80urn:lsid:arphahub.com:pub:3CDB4662-2EEC-55B0-878A-88F876A7F403Modern Electronic MaterialsMoEM2452-24492452-1779NUST MISiS10.3897/j.moem.7.3.7670076700Research ArticleComparison between results of optical and electrical measurements of free electron concentration in n-InAs specimensYugovaTatyana G.1p_yugov@mail.ruBelovAleksandr G.1KanevskiiVladimir E.1KladovaEvgeniya I.1KnyazevStanislav N.1Parfent’eva1Irina B.1Federal State Research and Development Institute of Rare Metal Industry (Giredmet JSC), 2 Elektrodnaya Str., Moscow 111524, RussiaFederal State Research and Develpment Institute of Rare Metal Industry ("Giredmet")MoscowRussia
Corresponding author: Tatyana G. Yugova (P_Yugov@mail.ru)
202130092021737984FE809321-3BC8-5A21-B2B1-81B229FB9FFF57511602308202118092021Tatyana G. Yugova, Aleksandr G. Belov, Vladimir E. Kanevskii, Evgeniya I. Kladova, Stanislav N. Knyazev, Irina B. Parfent'evaThis is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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 N_{opt} 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 (N_{opt}) and the conventional four-probe Hall method (the Van der Pau method, N_{Hall}). The reflecting surfaces of the specimens have been chemically polished or fine abrasive ground. The condition N_{opt} > N_{Hall} 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.
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.
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.
Parameters of test specimens
No.
Doping impurity
d (mm)
ν_{+} (cm^{–1})
N_{opt} (10^{18} cm^{–3})
N_{Hall} (10^{18} 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
Note: Reflection spectra for polished surfaces.
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 N_{Hall} and the electron mobility μ were calculated. The relative random error of N_{Hall} 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 ε_{1}(ν) and imaginary ε_{2}(ν) parts of the complex dielectric permeability ε = ε_{1} + iε_{2} and for the determination of the relationship between the imaginary part (–1/ε) and the wavenumber: f_{1}(ν) = Im(–1/ε) = ε_{2}/(ε_{1}^{2} + ε_{2}^{2}) [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, N_{opt} was found using a calculated calibration curve.
More detailed information on N_{opt} determination from the reflection spectrum was reported earlier for n-GaAs [1] and for n-InSb [22]. When plotting the N_{opt} = f_{2}(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 E_{g}, eV 0,36 [23]
HF dielectric permeability ε_{∞} 11,6 [24]
Wavenumber at longitudinal optical phonon frequency ν_{LO}, cm^{–1} 243 [24]
Wavenumber at transverse optical phonon frequency ν_{TO}, cm^{–1} 219 [24]
Valence and conduction band interaction matrix element P_{cv}, 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:
Here N_{opt} is in cm^{–3} and ν_{+} is in cm^{–1}.
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 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 N_{opt} is within ±0.6 %.
39D11935-5BC1-5F8E-8733-6A3BB752069F
Calculated electron concentration vs characteristic wavenumber calibration curve. The vertical dotted line is the edge of the Tensor-27 Fourier spectrometer operation range
https://binary.pensoft.net/fig/6176933. Results and discussion
Table 1 shows the parameters of the test specimens, i.e., thickness (d), ν_{+}, N_{opt}, N_{Hall} (specimens are arranged in order of increasing N_{opt}), as well as δ = 100 %(N_{opt} – N_{Hall})/N_{opt} characterizing the difference between the optical and Hall data.
As can be seen from Table 1 all the test specimens obey the relationship N_{opt} > N_{Hall}. 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} = 0.9002N_{opt} – 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 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 δ (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.
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 μm) so they became matted. Then we recorded reflection spectra and calculated N_{opt} following which we repeated electrical measurements.
