Corresponding author: Tatyana G. Yugova ( p_yugov@mail.ru ) © 2020 Tatyana G. Yugova, Aleksandr G. Belov, Vladimir E. Kanevskii, Evgeniya I. Kladova, Stanislav N. Knyazev.
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Citation:
Yugova TG, Belov AG, Kanevskii VE, Kladova EI, Knyazev SN (2020) Comparison between optical and electrophysical data on free electron concentration in tellurium doped nGaAs. Modern Electronic Materials 6(3): 8589. https://doi.org/10.3897/j.moem.6.3.64492

A theoretical model has been developed for determining free electron concentration in nGaAs from characteristic points in the far infrared region of reflection spectra. We show that when determining free electron concentration one should take into account the pasmon–phonon coupling, otherwise free electron concentration will be overestimated. We have calculated electron concentration N_{opt} as a function of characteristic wave number ν_{+} which is described by a second order polynomial.
Twentyfive tellurium doped gallium arsenide specimens have been tested for electron concentration using two methods, i.e., the conventional fourprobe method (Van der Pau) and the optical method developed by us (the measurements have been carried out at room temperature). We have used the experimental results to plot the dependence of electron concentration based on the Hall data (N_{Hall}) on electron concentration based on the optical data (N_{opt}). This dependence is described by a linear function. We show that the data of optical and electrophysical measurements agree if the electron concentration is N_{eq} = 1.07 · 10^{18} cm^{3}. At lower Hall electron concentrations, N_{Hall} < N_{opt}, whereas at higher ones, N_{Hall} > N_{opt}. We have suggested a qualitative model describing these results. We assume that tellurium atoms associate into complexes with arsenic vacancies thus reducing the concentration of electrons. The concentration of arsenic vacancies is lower on the crystal surface, hence the N_{opt} > N_{Hall} condition should be met. With an increase in doping level, more and more tellurium atoms remain electrically active, so the bulk concentration of electrons starts to prevail over the surface one. However with further increase in doping level the N_{Hall}/N_{opt} ratio starts to decrease again and tends to unity. This seems to originate from the fact that the decomposition intensity of the tellurium atom + arsenic vacancy complexes decreases with an increase in doping level.
gallium arsenide, electron concentration, Hall effect, reflection spectrum, plasmon–phonon coupling.
If measurements of the same parameter are conducted using different methods the value of the measurement data increases considerably. However one can then expect differences between such measurement data because different physical methods are used. These differences should be taken into account in each specific case because every measurement method has its specific applicability limits and errors (random and systematic).
Below we consider measurement data for free electron concentration N (cm^{3}) in heavily tellurium doped nGaAs specimens. The data were obtained using two methods, i.e., electrophysical measurements using the Van der Pau method (N_{Hall}) and measurements in the far infrared region of reflection spectra (N_{opt}). It should be noted that for Hall measurements the specimens are exposed to magnetic fields. This question has attracted serious attention in recent years. The magnetoplastic effect, i.e., the movement of dislocations in crystals due to the effect of a magnetic field, was found and thoroughly studied in many earlier works [
The test tellurium doped gallium arsenide single crystal specimens GaAs : Te were in the form of planeparallel square wafers with the (100) orientation, linear sizes of 6–10 mm and a thickness of 1–2 mm. (100) wafers were cut from Czgrown single crystal GaAs(Te) perpendicularly to the growth axis and then cut into test specimens. After cutting the planar surfaces of the test specimens were first mechanically ground and then chemically polished.
All the measurements were carried out at room temperature.
For the electrophysical measurements the contacts were soldered with tin at the specimen corners. The contact conductors were made of 0.05 mm diameter tincoated copper wires. The specimens were placed on a doubleside holder (one specimen at each side) and the wires were soldered to the respective contact pads of the holder.
The electrophysical measurements were conducted using the conventional fourprobe arrangement (the Van der Pau method). A holder with two test specimens was placed between the poles of electric magnet cores perpendicularly to the magnetic field induction vector. The measurements were carried out at a constant magnetic field induction (В = 0.5 T), and a 100 mA current was passed through the specimens. Then we calculated the electrical resistivity ρ, the free electron concentration N_{Hall} and the free electron mobility μ. The relative random error of N_{Hall} determination was within ±7%.
The reflection spectra of the specimens were recorded with a Tensor27 Fourier spectrometer in the ν = 340÷5000 cm^{1} wave number range. Then using the Kramers–Kronig dispersion relations we calculated the dependences of the real ε_{1} and imaginary ε_{2} components of dielectric permittivity (ε = ε_{1} + iε_{2}) on wave number ν and plotted the dependence
$f\left(v\right)=Im\left(\frac{1}{\epsilon}\right)=\frac{{\epsilon}_{2}}{{\epsilon}_{1}^{2}\xf7{\epsilon}_{2}^{2}}$.
This dependence has a distinctive bellshaped pattern with a prominent maximum. We determined the wave number ν_{+} corresponding to this maximum and then calculated the concentration of electrons N_{opt} on the basis of this wave number. It should be noted that when calculating N_{opt} on the basis of a specific wave number ν_{+} one should take into account the plasmonphonon coupling because materials with a significant ionic conductivity contribution (e.g. GaAs) have not only longitudinal collective oscillations of the free carrier (plasmon) system but also longitudinal oscillations of the crystal lattice (LO phonons). The frequency of the plasmon oscillations ωр depends on the free carrier concentration (in the case in question, electrons) N_{opt} through a simple relationship [
${\omega}_{\mathrm{p}}^{2}=\frac{4\pi {N}_{\mathrm{opt}}{e}^{2}}{{\epsilon}_{\infty}{m}^{*}}$. (1)
Here e is the electron charge, ε_{∞} is the highfrequency dielectric permittivity and m* is the effective electron mass.
