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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 CV1 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 CV1 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 CV1 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 CV1 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 positive and negative numbers in the range 103 038 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.

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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.

2.1. The Peripheral Correction

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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where q is the electronic charge, κ is the relative dielectric constant of the test specimen, Eo 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 CV1, 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).

2.2. The Back Depletion Correction

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

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As shown in figure 1, A is the depletion width on the heavily-doped side of the junction, B is the depletion width on the lightly-doped side of the junction, N(A) and N(B) are the respective net dopant densities at distances A and B from the junction, No is the dopant density at the surface of the diffused layer, and N is the estimated background dopant density in the diffused layer. The characteristic length of the Gaussian diffusion is

represented by L, and I is the area of either of the two shaded regions in figure 1. The lower case w and a are used as variables of integration; whereas, upper case W and A represent specific values of w and a. The quantities W, A and B are related by eq (11).

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In computer program CV1, the integral on the left in eq (9) is calculated by numerical integration of the N (W) vs. W data. Equation (7) is used in calculating the first step in the numerical integration and the trapezoidal approximation is used in subsequent steps. The value of this integral, I, is used to solve eq (10) for A. Knowing A, N(A) is calculated from eq (5), N(B) is calculated from eq (8) and B is calculated from eq (11). The desired profile is a plot or listing of N(B) vs. B.

Equation (12) below is used to calculate a true Gaussian profile for comparison with the experimentally determined profile.

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Experimental profiles sometimes show a decrease in N(B) for small B. A plot of the true Gaussian profile can be helpful in determining whether such an experimental observation is caused by the diffusion tail or by some other phenomenon. The true Gaussian plot is also helpful in evaluating the adequacy of the assumed N value. If the experimental plot and the true Gaussian plot do not asymptotically approach the same background density, the calculation should perhaps by repeated using a different assumed value.

2.3. The Complementary Error Function

The solution of eqs (7) and (10) calls for calculating the complementary error function and its inverse. The calculation of the complementary error function employs equations given by Stegun and Zucker [14] as

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Figure 1. Schematic representation of the depletion region of a p-n junction diode diffused in an epitaxial semiconductor. (The distances A and B have their origins at the junction. Note that the actual background dopant density, represented by the irregular line to the right of the junction, is assumed to be constant within the diffused layer to the left of the junction.)

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The reader is advised that eqs (13) to (15) and the software that implements them (see sec. 3.3.) have been thoroughly proven within the range of applicability. Other approximations to the error function and its complement should be used only with extreme caution. In program CV1, the power series representation of eq (13) is used when x ≤ 1, and the continued fraction of eq (15) is used when x > 1.

2.4. The Inverse Complementary Error Function

The inverse complementary error function is calculated by interpolation in a table of values. The equation to be solved is

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to yield a value of z from 0 to 100 corresponding to a value of x from 10-7 to 1. The transformation is illustrated in figure 2. The 107 lower limit was chosen so that eq (10) could be solved for all practical ratios of No to N The table of values (see listing in lines 150 to 188 of Appendix A) lists those values of y corresponding to integer values of z from 1 to 100. For example, the first value in the table y 3.74580, corresponds to z = 1 or x = 1.17490 × 10-7. To calculate y from a given x, z is first calculated and three data pairs (zı, yı), (z2, y2), and (z3, yз) are chosen from the table such that

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