Issue 
Eur. Phys. J. Appl. Phys.
Volume 84, Number 2, November 2018



Article Number  20101  
Number of page(s)  8  
Section  Semiconductors and Devices  
DOI  https://doi.org/10.1051/epjap/2018180156  
Published online  14 January 2019 
https://doi.org/10.1051/epjap/2018180156
Regular Article
The influence of a strong infrared radiation field on the conductance properties of doped semiconductors
^{1}
Wigner Research Centre for Physics of the Hungarian Academy of Sciences, Konkoly − Thege Miklós út 29  33, H1121 Budapest, Hungary
^{2}
ELIHU Nonprofit Kft., Dugonics Tér 13, H6720 Szeged, Hungary
^{3}
University of Pécs, Ifjúság útja 6, H7624, Pécs, Hungary
^{*} email: barna.imre@wigner.mta.hu
Received:
22
May
2018
Received in final form:
17
October
2018
Accepted:
6
November
2018
Published online: 14 January 2019
This work presents an analytic angular differential cross section formula for the electromagnetic radiation fieldassisted electron scattering on impurities in semiconductors. These impurities are approximated with various model potentials. The scattered electrons are described with the wellknown Volkov wave function, which has been used to describe strong laser field matter interaction for more than half a century, which exactly describes the interaction of the electron with the external oscillating field. These calculations show that the electron conductance in a semiconductor could be enhanced by an order of magnitude if an infrared electromagnetic field is present with 10^{11} W/cm^{2} < I < 10^{13} W/cm^{2} intensity.
© EDP Sciences, 2018
1 Introduction
The key issue in understanding the electric conduction phenomena in semiconductors is the study of the corresponding scattering processes of electrons by impurities. One possible way to evaluate the rate of scattering transition probabilities is to solve the oneelectron timedependent Schrödinger equation up to the first order (Born approximation). Numerous models exist to approximate the electronimpurity interaction via a central potential of U(r) the standard description can be found in text books [1–4]. This study extends this description to the case where an electromagnetic (EM) field is simultaneously present in addition to impurity induced scattering.
The main motivation for this work comes from the field of laser–matter interactions. In this topic the nonlinear response of atoms, molecules and plasmas can be investigated both theoretically and experimentally [5]. These result in many phenomena including high harmonic generations or plasmabased laserelectron acceleration. The original theory of potential scattering in external EM fields was developed about half a century ago and can be found in various papers of [6–13]. Numerous studies on laserassisted electron collisions on atoms are also available [14]. Kanya and Yamanouchi generalized the KrollWatson formula [15] for a singlecycle infrared pulse and applied it to timedependent electron diffraction. There are only two studies where heavy particles e.g., protons, are scattered by nuclei in strong electromagnetic fields [16,17]. Similar theoretical studies of solid states or semiconductors in such strong electromagnetic fields are rare, and recently, it has become possible to investigate the bandgap dynamics [18] and the strongfield resonant dynamics [19] of semiconductors in the attosecond (as) time scale.
This paper contains a selfcontained overview of electron conduction calculations in a doped semiconductor; the theory of laserassisted potential scattering; and the numerical calculation of the Lindhard dielectric function − all of which are essential tools for the presented theoretical description.
Finally, numerical calculations were performed for a model potential in infrared electromagnetic fields with intensities 10^{11} W/cm^{2} < I < 10^{13} W/cm^{2}. The photon energy of such fields are below 1 eV which is comparable to some semiconductor band gaps. It was shown by Kibis [20] that the backscattering of conduction electrons is suppressed by strong highfrequency electromagnetic field and this effect does not depend on the shape of the scattering potential. Later, Morina et al. [21] calculated the transport properties of a twodimensional electron gas which interacts with light. This can be considered as the precursor of this study.
This investigation shows that electrical conductivity of doped semiconductors can be changed by more than an order of magnitude in the presence of a strong infrared radiation field. This may open the way to a new forms of electronic gating. These kind of coherent infrared radiations will be soon available e.g., at the ELIALPS Research institute in Szeged, Hungary [22].
2 Theory
2.1 Electron scattering on impurities in semiconductors
A short overview of the derivation, using first quantum mechanical and statistical physical principles, of the electron conductivity without the laser field is now presented. A much detailed derivation can be found in many theoretical solid state physics textbooks e.g., [1–4].
