Experimental Results

4.5.2.1. Solubility and Distribution Coefficient of Na in Ge

Solubility of Na in Ge crystals was roughly evaluated by us in Ge:Na whose growth from the melt by Stepanov method or by horizontal directional crystallization was indicated above. Let us recall that a distinctive feature of the sodium doping process was that Na was introduced from the "unlimited" source and the amount of the introduced Na was not controlled, as distinct Ge doping with V group elements, such as Sb, which is introduced in a predetermined amount by adding, to the Ge melt, the GeSb pellets containing a known quantity of Sb.

While doping germanium raw material from the "unlimited" source of Na, in many years of our practice we have encountered with cases where the grown crystals had a resistivity above 40 Ohmꞏcm, i.e., they were close to Ge with the intrinsic conductivity whose resistivity equals to 47 Ohmꞏcm [52]. These failures we explained by random deviations of doping regime from the optimum one. However, never was the case when the resistivity of the grown crystals was below 5 Ohmꞏcm, which, according to [8], corresponds to the density of free electrons above 3×10¹⁴ cm⁻³.

The Ge:Na crystals used for measuring Na concentration by the neutron activation analysis (Table 4.3) and by glow discharge mass spectrometry (Table 4.4) had the resistivity about 15 Ohmꞏcm, i.e., the density of free electrons in the crystals was about 1×10¹⁴ cm⁻³ [8]. As seen from Tables 4.3 and 4.4, the density of free electrons and the Na content in the crystals are the same within one order of magnitude, although the Na content is slightly higher than the density of free electrons. This can be due, most likely, to formation of neutral complexes of Na ions with the negatively charged ions or vacancies, or to minor retrograde Na solubility. The solubility of such a nature is typical of many impurities in germanium [53]. In the case of crystal doping with such impurities, during cooling the crystal from the growth temperature to room temperature, the Ge-impurity solid solution becomes supersaturated and the excess impurity atoms would form the second phase and, hence, would become electrically inactive. But we have an argument against the hypothesis of Na second phase formation, at least, against the formation of micron-sized inclusions. Really, as evidenced by the results of our optical measurements (Table 4.2), infrared radiation scattering in Ge:Na crystals is usually lower than that typical of Ge:Sb crystals, in which, due to retrograde Sb solubility, second-phase inclusions of character sizes about 6-9 μm may be formed, which scatter the infrared radiation [14]. Thus, examining the results shown in Tables 4.3 and 4.4, it may be supposed that the retrograde solubility of Na in Ge, if any, is not essential.

Our conclusion that the content of the electrically inactive Na in Ge crystals is small, is inconsistent with the results obtained while studying Na diffusion-in from the Ge crystal surface, on the basis of which it was concluded that Na introduced in this way in the Ge surface layer, is electrically inactive [5, 54] or the content of electrically active Na ions introduced is extremely small as compared with electrically inactive Na impurity [6].

Based on quantitative data on the Na content in Ge crystals doped from the unlimited Na source shown in Tables 4.3 and 4.4 and above considerations, we concluded that the value of the maximum Na solubility in Ge crystals at room temperature may be evaluated as Сsmах = (0.3-1)×10¹⁵ сm^-3.

Since the value of the maximum solubility of the impurity in the solid phase Сsmах and distribution coefficient of this impurity k0 (i.e., the ratio of solubility in the solid phase to a solubility in the liquid phase) are correlated with each other, and in the case of Ge such a correlation is described by the relation Сsmах = 4.4×10²¹k0 [55], by using the above range of Сsmах values, we obtain that the distribution coefficient of Na in Ge equals k0 = (0.7 – 2.3) × 10⁻⁷. This value is very small but is not unrealistic since some other impurities in Ge, for instance, Ag, are characterized by approximately the same low values of the distribution coefficient as well [56]. A very low value of Na distribution coefficient in Ge agrees with our observation that the process of Ge zone-refining from Na is more efficient compared to most other impurities.

