Comparison between results of optical and electrical measurements of free electron concentration in n-InAs specimens

A theoretical model has been developed for determining the free electron concentration in n-InAs specimens from characteristic points in far IR reflection spectra. We show that this determination requires plasmon-phonon coupling be taken into account, otherwise the measured electron concentration proves to be overestimated. A correlation between the electron concentration Nopt and the characteristic wavenumber ν+ has been calculated and proves to be well fit by a third order polynomial. The test specimens have been obtained by tin or sulfur doping of indium arsenide. The electron concentration in the specimens has been measured at room temperature using two methods: the optical method developed by the Authors (Nopt) and the conventional four-probe Hall method (the Van der Pau method, NHall). The reflecting surfaces of the specimens have been chemically polished or fine abrasive ground. The condition Nopt > NHall has been shown to hold for all the test specimens. The difference between the optical and the Hall electron concentrations is greater for specimens having polished reflecting surfaces. The experimental data have been compared with earlier data for n-GaAs. A qualitative model explaining the experimental data has been suggested.


Introduction
The information value of experimental data increases greatly if the target parameter can be measured using different methods. It is recommendable to compare the different measurement data taking into account that each method has its specific features. For example the free carrier concentration in semiconductor specimens is typically measured using the Hall effect, either its classic six-probe variant or the more convenient four-probe Van der Pau modification.
Along with the Hall method the free carrier concentration in heavily doped semiconductors is often measured using the so-called plasma reflection method which is contactless and nondestructive unlike the Hall one. The spectral dependence of the reflection coefficient is recorded in the far IR region and the free carrier concentration is determined from the positions of the characteristic points.
It should be noted that free carrier concentration data obtained by electrical measurements represent the whole specimen bulk while those obtained by optical measurements only refer to the narrow superficial layer of the specimen. For this reason data obtained with these methods may differ. It was shown [1] that the free electron concentrations in n-GaAs calculated on the basis of Hall effect (N Hall ) may be either lower or higher than those determined from far IR reflection spectra (N opt ).
By analogy with the cited work [1] in which tellurium-doped gallium arsenide specimens were studied we present below the optical and electrical measurement data for sulfur-and tin-doped n-InAs specimens.

Experimental
The measurements were conducted for 21 n-InAs specimens 16 of which were sulfur-doped and 5 were tindoped. The specimens were in the form of plane-parallel 6-10 mm side square wafers 1.03-2.26 mm in thickness (Table 1). ~2 mm thick wafers were cut from Cz-grown (100) single crystal indium arsenide ingots perpendicularly to the growth axis following which these wafers were cut into the specimens as described above. The specimens were then mechanically ground and chemically polished. The measurements were conducted at room temperature.
The contact material for electrical measurements was indium. Two specimens were placed on a holder one at each side and tinned copper contact wires were soldered to the holder outputs. The holder with the specimens was placed in the gap between electric magnet core poles perpendicularly to the magnetic field induction vector. The measurements were conducted at magnetic induction B = 0.5 Tl and a 200 mA current passing through the specimen. Then the electrical resistivity r, the free electron concentration N Hall and the electron mobility m were calculated. The relative random error of N Hall measurement was within ± 10 %.
We show that the plasmon-phonon coupling being disregarded, N opt proves to be overestimated but this difference near the edge of the Tensor-27 Fourier spectrometer operation range is within 10% and decreases further with an increase in n. Note that the respective differences are 20% for n-GaAs [1] and 30% for n-InSb [22]. Since the absolute random error of n + is only controlled by spectral instrument resolution and is within ±2 cm -1 , the relative random error of N opt is within ±0.6 %. Table 1 shows the parameters of the test specimens, i.e., thickness (d), n + , N opt , N Hall (specimens are arranged As can be seen from Table 1 all the test specimens obey the relationship N opt > N Hall . The parameters d is the highest for Specimen No. 10 (19.8%) and the lowest for Specimen No. 7 (5.3%). Figure 2 shows the relationship between the Hall and optical electron concentrations which is described by the following linear function:

