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LIST OF FIGURES

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Figure 1.

Schematic representation of the depletion region of a p-n junction diode
diffused in an epitaxial semiconductor

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Figure 5.

A plot artificially constructed to show the six types of printed output of
the PLOT subroutine

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Figure 6. Summary flow chart of program cvi

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PREFACE

This work was carried out as a part of the Semiconductor Technology Program in the Electronic Technology Division at the National Bureau of Standards. The Semiconductor Technology Program serves to focus NBS efforts to enhance the performance, interchangeability, and reliability of discrete semiconductor devices and integrated circuits through improvements in measurement technology for use in specifying materials and devices in national and international commerce and for use by industry in controlling device fabrication processes. The Program receives direct financial support principally from three major sponsors: The Defense Advanced Research Projects Agency (ARPA), The Defense Nuclear Agency (DNA) and the National Bureau of Standards. The specific work reported herein was supported by ARPA. *

The computer program cvi, the subject of this report, is derived from a program reported by D. B. DeVries, G. Lee, and S. Watelski in Integrated-Circuit Process Control and Development, Technical Report AFAL-TR-73-268, August 1973. The present program contains improvements in the peripheral capacitance correction, the error function calculations, the numerical integration, the plot routine, and data input/output. The present report is intended as a guide for persons using the program described herein. It is not intended to provide a detailed comparison of the present and the original programs.

The authors are indebted to several persons who assisted in the editing and preparation of this report. Gerard N. Stenbakken performed a critical and constructive reading of the text. The several figures were prepared by Edgar C. Watts and Leo R. Williams. The typing of the final draft was done by Frances C. Butler.

*

Through ARPA Order 2397, Program Code 4010.

A BASIC Program for Calculating Dopant Density

Profiles from Capacitance-Voltage Data

by

Richard L. Mattis and Martin G. Buehler

Abstract: A computer program is presented which is suitable for calculating dopant density vs. depth profiles from capacitance-voltage data for the case of a Gaussian-diffused p-n junction diode. The program includes corrections for peripheral capacitance of round or rectangular diodes and back depletion of the space-charge region into the diffused layer. Inputs to the program consist of the surface dopant density, the junction depth, the background dopant density in the diffused layer, the junction diameter, three scaling parameters, and the capacitance-voltage data pairs. Output from the program is in the form of a plot and an optional listing of dopant density as a function of depth. The equations underlying the program are given and are related to the program whose operation is described in detail. A second program, for generating idealized capacitancevoltage data for a Gaussian-diffused diode on material with a constant dopant density is also included.

Key Words: BASIC; capacitance-voltage measurements; computer programs; dopant profiles; error function; Gaussian diffusion; plotting, computer; semiconductors; silicon.

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The capacitance-voltage (C-V) method is widely used to measure dopant density versus depth profiles of semiconductor specimens (1-4). The basic equations for calculating dopant density N(W) as a function of depletion width W were derived by Schottky (5) and are suitable for the case of a large area one-sided abrupt junction diode under reverse bias conditions. However, in many cases the dimensions of the diode are not large compared with the depletion width, so the peripheral capacitance can cause a significant error in the calculated dopant density profile. Similarly, when the diffused layer dopant density is not large compared to the dopant density of the region to be profiled, back depletion into the diffused layer can cause significant error in the calculated dopant density profile [6].

A computer program, henceforth denoted cvi for convenience and listed in Appendix A, is presented. It is suitable for calculating dopant density versus depth profiles from C-V data for the case of a Gaussian-diffused p-n junction diode. The case of a Gaussian diffusion is treated because of its common usage in the semiconductor industry. Program cvi is not intended for profiling junctions which are part of transistors or other multijunction structures. The program has not yet been satisfactorily tested on diodes diffused in epitaxial material in which the layer and substrate are of opposite conductivity type; the program has been proven, however, on diodes diffused in epitaxial material in which the layer and substrate are of the same conductivity type and on diodes diffused in bulk material. Program cvi is not intended to take into account the effects of diffusion capacitance which occur under heavy forward bias conditions; it is recommended that caution be exercised in interpreting any data taken in forward bias conditions. The program includes corrections for peripheral capacitance of round and rectangular diodes and back depletion of the spacecharge region into the diffused layer. Inputs to the program consist of the surface dopant density, the junction depth, the background dopant density in the diffused layer, the junction diameter, three scaling parameters, and the C-V data pairs. Output from the program is in the form of a plot and an optional listing of dopant density as a function of depth.

The equations underlying program cvl are given in section 2 along with a somewhat expanded discussion relating to the calculation of the complementary error function and its inverse. The program is described in detail in section 3. In section 4, program modifications and check-out are presented, including the discussion of a second computer program which generates idealized C-V data for the case of a Gaussian-diffused junction diode fabricated in material of constant background dopant density. For convenience, this second program is henceforth denoted CV2.

The programs described in this report are written in the BASIC language. A description of the BASIC language can be found in several books (7-9). However, when using the programs described below the reader must be alert to the particular characteristics of the BASIC he may be using. The particular type of BASIC employed in the programs described in this report is applicable to a time-sharing system, has a six decimal place precision, can handle

38 positive and negative numbers in the range 103 to 10-38, and can accommodate programs as long as 256 lines plus comment statements. This particular BASIC is compatible with most of the BASIC in use. However, some of its characteristics are worthy of comment to avoid possible confusion. These are described in Appendix B.

2. EQUATIONS

In this section, the equations which are employed to calculate the dopant density profile from the experimental C-V data are given. These equations relate to the peripheral correction, the back depletion correction and the calculation of the complementary error function and its inverse.

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The Schottky equations for calculating dopant density N(W) and depletion width W have been referred to above and are given below as eqs (1) and (2),

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m

where q is the electronic charge, k is the relative dielectric constant of the test specimen, €0 is the permittivity of free space, A, is the area of the diode, C is the measured capacitance and V is the applied voltage. In program cvi, the peripheral"capacitance is first

, substracted from the measured capacitance (10-13). The peripheral capacitance calculation for a circular diode can be written as

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Equation (3) was derived by assuming that the peripheral region is a one sided junction. After the plane capacitances have been determined, an apparent profile N(W) vs. W is calculated using eqs (1) and (2).

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The back depletion correction (10) is based on eqs (5) through (10) below:

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