holder centers the slice with sufficient accuracy for the measurement. Each knife edge, K, has a single leaf spring behind it which permits enough freedom of movement to allow each to make contact with the circumference of the slice, W, at the same time. The four knife-edge holders, E, each holding two knife edges, move in concert at the ends of the levers, J. The levers are each pivoted at points, P, at 90-deg intervals from one another on the circumference of a common circle. As the adjustment ring, A, is moved by means of the locking knob, N, the pin, 1, rides in a notch, T, of each lever which forces the four holders in or out together. Note that the knife-edge holders do not rotate at the ends of the levers. Thus, the four knife edges do not strike the wafer exactly at the ends of perpendicular diameters for slices of all diameters; however, the variation about the design slice diameter permitted by the SEMI specifications [17] is small enough that the locations at which the knife edges make contact do not change enough to affect the measurements significantly.

An automated data collection and analysis system was designed and assembled. A block diagram of this system is shown in figure 5c. The slice under test, W, is mounted in the holder, H, described above. For clarity, only two of the knife edges, K, are shown at opposite ends of a diameter. The holder is oriented on two stages so that the slice is concentric with the platform of the rotary stage, R, and so that the movement of the linear stage, L, is parallel to one of the diameters determined by the knife edges. rotary stage has a rotational resolution of 2 min of arc and bidirectional rotational repeatability of 0.2 min of arc. The linear stage has a total travel of 85 mm, a linear resolution of 10 um, and bidirectional linear repeatability of 5 μm.

The

Photo-excitation at a wavelength of 1.15 um is furnished by a 2-mW helium-neon laser, which is mounted horizontally. The laser beam is directed to the center of the rotary stage by a mirror, M. The slice is scanned by moving the stage along a slice diameter under the stationary laser beam. The control card, C, for the stepping-motor in the multiprogrammer, operating under the direction of the calculator/controller, generates a pulse train of the appropriate length to move the stage the desired number of 10-um steps in either direction; the pulse train is supplied to the motor windings through the selector switch, S, and the translator. End-point protection is provided by microswitches and

appropriate relays at both ends of the stage travel.

During the first scan, the photovoltage is measured at predetermined points along the diameter. To improve the signal-to-noise ratio, the beam is chopped and the photovoltage is measured by a lock-in amplifier which is synchronized to the frequency of the chopped light beam. The chopping frequency is adjusted so that the on time of the laser beam is long enough to achieve a steady-state excess-carrier distribution. Following the first scan, a second scan along the same diameter is made; during this scan the photoconductivity is measured at the same points along the diameter as before. This measurement requires that a small current be passed along the diameter of the slice (using one pair of knife edges) while the potential drop across the diameter is measured as a function of light position (using the other pair of knife edges). The calculator/controller, which together with its associated multipro grammer and printer/plotter is compatible with standard interface requirements [18], records the data, performs the calculations required to obtain the resistivity, and plots the results. After measurements along the first diameter are completed, the wafer can be rotated 90 deg and the measurements repeated. (G. J. Rogers and D. L. Blackburn)

3.5. Reevaluation of Irvin's Curves

Additional room-temperature resistivity and dopant density measurements were made on silicon wafers doped with phosphorus as part of the continuing experimental redetermination of the resistivity-dopant density relation first reported by Irvin [19]. In addition, initial measurements were made on silicon wafers doped with boron.

The measurements were made on test structures included in test pattern NBS-3 [20]. Dopant density values were obtained from measurements on structure 3.10, base-collector diode (NBS Spec. Publ. 400-17, pp. 27-28), by the junction capacitance-voltage (C-V) method. The gate on the structure was biased at the flat-band voltage as determined from measurements on structure 3.8, MOS capacitor over collector (NBS Spec. Publ. 400-17, pp. 26-27). Instead of calculating discrete dopant density values from the C-V data as has been done previously (NBS Tech Note 788, pp. 9-11), a new data reduction procedure was developed in which a Gaussian shape is assumed for the base diffusion near the junction and the sur

This

face concentration of the diffusion and the background dopant density are adjusted to give a best fit to all the C-V data. procedure has made it possible to obtain results on wafers with dopant densities greater than 1017 cm-3 which could not be measured by the previous approach.

