# Shubnikov–de Haas Oscillations in Semiconductors at the Microwave-Radiation Absorption

Authors: G. Gulyamov; U. I. Erkaboev; A. G. Gulyamov
Journal: Advances in Condensed Matter Physics
 (2019)

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## Abstract

Mathematical models for the Shubnikov-de Haas oscillations in semiconductors are obtained at the microwave-radiation absorption and its temperature dependence. Three-dimensional image of microwave magnetoabsorption oscillations in narrow-gap semiconductors is established. Using a mathematical model, the oscillations of the microwave magnetoabsorption are considered for different values of the electromagnetic field. The results of calculations are compared with experimental data. The proposed model explains the experimental results in HgSe at different temperatures.


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## Body

## 1. Introduction

In recent years, quantum oscillation phenomena in the presence of a strong electromagnetic field and temperature have become a subject of intense experimental [1–5] and theoretical research [6–8]. With the help of such phenomena, some basic physical quantities can be determined (effective mass of charge carriers, magnetoresistance, magnetic susceptibility, etc.) and the band energy spectra of electrons in new semiconductor materials can be studied.The Shubnikov-de Haas and de Haas-van Alphen oscillations and the quantum Hall effect were detected at ultralow temperatures and ultrastrong magnetic fields [9]. In this case, oscillation phenomena were observed in massive semiconductors and metals.Currently, interest in the oscillation of Shubnikov-de Haas in semiconductors is increased at the microwave-radiation absorption. In particular, the work in [10] considered Shubnikov-de Haas oscillations in HgSe samples with different concentrations of impurities of Fe at low temperatures and strong microwave fields. In this work, dependence oscillations of the derivative power microwave-radiation absorption P by the magnetic field strength H (dP/dH) on the reverse magnetic field 1/H in narrow-gap semiconductors were observed at values of the impurity concentrations.As is known, all quantum oscillation phenomena strongly depended on the spectral density of states in semiconductors. In [11–13], temperature dependence of the spectral density of energy states in narrow-gap semiconductors was investigated in the quantizing magnetic fields. But, in these works, the influences of absorption of microwave on quantum oscillations phenomena in semiconductors are not considered.The aim of the present article is to develop mathematical modeling of the temperature dependence of magnetoabsorption microwave oscillations in semiconductors and into the influence temperature of a sample on the results of experimental data processing.


## 2. The Theoretical Part

### 2.1. Calculation of Temperature Dependence of the Spectral Density of States Oscillations with the Help of Gaussian Distribution

In [14], spectral density of energy states oscillations in narrow-gap semiconductors is investigated. The following expression is received for the density of states:(1)NkE,H=m3/221/2π2ħ3ħeH2mc∑N=0Nmax2E/Eg+1E2/Eg+E-N+1/2ħeH/mcHere,NkE,H is spectral density of states for the conduction band with the Kane dispersion law, H is magnetic field strength, Е is energy of a free electron and hole in a quantizing magnetic field, N is number of Landau levels, and Eg is band gap of semiconductor. This formula is applicable only for narrow-gap (Eg<0.6eV) materials.If the energy spectrum of electrons is discrete, then the density of energy states is equal to the sum ofδ-functions, concentrated at the points of the spectrum ЕN (N is number of discrete levels) [15–17]:(2)NSE=∑NNSδE-ENIn the general case, the energy density of states is a set ofδ-functional peaks, located at ħωc from each other in a quantizing magnetic field [16]. In this expression, the temperature-induced broadening of the energy levels is disregarded. In order to take into account the temperature dependence of the density of states, we expand Ns(E,T) into series with respect to Gauss functions. We then obtain the density of energy states depending on temperature. The Gaussian distribution for the energy levelsEN is determined by the following expression [9]:(3)GaussE,T=1kT·exp⁡-E-EN2kT2Here,GaussE,T is Gaussian distribution function.At low temperatures (T→0), the Gauss function will turn into a delta Dirac function. Thermal broadening of the levels in a magnetic field gives rise to the smoothing of discrete levels. Thermal broadening is to be taken into account using a Gauss function. The density of states depends on temperature; as in [11–14], we expand Ns(E,T) into series with respect to Gauss functions. Using expressions (1), (2), and (3), we obtain the following expressions for the temperature dependence of the spectral density of states in narrow-gap semiconductors:(4)NkE,H,T=m3/221/2π2ħ3ħeH2mc∑N=0Nmax2E/Eg+1E2/Eg+E-N+1/2ħeH/mc·GaussE,EN,TMaking a summation using formula (4), we obtain Ns(E,H,T) depending on temperature. For narrow-gap semiconductors, this expression at T→0 turns into (1). In this case, Landau levels appear sharply. With increasing temperature, sudden bursts begin to smooth out. At high temperatures, the density of states turns into a continuous spectrum and the influence of a magnetic field will not be felt. This allows us to obtain a density of states, which depends on temperature.Using this formula, the temperature dependence of the quantum oscillations phenomena in narrow-gap semiconductors can be explained.


