermal oxidation process for the formation of gate dielectrics, device isolation regions, spacer regions, and ion implantation mask regions. It is well known that silicon dioxide (SiO2 ) formation on highly-doped n-type and p-type substrates can be enhanced compared to SiO2 formation on lightly-doped silicon substrates [1-6]. Of prime importance is the precise control of the oxide thickness as device geometries continue to scale to deep sub-micron dimensions. The general-purpose process simulator ATHENA includes numerical thermal oxidation models, allowing the substrate dopants influence on the oxidation kinetics to be simulated. This article is intended to review the physically-based dopant-dependent oxidation models implemented in ATHENA and to demonstrate the compatibility of thermal oxidation simulation results to measured data.

Model Description

The oxidation process is modeled by considering three steps: transport of oxidant across the ambient/SiO2 interface, diffusion of oxidant molecules across the growing SiO2 layer, and reaction of oxidant at the silicon/SiO2 interface [7]. The numerical oxidation models (COMPRESS and VISCOUS) implemented in ATHENA solve the oxidant diffusion equation at incremental times steps at discrete grid points in the growing SiO2 layer. The diffusion equation is given by

(1)

where C is the oxidant concentration in SiO2, t is the oxidation time, and F is the oxidant flux (the number of oxidant molecules crossing a unit surface area of SiO2 in unit time). To solve equation (1), the oxidant flux needs to be specified in the growing SiO2 layer and at material interfaces with SiO2.

At the ambient/SiO2 interface oxidant is transported across the interface and the corresponding flux is given by

(2)

where h is the gas-phase mass transport coefficient, C* is the equilibrium oxidant concentration in SiO2, C0 is the oxidant concentration in SiO2 at the ambient/SiO2 interface, and nout is a unit vector normal to the ambient/SiO2 interface directed from silicon to SiO2.

The diffusion of oxidant molecules in SiO2 is driven by a concentration gradient and is expressed as

(3)

where D is the oxidant diffusivity in the growing SiO2 layer, and C was defined in equation (1).

At the silicon/SiO2 interface the oxidant reacts with silicon atoms to form a new layer of SiO2 and the corresponding flux is given by

(4)

where ks is the reaction rate constant, Ci is the oxidant concentration in SiO2 at the silicon/SiO2 interface, and nin is a unit vector normal to the silicon/SiO2 interface directed from SiO2 to silicon. At other material interfaces with SiO2 the oxidant flux is set equal to zero. Note that the equations describing volume consumption of silicon and volume expansion of SiO2 have not been presented here. Detailed description of these equations can be found in SILVACO literature.

From equations (1-4) and considering steady-state conditions, the familiar Deal-Grove linear-parabolic growth law [7] can be derived. Silicon dioxide growth on extrinsic silicon substrates can be modeled [1] by a modification of the Deal-Grove linear-parabolic growth law. The dependence of silicon dioxide growth kinetics on doping concentration is manifested as part of the linear rate constant, where the physical significance of the high doping levels has been explained primarily as an electrical effect [1-3]. The modified linear rate constant, including the doping dependence, becomes [1]

(5)

where (B/A)i is the linear rate constant on intrinsic silicon, and

(6)

where V is the vacancy concentration in silicon at the silicon/SiO2 interface, Vi* is the equilibrium vacancy concentration in intrinsic silicon, and K is an Arrhenius experimentally-determined coefficient.

In general, the doping type and concentration level in the silicon substrate cause a variation in the location of the Fermi level. A shift in the Fermi level alters the equilibrium vacancy concentration in the silicon substrate [8, 9]. Figure 1 shows the functional dependence of the equilibrium vacancy concentration at 950 oC versus doping concentration for commonly used silicon dopants. The physical significance of an increase in the vacancy concentration is an increase in the number of available reaction sites for the incoming oxidant, which in turn enhances the oxidation rate.

Figure 1: Equilibrium Vacancy
Concentration in Silicon Versus Doping

Concentration for Common Silicon Dopants at 950 C.

The influence of doping concentration on SiO2 thickness can be seen in Figure 2, where a large enhancement (with respect to the lower doping concentrations) in SiO2 thickness is observed on highly-doped n-type substrates, and less enhancement for the p-type dopant. This trend is consistent with Figure 1, where the equilibrium vacancy concentration at high doping levels is larger for the n-type dopants than the p-type dopant.

Figure 2: Simulated Silicon Dioxide
Thickness Versus Doping

Concentration for Common Silicon Dopants.

For completeness, Figures 3-6 show plots of SiO2 thickness versus doping concentration for boron, phosphorus, arsenic, and antimony substrates respectively, with temperature as a parameter. These four figures can be used to easily make "eye-ball" predictions of the SiO2 thickness dependence on temperature and doping concentration for the given oxidation conditions.

**Figure 3: Simulated Silicon
Dioxide Thickness Versus Boron
Concentration with Temperature as a Parameter.**

**Figure 4: Simulated Silicon
Dioxide Thickness Versus Phosphorus
Concentration with Temperature as a Parameter.**

**Figure 5: Simulated Silicon
Dioxide Thickness Versus Arsenic
Concentration with Temperature as a Parameter.**

**Figure 6: Simulated Silicon
Dioxide Thickness Versus Antimony
Concentration with Temperature as a Parameter.**

Simulation and Experimental Results

A number of experiments have previously been performed in order to better understand the physics of dopant-dependent oxidation on <111> silicon [3, 4] and <100> silicon [5, 10]. In this section, results predicted by the model will be compared to measured data from oxidation experiments on <100> silicon.