4.5.2.2. Na Diffusion in Ge Crystals

Na diffusion in optical-grade Ge:Na single crystals grown from the melt by Stepanov method, has been investigated in 10 cuboid-shape samples, 54×4×4 mm in size, cut along

the (111) axis. Before the experiments, the samples were subjected to mechanical grinding, polishing and then to chemical etching in HCl solution for removing the damaged layer. Low-resistance indium contacts, 9-10 μm in thickness, were deposited on the opposite edges of the samples (along their length) by thermal evaporation. All samples had almost the same resistivity over their volume, approximately 15 Ohmꞏcm.

For comparative measurements, the single-crystalline Ge:Sb samples of the same geometric size, crystallographic orientation and resistivity were prepared.

The sample under investigations was placed in a thermostat and, via deposited contacts, DC voltage of 35 V was applied, i.e., the initial field strength along the sample was about 7 V/cm. The time dependences of the electrical current passing through the Ge:Na sample, were measured at 5 different temperatures in the range from 271 К to 296 K. It was found that the current first increased, then decreased gradually and finally stabilized at a value about 10 mA. The duration of this process depended on temperature T and varied from 800 min at T=271 K to 240 min at T=296 K (Fig. 4.10). After current stabilization, the resistivity distribution along the sample was measured at room temperature by Van der Pauw method [57]. It turned out that this distribution is almost the same for all samples (Fig. 4.11). As shown in Fig. 4.12, after finishing the process of current stabilization in Ge:Na crystals, the resistivity in the anode-adjacent region, about 2 cm in length, was about 46 Ohmꞏcm, i.e., was very close to the resistivity of intrinsic Ge. When approaching the cathode, the resistivity decreased gradually to about 1 Ohmꞏcm, which corresponded to the free electrons density about 1×10¹⁵ сm^-3 [8].

Such distribution of resistivity was obtained at all temperatures used, but the period of time required to achieve the current stabilization, was different at different temperatures.

Fig. 4.10. Time dependences of the current passing through the Ge:Na sample at 5 different temperatures (from 271 K to 296 K) and through the Ge:Sb sample at room temperature.

Fig. 4.11. Distribution of resistivity along the Ge:Na crystal after current stabilization.

The left side corresponds to the anode-adjacent region of the crystal, the right side – to the cathode-adjacent region. T = 300 K.

From our point of view, it is natural to suppose that the changes in resistivity along the Ge:Na samples are due to Na ions drift toward the cathode under the influence of DC electric field. Similar changes in the resistance distribution along the semiconductor crystals as well as the processes of electric contacts "formation" during the initial period of time after applying DC voltage were observed by us previously in studies the drift of shallow mobile donors in II-VI compounds [22, 58].

The distribution of resistivity along the Ge:Na sample length obtained as described above, proved to be very stable: it remained unchanged during the entire observation period of sample storage at room temperature, which lasted for more than 2 years.

The resistivity distribution along the Ge:Na sample could be easily reversed if one swapped anode and cathode and repeated the experiment. In this case, the distribution of resistivity along the sample changed to the "mirror" one, in accordance with the new direction of ions drifting. If, after carrying out the drift process, to anneal the crystal at temperature about 480 °C for 3-3.5 h, it becomes uniform in resistivity again at the same value of resistivity about 15 Ohmꞏcm as before the experiment.

The observed shape of the time dependences of the current may be clearly explained if one presents the equivalent circuit model of the sample as two series-connected resistances R1 and R2, where R1 is the resistance of the anode-adjacent region and R2 is the resistance of the cathode-adjacent region. The changes in the current passing through the Ge:Na samples are due to the fact that the resistance R1 gradually increases since the donor ions gradually move from the anode-adjacent region toward the cathode-adjacent region. Simultaneously, the resistance R2 decreases for the same reasons. Since the dependence of the resistivity on the carrier concentration is not linear [8], it is difficult to predict in advance how the current will change with time. But it should be noted that the change of resistance R1 is limited since, as noted above, the maximum resistivity of Ge at room temperature cannot exceed 47 Ohmꞏcm (the resistivity of intrinsic Ge) [52].

However, the resistance R2 can be reduced to rather low values [8], depending on the Na ions density in the cathode-adjacent region of the sample. Reduction in the current value,

which begins 30 min after the start of the experiment, shows that the length of the depleted region (R1) begins to increase as Na ions drift from this area toward the cathode.