Figure 3a shows the reflection spectra of Specimen No. 14 (Table 1) for the polished surface (Curve 1) and for the ground surface (Curve 2). Figure 3b shows the f_{1}(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. 3a 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 (ν) curve and a decrease in the absolute values of the reflection coefficient. Therefore the f_{1}(ν) = Im(–1/ε) function is also smeared and its maximum also shifts towards lower ν. 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 ν_{+} of polished surface is greater than that of ground surface, indicating a decrease in N_{opt} 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 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.
C173D9BE-3673-594C-8F5B-D6497D62CAF1
Hall vs optical electron concentration: white circle is sulfur-doped specimens and blue circle is tin-doped specimens.
https://binary.pensoft.net/fig/617694
Optical and electrical measurement data on electron concentration for polished and ground reflecting surfaces of specimens
No.
Reflecting surface treatment
d (mm)
n_{+} (cm^{–1})
N_{opt} (10^{18} cm^{–3})
N_{Hall} (10^{18} cm^{–3})
DN (10^{17} 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
Note: DN = N_{opt} – N_{Hall}.
0D4B4812-1AB3-5828-A012-949658F01ABF
(a) reflection spectra of Specimen No. 14 for the polished surface (Curve 1) and the ground surface (Curve 2) and (b) f_{1}(n) = Im(–1/e) functions for reflection spectra of Curves 1 and 2 of the same specimen
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.
References1 Yugova T.G., Belov A.G., Kanevskii V.E., Kladova E.I., Knyazev S.N. Comparison between optical and electrophysical data on free electron concentration in tellurim doped n-GaAs. Modern Electronic Materials, 2020; 6(3): 85–89. https://doi.org/10.3897/j.moem.6.3.644922 Galkin G.N., Blinov L.M., Vavilov V.S., Solomatin A.G. Plasma resonance on nonequilibrium carriers in semiconductors. Pis’ma v zhurnal tekhnicheskoi fiziki, 1968; 7(3): 93–96. (In Russ.)3 Belogorokhov A.I., Belov A.G., Petrovitch P.L., Rashevskaya E.P. Determination of the concentration of free charge carriers in Pb_{1-}_{x}Sn_{x}Te taking into account the damping of plasma oscillations. Optika i spectroskopiya, 1987; 63(6): 1293–1296. (In Russ.)4 Belogorokhov A.I., Belogorokhova L.I., Belov A.G., Rashevskaya E.P. Plasma resonance of free charge carriers and estimation of some parameters of the band structure of the material Cd_{x}Hg_{1–}_{x}Te. Fizika i tekhnika poluprovodnikov, 1991; 25(7): 1196–1203. (In Russ.). URL: https://journals.ioffe.ru/articles/viewPDF/234915 Sharov M.K. Plasma resonance in Pb_{1-}_{x}Ag_{x}Te alloys. Semiconductors, 2014; 48(3): 299–301. https://doi.org/10.1134/S10637826140302456 Rokakh A.G., Shishkin M.I., Skaptsov A.A., Puzynya V.A. On the possibility of the plasma resonance in CdS–PbS films in the middle infrared region. Prikladnaya Fizika. 