Free electrons are considered in three dimensions with the initial and final states defined as plane waves with the following form(1)
The considered perturbation is simply due to the extra potential energy of the impurity U(r) and therefore the transition rates can be evaluated as(2)
This is the twodimensional Fourier transform of the scattering potential and q = p _{ i } − p _{ f } is the momentum transfer of the scattering electron, where p _{ i } and p _{ f } stand for initial and final electron momenta, respectively. The differential Born cross section of the corresponding potential is given by . The wellknown total scattering cross section σ _{ T } for the elastic process can be calculated from the differential scattering cross section via an angular integration, where the back scattered electrons gives significant contributions therefore a [1 − cos(θ)] factor appears(3)
The relaxation time or the τ singleparticle life time against impurity scattering is defined in terms of the total scattering cross section by multiplying with the number of impurities n _{ imp } (4)
Finally, the electron mobility and the conductivity are defined by
μ = eτ/m _{ e }, G = eμn _{ e }, (5) where e, m _{ e }, n _{ e } are the elementary charge, effective mass and the number of the scattered electrons, respectively. Further technical details including references and an overview over various additional methods e.g., the derivation of equation (3) from Boltzmann equations, are given in the review of Chattopadhyay [23]. This model is only valid for “dilute” semiconductors where the concentration of the doping atoms is below a given threshold and thus, the effects of multiple scattering can be neglected. For silicon this value lies around 10^{15} cm^{−3}, the degeneracy level is at 10^{18} cm^{−3}.
2.2 Electromagnetic fieldassisted potential scattering
The following section contains a detailed summarization of the nonrelativistic quantum mechanical description of this system. The coherent infrared field is treated semiclassically via the minimal coupling. The IR beam is taken to be linearly polarized and the dipole approximation is used. The nonrelativistic description in dipole approximation is only valid if the dimensionless intensity parameter (or the normalized vector potential) of the external field is less than unity. A laser wavelength of 3 μm correlates to a critical intensity of I = 1.52 ⋅ 10^{17} W/cm^{2}, however, much smaller laser intensities will be assumed and a moderate electron kinetic energy below 1 eV will be considered.
To avoid the ionization of the lattice atoms, silicon will be considered to have a band gap of 1.12 eV and a 3 μm infrared (IR) electromagnetic field has a photon energy of 0.41 eV. Therefore, the ionization of the highest energy bound valence electron would require a three photon absorption process in the perturbative regime. The probability of absorbing N photons depends on the laser intensity, I, as I ^{ N }. The characteristic field strengths in an atom are rather high and correspond to a laser intensity of 3.5 ⋅ 10^{16} W/cm^{2}. In this work, it is considered that the intensity of the IR field is much lower than the characteristic atomic intensitites.
The following Schrödinger equation has to be solved in order to describe the nonrelativistic scattering process of an electron on an impurity by an external EM field(6)where is the momentum operator of the electron and U(r) represents the scattering potential of the impurity atom, A(t) = A _{0} ecos(ωt) is the vector potential of the radiation field with unit polarization vector of e. Figure 1 shows the geometry of the scattering process. Without the scattering potential U(r) the particular solution of (6) can be immediately written as nonrelativistic Volkov states ϕ _{ p }(r, t) which analytically incorporate the interaction with the EM field,
The Volkov states are modulated de Broglie waves; parametrized by momenta p and form an orthonormal and complete set(7) (8)
To solve the original problem of equation (6) the exact wave function is written as a superposition of an incoming Volkov state and a correction term, which vanishes at the beginning of the interaction (in the t _{0}→ − ∞ limit). The correction term can also be expressed in terms of the Volkov states as these form a complete set (see Eq. (8)),(9)
It is clear that the unknown expansion coefficients a _{ p }(t) describe the nontrivial transition symbolized as p _{ i } → p, from a Volkov state of momentum p _{ i } to another Volkov state with momentum p. The projecton of some Volkov state ϕ _{ p }(t) to Ψ results in(10)
The insertion of Ψ of equation (9) into the complete Schrödinger equation (6) results in the following integrodifferential equation for the coefficients a _{ p }(t), (11)where the scalar product was taken with on both sides of the resulting equation and the orthogonality relation of the Volkov sates was taken after all (see the first Eq. of (8)). The initial condition a _{ p }(t _{0}) = 0, already shown in (8), means that the formal solution of (6) can be written as (12)
In the spirit of the iteration procedure used in scattering theory, the (k + 1)th iterate of a _{ p }(t) is expresses by the kth iterate on the righthand side in (12) as (13)
In the first Born approximation, where the transition amplitude is linear in the scattering potential U(r), the transition amplitude has the form(14)
The A ^{2} term drops out from the transition matrix element (14), because it represents a uniform timedependent phase. By taking the explicit form of the Volkov states (7) with the vector potential A(t) = e A _{0}cos(ωt) means that T _{ fi } becomes(15)
Before time integration, the exponential expression can be expanded into a Fourier series with the help of the JacobiAnger formula [24]. This results in the next formula including the Bessel functions of the first kind(16)
The U(q) is the Fourier transform of the scattering potential with the momentum transfer of q ≡ p _{ i } − p _{ f } where p _{ i } is the initial and p _{ f } is the final electron momenta, respectively, and its absolute value is . In the case where electrons have 0.1–1 eV energy in the n = 0 channel (which means elastic scattering), the following approximation is valid q ≈ 2p _{ i }sin(θ/2).
