IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-32, NO. 9, SEPTEMBER 1985 1889

TABLE I DIFFUSIVITY (D), CRITICAL TIME-STEP SIZES (A&), AND THE

DIFFUSION LENGTH (-2 FOR VARIOUS PREDEPOSITION PROCESSES

Diffusivity (D) JD.AtCri, Critical time step (Atcrit)

Poron 950°C 1 , 5 x lE1:m2;;ec 1 20 sec 1 Boron l0OO0C 1 x 10 sec

Phosphorus 95OOC 5 x 1CL2 10 sec

Phosphorus 1000’C 1 x IC1’ 6 sec

2 0.03 yrn

2 0.07.5 vm

values within 1 percent. (For assessing the accuracy of the impurity profile in terms of percentage error, n ( C ) is more appropriate than C(x) , where x denotes distance from surface and C denotes con- centration because C is w y i n g in a much wider dynamic range than x does.)

The diffusivity values in Table I are for the surface electron and hole concentration values of 8 X lOZ0/cm3 and 2 X 1020/cm3 for phosphorus and boron, respectively. It was observed that the

values for boron and phosphorus were approximately 0.03 and 0.075 p m , respectively. For both cases, it was shown that the well-known criterion for selecting A t and h, i.e., h2 D - At, approximately holds. (h = 0.05 pm in our case.)

m. CONCLUSION

A 2-D numerical scheme combining AD1 with Gaussian elimi- nation was presented. It showed a significant CPU time reduction (about a factor of two or three in our examples) compared to the

’ conventional iterative schemes. Memory requirement is also mini- mal. The 2-D program result was calibrated with the I-D program result, which was, in turn, in good agreement with the experimental results under various process conditions for boron and phosphorus.

~ F E R E N C E S

[l] D. A. Antoniadis and R. W. Dutton, “Models for computer simulation of complete IC fabrication processes,” IEEE Trans. Electron Devices,

[2] B. R. Penumalli, “Comprehensive two-dimensional VLSI process sim- ulation program, BICEPS,” IEEE Trans. EZectron Devices, vol. ED- 30, p. 986, Sept. 1983.

131 A. Seidl, “A multigrid method for solution of the diffusion equation in VLSI process modeling,” IEEE Trans. Electron Devices, vol. ED-30, pp. 999-1004, Sept. 1983.

[4] K. A. Salsburg and H. H. Hansen, “FEDSS: Finite-element diffusion- simulation system,” IEEE Trans. Electron Devices, vol. ED-30, pp. 1004-1011, Sept. 1983.

[5] H. L. Stone, “Iterative solution of implicit approximations of multi- dimensional partial differential equations,” SIAMJ. Nurner. &tal., vol.

[6) J. H. Ferziger, Numerical Methods for Engineering Application. New

V O ~ . ED-26, p. 490, 1979.

5, pp. 530-558, 1968.

York Wiley-Interscience, 1981, ch. 4.

The Modeling of Ion Implantation in a Three-Layer Structure Using the Method of Dose Matching

G . A. J. AMARATUNGA, K. SABINE, AND A. G . R. EVANS

Abstmcf-The method of modeling ion implantation in a multilayer target using moments of a statistical distribution and numerical inte-

Manuscript received October 9, 1984; revised March 25, 1985. The authors are with the Department of Electronics and Information En-

gineering, University of Southampton, SO9 5NH, U.K.

gration for dose calculation in each target layer is applied to the mo- delling of As’ in poly-SilSi0,lSi. Good agreement with experiment is obtained.

In VLSI technology it is important to accurately predict the im- planted profiles of impurity concentration in a target comprised of different material layers with different stopping powers. The SiOZ/ Si two-layer structure associated with implantation through a gate oxide and the poly-SilSi02/Si three-layer self-digned structure used to mask the channel during source-drain implantation are two ex- amples widely encountered in MOS device fabrication.

The Si021Si target has been studied closely using the Boltzmann transport equation [l]. This method gives good results for second- ary implantation phenomena caused by the recoiling of oxygen from Si02 into Si. A drawback of this method, however, is the inability to take into account channeling in a crystalline target due to the assumption of a completely amorphous target. A simpler method using moments’ of a statistical distribution and dose conservation was proposed by Ryssel [2] for constructing As-implanted profiles in a Si02/Si target. This method does not take into account recoil implantation but can simulate channeling behavior. We have ex- tended this latter model for the case of implantatibn in a three-layer structure. Good agreement between experimental and modeled pro- files have been obtained for As’ implanted in a poly-SilSi0,lSi tar- get. The steps used to calculate the profile are outlined in the fol- lowing.

The following steps are required for modeling an implant of total dose D at an energy E.

Step 1. The distribution of the implant from 0 - y1 in Fig. 1 is that of implant of dose 0 and energy E in MAT 1. Denoted as Plf,( y) , where Pl is the peak concentration and f , ( y ) the statistical distri- bution as a function of depth y . The number of implanted ions in MAT I is dl where

(1)

Step 2. Assuming the implant D, E was directly into MAT 2, the

Yi

dl = d2 = Io P2f2(y) du. (2)

depth y1 which contain dl implanted ions is calculated as

This is equivalent to replacing the MAT 1 layer by a MAT 2 layer of thickness y I.