### 2.2. Mathematical Modeling of Shubnikov-de Haas Oscillations in Narrow-Gap Semiconductors at the Temperature and Microwave-Radiation Absorption

Consider the Shubnikov-de Haas oscillations in narrow-gap semiconductors at the temperature and a strong electromagnetic field. Per unit volume, the power of microwave field is determined by the following expression [18]:(5)P=σ·EE2Here,σ is conductivity of semiconductor; EE is electric field strength of a wave.Consider the dependence of the longitudinal resistance on the magnetic field and temperature in semiconductors. In this case, for each Landau level, you can enter your Boltzmann distribution function and your relaxation timeτN. The distribution function satisfies the kinetic equation [19]:(6)∂fN∂t+eEzħ∂fN∂kz=-fN-f0NτN.The current associated with the Nth quantum level can be calculated in the usual way:(7)jzN=-e2Ezπm∫kzτN∂f0N∂kzdkz.σzz=-2m1/2e2π2ħ3ħωc∫ħωc/2∞∑NτENsE,EN,H∂f0E∂EdE.Here,τ(E) is transport relaxation time. The transport relaxation time is taken as follows: τ=τ0Er. The exponent r has different values for different scattering mechanisms. For example, in the case of scattering on acoustic vibrations and impurity ions, the exponent is-1/2 and3/2 [20]. τ0- is not dependent on the electron energy. To study the temperature dependence of the Shubnikov-de Haas oscillations, ∂f0(E)/∂E can be replaced by a Gaussian distribution function. From here, we determine the temperature dependence of the Shubnikov-de Haas oscillations in a quantizing magnetic field:(8)σzzE,EN,H,T=-2m1/2e2π2ħ3ħωc∫ħωc/2∞∑NτENsE,EN,H,TdEFor unit volume of semiconductors the following condition is performed:ρzz≈Rzz=1/σzz. Here, ρzz is specific longitudinal magnetoresistance.Substituting expressions (4) and (8) in (5), we obtain the following expression:(9)PE,EN,H,T,EE=σzzE,EN,H,T·EE2Differentiating (9) byH, that is, dPkH,T,E,EN,EE/dH, we obtain expressions for the dependence of the Shubnikov-de Haas oscillations on the microwave-radiation absorption and temperature in narrow-band semiconductors:(10)dPkH,T,E,EN,EEdH=C∑N=0max2E/Eg+1E2/Eg+E-N+1/2AH·1+AHN+1/22E2/Eg+E-N+1/2AH·GaussE,EN,T·τE·EE2A=ħemc;C=m3/223/2π2ħ3Here,Pk is power of microwave field for the model of Kane.Thus, we have the opportunity calculation oscillationsdPkH,T,E,EN,EE/dH at the strong electromagnetic field and at different temperatures, with the help of formula (10).Let us consider the analysis of quantum oscillation phenomena in semiconductors, with the help of reduced model. In particular, we obtain the graphdPkH,T,E,EN,EE/dH oscillations for InSb, using formula (10). In Figure 1, dependence of dP/dH on the magnetic field strength H in InSb (Eg(0)=0.234) [21] at Т=3КиEE=103(V/sm) is shown. In this case, EE=const. At the increased power of electromagnetic wave amplitude dP/dH oscillation is increased.Figure 1
Microwave-radiation absorption oscillations (dP/dH) in InSb at the temperature T = 3K and electromagnetic field strength EE=103(V/sm), with the help of formula (10) being calculated.This is evident from Figure2, where dP/dH oscillations are given for two different values power of the absorbed electromagnetic wave. In Figure 2, the microwave-radiation absorption oscillations at different electromagnetic field strengths and constant low temperatures are shown, where EE1=103(V/sm) and EE2=2·103(V/sm). As seen in Figure 2, amplitude of quantum phenomena oscillations can be controlled, with the help of power of electromagnetic wave.Figure 2
Microwave-radiation absorption oscillations (dP/dH) in InSb at different electromagnetic field strengths. 1-EE1=103(V/sm); 2-EE2=2⋅103(V/sm).Now, consider the microwave-radiation absorption oscillations in InSb at different temperatures. Thermal broadening of oscillations will be taken into account using the Gaussian distribution function. We investigated spectral density of states with the help of expansion in a series of Gauss functions. Then temperature dependence of microwave-radiation absorption oscillations in semiconductors at the constant electromagnetic field strength can be explained. In Figure3, temperature dependence of microwave-radiation absorption oscillations in semiconductors at constant electromagnetic field strength is given. As can be seen from these figures, at T = 3K, with increasing magnetic field, oscillations amplitude of the microwave-radiation absorption is increased. Therefore, in this case, thermal broadening is very weak; that is, kT<<ħωc (ωc is cyclotron frequency). Every peak of oscillations corresponds to one discrete Landau level. With increasing temperature, the amplitude oscillation is decreased; at sufficiently high temperatures, discrete energy spectrum of the zone becomes continuous (Figure 3). This leads to the smoothing of oscillations dP/dH.Figure 3
The temperature dependence of microwave-radiation absorption oscillations in semiconductors. 1- T = 3 K; 2- T = 25 K; 3- T = 77 K.In Figure4, dependence of microwave-radiation absorption oscillations on the temperature and magnetic field is given. This three-dimensional image is obtained in semiconductors with the Kane dispersion law. Using formula (10), we can obtain the same three-dimensional graphics for the narrow band gap semiconductors. As can be seen from these figures, with increasing temperature, oscillation amplitude is gradually smoothed. At temperature T = 77K, microwave-radiation absorption oscillation amplitudes become virtually invisible and coincide with oscillations in the absence of a magnetic field at the interval H = 10 ÷ 25 kOe.Figure 4
OscillationsdP/dH dependence on the magnetic field and temperature in semiconductors with the Kane dispersion law.


