G1 minimum value of dopant density to be plotted, cm 3 tick marks and not the set of tick marks denoting the scale on the dopant density axis as shown in figure 5. There are six possible ways in which the data point may appear in the plot. = (1) If Y2 < 3, the data point is off scale at the low end (dopant density too small); this is represented by the symbol < to the left of the tick mark preceded by the value of Z to indicate how many data points are off scale at that depth value (lines 2260 and 2310). (2) If Y2 > 71, the data point is off scale at the high end (dopant density too large); this is represented by the symbol > to the left of the tick mark preceded again by the value of Z (lines 2300 and 2370). (3) If a single X1, Y1 pair (for which Z = 1) falls on the depth axis, an asterisk* is printed on the axis at that depth value in place of the usual + sign (lines 2270 and 2330). (4) If an average of several X1, Yl pairs (Z > 1) falls on the depth axis, a 0 is printed on the depth axis at that depth value in place of the usual + sign and the value of Z is printed to the left of the 0 (lines 2270, 2329, and 2342). (5) If a single X1, Y1 pair (Z * 1) falls within the range of the plot but not on the depth axis, a is printed at the proper number of carriage spaces at the proper depth position (lines 2290 and 2350). (6) If more than one X1, Y1 pair (Z > 1) falls at the same position along the depth axis and Y2 is such that it is within the range of the plot but not on the depth axis, a 0 is printed at the proper number of carriage spaces at the proper depth position, and the value of Z is printed to the left of the tick mark (lines 2290, 2349, and 2362). After the point is plotted XO is incremented (that is, the depth axis is advanced one line), Z is reset to 1, X1 and Y1 are changed to X2 and Y2, and a new X1, Y1 pair is sought (lines 2380 to 2406). The process continues until either all the C-V data have been processed (lines 3001, 3290, 2155, 2385) or the true Gaussian plot has been completed. Note that the quantities Z3, Z6, and XO are reset to zero before the start of the second plot (lines 4000 to 4002). The flag Z7 at line 3282 determines whether experimental or calculated values of X1 and Y1 are sent to the PLOT subroutine. An example plot, showing the six possible print-outs discussed above, is shown in figure 5. In practice it is desirable to choose G1 and G2 so that as few points as possible are off scale. It is also desirable to choose G3 so that no more X1, Y1 pairs than necessary are averaged and represented by a single data point on the plot. resolution. Averaged points mean reduced It should be noted that the PLOT subroutine breaks down when 10 or more X1, Y1 pairs are averaged to produce one printed data point. Because Z is printed in 12 format (line 2312), and because one of those two spaces is reserved for the sign (even though the sign is positive and not printed), a two-digit number cannot be represented, and the print-out reverts to exponential format. The data point may or may not be plotted properly depending on whether it lies outside or inside the field of the exponential number and tick mark. This problem could be resolved by printing Z in 13 format and adjusting the TAB (x) arguments accordingly. A carriage space otherwise available for plotting would be sacrificed, however. As one further comment, it is possible to nearly disable the PLOT subroutine by a proper choice of scale factor G3 when plots are not desired. If G3 is made sufficiently small, for example, 0.01, all dopant density values corresponding to distances of less than 50 μm are averaged and plotted as a single point. In addition the true Gaussian plot will probably not appear at all since, for such small values of G3, X1 is always 1 (line 3270). The FORNEXT loop (lines 4010 to 4050) is therefore completed for all necessary B values before any points are plotted, and the program advances to line 4060 and exists. Figure 5. A plot artificially constructed to show the six types of printed out- 3.7. The Main Program The main portion of the program consists of lines 098 to 100, 210 to 570, and 2500 to 4060 as listed in Appendix A. A summary flow chart is given in figure 6. In lines 210 to 220, the values stored in TABLE are read for later use by the INERF subroutine. Flags and other quantities are assigned initial values in lines 2500 to 2508. Input of scaling and diffused layer parameters occurs in lines 2514 to 2585 as discussed in section 3.1. In lines 2589 to 2682, several constants are entered or calculated for later use. The constants are (1) K, the dielectric constant of silicon, (2) P, the circumference of the diode in mils, (3) L, the characteristic length of the diffused layer, as defined in eq (6) and used in several equations, (4) K3, equal to erfc (x/L), and used to solve eq (10), (5) K1, the coefficient of W in the second term on the right-hand side of eq (7), (6) K5, which has = the value к¤¡Aj where A¡ TD2/4, used in solving eq (2), where the factor 4.48649E-03 includes the conversion of D from mils to micrometres and assumes £0 8.8542 × 10-14 F/cm, = = (7) K7, which is 2 times K5 and is used to solve eq (2) for a single W value from two successive capacitance values (line 3090), (8) K6, which has the value 2/qK0A3 where Aj TD2/4, used in solving eq (1), where the factor 5.49166E+18 includes the conversion of D from mils to centimetres, and assumes q = 1.602 × 10 ̄19 C, (9) K8, which is π times 25.4, the conversion factor for converting mils to micrometres, used in eq (3), and (10) K9, which has the value кɛo2D/2, including the appropriate conversion factor from mils to centimetres, and used to solve eq (3). These relationships are summarized in table 2. After these constants are calculated, the program number Z91, values of P, K and L, and a heading to the plot are printed in lines 2690 to 2750. The C-V data are read from file CVIN in line 3000. The peripheral correction is implemented in lines 3005 to 3014. The measured capacitance C3 is separated