In general, the Dirac delta function describes photon absorptions (n < 0) and emissions (n > 0). J _{ n }(z) is the Bessel function of the first kind with the argument depending on the parameters of the laser field, the intensity and the frequency where a _{0}, q, χ are the dimensionless intensity parameter, the momentum transfer of the electron, and the angle between the momentum transfer and the polarization direction of the EM field, respectively.
The general differential cross section formula for the laserassisted collision with simultaneous nthorder photon absorption and stimulated emission processes are(17)
The is the usual Born cross section for the scattering on the potential U(r) alone, without the external EM field. The expression equation (17) was calculated by several authors using different methods [6–13] and if the Born cross section is exactly known, equation (17) can be substituted in equation (3) and the singleparticle lifetime can be easily calculated.
Fig. 1
The geometry of the scattering process. The impurity atom is located in the center of the circle, p _{ i } and p _{ f } are the initial and final scattered electron momenta, θ is the electron scattering angle, the EM pulse propagates parallel to the xaxis and is linearly polarized in the z direction. The χ angle is needed for calculating the EMelectron momentum transfer. 
2.6 Scattering model potentials in semiconductors
Different kinds of analytic model potentials are available to model the electron scattering on impurities in a semiconductor and four are presented within this paper. The simplest model is the “box potential” which is wellknown from quantum mechanics textbooks. It is capable to mimic the twodimensional impurity scattering provided by a cylindricalbarrier of radius a. It can be used to describe a neutral impurity such as an Aluminum atom that has diffused from a barrier into a GaAs well [3].
Mathematically, U(r) = U _{0} if r ≤ a and U(r) = 0 if r > a, where U _{0} is the depth of the potential well in eV and a is the radius in nm. The two dimensional Fourier transformation of the potential yields the firstorder Bessel function of the form of . The infinite range Coulomb potential has an infinite total scattering cross section in the firstorder Born approximation. However, considering a maximal limiting impact parameter due to electron screening in semiconductors the isotropic elastic cross section can be evaluated (3) and has been done by Conwell and Weisskopf [25]. This is the BrooksHerring (BH) model [26–28] and it is applicable to describe electron scattering on an ionized impurity atom(18)where is the Debye screening length, ϵ _{0} and ϵ _{ r } are the vacuum permittivity and the dielectric constant of the present media. To avoid confusion ϵ is used for the charge of the ionized impurity atom instead of q which is fixed for the momentum transfer of the electrons. The Debye screening is just the solution of the linearized PoissonBoltzmann theory and is the simplest way to treat the problem. More general theory that describe screening would be Landau's approach known as Fermi liquid theory where the electron–electron interaction has been taken into account quantitatively [29].
The Fourier transform of this potential is as follows [2],(19)
More realistic screening lengths can be calculated with the Friedel sum and the phase shift analysis of the potential [23]. There are numerous models available from the original BH interaction which include various additional effects e.g., dielectric of ThomasFermi screening, electron–electron interaction [23]. The original BH can also be calculated from a realistic electron concentration of the ionized impurity containing the FermiDirac integral [30].