## 2.1. Calculation of Temperature Dependence of the Spectral Density of States Oscillations with the Help of Gaussian Distribution

In [14], spectral density of energy states oscillations in narrow-gap semiconductors is investigated. The following expression is received for the density of states:(1)NkE,H=m3/221/2π2ħ3ħeH2mc∑N=0Nmax2E/Eg+1E2/Eg+E-N+1/2ħeH/mcHere,NkE,H is spectral density of states for the conduction band with the Kane dispersion law, H is magnetic field strength, Е is energy of a free electron and hole in a quantizing magnetic field, N is number of Landau levels, and Eg is band gap of semiconductor. This formula is applicable only for narrow-gap (Eg<0.6eV) materials.If the energy spectrum of electrons is discrete, then the density of energy states is equal to the sum ofδ-functions, concentrated at the points of the spectrum ЕN (N is number of discrete levels) [15–17]:(2)NSE=∑NNSδE-ENIn the general case, the energy density of states is a set ofδ-functional peaks, located at ħωc from each other in a quantizing magnetic field [16]. In this expression, the temperature-induced broadening of the energy levels is disregarded. In order to take into account the temperature dependence of the density of states, we expand Ns(E,T) into series with respect to Gauss functions. We then obtain the density of energy states depending on temperature. The Gaussian distribution for the energy levelsEN is determined by the following expression [9]:(3)GaussE,T=1kT·exp⁡-E-EN2kT2Here,GaussE,T is Gaussian distribution function.At low temperatures (T→0), the Gauss function will turn into a delta Dirac function. Thermal broadening of the levels in a magnetic field gives rise to the smoothing of discrete levels. Thermal broadening is to be taken into account using a Gauss function. The density of states depends on temperature; as in [11–14], we expand Ns(E,T) into series with respect to Gauss functions. Using expressions (1), (2), and (3), we obtain the following expressions for the temperature dependence of the spectral density of states in narrow-gap semiconductors:(4)NkE,H,T=m3/221/2π2ħ3ħeH2mc∑N=0Nmax2E/Eg+1E2/Eg+E-N+1/2ħeH/mc·GaussE,EN,TMaking a summation using formula (4), we obtain Ns(E,H,T) depending on temperature. For narrow-gap semiconductors, this expression at T→0 turns into (1). In this case, Landau levels appear sharply. With increasing temperature, sudden bursts begin to smooth out. At high temperatures, the density of states turns into a continuous spectrum and the influence of a magnetic field will not be felt. This allows us to obtain a density of states, which depends on temperature.Using this formula, the temperature dependence of the quantum oscillations phenomena in narrow-gap semiconductors can be explained.


