Research Article 
Corresponding author: Aleksandr G. Belov ( iadenisov@giredmet.ru ) © 2023 Aleksandr G. Belov, Vladimir E. Kanevskii, Evgeniya I. Kladova, Stanislav N. Knyazev, Nikita Yu. Komarovskiy, Irina B. Parfent'eva, Evgeniya V. Chernyshova.
This 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.
Citation:
Belov AG, Kanevskii VE, Kladova EI, Knyazev SN, Komarovskiy NY, Parfenteva IB, Chernyshova EV (2023) Comparison between optical and electrical data on hole concentration in zincdoped pGaAs. Modern Electronic Materials 9(2): 6976. https://doi.org/10.3897/j.moem.9.2.109743

The optical and electrical properties of zincdoped Cz pGaAs have been studied. Reflection spectra of ten pGaAs specimens have been taken in the midIR region. Van der Pau galvanomagnetic, electrical resistivity and Hall coefficient measurements have been carried out for the same specimens (all the measurements were carried out at room temperature). The reflection spectra have been processed using the Kramers–Kronig relations, spectral dependences of the real and imaginary parts of the complex dielectric permeability have been calculated and loss function curves have been plotted. The loss function maximum position has been used to calculate the characteristic wavenumber corresponding to the highfrequency plasmonphonon mode frequency. Theoretical calculations have been conducted and a calibration curve has been built up for determining heavy hole concentration in pGaAs at T = 295 K based on known characteristic wavenumber. Further matching of the optical and Hall data has been used for determining the light to heavy hole mobility ratio. This ratio proves to be in the 1.9–2.8 range which is far lower as compared with theoretical predictions in the assumption of the same scattering mechanism for light and heavy holes (at optical phonons). It has been hypothesized that the scattering mechanisms for light and heavy holes differ.
gallium arsenide, electron concentration, Hall effect, reflection spectrum, plasmonphonon interaction
This paper is a continuation of our earlier works at Giredmet JSC aimed at developing a contactless nondestructive optical method of free carrier concentration measurement in heavily doped semiconductors. The basic principle of the method is as follows. A reflection spectrum is taken from the test specimen in the midIR region. The reflection spectrum is analyzed using the Kramers–Kronig relations in order to determine the characteristic wavenumber based on which the free carrier concentration (FCC) is calculated.
This approach has a number of advantages over the conventional Hall method: it does not require attaching contacts to the specimen and is streamlined and local (the probing area depends on the light spot dimensions). Furthermore, by moving the specimen relative to the light spot one can take reflection spectra at different points on the test specimen and thus to have an overview of the FCC distribution over its area.
The required calculations were made for nInSb [
In this work we attempted to apply the above described approach to a ptype conductivity material. We studied zincdoped pGaAs specimens. Since this material has two types of holes (light and heavy), the earlier approach required a substantial correction, and that is what this work deals with. All the measurements were carried out at room temperature.
The aim of the work was to build up a calibration curve for determining the heavy hole concentration in pGaAs based on the characteristic wavenumber, carry out optical measurements, calculate the heavy hole concentration and compare the data with electrical measurement data obtained for the same specimens.
The test materials were zincdoped single crystal Cz GaAs ingots. (100) oriented wafers were cut from the ingots perpendicular to the growth axis. The wafers were cut into specimens sized 6–10 with the thickness d = 1–2 mm. After cutting the flat specimen surfaces were mechanically ground and then chemomechanicaly polished.
The free carrier concentration was measured by analyzing the midIR region reflection spectra of the specimens R (ν) recorded with a Tensor27 Fourier spectrometer in the ν = 340÷2000 cm^{1} wavenumber range with a 2 cm^{1} resolution. The light spot diameter was 4.5 mm. The reflection spectra were processed using the Kramers–Kronig relations in order to calculate the real ε_{1} and imaginary ε_{2} parts of the complex dielectric permeability ε = ε_{1} + iε_{2} as a function of wavenumber and to plot the socalled loss function:
$LF=lm\left(\frac{1}{\epsilon}\right)=\frac{{\epsilon}_{2}}{{\epsilon}_{1}^{2}+{\epsilon}_{2}^{2}}$
This function has a typical bellshaped pattern with its maximum corresponding to the characteristic wavenumber. Then this wavenumber was used for calculating the FCC with a special calculation formula taking into account the effect of plasmonphonon interaction.
