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Comparison between results of optical and electrical measurements of free electron concentration in n-InAs specimens
expand article infoTatyana G. Yugova, Aleksandr G. Belov, Vladimir E. Kanevskii, Evgeniya I. Kladova, Stanislav N. Knyazev, Irina B. Parfent'eva
‡ Federal State Research and Develpment Institute of Rare Metal Industry ("Giredmet"), Moscow, Russia
Open Access

Abstract

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.

Keywords

indium arsenide, electron concentration, Hall effect, reflection spectrum, plasmon-phonon coupling

1. 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 (NHall) may be either lower or higher than those determined from far IR reflection spectra (Nopt).

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.

2. Experimental

The measurements were conducted for 21 n-InAs specimens 16 of which were sulfur-doped and 5 were tin-doped. 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.

Table 1.

Parameters of test specimens

No. Doping impurity d (mm) ν+ (cm–1) N opt (1018 cm–3) N Hall (1018 cm–3) δ (%)
1 S 1.30 369 0.660 0.564 14.5
2 S 1.50 457 1.19 1.04 12.6
3 S 1.74 491 1.43 1.28 10.5
4 Sn 1.03 496 1.48 1.27 14.2
5 S 2.26 532 1.78 1.59 10.7
6 S 1.90 538 1.83 1.59 13.1
7 S 1.77 545 1.89 1.79 5.3
8 S 1.93 573 2.15 1.93 10.2
9 Sn 1.43 591 2.34 2.14 8.5
10 S 1.39 600 2.43 1.95 19.8
11 Sn 1.24 617 2.63 2.32 11.8
12 S 1.01 635 2.84 2.44 14.1
13 S 1.57 639 2.89 2.50 13.5
14 S 2.12 656 3.09 2.79 9.7
15 S 1.47 668 3.25 2.95 9.2
16 S 1.75 673 3.31 3.11 6.0
17 Sn 1.10 677 3.36 2.98 11.3
18 S 2.17 678 3.38 3.05 9.8
19 S 1.79 683 3.44 3.09 10.2
20 S 1.92 684 3.46 3.13 9.5
21 Sn 1.32 684 3.46 2.92 15.6

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 ρ, the free electron concentration NHall and the electron mobility μ were calculated. The relative random error of NHall measurement was within ± 10 %.

N opt was calculated from far IR reflection spectra (plasma resonance) [2–6] taking into account the interaction of plasma oscillations with longitudinal optical phonons (the plasmon-phonon coupling) [7–17]. The wavenumber dependence of the reflection spectrum R (ν) was recorded in the 340–1000 cm–1 spectral range with a Tensor-27 Fourier spectrometer. Then the reflection spectra were processed using the Kramers–Kronig dispersion relations for the calculation of the real ε1(ν) and imaginary ε2(ν) parts of the complex dielectric permeability ε = ε1 + iε2 and for the determination of the relationship between the imaginary part (–1/ε) and the wavenumber: f1(ν) = Im(–1/ε) = ε2/(ε12 + ε22) [18]. This relationship has a typical bell shape with a clear maximum [18–21]. Then the characteristic wavenumber ν+ corresponding to that maximum was determined and after that, based on the ν+ obtained, Nopt was found using a calculated calibration curve.

More detailed information on Nopt determination from the reflection spectrum was reported earlier for n-GaAs [1] and for n-InSb [22]. When plotting the Nopt = f2(ν) calibration curve for n-InAs the non-parabolic shape of the conduction band of InAs was taken into account. Below are parameters included into the calculation formulae:

Band gap Eg, eV 0,36 [23]

HF dielectric permeability ε 11,6 [24]

Wavenumber at longitudinal optical phonon frequency νLO, cm–1 243 [24]

Wavenumber at transverse optical phonon frequency νTO, cm–1 219 [24]

Valence and conduction band interaction matrix element Pcv, eV · cm 8.7 · 10–8 [24]

As a result the electron concentration vs characteristic wavenumber calibration curve was plotted (Fig. 1) which is described adequately well by a third order polynomial:

N opt = 1.05 · 1010+)3 – 3.15 · 1012+)2 + 3.22 · 1015+) – 6.27 · 1017. (1)

Here Nopt is in cm–3 and ν+ is in cm–1.