There are two additional potentials which are widely used to model impurities in semiconductors. The first has been developed to investigate the electron charged dislocation scattering in an impure electron gas. The derivation of the formula can be found in [1]. The radial potential has the form of where K _{0} is the zerothorder modified Hankel function, ϵ, ϵ _{ r }, 1/c, λ _{ D } are the elementary charge, dielectric constant, linear charge density and the Debye screening length, respectively. This dislocation is a twodimensional interaction and has a cylindrical symmetry. The Fourier transform of the potential is Jena and Mishra [31] used this interaction to evaluate the quantum and classical scattering times due to charged dislocation in an impure electron gas. Half a century earlier, Pödör [32] gave an analytic formula for relaxation time and investigated the electron mobility in plastically deformed germanium. These are remarkably similar to the threedimensional BH potential.
The last most advanced model is dipole scattering in polarizationinduced twodimensional electron gas. This considers the electrical field of a dipole above a plane [33]. In the following, the BH model will be analyzed in details.
2.7 Generalized fieldassisted potential scattering in a media
The previously outlined laserassisted potential scattering model with the listed potentials is not sufficient for a realistic model to evaluate electron conduction in a solid at finite temperature. Therefore, two additional improvements are considered.
Firstly, the scattering electrons now move in a media (doped semiconductor) instead of a vacuum, therefore the effect of the media, the dielectric response functions, has to be taken into account. Equation (17) is modified and generalized and the numerically evaluated Lindhard dielectric function [34] is included in the scattering potential. It can be shown, using quantum Vlasov theory, in the first Born approximation using the Wigner representation of the density matrix of the electron [35] that the total interaction potential in the frequency domain is equivalent to the Fourier transform of the interaction potential in vacuum multiplied by the Lindhard dielectric function. Therefore, the final differential cross section formula is(20)
The dielectric function now depends on the angular frequency of the external applied field, the coherent IR field, and the wave vector of the scattering electron. The correct form of the interaction for the BH model is(21)where ϵ in the numerator is the charge of the impurity. The next technical step in the model is to calculate the Lindhard dielectric function. For a fermion gas with electronic density n at a finite temperature, T, the form can be expressed in terms of real and imaginary part [36](22)
At finite temperatures, the dielectric function contains singular integrals of the Fermi function where the singularities can be eliminated with various mathematical transformations. According to [37], the following expressions have to be evaluated:(23)and(24)
Here the Fermi wave number is k _{ F } = [3π ^{2} n]^{1/3}, the reduced temperature is t = T/T _{ F }, the Fermi energy is (where the electron mass and ℏ were set to unity). The reduced variables u and κ introduced by Lindhard are defined as(25)where ω is the angular frequency of the IR field and k is the wave vector or the scattered electron in the model. First, the reduced chemical potential α(t) = μ/E _{ f } has to be evaluated at a finite temperature from the integral of(26)
After determining the chemical potential, the function g _{ t }(λ) can be calculated via an integral where the usual singularity is successfully eliminated by a proper mathematical transformation(27)where A = λ/t and B = α(t)/t. Exhaustive technical details can be found in the original paper [37].
At this point, the static ϵ(q, ω → 0) and the long wavelength limit ϵ(q → 0, ω) of the Lindhard function can be reduced to analytic formulas [29,34]. For the static limit , the 3D screening wave number, κ, (3D inverse screening length) is defined as where n, μ, ϵ are the particle density N/L ^{3}, the chemical potential and the charge, respectively. However, in the long wavelength limit in 3D, where the angular frequency of the plasma reads .
At a finite temperature in a realistic semiconductor, the scattering electrons are not monoenergetic and thus an averaging over the distribution has to be evaluated(28)
This means that there is an additional numerical integration of the total cross section multiplied by the FermiDirac distribution function f(E) (for nondegenerate electrons) times the density of states g(E) according to Shang [2],(29)with being the energy of the electrons. In this representation, the integration can be traced back to an integral over k. The numerical value of the density of state function is wellknown for one, two or threedimensional solids. A twodimensional system is independent of the electron energy. In three dimensions, considering BH potential, . The charged dislocation potential mentioned above corresponds to a two dimensional model.