The optical data were compared with the electrical measurement results. ntype conductivity materials were studied in earlier works [
For electrical measurements the contacts were tinsoldered to the edges of specimen buttends. Two specimens were placed on a twoside holder one at each side, and the contact wires were soldered to the matching contact pads of the holder. The holder with the specimens was placed in the gap between the poles of an electric magnet core perpendicular to the magnetic field induction vector. The measurements were carried out using a standard fourprobe setup (the Van der Pau method). The electrical resistivity was measured without magnetic field, and the Hall coefficient, in a B = 0.5 T field with a 200 mA current through the specimen.
It is wellknown (see e.g. overview [
In other words, ptype conductivity GaAs contains two types of holes: light and heavy, and hence treatise of optical and electrical measurement data is much complicated as compared with an ntype material which contains only electrons. The plasma frequency ω_{p} formula for the case of two types of holes is as follows [
${\omega}_{p}^{2}=\frac{4\pi {e}^{2}}{{\epsilon}_{\infty}}\left(\frac{{p}_{h}}{{m}_{{p}_{h}}}+\frac{{p}_{1}}{{m}_{{p}_{1}}}\right)=\frac{4\pi {e}^{2}{p}_{h}}{{\epsilon}_{\infty}{m}_{{p}_{h}}}\left(1+\frac{{p}_{1}{m}_{{p}_{h}}}{{p}_{h}{m}_{{p}_{1}}}\right).$ (1)
Here p_{h} and p_{l} are the heavy and light hole concentrations, respectively, m_{p}_{h} and m_{p}_{l} are their effective optical masses, respectively, ε_{∞} is the highfrequency dielectric permeability, e = 4.8 ∙ 10^{10} CGS u. is the electron charge.
The concentrations of heavy and light holes obey the following relationship:
${p}_{h}=\frac{8\pi}{3{h}^{3}}{\left(2{m}_{{p}_{h}}kT\right)}^{3/2}{F}_{3/2}\left(\eta \right);$ (2)
${p}_{1}=\frac{8\pi}{3{h}^{3}}{\left(2{m}_{{p}_{1}}kT\right)}^{3/2}{F}_{3/2}\left(\eta \right).$ (3)
Here h = 6.62 ∙ 10^{27} erg ∙ s is Planck’s constant, k = 1.38 ∙ 10^{16} erg/K is the Boltzmann constant (for T = 295 K, kT = 25.4 meV) and F_{3/2}(η) is the oneparameter Fermi integral:
${F}_{3/2}\left(\eta \right)=\underset{0}{\overset{\infty}{\int}}\left(\frac{\partial {f}_{0}}{\partial x}\right){x}^{3/2}dx$ (4)
where f_{0}(x, η) = [1 + exp (x – η)^{–1}; η = E_{F}/kT is the reduced Fermi level (counted down from the valence band ceiling at the Γpoint).
It can be seen from Eqs. (2) and (3) that the heavy to light hole concentration ratio does not depend on the reduced Fermi level and equals the heavy to light hole effective mass ratio to a power of 3/2:
$\frac{{p}_{h}}{{p}_{1}}={\left(\frac{{m}_{{p}_{h}}}{{m}_{{p}_{1}}}\right)}^{3/2}.$ (5)
It should also be noted that since the heavy and light hole subbands have parabolic shapes and are isotropic, the effective masses of the density of states in Eqs. (2) and (3) equal the effective optical masses in Eq. (1).
Since gallium arsenide is a semiconductor with a substantial fraction of ionic bond, one should take into account the interaction between plasmon oscillations and longitudinal optical phonons (the socalled plasmonphonon interaction).
In other words, the test material contains not purely plasma oscillations but some combined plasmonphonon modes [
In the case in question, neglect of plasmonphonon interaction can cause a tangible systematic error in FCC measurement.
To calculate the frequencies of the combined plasmonphonon modes, we use the following relationship:
$\epsilon \left(\omega \right)={\epsilon}_{\infty}\left[1{\left(\frac{{\omega}_{p}}{\omega}\right)}^{2}\right]+\left({\epsilon}_{0}{\epsilon}_{\infty}\right){\left[1\frac{{\epsilon}_{0}}{{\epsilon}_{\infty}}{\left(\frac{\omega}{.{\omega}_{LO}}\right)}^{2}\right]}^{1},$ (6)
where ε_{0} is the static dielectric permeability and ω_{LO} is the longitudinal optical phonon frequency. Equation (6) does not take into account the extinction of plasmons and longitudinal optical phonons and therefore the dielectric permeability is not a complex but a natural function of the frequency ω. This approximation is very rough but it provides the desired result.