into its plane and peripheral components by the iterative loop in lines 3006 to 3014. In line 3005, a first estimate WO of the depletion width W is calculated from eq (2). The peripheral capacitance C5 is calculated from eq (3) in line 3006, and a new value of depletion width Wl is calculated from eqs (4) and (2). The new value W1 is renamed WO and is inserted into eq (3). Successive iterations are made until the relative difference between W1 and WO is 10-6 or less. The plane capacitance C1 is then used in subsequent calculations. In lines 3030 to 3070, a second C-V pair is read if the previous C-V pair was the first. Two C-V pairs are required to calculate the derivative in eq (1). The quantities C2 and V2 thenceforth denote the previous C-V pair (see also lines 3100 and 3110). Flag 22 is zero when the first data pair is read in line 3000; thereafter Z2 = 1. An apparent profile N(W) versus W is calculated in lines 3080 and 3090 by eqs (1) and (2) where plane capacitances rather than measured capacitances are operated on. The taking of the absolute value in line 3080 permits the voltage V1 to be measured with either polarity, although reverse bias is generally assigned positive values for convenience. The back depletion correction is performed in lines 3120 to 3230. The heart of the calculation is the iteration which computes A in lines 3140 to 3175. A numerical integration is performed to evaluate the integral of eq (9). Although the trapezoidal approximation is employed after the first step (line 3130), the first step is calculated assuming that the Gaussian form of the diffusion continues past the junction to the depth represented by the first N(W) versus W point. Any value of 29 can be entered in line 098. The authors find it helpful to increment 29 by 0.01 each time the program is altered. The first step numerical itegration is carried out in lines 500 to 570. Referring to eq (7) a first estimate AO = 0 is made for A in line 500. The right-hand side of eq (7) is calculated in lines 505 to 515. A second estimate Al is found by calculating A on the left-hand side of eq (7) in lines 520 to 525. The second estimate is compared with the first estimate; if the relative difference is greater than 10-5 further estimates are calculated by repeatedly placing the value of Al calculated in line 525 back into the right-hand side of eq (7) (lines 530 to 545). When a suitable value of Al is obtained, it is used in eq (10) to calculate the desired integral (line 550 to 560). Having thus calculated the integral in eq. (10), the same equation is effectively rearranged in lines 3135 and 3145 so that the expression erfc (x - A)/L is on the left-hand side of the equation and everything else is on the right-hand side. An initial value of A, AO, is placed in the right-hand side and a corrected value Al is calculated on the left-hand side in lines 3140 to 3155. The values Al and AO are compared in line 3160; if their relative difference is greater than 10-5, the value Al becomes the initial value AO and a new corrected value Al is calculated (lines 3165 and 3170). The iteration continues until sufficient agreement between AO and Al is achieved, and A is set equal to Al. Using eqs (5), (8), and (11), N(A), N(B) and B are calculated in lines 3190 to 3210. The N(B) versus B pairs are written into file NBVSB in line 3215 for later listing if desired. The values of N(W) and W are saved as N7 and X7 in lines 3220 and 3230 for subsequent use in the trapezoidal numerical integration. Scaling and plotting is then done in lines 3240 to 3280. After all the C-V data have been processed and the print-out of line 3990 is completed, a true Gaussian dopant density profile is generated in lines 4010 to 4050. Equation (12) is solved in line 4020 for a set of B values given in line 4010. The N(B) versus B values are written into IDEAL for later listing if desired in line 4025 and plotted in line 4040. This represents the profile that would be calculated if (1) the diffused layer were a pure Gaussian having parameters No, N and x. which were accurately known, (2) the dopant density of the specimen were uniform at å level N, and (3) various assumptions such as zero diffusion capacitance and abrupt space charge řegion boundaries were valid. j. A sample calculation using program CV1 is given in Appendix C. Program CV1 contains 249 statements (exclusive of comment statements). As stated in section 1, the maximum number of statements permitted by BASIC is 256. The user must therefore exercise caution in attempting to expand the program to meet his individual needs. Additional space could be created by compressing the TABLE data file to more than five entries per line. Even more space could be made available by removing statements relating to the true Gaussian plot if that is not needed. This could be done by deleting lines 2504, 3282 and 4000 through 4050 and deleting IDEAL from line 099. The problem of program length discussed above can be solved and other advantages can be gained by separating one or all of the ERF, INERF and PLOT subroutines and the TABLE data file from the main program and creating each as an independent program or file. These subprograms can then be executed by the main program using CALL statements in place of the present GOSUB statements. When this is done, substantial additions can be made to the main program before the limit on program length is reached. This procedure has the advantage of making the subprograms available to be called by other programs. The authors have employed the ERF subroutine in this manner. If the INERF subroutine is to be set up as an independent subprogram, a means for reading in the TABLE values, presently done in lines 100 and 210 to 220, must be included in the program which calls INERF. In all such subprograms, an END statement must be included as the last statement. In some applications it may be desirable or necessary to measure dopant density profiles using rectangular rather than circular diodes. The equation for the peripheral correction |