2014; (5): 58–60. (In Russ.)7 Varga B.B. Coupling of plasmons to polar phonons in degenerate semiconductors. Phys. Rev., 1965; 137(6A): 1896–1901. https://doi.org/10.1103/PhysRev.137.A18968 Singwi K.S., Tosi M.P. Interaction of plasmons and optical phonons in degenerate semiconductors. Phys. Rev., 1966; 147(2): 658–662. https://doi.org/10.1103/PhysRev.147.6589 Shkerdin G., Rabbaa S., Stiens J., Vounckx R. Influence of electron scattering on phonon-plasmon coupled modes dispersion and free electron absorption in n-doped GaN semiconductors at mid-IR wavelengths. Phys. Status Solidi (b), 2014; 251(4): 882–891. https://doi.org/10.1002/pssb.20135003910 Ishioka K., Brixius K., Höfer U., Rustagi A., Thatcher E.M., Stanton C.J., Petek Hr. Dynamically coupled plasmon-phonon modes in GaP: an indirect-gap polar semiconductor. Phys. Rev. B. 2015; 92(20): 205203. https://doi.org/10.1103/PhysRevB.92.20520311 Volodin V.A., Efremov M.D., Preobrazhensky V.V., Semyagin B.R., Bolotov V.V., Sachkov V.A., Galaktionov E.A., Kretinin A.V. Investigation of phonon-plasmon interaction in GaAs/AlAs tunnel superlattices. Pis’ma v zhurnal tekhnicheskoi fiziki. 2000; 71(11): 698–704. (In Russ.)12 Kulik L.V., Kukushkin I.V., Kirpichev V.E., Klitzing K.V., Eberl K. Interaction between intersubband Bernstein modes and coupled plasmon-phonon modes. Phys. Rev. B, 2000; 61(19): 12717–12720. https://doi.org/10.1103/PhysRevB.61.1271713 Mandal P.K., Chikan V. Plasmon-phonon coupling in charged n-type CdSe quantum dots: a THz time-domain spectroscopic study. Nano Lett., 2007; 7(8): 2521–2528. https://doi.org/10.1021/nl070853q14 Stepanov N., Grabov V. Optical properties of Bi_{1-}_{x}Sb_{x} crystals, related electron-plasmon and plasmon-phonon interactions. Izv. RGPU im. Gertsena, 2004; 4(8): 52–64. (In Russ.)15 Trajic J., Romcevic N., Romcevic M., Nikiforov V.N. Plasmon-phonon and plasmon-two different phonon interaction in Pb_{1-}_{x}Mn_{x}Te mixed crystals. Mater. Res. Bull., 2007; 42(12): 2192–2201. https://doi.org/10.1016/j.materresbull.2007.01.00316 Chudzinski P. Resonant plasmon-phonon coupling and its role in magneto-thermoelectricity in bismuth. Eur. Phys. J. B, 2015; 88(12): 344. https://doi.org/10.1140/epjb/e2015-60674-317 Belov A.G., Denisov I.A., Kanevskii V.E., Pashkova N.V., Lysenko A.P. Determining the free carrier density in Cd_{x}Hg_{1-}_{x}Te solid solutions from far-infrared reflection spectra. Semiconductors, 2017; 51(13): 1732–1736. https://doi.org/10.1134/S106378261713004818 Yu P. Y., Cardona M. Fundamentals of Semiconductors. Berlin; Heidelberg: Springer-Verlag 2010, 778 p. https://doi.org/10.1007/978-3-642-00710-119 Vinogradov E.A., Vodopyanov L.K. Graphical method for determining phonon frequencies from reflection spectra of crystals in the far infrared region of the spectrum. Kratkie soobtsheniya po fizike, 1972; (11): 29–32. (In Russ.)20 Belogorokhov A.I., Belogorokhova L. I. Optical phonons in cylindrical filaments of porous GaP. Fizika tverdogo tela, 2001; 43(9): 1693–1697. (In Russ.). URL: https://journals.ioffe.ru/articles/viewPDF/3832021 Yugova T.G., Belov A.G., Knyazev S.N. Magnetoplastic effect in Te-doped GaAs single crystals. Crystallography Reports, 2020; 65(1); 7–11. https://doi.org/10.1134/S106377452001027722 Belova I.M., Belov A.G., Kanevskii V.E., Lysenko A.P. Determining the concentration of free electrons in n-InSb from far-infrared reflectance spectra with allowance for plasmon-phonon coupling. Semiconductors, 2018; 52(15): 1942–1946. https://doi.org/10.1134/S106378261815003423 Pankove J.I. Optical processes in semiconductors. New York: Prentice Hall, 1971, 422 p.24 Madelung O. Physics of III-V compounds. New York: John Wiley & Sons, Inc., 1964, 409 p.25 Bublik V.T., Milvidsky M.G. Intrinsic point defects, nonstoichiometry and microdefects in A^{3}B^{5} compounds. Materialovedenie, 1997; (2): 21–29. (In Russ.)