The bulk resistivity of the wafers was determined from measurements on structure 3.17, four-probe collector resistor (NBS Spec. Publ. 400-17, pp. 25-26) [21]. For a few wafers, the resistivity used for calculating mobility was that determined prior to processing by mechanical four-probe measurements [2]. These were either wafers for which the emitter diffusion was omitted in order to obtain a shallow base junction depth or wafers processed with the large-pipe base mask which introduced a significant uncertainty in the pipe spacing [21].

where N is the carrier, or net dopant, density, q is the electronic charge, and p is the resistivity corrected to 300 K. To minimize the influence of resistivity variations over the wafer the average of the resistivities measured on the two collector resistors¶ on either side of the diode used to determine N was used for p in eq (5). The procedure was repeated for a number of good diodes in the same general area of the wafer, usually near the center, to arrive at an average mobility as listed in tables 4 and 5. The uncertainty in the mobility for each wafer is the sample standard deviation calculated from the individual mobility values. Table 4 contains revised electron mobility values based on repeated and additional measurements for wafers reported on previously (NBS Spec. Publ. 40019, p. 25). In general the electron mobility values lie above those which would be predicted using the Caughey-Thomas fit [22] to Irvin's data (see sec. 3.6.), and the hole mobility values lie below those which would be predicted using Wagner's empirical relation [12]. (W. R. Thurber, R. L. Mattis, Y. M. Liu, M. L. Doggett, and M. G. Buehler)

Initially the lattice mobility, taken from the work of Norton et al. [25], was combined with the ionized impurity mobility, as calculated by Long [26], according to the sineand-cosine-integral method [27] to obtain the mixed scattering mobility. This was combined reciprocally with the neutral impurity mobility, as calculated by Sclar [28], to obtain the total electron mobility. The mobility calculated in this way for 300 K is shown as curve 1 of figure 6. For comparison, the Caughey-Thomas fit [22] to Irvin's data [19] is shown as curve 2 and more recent experimental results are shown as data points; the solid dots are taken from the mobility data of Mousty et al. [29] and Baccarani and Ostoja [30], corrected to 300 K, and the open circles are NBS data obtained on processed phosphorus-doped silicon wafers (see sec.

3.5.).

It can be seen that the calculated result fits the more recent experimental data for dopant density less than 2 x 1016 cm-3. However, at higher dopant density, the calculated curve is higher than the experimental data.

While electron-electron scattering does not affect the current density directly since it cannot alter the total momentum, it tends to randomize the way in which this total momentum is distributed among electrons with different energy. When the scattering mechanism is such as to lead to a nonuniform distribution, electron-electron scattering gives rise to a net transfer of momentum from electrons which dissipate momentum less efficiently to those which dissipate momentum more efficiently, resulting in an overall greater rate of momentum transfer and lower mobility. The size of the effect of electronelectron scattering on the mobility is a function of the energy dependence of the relaxa

When the resistivity was measured by means of a mechanical four-probe array, the resistivity value for the location was based on radial profiles along two diameters of the wafer.

RESISTIVITY (⋅cm)

No =1.2 x1014 cm-3

1017

TOTAL DONOR DENSITY (cm3)

Figure 6. Conductivity mobility of electrons. at 300 K as a function of total donor density for phosphorus-doped silicon. (Curve 1: Theoretical, not including electron-electron scattering; Curve 2: Caughey and Thomas representation of Irvin's curve [22]; Curve 3: Theoretical, including maximum electron-electron scattering over entire range; Curve 4: Theoretical, including electron-electron scattering in intermediate and high dopant density ranges; open data points: experimental, this work; solid data points: experimental [29,30].)

10

100 150 200 250 300 350 400 450 500 TEMPERATURE (K)

Figure 7. Resistivity as a function of temperature for seven phosphorus-doped silicon slices. (Solid points are measured values and curves are calculated assuming the phosphorus density, No, as indicated on the curves.)