As can be seen from Eq. (1) the frequency of plasmon oscillations for a specific material depend only on the concentration of electrons N_{opt}, by varying which one can change ω_{р}. If the plasmon oscillation frequency wр and the longitudinal optical phonon oscillation frequency ω_{LO} differ significantly then the two abovementioned types of longitudinal oscillations exist independently. The longitudinal optical phonon oscillation frequency ω_{LO} is determined by the elastic properties of the semiconductor crystal lattice and does not depend on the doping level. On the contrary, ω_{р} can be easily controlled by varying N_{opt} (see Eq. (1)).
If ωр and ω_{LO} are close then the independent plasmons and longitudinal optical phonons which existed earlier are replaced by coupled plasmonphonon modes [
${\omega}_{\pm}^{2}=\frac{1}{2}\left[\left({\omega}_{\mathrm{p}}^{2}+{\omega}_{\mathrm{LO}}^{2}\right)\pm \sqrt{{\left({\omega}_{\mathrm{p}}^{2}+{\omega}_{\mathrm{LO}}^{2}\right)}^{2}4\frac{{\epsilon}_{\infty}}{{\epsilon}_{0}}{\omega}_{\mathrm{p}}^{2}{\omega}_{\mathrm{LO}}^{2}}\right]$. (2)
Here ε_{0} is the static dielectric permittivity.
Thus the reflection spectrum of the material will contain two minima corresponding to the frequencies of the coupled plasmonphonon modes, i.e., ω_{+} and ω–. To calculate N_{opt} one can use any of these mode frequencies, the choice being determined by the capacity of the spectrometer (in this work we used ω_{+}).
The basics of the ω_{+} and ω– frequencies calculation method and the respective wave numbers ν_{+} and ν– were described in detail for the InSb semiconductor earlier [
Figure
Calculated dependences of the electron concentration Nopt on the characteristic wave number ν_{+} taking into account (1) and disregarding (2) the plasmonphonon coupling. Vertical line is the edge of the working range of the Tensor27 Fourier spectrometer (340 cm^{1}).
Thus using Curve 1 as a calibration dependence and using known ν_{+} (in cm^{1}) one can calculate N_{opt} (in cm^{3}). This dependence is described by a second order polynomial as follows:
N _{opt} = 6.33 ∙ 10^{12} (ν_{+})^{2} + 2.11 ∙ 10^{15} (ν_{+}) – 6,81 ∙ 10^{17}. (3)
A typical reflection spectrum R (ν) of the nGaAs specimens (Curve 1) is shown in Fig.
Typical reflection spectrum of (1) nGaAs sample and (2) f (ν) = Im (–1/ε) dependence. Vertical line marks the value of the characteristic wavenumber ν_{+}.
Figure
It can be seen from Fig.
N _{Hall} = 1.1973N_{opt} – 2.1033. (4)
Eq. (4) shows that the equality of the concentrations N_{Hall} and N_{opt} is achieved at N_{eqt} ≈ 1.07 ∙ 10^{18} cm^{3}. At lower Hall concentrations N_{Hall} < N_{opt} whereas at higher Hall concentrations, N_{Hall} > N_{op}.
Figure
It can be seen from Fig.
$\frac{{N}_{\text{Hall}}}{{N}_{\text{opt}}}=0.0005{N}_{\text{Hall}}^{2}+0,028{N}_{\text{Hall}}+0.7442$. (5)
Studying the magnetoplastic effect in tellurium doped GaAs single crystals we noted the systematic disagreement between the free electron concentration data, with the inequality N_{Hall} > N_{opt} being obeyed [
Thus one can state the disagreement between N_{Hall} and N_{opt} and that the inequality of these parameters is not unilateral, i.e., N_{Hall} is smaller than N_{opt} in one range of electron concentrations while this is on the contrary in another range. Furthermore there is no random factor (scatter of parameters to one or another side relative to a certain average value).
It should also be borne in mind that the information obtained from the reflection spectra relates to a narrow (several tenths of a micron) surface layer of the specimens.
On the contrary electrophysical measurements cover the entire specimen bulk. Then the systematic disagreement between N_{Hall} and N_{opt} could be accounted for by the differences between the physical properties of the surface and the bulk of the specimen. Another factor contributing to the systematic disagreement between the data obtained using optical and electrophysical methods could be the imperfection of the mathematical model used for the calculation of N_{opt}. However the fact that the difference between N_{Hall} and N_{opt} is not unilateral requires separate investigation.
It was hypothesized [
It should be noted that the concentration of electrons N_{opt} on specimen surfaces did not change after magnetic treatment. This result is in agreement with the assumption that the concentration of tellurium atom + arsenic vacancy complexes is the lowest on the specimen surface.
We developed a theoretical model for determining free electron concentration N_{opt} from characteristic points in far infrared reflection spectra.
We showed that when determining free electron concentration N_{opt} one should take into account the pasmon–phonon coupling, otherwise the free electron concentration N_{opt} will be overestimated (by up to 20%).
We measured the free electron concentration based on the reflection spectrum (N_{opt}) and using the conventional fourprobe method (Van der Pau) (N_{Hall}). For electron concentrations N_{eq} ≈ 1.07 ∙ 10^{18} cm^{3}, the ratio N_{Hall}/N_{opt} = 1; at lower Hall electron concentrations, N_{Hall} < N_{opt} whereas at higher Hall electron concentrations, N_{Hall} < N_{opt}. The difference of the values is greater than the measurement error.
We showed that the N_{Hall}/N_{opt} = f (N_{Hall}) dependence is described adequately well by a second order polynomial.
We suggested a model that explains the experimental data by decomposition of tellurium atom + arsenic vacancy complexes as a result of magnetic treatment of gallium arsenide specimens.