Results and discussion
When doing RMS linear approximation one should estimate the quality of the fit between the experimental points and the linear function. The criterion is the parameter R 2 : the closer R 2 to unity the better the approximation. In the case considered R 2 = 0.9896 as calculated by the software along with the other approximation parameters.
It can be seen from Fig. 2 that both the sulfur-and the tin-doped specimens obey a similar regularity. The same is true for d (Table 1).
One can therefore safely assure that there is a difference between N Hall and N opt and this difference is unilateral, i.e., N Hall anis always lower than N opt . The random factor (scatter about a certain mean value) is also absent.
As noted above the result was different for n-GaAs: N Hall could be lower or higher than N opt . The concentrations measured by the two methods were equal at N eq = 1.07 · 10 18 cm -3 [1]. For the low-doped material this ratio was < 1 but with an increase in the tellurium concentration the relation became > 1.
This difference in the behavior of doping impurities in GaAs and InAs single crystals can be accounted for by the difference in the homogeneity ranges of these compounds. Excess gallium or indium controls the bulk concentration of arsenic vacancies. As reported earlier [25] the homogeneity range of GaAs is substantially broader than that of InAs. In other words, the concentration of arsenic vacancies in InAs is much lower than in GaAs. Note: Reflection spectra for polished surfaces. The lower arsenic vacancy concentration determines the smaller fraction of the electrically neutral doping impurity in the bulk which forms complexes with arsenic vacancies. During Hall measurements the magnetic field destroys these complexes to transfer the impurity to an electrically active state thus increasing the N Hall concentration [21]. However since the fraction of the electrically neutral doping impurity is but low, the N Hall concentration unlike that for tellurium-doped GaAs crystals cannot become higher than the N opt concentration which is confirmed by the experiment. Furthermore the arsenic vacancy concentration on the wafer surface is very low and as a result almost all the doping impurity is in an electrically active state thus ensuring the highest N opt concentration.
As we showed earlier [1] the N Hall /N opt vs N Hall dependence for GaAs is described by a parabolic function and this ratio tends to unity with an increase in N Hall . Unfortunately InAs exhibits a significant scatter of experimental points and therefore one cannot trace any regularity. However the tendency of decreasing the difference between the optical and Hall measurement data with an increase in N Hall persists.
It is a well-known practice for optical measurements to thoroughly polish the reflecting surface of the test specimen to a mirror-like condition. The question arises, what if the quality of the reflecting surface is intentionally degraded, i.e., the specimen is abrasive ground so the reflecting surface becomes matted? How strongly will the optical properties of the specimen and hence N opt change?
With this task in mind, we conducted the following experiment: we ground the reflecting surfaces of four specimens out of those listed in Table 1 with M10 grinding powder (grain size 10 mm) so they became matted. Then we recorded reflection spectra and calculated N opt following which we repeated electrical measurements. Figure 3 a shows the reflection spectra of Specimen No. 14 ( Table 1) for the polished surface (Curve 1) and for the ground surface (Curve 2). Figure 3 b shows the f 1 (n) = Im(-1/e) functions for the reflection spectra of Curves 1 and 2 of the same specimen in the same scale.
As can be seen from Fig. 3 a degradation of the reflectivity of the specimen surface causes a shift of the reflection spectrum towards lower wavenumbers indicating a decrease in N opt , smearing of the leading edge of the R(n) curve and a decrease in the absolute values of the reflection coefficient. Therefore the f 1 (n) = Im(-1/e) function is also smeared and its maximum also shifts towards lower n. When recalculated to N opt , the decrease proves to be but minor (Table 2).   Table 2 shows optical and electrical measurement data for four specimens out of those listed in Table 1 (the original specimen numbers are kept).
It can be seen from Table 2 that n + of polished surface is greater than that of ground surface, indicating a decrease in N opt for lower surface quality. DN decreases also, i.e., the optical data for ground surface are in a better agreement with the Hall data.
Furthermore the N opt concentration in all the test specimens decreases after surface grinding and becomes closer to the N Hall concentration ( Table 2). The origin of this change in the N opt concentration is not quite clear but one can assume that grinding of specimen surface uncovers the crystal bulk where the vacancy concentration is higher than at the wafer surface and the doping impurity forms complexes with arsenic vacancies. These complexes reduce the fraction of the electrically active doping impurity in the crystal bulk and hence lead to a decrease in N opt . However N opt anyway remains higher than N Hall which is not quite clear either.
Thus one can assert that there is a systematic difference between the N opt and N Hall concentrations, the former concentration always being higher. This difference is smaller for ground specimen surface.
Note that this study was undertaken in order to explore the possibility of transition from the conventional Hall method of free electron concentration measurement to a more convenient optical one. The experimental results reported in this work should be taken into account.

Conclusion
A theoretical model was developed for determining the free electron concentration in n-InAs specimens (N opt ) from characteristic points in far IR reflection spectra. We showed that N opt determination requires plasmon-phonon coupling be taken into account, otherwise the measured N opt electron concentration proves to be 10% overestimated. A correlation between the electron concentration and the characteristic wavenumber was calculated and proves to be well fit by a third order polynomial.
The free electron concentration in the specimens was measured using two methods: based on reflection spectra (N opt ) and using the conventional Van der Pau method (N Hall ), for different treatment of specimen reflection surface: chemical polishing and fine abrasive grinding. The condition N opt > N Hall was shown to hold for all the test specimens. The difference between the optical and the Hall electron concentrations proved to be greater for specimens having polished reflecting surfaces.
The experimental data were compared with earlier data for n-GaAs. A qualitative model explaining the experimental data was suggested.