Figures 7-9 show a comparison between simulated
and measured [5] SiO2 thicknesses as a function of time for three
temperatures, where the substrate doping conditions were B = 1x10^{16}
cm^{-3}, B = 6x10^{19}
cm^{-3}, and P = 8x10^{19}
cm^{-3}, respectively. The agreement
between the simulated and measured SiO2 thicknesses
is very reasonable over the time and temperature range considered.
The simulations were performed with the pre-exponential of K in
equation (6) equal to one-fourth of its reported value [1]. Decreasing
the value of K might be justified by considering that the original
reported value was extracted from oxidation experiments on <111>
silicon; whereas the simulation results in figures 7-9 are for oxidations
on <100> silicon.

**Figure 7: Comparison of Simulated
and Measured Silicon Dioxide Thicknesses for
Lightly-doped Boron (B = 1x10 ^{16}
cm^{-3}) Substrates.**

**Figure 8: Comparison of Simulated
and Measured Silicon Dioxide Thicknesses for
Heavily-doped Boron (B = 6x10 ^{19}cm^{-3})
Substrates.**

**Figure 9: Comparison of Simulated
and Measured Silicon Dioxide Thicknesses for
Heavily-doped Phosphorus (P = 8x10 ^{19}cm^{-3})
Substrates.**

Figure 10 shows simulated and measured [10] SiO2
thicknesses versus time for oxidation of heavily-doped phosphorus
substrates (P = 1.8x1020 cm^{-3})
for four different temperatures (800 C,
850 C, 900 C,
and 950 C). These simulations
used the same coefficients as those for Figures 7-9. The agreement
shown for each temperature in Figure 10 helps to justify decreasing
the value of K for dopant-dependent oxidation on <100> silicon.

Figure 10: Comparison of Simulated
and Measured Silicon Dioxide Thicknesses for

Heavily-doped Phosphorus (P = 1.8x10
cm^{-3}) Substrates.

Conclusion

The dopant-dependent oxidation models implemented in ATHENA have been reviewed and it has been shown that the SiO2 thicknesses predicted by the oxidation models agree reasonably with experimentally-determined SiO2 thicknesses. It might seem feasible to decrease the value of K in equation (6) for oxidation simulations on <100> silicon because the default value of K was extracted from oxidation experiments on <111> silicon. Further, the use of simulation for silicon thermal oxidation can further aid engineers in determining process conditions for deep sub-micron devices.

Parameters Used in This Work

The tuned parameters used to match the measured data presented here will be available by default in the next release. Users of the current ATHENA release can reproduce the results by introducing the following syntax:

oxide silicon baf.pe = -0.46 baf.ppe = -1.0 \ baf.ne = -0.145 baf.nne = -0.62 \ baf.k0 = 6.5e2

References

[1] C. P. Ho, and J. D. Plummer, "Si/SiO2 Interface Oxidation Kinetics: A Physical Model for the Influence of High Substrate Doping Levels, I. Theory, " J. Electrochem. Soc., Vol. 126, No. 9, pp. 1516-1522, 1979.

[2] C. P. Ho, and J. D. Plummer, "Si/SiO2 Interface Oxidation Kinetics: A Physical Model for the Influence of High Substrate Doping Levels, II. Comparison with Experiment and Discussion," J. Electrochem. Soc., Vol. 126, No. 9, pp. 1523-1530, 1979.

[3] C.P. Ho, J. D. Plummer, and J. D. Meindl, "Thermal Oxidation of Heavily Phosphorus-Doped Silicon," J. Electrochem. Soc., Vol. 125, No. 4, pp. 665-671, 1978.

[4] B.E. Deal and M. Sklar, "Thermal Oxidation of Heavily Doped Silicon, " J. Electrochem. Soc., Vol. 112, No. 4, pp. 430-435, 1965.

[5] E. A. Irene and D. W. Dong, "Silicon Oxidation Studies: The Oxidation of Heavily B- and P-Doped Single Crystal Silicon," J. Electrochem. Soc., Vol. 125, No. 7, pp 1146-1151, 1978.

[6] S. M. Sze, VLSI Technology, Ch 3, McGraw-Hill, New York, 1988.

[7] B. E. Deal, and A. S. Grove, "General Relationship for the Thermal Oxidation of Silicon," J. Appl. Phys., Vol. 36, No. 12, pp. 3770-3778, 1965.

[8] W. Shockley and J. L. Moll, "Solubility of Flaws in Heavily-Doped Semiconductors, " Physical Review, Vol. 119, No. 5., pp. 1480-1482, 1960.

[9] J. A. Van Vechten, and C. D. Thurmond, "Entropy of Ionization and Temperature Variation of Ionization Levels of Defects in Semiconductors," Physical Review B, Vol. 14, No. 8, pp. 3539-3550, 1976.

[10] H. Z. Massoud, "Reverse Dopant Redistribution during the Initial Stages of the Oxidation of Heavily Doped Silicon in Dry Oxygen," Appl. Phys. Lett., Vol. 53, No. 6, pp. 497-499, 1988.