If the number of donors are enhanced, the Fermi level will rise towards the conduction band. At some stage the approximations will no longer hold because a larger part of the Fermi Dirac function overlaps with the band edge. The approximations break down when the Fermi level is closer than 3k _{ B } T to one of the band edges. This is approximately 75 meV at room temperature. In this case, the semiconductor becomes degenerate and the Boltzmann distribution function has to be applied, instead of the FermiDirac function. According to our best knowledge these two completions were never added to the general laserassisted potential scattering to model electron scattering in realistic solid states.
In practical calculations, the upper limit of the integral can be cut at the Fermi energy which is about 1 eV at room temperature for semiconductors. Numerical values obtained from (29) with or without external electromagnetic field can be directly compared in the future to experimental data.
3 Results
Doped silicon has been considered as a semiconductor with a Fermi energy of 1 eV and the external EM field has been considered as a coherent IR field with wavelengths in the λ = 1–5 μm regime. According to the ELIALPS' white book [38], a λ = 3.2 μm wavelength midIR laser will operate, which will be similar to the one presented in [39]. An intensity range of 10^{12} W/cm^{2} < I < 10^{13} W/cm^{2} is used to avoid laser damage of the silicon sample. Stuart et al. [40] and Tien et al. [41] published experimental results for radiation damage of silica for λ = 1 μm wavelength laser pulses with different pulse length and found that the threshold lies at 10^{13} W/cm^{2} for 100 fs pulse duration. Unfortunately, no experimental measurement values for λ = 3μm could be found. However, there is an empirical power law dependence for damage threshold for silica glass at various wavelengths I _{ th }(λ) = 1.55 × λ ^{0.43} × I _{ th }, where the wavelength and the intensities should be given in μm and W/cm^{2} units, respectively [42]. This means that the threshold at 3 μm should be 2.5 × 10^{13} W/cm^{2}.
According to [23], electron mobility versus electron concentration measurement were presented and compared within the confines of BH model on a logarithmiclinear scale below 200 K giving discrepancy of a factor of 2−5 which validates the original electron conduction model. These results encouraged to develop the model mentioned above. The lowest level of the kinetic energy of the electrons considered was thermal noise E = k _{ B } T which is 0.025 eV at room temperature. The parameters of the BH potential are the following: the screening range λ _{ d } = 30 nm, the dielectric constant ϵ _{ r } = 35, and the charge ϵ = 1.
In the following, calculations for two distinct models are presented. The first system simply considers a semiconductor media with a dielectric constant. The literature value is 35 for semiconductor Silicon. The second more realistic model fully includes the frequency dependent Lindhard function, see equation (22). This model even includes the energy dependence of the scattering electrons. The two field independent cross sections are σ _{ T,ϵ=35} = 1.9 nm^{2} and σ _{ T,ϵ(ω,k)} = 39.6 nm^{2}, respectively. The ϵ(k) dependence and averaging over the electron energy makes the ratio less than 35 – the numerical value is 20.8.
Figure 2 shows the averaged cross sections as the function of the external field intensity for λ = 3 μm wave length. The more realistic model gives larger cross sections which means that there is a smaller electron conductance. By considering the field independent cross section σ _{ T,ϵ=35} = 1.9 nm^{2} as a standard value gives the suppression of the electron conductance by the external field by a factor of 15.
Figure 3 presents the ratio of the two models as the function of the field intensity with values between 21 and 26. At larger field intensities the gradient of the ratio is reduced.
Figure 4 shows the averaged cross sections as the function of the external field wavelength for the intensity of I = 10^{12} W/cm^{2}. The cross sections obtained from the model incorporating the Lindhard function are still larger than the simpler model. The cross sections of both models decay at large intensities. Note, that for 1 μm wavelength, the ratio of the original cross section (1.9 nm^{2}) goes up to 310 nm^{2} a gain factor of 155. Calculations below 1 μm wave length were not performed because such fields may excite valence electrons into the conductance band and that would lie out of the scope of this elastic scattering model. Figure 5 shows the wavelength dependence of the ratio of these two models. Larger wavelengths correspond to smaller cross sections or larger conductance. These results are in full agreement with the general theory of Kibis [20]. The ratio between the two models still lie between a factor of 23–25. Multiplying with the remaining constants of where e is the charge of the electron, m _{ e } is the effective mass of an electron in a semiconductor and is about 0.5 × 9.0 × 10^{−31} kg and n_{imp} the number of the impurities per cm^{3} lies between 10^{9} and 10^{16}, therefore, the obtained final conductance values would lie between 10^{−6} and 10^{5} S/cm [43]. This is a very broad range of conduc tance, hence not reporting exact numerical conductance values. However, these ratio of the conductances with or without strong external IR fields can vary by more than an order of magnitude.
