Physical review volume 05, number january 15, 1957

I. DIFFUSION AND SOLUBILITY OF Zn IN Si
Preliminary experiments showed that Zn provides at least one acceptor level² in silicon and in this respect behaves like Zn in Ge.³ In order to investigate the diffusion in Si more completely the method of fractional saturation was employed, the details of which are de­scribed below.

1. 
   Basis of the Method
   

When diffusion takes place into a slab whose broad faces are large in dimensions compared to its thickness, it can be easily shown^{4 5} that the amount of diffusant entering the slab in a time I, divided by the amount which would enter in infinite time, i.e., the fractional saturation, is simply related to the diffusion coefficient. The relation is
F=2.26(Z>Z)*/a, (1)
where F is the fractional saturation in time t, D is the diffusion coefficient in cm²/sec, and a is the thickness of the slab in cm. This relation is based on the error function solution and holds only in the case where no appreciable overlapping of the penetration curves from the two sides of the slab occurs, i.e., for fractional saturations in the range up to about 0.6. In (1) it is assumed that the surface concentration is constant all during the diffusion, that the diffusion coefficient is constant independent of concentration, that Fick’s laws hold, and that the initial concentration is_uniform or zero.⁶
The same relationship holds for the diffusion of a volatile diffusant out of a slab in which case one measures the fractional amount of diffusant removed from the slab, i.e., the amount removed or lost in time t divided by the amount which would be lost in infinite time. In this case, the concentration of the slab surface is assumed to be zero or some constant value less than the uniform initial concentration all during the process.
In the present work the amount of diffusant, in this case Zn, entering or leaving the slab was measured by the change in the electrical conductance (measured at room temperature) after successive anneals for known times at fixed temperatures. Since the conductance of the slab is proportional to the number of carriers present multiplied by their mobility, a simple electrical measurement made initially and after a given time of anneal suffices to determine the diffusion coefficient D. Mobilities corresponding to the average resistivities of the specimens have been used in each instance. No corrections for mobility variation over the penetration curve have been made since these are small because of the low Zn concentrations.
The use of conductivities to determine changes proportional to the Zn concentrations requires that a constant number of holes is contributed by each added Zn atom. To insure complete ionization of the Zn, w-type silicon sufficiently doped with arsenic so that it remained n type after saturation was employed. It
⁶ It is also assumed that diffusion potential and surface po­tential effects are negligible. This requirement is met since at the high temperatures employed in this work such fields are swamped out by intrinsic carriers.

Fig. 1. Method of obtaining precise control of Zn vapor pressure.

appears from Hall measurements (see below) that one hole is provided by each Zn atom under these conditions.
The relation employed to calculate D for the inward and outward diffusions is as follows:
D=a²(a₀-^)²/5.H/(ao-a₀₀)². (2)
Here a is the slab thickness in cm, tro is the conductivity at zero time, a_t that at time /, and a_x at infinite time, i.e., the conductivity corresponding to saturation at the temperature in question for inward diffusion and to entire absence of Zn for the outward diffusion.

2. 
   Procedure
   

The method employed consisted in heating silicon wafers 1.OX0.4 cm by 0.060-0.070 cm thick in Zn vapor at temperatures between 900 and 1360°C. The Si was arsenic-doped during the crystal growing and ranged in resistivity from about 1.0 to 0.04 ohm cm n type. The diffusions were conducted in sealed quartz tubes which contained the Si specimens together with pure Zn.⁶ The latter was regulated in amount to give between 0.5 and 1.5 atmospheres pressure at the diffusion temperature. More precise control of the Zn vapor pressure was obtained in a number of runs by maintaining a temperature gradient in the tubes by an arrangement similar to that shown in Fig. 1. No effects attributable to a change in the Zn vapor pressure were found in these runs, even though the pressure was varied from 7 to 760 mm of Hg. Possible reasons for this are discussed below. Temperatures were controlled to within ±3°C and residual air, He, and H2 were all separately employed as ambients (at 0.001 mm pressure at 25 °C) without finding any effects of the different gases on the results. Outward diffusions were conducted in flowing H2 or flowing He as well as under continuous pumping at 0.001 mm pressure. Similar diffusion results to those observed in sealed tubes were observed under these conditions. No evidence of a reaction of Zn with the quartz containers was evident even at the highest diffusion temperatures. This is to be ex­pected because of the free energy of formation of the respective oxides.⁷
Saturation runs were made at several temperatures in an attempt to determine a solubility-temperature curve for Zn. Inasmuch as the duplicate runs differed considerably, even under carefully controlled conditions of temperature and pressure, it was possible to secure only an approximate solubility curve.
All conductivities were determined by means of a two-point probe having 0.635-cm separation. A precision of at least ±1% was obtainable in this measurement.