We show that the plasmon-phonon coupling being disregarded, Nopt 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 ν. Note that the respective differences are 20% for n-GaAs [1] and 30% for n-InSb [22]. Since the absolute random error of ν+ is only controlled by spectral instrument resolution and is within ±2 cm–1, the relative random error of Nopt is within ±0.6 %.

Figure 1.

Calculated electron concentration vs characteristic wavenumber calibration curve. The vertical dotted line is the edge of the Tensor-27 Fourier spectrometer operation range

3. Results and discussion

Table 1 shows the parameters of the test specimens, i.e., thickness (d), ν+, Nopt, NHall (specimens are arranged in order of increasing Nopt), as well as δ = 100 %(NoptNHall)/Nopt characterizing the difference between the optical and Hall data.

As can be seen from Table 1 all the test specimens obey the relationship Nopt > NHall. The parameters δ 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:

N Hall = 0.9002Nopt – 0.0309. (2)

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 R2: the closer R2 to unity the better the approximation. In the case considered R2 = 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 δ (Table 1).

One can therefore safely assure that there is a difference between NHall and Nopt and this difference is unilateral, i.e., NHall anis always lower than Nopt. The random factor (scatter about a certain mean value) is also absent.

As noted above the result was different for n-GaAs: NHall could be lower or higher than Nopt. The concentrations measured by the two methods were equal at Neq = 1.07 · 1018 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.

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 NHall concentration [21]. However since the fraction of the electrically neutral doping impurity is but low, the NHall concentration unlike that for tellurium-doped GaAs crystals cannot become higher than the Nopt 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 Nopt concentration.

As we showed earlier [1] the NHall/Nopt vs NHall dependence for GaAs is described by a parabolic function and this ratio tends to unity with an increase in NHall. 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 NHall 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 Nopt 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 μm) so they became matted. Then we recorded reflection spectra and calculated Nopt 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 f1(ν) = Im(–1/ε) 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 Nopt, smearing of the leading edge of the R (ν) curve and a decrease in the absolute values of the reflection coefficient. Therefore the f1(ν) = Im(–1/ε) function is also smeared and its maximum also shifts towards lower ν. When recalculated to Nopt, 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 ν+ of polished surface is greater than that of ground surface, indicating a decrease in Nopt for lower surface quality. ΔN decreases also, i.e., the optical data for ground surface are in a better agreement with the Hall data.

Furthermore the Nopt concentration in all the test specimens decreases after surface grinding and becomes closer to the NHall concentration (Table 2). The origin of this change in the Nopt 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 Nopt. However Nopt anyway remains higher than NHall which is not quite clear either.

Thus one can assert that there is a systematic difference between the Nopt and NHall 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.

Figure 2.

Hall vs optical electron concentration: white circle is sulfur-doped specimens and blue circle is tin-doped specimens.

Table 2.

Optical and electrical measurement data on electron concentration for polished and ground reflecting surfaces of specimens

No. Reflecting surface treatment d (mm) ν+ (cm–1) N opt (1018 cm–3) N Hall (1018 cm–3) ΔN (1017 cm–3)
1 Polished 1.30 369 6.60 5.64 0.96
Ground 1.14 364 6.35 5.67 0.68
10 Polished 1.39 600 2.43 1.95 4.8
Ground 1.42 597 2.40 2.21 1.9
14 Polished 2.12 656 3.09 2.79 3.0
Ground 2.05 639 2.89 2.69 2.0
18 Polished 2.17 678 3.38 3.05 3.3
Ground 2.00 667 3.24 3.05 1.9
Figure 3.

(a) reflection spectra of Specimen No. 14 for the polished surface (Curve 1) and the ground surface (Curve 2) and (b) f1(n) = Im(–1/e) functions for reflection spectra of Curves 1 and 2 of the same specimen

4. Conclusion

A theoretical model was developed for determining the free electron concentration in n-InAs specimens (Nopt) from characteristic points in far IR reflection spectra. We showed that Nopt determination requires plasmon-phonon coupling be taken into account, otherwise the measured Nopt 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 (Nopt) and using the conventional Van der Pau method (NHall), for different treatment of specimen reflection surface: chemical polishing and fine abrasive grinding. The condition Nopt > NHall 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.

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