## 2.2. Mathematical Modeling of Shubnikov-de Haas Oscillations in Narrow-Gap Semiconductors at the Temperature and Microwave-Radiation Absorption

Consider the Shubnikov-de Haas oscillations in narrow-gap semiconductors at the temperature and a strong electromagnetic field. Per unit volume, the power of microwave field is determined by the following expression [18]:(5)P=σ·EE2Here,σ is conductivity of semiconductor; EE is electric field strength of a wave.Consider the dependence of the longitudinal resistance on the magnetic field and temperature in semiconductors. In this case, for each Landau level, you can enter your Boltzmann distribution function and your relaxation timeτN. The distribution function satisfies the kinetic equation [19]:(6)∂fN∂t+eEzħ∂fN∂kz=-fN-f0NτN.The current associated with the Nth quantum level can be calculated in the usual way:(7)jzN=-e2Ezπm∫kzτN∂f0N∂kzdkz.σzz=-2m1/2e2π2ħ3ħωc∫ħωc/2∞∑NτENsE,EN,H∂f0E∂EdE.Here,τ(E) is transport relaxation time. The transport relaxation time is taken as follows: τ=τ0Er. The exponent r has different values for different scattering mechanisms. For example, in the case of scattering on acoustic vibrations and impurity ions, the exponent is-1/2 and3/2 [20]. τ0- is not dependent on the electron energy. To study the temperature dependence of the Shubnikov-de Haas oscillations, ∂f0(E)/∂E can be replaced by a Gaussian distribution function. From here, we determine the temperature dependence of the Shubnikov-de Haas oscillations in a quantizing magnetic field:(8)σzzE,EN,H,T=-2m1/2e2π2ħ3ħωc∫ħωc/2∞∑NτENsE,EN,H,TdEFor unit volume of semiconductors the following condition is performed:ρzz≈Rzz=1/σzz. Here, ρzz is specific longitudinal magnetoresistance.Substituting expressions (4) and (8) in (5), we obtain the following expression:(9)PE,EN,H,T,EE=σzzE,EN,H,T·EE2Differentiating (9) byH, that is, dPkH,T,E,EN,EE/dH, we obtain expressions for the dependence of the Shubnikov-de Haas oscillations on the microwave-radiation absorption and temperature in narrow-band semiconductors:(10)dPkH,T,E,EN,EEdH=C∑N=0max2E/Eg+1E2/Eg+E-N+1/2AH·1+AHN+1/22E2/Eg+E-N+1/2AH·GaussE,EN,T·τE·EE2A=ħemc;C=m3/223/2π2ħ3Here,Pk is power of microwave field for the model of Kane.Thus, we have the opportunity calculation oscillationsdPkH,T,E,EN,EE/dH at the strong electromagnetic field and at different temperatures, with the help of formula (10).Let us consider the analysis of quantum oscillation phenomena in semiconductors, with the help of reduced model. In particular, we obtain the graphdPkH,T,E,EN,EE/dH oscillations for InSb, using formula (10). In Figure 1, dependence of dP/dH on the magnetic field strength H in InSb (Eg(0)=0.234) [21] at Т=3КиEE=103(V/sm) is shown. In this case, EE=const. At the increased power of electromagnetic wave amplitude dP/dH oscillation is increased.Figure 1
Microwave-radiation absorption oscillations (dP/dH) in InSb at the temperature T = 3K and electromagnetic field strength EE=103(V/sm), with the help of formula (10) being calculated.This is evident from Figure2, where dP/dH oscillations are given for two different values power of the absorbed electromagnetic wave. In Figure 2, the microwave-radiation absorption oscillations at different electromagnetic field strengths and constant low temperatures are shown, where EE1=103(V/sm) and EE2=2·103(V/sm). As seen in Figure 2, amplitude of quantum phenomena oscillations can be controlled, with the help of power of electromagnetic wave.Figure 2
Microwave-radiation absorption oscillations (dP/dH) in InSb at different electromagnetic field strengths. 1-EE1=103(V/sm); 2-EE2=2⋅103(V/sm).Now, consider the microwave-radiation absorption oscillations in InSb at different temperatures. Thermal broadening of oscillations will be taken into account using the Gaussian distribution function. We investigated spectral density of states with the help of expansion in a series of Gauss functions. Then temperature dependence of microwave-radiation absorption oscillations in semiconductors at the constant electromagnetic field strength can be explained. In Figure3, temperature dependence of microwave-radiation absorption oscillations in semiconductors at constant electromagnetic field strength is given. As can be seen from these figures, at T = 3K, with increasing magnetic field, oscillations amplitude of the microwave-radiation absorption is increased. Therefore, in this case, thermal broadening is very weak; that is, kT<<ħωc (ωc is cyclotron frequency). Every peak of oscillations corresponds to one discrete Landau level. With increasing temperature, the amplitude oscillation is decreased; at sufficiently high temperatures, discrete energy spectrum of the zone becomes continuous (Figure 3). This leads to the smoothing of oscillations dP/dH.Figure 3
The temperature dependence of microwave-radiation absorption oscillations in semiconductors. 1- T = 3 K; 2- T = 25 K; 3- T = 77 K.In Figure4, dependence of microwave-radiation absorption oscillations on the temperature and magnetic field is given. This three-dimensional image is obtained in semiconductors with the Kane dispersion law. Using formula (10), we can obtain the same three-dimensional graphics for the narrow band gap semiconductors. As can be seen from these figures, with increasing temperature, oscillation amplitude is gradually smoothed. At temperature T = 77K, microwave-radiation absorption oscillation amplitudes become virtually invisible and coincide with oscillations in the absence of a magnetic field at the interval H = 10 ÷ 25 kOe.Figure 4
OscillationsdP/dH dependence on the magnetic field and temperature in semiconductors with the Kane dispersion law.