3. 
   Results
   

The results of the saturation runs are shown in Fig. 2 which also plots data from Part II to be discussed. Each data point in this figure is the average of 3 to 9 determinations whose spread is indicated by the range line at that point. In calculating the number of holes added (equal to the number of electrons removed), the mobilities for electrons were taken from the work of Prince⁸ for the higher resistivities and of Morin and Maita⁹ for the lower resistivities. The diffusion results are too scattered to be reported in detail and will be summarized below.

4. 
   Discussion—Diffusion Results
   

All of the diffusion coefficients, whether determined by inward or outward diffusion, and more or less

Fig. 2. Free-hole concentrations as a measure of Zn solubility as it depends upon reciprocal temperature.
independently of temperature, fall between about 10~⁶ and 10^-7 cm² per sec. The average of 30 results for inward diffusion is 6.1X10"⁷ cm²/sec and of 8 results for outward diffusion 4.3X10~⁷ cm²/sec. There appears to be little definite trend with temperature although there is a correlation of higher temperatures with higher diffusion constants. We do not have an explanation for this or for the variability in the diffusion values. That the presence of oxide films on the silicon surface is one factor of importance is indicated by the fact that alloy­ing of the Zn with the silicon specimens was observed in only one instance.¹⁰ This occurred after sealing and dur­ing the high-temperature heating period when it is be­lieved one of the specimens fractured, since alloying was observed on the freshly fractured surface only. A variable amount of surface oxide (SiO₂) is also suggested by the outward diffusion experiments in which induction periods for diffusion were observed, as if surface films had to be removed or penetrated before rapid outward diffusion could take place. In view of the inability of Zn to displace oxygen from SiO₂,⁷ and as evidenced by the fact that no visible attack of Zn on the quartz containers occurred, even after hundreds of hours heating at 1250°C, it is perhaps not surprising that surface barriers to Zn diffusion are present on silicon.
Another observation which appears to be significant is that the alloying of Zn occurs much more readily on crystals grown with slow-rotation (1.5 rpm) than on those rotated rapidly (100 rpm) during growth. This suggests that the supposed higher oxygen content of the rotated crystals¹¹’⁷ may be the cause. Perhaps the latter form surface oxide barriers more readily under the conditions of our experiments than do the slow- rotated crystals. However, no significant difference in diffusion constant or in the consistency of the results was evident for the slow-rotated silicon crystals. Further evidence of a surface barrier for diffusion is provided by the lack of dependence of the diffusion rate on the Zn vapor pressure mentioned earlier.¹²
In most of the diffusion experiments care was taken to avoid errors due to precipitation of Zn by rapid quenching and by diffusing out at temperatures equal to or above those where saturation was carried out. In some experiments in which Zn was diffused out, saturation was at higher temperatures. These runs would be expected to result in lower D values. The large variability in D, however, even for runs in which no precipitation could have occurred, obscures any effect on this kind if it is present. That Zn does not precipitate from silicon at room temperature is shown by the fact that no resistivity changes occur on standing. Even at 450°C little or no precipitation of Zn occurs.
The diffusion rate of Zn in silicon would therefore appear to be low at this temperature.
Summarizing, we may say that the diffusion of Zn in silicon is complicated by the presence of surface barriers, probably SiO₂. The largest diffusion constants are of the order of 10~¹³ cm²/sec, although even these may be subject to some surface restriction. Since values of D= 10~⁶ occur at lower temperatures as well as high (although less frequently), no activation energy for the diffusion can be estimated.