It is wellknown that longitudinal oscillations (and that is what the combined plasmonphonon modes are) can exist in a media only if its dielectric permeability falls down to zero. Bringing Eq. (6) to zero, solving the biquadratic equation for the frequency of the combined modes ω– (the lowfrequency one) and ω_{+} (the highfrequency one) and transiting from frequencies ω to wavenumbers ν we obtain the following expression:
${\nu}_{\pm}^{2}=\frac{1}{2}\left[\left({\nu}_{p}^{2}+{\nu}_{LO}^{2}\right)\pm \sqrt{{\left({\nu}_{p}^{2}+{\nu}_{LO}^{2}\right)}^{2}4\frac{{\epsilon}_{\infty}}{{\epsilon}_{0}}{\nu}_{p}^{2}{\nu}_{LO}^{2}}\right].$ (7)
We will hereinafter deal only with the highfrequency mode ν_{+}.
The calculation algorithm is as follows:
– set the value of η and calculate p_{h} and p_{l} using Eqs. (2) and (3);
– substitute the result into Eq. (1) and calculate ω_{p};
– substitute the result into Eq. (7) and calculate ν_{+};
– change the value of η and repeat the above operations;
– build up a calibration curve of heavy hole concentration as a function of characteristic wavenumber p_{h} = f (ν_{+}).
The calculations were carried out with the following parameters borrowed from earlier review [
Then Eqs. (2) and (3) take on as follows:
p _{h} = 6.696 ∙ 10^{18} F_{3/2}(η); (8)
p _{l} = 4.322 ∙ 10^{17} F_{3/2}(η). (9)
Transiting to wavenumbers one can transform Eq. (1) as follows:
ν_{p} = 15.02 ∙ 10^{8}$\sqrt{{p}_{h}}$ (10)
Table
η  F _{3/2}(η)  p _{h} (10^{18} cm^{3})  ν_{p} (cm^{1})  ν_{+} (cm^{1}) 
–1  0.436  2.918  256.6  325.8 
–0.5  0.675  4.518  319.2  361.8 
0  1.017  6.811  392.0  419.3 
0.5  1.485  9.946  473.7  492.9 
1  2.095  14.025  562.5  577.2 
1.5  2.851  19.092  656.3  668.1 
2  3.754  25.135  753.0  762.9 
2.5  4.795  32.107  851.1  859.6 
3  5.966  39.945  949.3  956.8 
As can be seen from Table
The data presented in Table
Calculated calibration curve for heavy hole concentration as a function of characteristic wavenumber
p _{h} = 3.937 ∙ 10^{13}(ν_{+})^{2} + 7.635 ∙ 10^{15}(ν_{+}) –
– 3.495 ∙ 10^{18}. (11)
If there are two types of holes the Hall coefficient R_{H} and the electrical resistivity r can be described with the following relationships:
${R}_{H}=\frac{1}{e}\frac{\left({p}_{1}{\mu}_{{p}_{1}}^{2}+{p}_{h}{\mu}_{{p}_{h}}^{2}\right)}{{\left({p}_{1}{\mu}_{{p}_{1}}+{p}_{h}{\mu}_{{p}_{h}}\right)}^{2}};$ (12)
${p}^{1}=e\left({p}_{1}{\mu}_{{p}_{1}}+{p}_{h}{\mu}_{{p}_{h}}\right).$ (13)
Introducing the dimensionless parameter equalling the light to heavy hole mobility ratio b = μ_{pl}/μ_{ph}, one can transform Eqs. (12) and (13) as follows:
${R}_{H}=\frac{1}{e}\frac{\left({p}_{1}{b}^{2}+{p}_{h}\right)}{{\left({p}_{1}b+{p}_{h}\right)}^{2}};$ (14)
${p}^{1}=e{\mu}_{{p}_{h}}\left({p}_{1}b+{p}_{h}\right).$ (15)
Taking into account that p_{h} = 15.51p_{l}, we finally obtain:
${R}_{H}=\frac{1}{e{p}_{h}}\frac{\left(1+0.06447{b}^{2}\right)}{{\left(1+0.06447b\right)}^{2}};$ (16)
${p}^{1}=e{\mu}_{{p}_{h}}{p}_{h}\left(1+0.06447b\right).$ (17)
Thus, having determined p_{h} from optical data using the calibration function as per Eq. (11) and knowing R_{H}, one can use Eq. (16) to calculate the parameter b and then use Eq. (17) to calculate μ_{ph} based on known ρ. To the best of our knowledge, this approach is used for the first time.