## 3. Comparison of Theory with Experimental Results

Let us analyze the microwave-radiation absorption oscillations in semiconductors at the temperature and external fields. In [10], microwave magnetoabsorption oscillations in semiconductors are observed. In Figure 5, microwave magnetoabsorption oscillations in samples HgSe at a temperature T = 2.7 K and constant power of electromagnetic wave over the entire measurement range are shown [10]. Here, the dependence of dP/dH oscillations on the inverse magnetic field at low temperatures is shown. In this work, oscillations were obtained for a narrow-gap semiconductor. From here, using formula (10), we determine dP/dH oscillations at a temperature T = 2.7 K (Figure 6).Figure 5
Shubnikov-de Haas oscillations in samples HgSe at T = 2.7 K [10].Figure 6
Shubnikov-de Haas oscillations in samples HgSe at T = 2.7 K, calculated using formula (10).In Figure6, microwave magnetoabsorption oscillations in HgSe are shown. In this case, quantum oscillations amplitude begins abruptly. Using formula (10), experimental results can be explained at different temperatures (Figure 7). In Figure 7, the dependence of dP/dH on 1/H for temperatures of 2.7 K, 50 K, and 100 K is shown. With increasing temperature, dP/dH oscillation amplitudes are decreased and at values of H in the interval 10/H=1,5÷2,5kOe-1 effects of magnetic field are not felt. As can be seen from this figure, at temperature T = 100 K, dP/dH oscillations are smoothed out.Figure 7
Influence of temperature on microwave magnetoabsorption oscillations in HgSe. 1- T = 2.7 K, 2- T = 20 K, and 3- T = 100 K.


## 4. Conclusion

Based on the study, the following conclusion can be made: model of determination of microwave magnetoabsorption oscillations in semiconductors is developed at different electromagnetic fields and temperatures. Formula for the dependence ofdP/dH oscillations on the electric field strength of an electromagnetic wave and temperature is obtained with the parabolic and Kane dispersion law. Using the proposed model, the experimental results of HgSe were investigated. Using formula (10), experimental oscillations dP/dH in the narrow-gap semiconductor HgSe are explained at different temperatures.



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