5. 
   Discussion—Solubility Results
   

The curve showing the solubility of Zn as a function of temperature (Fig. 2) appears to have significance in spite of the large errors of the individual determinations. This curve represents the approximate trend of all the data including that described in Part II below. There is a suggestion that a ^maximum in the solubility curve occurs at about 1300°C corresponding to 10¹⁷ atoms/cm³ if one acceptor level per atom is assumed.
We have no explanation for the wide variations in solubility for identical temperatures and saturation conditions. It was at first believed that this was a result of variable vapor pressure of the Zn. Several experiments were therefore undertaken in which the vapor pressure was controlled at different levels dur­ing diffusion at 1185°C. However, inconsistent results were obtained in these runs. No relation between Zn vapor pressure and solubility could be established although at pressures near 1 atmosphere fairly good duplication was possible. Barrier films on the silicon may again be a factor. Perhaps more careful tests under high-vacuum conditions will show the expected pressure dependence.
Another possible cause of variation may be an interstitial-substitutional equilibrium such as has been proposed for the state of Cu in germanium.^{14 15 16} This might be expected to depend on the rate of quenching from high temperature. However, no correlation with quench rate, within our ability to control it, could be established, but we were unable to investigate quench times of less than about 5 sec, since these caused fracture of the specimens. The random nature of the solubility variations also suggests that an interstitial-substi­tutional Zn equilibrium may be a more realistic ex­planation of their origin.
In connection with Fig. 2 it is interesting to note that the data fit (in the linear portion) the equation
/>=7.3X10²¹ exp- (1.5ДТ), (3)
where k is expressed in ev per degree. Since, under the conditions of the experiment, the number of holes is equal to the number of Zn atoms if we assume a single acceptor level (see Part II), relation (3) also gives the


(C)

Fig. 3. Schematic diagrams for (a) interstitial Zn atom as a donor, (b) substitutional Zn atom as an acceptor, (c) Zn В com­pound as an acceptor.


tutional Zn atom which might be expected to introduce two acceptor levels into Si. Thus Zn could conceivably introduce both donors and acceptors into pure Si. Furthermore, in the light of our experience in diffusing Li into Ge,¹⁶ we might expect Zn diffusion into doped Si to produce ion pairs and compounds. Figure 3(c) illustrates a ZnB compound. The compound has empty orbitals and so might act as an acceptor.
Results
Hall effect and conductivity measurements as functions of temperature were made on the samples listed in Table I. Typical carrier concentration curves for some ^>-type samples are shown in Fig. 4. The samples were diffused to saturation with Zn using the method of Part I. Control sample 347 was known to contain approximately 7X10¹⁴ cm^-3 boron before heating as shown in Table I. It was found to contain 8X10¹⁴ cm^-3 boron after heating. This indicates that the heating process alone did not introduce an im­portant amount of electrically active impurities.

1. 
   Deep Level
   

The conductivity results of Part I suggest that a single deep acceptor and no donors are produced when Zn is diffused into Si. Hall effect measurements on As- doped samples 338, 339, and 351 (Group A, Table I)

Fig. 4. Carrier concentration curves for a series of silicon samples containing В or As with diffused Zn.

¹⁶ Reiss, Fuller, and Morin, Bell System Tech. J. 35, 535 (1956).


solubility of Zn as a function of temperature. The energy of solution of Zn in Si is then 1.51 ev or 35 kcal. This value lies close to that (27 to 35 kcal) found for Cu in Ge¹⁴ suggesting an analogous solution behavior for these two systems. The dashed line in Fig. 2 represents the extrapolation based on the expected behavior of the Zn-Si liquidus and again is analogous to the curve for Cu in Ge which likewise shows retrograde solubility.
Several saturations were carried out at 1350°C using />-type silicon of different resistivities. The changes observed in the resistivities (measured at room tem­perature) were small and consistent with a single deep-lying acceptor level (see Part II). These experi­ments appear to exclude the presence of any appreciable number of donor levels in the upper half of the forbidden gap and would argue against all but small concentrations of interstitial Zn.

11.