A typical room temperature reflection spectrum of zincdoped pGaAs specimens is shown in Fig.
Specimen  d (mm)  ρ (Ohm ∙ cm)  R _{H} (cm^{3}/Cl)  ν_{+} (cm^{1})  p _{h} (10^{18} cm^{3})  b  μ_{p}_{h} (cm^{2}/(V ∙ s)) 
1  1.87  0.015  +1.5  359  4.32  2.5  90 
2  1.66  0.013  +1.4  374  4.87  2.8  80 
3  1.36  0.012  +1.12  395  5.66  2.7  80 
4  1.33  0.010  +0.91  420  6.66  1.9  80 
5  1.09  0.010  +0.93  429  7.03  2.5  80 
6  1.55  0.0090  +0.80  444  7.66  1.9  80 
7  1.64  0.0091  +0.80  456  8.17  2.5  70 
8  1.72  0.0080  +0.66  488  9.61  2.3  70 
9  1.28  0.0063  +0.50  551  12.7  2.2  80 
10  1.36  0.0056  +0.39  615  16.1  2.3  60 
(1) reflection spectra and (2) loss function curves for (a) ptype and (b) ntype GaAs specimens
The minimum in the reflection spectrum of the ptype specimen (Fig.
Reflection spectrum pattern is known to depend on reflecting surface treatment quality. We specially studied earlier [
However this explanation is not applicable to the case considered since the reflecting surfaces of the n and ptype GaAs specimens were similar. It is safe to assume that the difference between the reflection spectra in Fig.
We will now dwell upon the experimental results. Table
The random relative errors of the parameters measured with the confidence probability P = 0.95 are not greater than (according to earlier special metrological studies):
– ±3 % for the electrical resistivity;
– ±6 % for the Hall coefficient;
– ≤ ±0.6 % for the characteristic wavenumber (depends on spectrometer resolution, i.e., 2 cm^{1}, thus p_{h} will have the same relative error).
The absolute random error of specimen thickness measurement is half of the measurement head scale unit (0.005 mm).
As for the relative random error of the b parameter, it can be evaluated by calculation. Expressing b from Eq. (16) and assuming that this parameter is a random function of two independent variables (R_{H} и p_{h}), then, knowing the relative random errors of these parameters one can calculate the relative random error of the b parameter which proves to be not greater than ±15 % (this is an overestimate). By analogy, using Eq. (17) one can find the relative random error of μ_{p}_{h} to be not greater than ±20 %.
It can be seen from Table
The classical book by O. Madelung [
Noteworthy, earlier works on the electrical properties of pGaAs (see e.g. [
The same is true for some later works [
The authors of that work [
It was also approved [
The introduction of the “effective Hall factor” formally simplifies experimental data processing, but the physical sense of this parameter remains unclear. As for the b parameter, it is generally out of research interest, it being a priori assumed that this parameter is the inverse efficient carrier mass ratio to a power of 3/2 (see above). Then the b parameter should be 8.57 based on earlier data [
Our results confront the literary accepted model according to which light and heavy holes are scattered similarly (at optical phonons). Since the b parameter proved to be 5–6 times smaller than expected, it has to be admitted that the accepted model does not work. In other words, light and the heavy holes should are scattered in different ways. The available data do not allow to judge on how they scattered, and this is to be investigated in separate work.
The plasmon scattering frequency and highfrequency combined plasmonphonon mode frequency were theoretically calculated as a function of heavy hole concentration for pGaAs at T = 295 K. A calibration curve for heavy hole concentration as a function of characteristic wavenumber ν_{+} was built up (described by a secondorder polynomial).
The room temperature reflection spectra of 10 zincdoped pGaAs specimens were studied. The spectral dependences of the real and imaginary parts of the complex dielectric permeability were calculated using the Kramers–Kronig relations, and the loss function was built up. The characteristic wavenumbers were determined from the positions of the loss function maxima, and the heavy hole concentrations were found from the calculated calibration curve.
Electrical measurements (Van der Pau method) were carried out for the same specimens, and the electrical resistivity and the Hall coefficients were determined.
Comparison of the optical and electrical data allowed calculating the heavy to light hole mobility ratio and the heavy hole mobilities (this approach was used for the first time). We showed that 1.9 ≤ b ≤ 2.8 which is quite smaller than the theoretically predicted figure (15.51). It was hypothesized that the light and heavy hole scattering mechanisms differ.