The fact that the value of the current becomes constant finally, indicates that the movement of Na ions to the cathode virtually stopped, which can be explained by the facts that the value of the external electric field is already insufficient for further ions drifting or that Na ions have moved away from a significant part of the sample.

A similar experiment was also carried out on Ge:Sb samples. It was found that, over the same temperature range, after applying the same, as to Ge:Na samples, electric field, the current first also increased (due to the above-mentioned process of contact "formation"), but then the value of the current rapidly stabilized (see curve for Ge:Sb in Fig. 4.10) and then remained unchanged with time, at least during the observation time, which lasted 24 h. This effect we have already described above in Section 4.2.3, it can be explained by the fact that Sb in Ge is substitutional impurity and hence is characterized by a very low diffusion coefficient (1.1×10^-11 cm²/s) [53, 56]. Because of this, under conditions of the experiments described, drift of Sb ions seems to be insignificant and the resistivity of Ge:Sb samples remains unchanged, unlike the Ge:Na samples.

Using the data shown in Fig. 4.10, a dependence lg Δt (10³/T) was constructed graphically, where Δt is time interval from the start of electric field application to current stabilization at the value of 10 mA, T – temperature (Fig. 4.12). From the slope of the dependence obtained, the activation energy Q for Na ions diffusion in Ge:Na crystals at temperatures close to room temperature was evaluated [59]. It appeared that Q = 0.33 eV.

Fig. 4.12. Dependence used for calculation of the activation energy for Na ions diffusion in Ge:Na crystals. The time interval ∆t was measured in seconds.

To evaluate the value of the diffusion coefficient D for Na ions in Ge crystals at room temperature, the Einstein relation u/D = q/kT was used, where u is the mobility of ions, q is the electron charge, k is the Boltzmann constant. Under conditions of electric field application, the mobility of ions u = x/EΔt, where x is the length of high-resistance part of the sample, E is the electric field applied, the determination of Δt was given above.

Hence, from the above two formulas one obtains that D = kTx/qEΔt. By means of this

relation the coefficient D of Li diffusion into Ge crystals was previously determined [59]

and then this method was used, in particular, in our laboratory in numerous studies of ion diffusion in crystals of II-VI compounds (see, for example, [60]). It was shown (see [59]

and Table 5.6 in [61]) that the values of D obtained by this method, are in good agreement with the results of thermal measurements of D carried out in the absence of the electric field. In general, the possibility of measuring the diffusion coefficient by the method under discussion was considered in some authoritative monographs on diffusion in semiconductors ([61, 62] and others) and was never in doubt.

Our calculations of coefficient D of Na ions diffusion in Ge crystals at room temperature made by using the relation (1), have shown that D = 3.6×10⁻⁷ cm²/s.

Evaluation of the pre-exponential factor made from the ratio D = D0еxp(-Q/kT) [59]

showed that D0 = 0.13 cm²/s.

For independent verification of the obtained parameters for Na diffusion in Ge, we have used the Dashman-Langmuir equation [63]. This equation, published in 1922, was cited in many monographs and articles on diffusion in solids [62, 64, 65] and, in spite of the empirical nature, its outstanding applicability for diffusion in many solids was often emphasized. According to this equation, D0 = d²Q/N0h, where d is the lattice constant, h is the Plank's constant, and N0 is the Avogadro number. Using the obtained value of Q, we calculated that D0calc = 0.026 cm²/s. This value is 5 times different from the above evaluated value D0 = 0.13 cm²/s. However, taking into account that the values of D0 for different impurities in Ge may vary within 12 orders of magnitude (see Table 4.6), the coincidence between the calculated and measured values of D0 can be considered quite satisfactory.

It should be noted that the fact that our results regarding the diffusion parameters of Na in Ge, significantly different from the data given in the literature and obtained mainly by studying Na diffusion-in from the Ge surface, does not lead us to believe that all previously obtained values are erroneous. Much more likely that, as it was shown while studying Na diffusion in Si [3], the measured diffusion parameters may depend on how the Na impurity is introduced into the host lattice.