Step 3. The implant distribution from yI -+ y1 + t is taken to be implant D, E directly in MAT 2 from y i + y + t. The number of ions in the MAT 2 layer of thickness t is d3, where

(3)

Step 5. The implant distribution ffrom y1 + t 4 a! is’taken to be that of implant D, E in MAT 3 from y -+ a. ,

The model was applied to predict As’ penetration through the poly-Si gate and gate oxide into the channel of an NMOS device during source/drain implantation. The profiles obtained for a 5.1015 atmlcm’ dose at 175, 160, and 135 keV are shown in Fig. 2. A Pearson type 1V statistical distribution was used for As‘ in Si and Si02, and for implantation purposes poly-Si was taken as being identical to Si. T$is assumption is valid provided the poly-Si grain size is 2 1000 A and smaller than the poly-Si layer thickness. Hence channeling behavior of As’ in Si could also be applied over the entire poly-Si area.

0018-9383/85/0900-1SS9$01.00 0 1985 IEEE

IEEE 1’WANSACTIONS ON ELECTRON DEVICES, VOL. ED-32, NO. 9, SEPTEMBER 1985

0 - Y, yt t Y--oc Y; Y;“‘

Y;

, Fig. 1. A general three-layer structure

w x l o - ~

Fig. 3. Modeled implants of As* in Si using a Pearson IV (solid line) and Gaussian (broken line) distributions for 5 X atmicm’ dose at 135, 160, and 175 keV energies.

decades are required, and the experimentally observed channeling effects of As’ in Si [5], [6] have to be included in the modeled profiles. This is further illustrated in Fig. 3 where modeled profiles using a pure Gaussian distribution for an amorphous Si target are compared with those obtained using a Pearson IV distribution for the same dose and energies as in Fig. 2. Clearly if the Gaussian profile is used As’ penetration through the SiOz layer into the sin- gle-crystal Si will not be predicted for the preceding example.

x10-1 pJM

Fig. 2. The modeled and experimental results of As+ implanted into a poly- SifSiOJSi target.

The range and standard deviation for implanted As ‘-in Si and SiOz were obtained from the empirical expressions given in [3], the skewness from [4], and kurtosis from the empirical formula given in [Z]. All the integrations for dose calculation were done numeri- cally using Simpson’s method. The discontinuities in the profiles are caused by the matching of dose in materials of different stop- ping powers. To evaluate the accuracy of the dose-matching method, the As profile in single-crystal Si, where the dose levels are 0.0001 times the original implanted dose, were measured. The computer time taken to obtain a profile to this accuracy was in the region of 20-s CPU on a Prime 9950 computer. The measurements were car- ried out by the C-V method using Zeigler analysis after the im- planted samples were annealed at 950°C for 15 min. The diffusion of As in the single-crystal Si due to heat treatment was simulated numerically and found to cause an insignificant shift from the im- planted profile except in the 20 nm adjacent to the SiOz interface. As the C-V method cannot be used for measurement in this ,region, the simulated implant profiles and the electrically measured profiles in the single-crystal Si for the 175 and 160 keV cases are shown in Fig. 2 (the background concentration has been subtracted from the measured points).

The method of dose matching has been applied to the modeling of As’ implantation in a three-layer structure and shown to give satisfactory results at does levels four orders of magnitude below the total implanted dose. Provided accurate implant parameters for the specific combination of implant ion and target materials exist, the method of dose matching could be applied to determine the implant profiles in targets of many layers. It is also shown that a Pearson type IV statistical distribution gives good results for As+ in Si and SiO, when the profiles of concentration over six or seven

REFERENCES L. A. Christel, J. F. Gibbons, and S . Mylroie, “An application of the Boltzmann transpofi equation to ion range and damage distributions in multilayered targets,” J. Appl. Phys., vol. 51, no. 12, pp. 6176-6182, 1980. H. Ryssel ad K. Hoffmann, “Ion Implantation, ” in Process and Device Simulation for MOS-VLSI Circuits, NATO ASI no. 62. Martinus Nighoff, 1983. J. Nakata and K. Kajiijarna, “Precise profiles for As implanted in Si and Si02,” Japan. J. Appl. Phys., vol. 21, no. 9, pp. 1363-1369, 1982. S. F. Gibbons, W. S . Williams, and S. Mylroie, Projected Range Stu- tistics, 2nd ed. New York: Wiley, 1975. J. M. Fairfield and B. L. Crowder, “Ion implantation doping of silicon for shallow junctions,” Trans. Met. Sac. AIME, vol. 245, p p . 469-473, 1968. D. E. Davies, “The implanted profiles of B, P, and As in Si from junc- tion depth measurements,” Solid-State Electron., vol. 13, pp. 229-237, 1970.

Andytical Model for Predicting Threshold Voltage in Submicrometer-Channel MOSFET’s

Ahtract-A quasi-two-dimensional analytical closed-form determi- nation of the threshold voltage has been derived for submicrometer- channc!l-length MOSFET’s. The invalid assumption of uniform

Manuscript received November 27, 1984; revised May 1, 1985. T. ring and D. Navon are with the Department of Electrical and Com-

puter Exgineering, University of Massachusetts, Amherst, MA 01003. Q. Zhang is with the Electronic Engineering Department, Fudan Uni-

versity, People’s Republic of China.

0018-9383/85/0900-1890$01.0~1 0 1985 IEEE