A doped semiconductor has a complex nature and the physical value of the resistivity varies in an external IR field can of course only be investigated in a real physical experiment but these calculation shows that it would be an interesting project.
These models only include elastic scattering processes, without any photon absorption or emission. The inclusion of one photon absorption only requires changing the zerothorder Bessel function to the firstorder term, otherwise the process and the way of calculation are the same. Note, that the corresponding cross sections or the probabilities of a first–order process are at least one order of magnitude lower than elastic ones.
Our calculations cannot include additional effects coming from the complex nature of a real solid state like, valence dielectric screening, bandstructure details, electronelectron scattering, nonlinear screening, multiple electron scattering, and impurity dressing − all of which are mentioned reference [23].
The twofold numerical integrations of (29) for various laser parameters were evaluated with Wolfram Mathematica (Copyright 1988 − 2012 Wolfram Research, Inc.) where the global adaptive integration builtin method was used with recursion number of 300. The numerical tolerances, i.e. the absolute and relative error have both set to 10^{−10}.
Fig. 2
The averaged cross section ⟨σ _{ T }⟩ (Eq. (29)) as the function of the field intensity for a λ = 3 μm laser wavelength at room temperature. The upper curve corresponds to the frequencydependent Lindhard dielectric function. The lower curve corresponds to the ϵ _{ r } = 35 dielectric constant case. 
Fig. 3
The intensity dependence of the ratio of the two models at λ = 3 μm. 
Fig. 4
The averaged cross section ⟨σ _{ T }⟩ (Eq. (29)) as the function of the wavelengths for I = 10^{12} W/cm^{2} field intensity at room temperature. The upper curve corresponds to the frequencydependent Lindhard dielectric function. The lower curve corresponds to the ϵ _{ r } = 35 dielectric constant case. 
Fig. 5
The wavelength dependence of the ratio of the two models at I = 10^{12} W/cm^{2} intensity. 
4 Summary
A formalism based on an interrelated model to calculate electron conductance in a doped semiconductor in strong external IR fields with intensities lying between 10^{11} and 10^{13} W/cm^{2} with the wavelength ranging from 1–5 μm. The mathematical description of multiphoton processes was coupled to the wellestablished potential scattering model based on the firstBorn approximation. An application of this general formalism in the present paper has been the modification of the scattering elastic cross sections in the elastic channel.
In solidsate physics, the elastic scattering of electrons on impurities modeled by the BH potential can model the electric conductance up to a factor of 2–5 [23]. Electron scattering has been treated in a perturbative manner whilst the influence of the external strong radiative IR field (which can cause photon absorption) has been treated nonperturbatively. The solidstate scattering model has been improved in two areas. A frequencydependent Lindhard function, which mimics the response of the solid to a quick varying external field has replaced the inclusion of dielectric constant, which models the semiconductor. The second improvement is that the final electron energy distribution above the Fermi function at room temperature is averaged. These two improvements give at least a factor of 15 suppression in the final conductance. It has been demonstrated that due to the joint interaction of the conduction electrons with the impurity scattering potential and the laser field that there could be a considerable change in the conduction as was expected at the beginning of this studies. These theoretical results have inspired our experimental colleagues in ELIALPS to perform measurements on silicon samples at 3 μm wavelength. Work is in progress to build up a setup to measure the prognosticated change in the conductivity. If the change of the conductivity lies in the same order of timescale as the duration of the midIR laser pulse (tens of femtoseconds) then only pumpprobe type measurements can be applied to measure the change of the optical properties of the sample [44]. If the experiments verify these theoretical predictions then it may be possible to start speculating about possible physical application of the phenomena, like a quick gating, a quick moldulator, or even a midIR light intensity sensor.
Acknowledgments
The authors would like to thank Dr. Ugo Ancarani for the useful discussions about numerical problems that arose from evaluating Lindhard functions. The ELIALPS project (GINOP2.3.615201500001) is supported by the European Union and cofinanced by the European Regional Development Fund.
Author contribution statement
IFB is the main author of the paper and gave the idea for this paper. He is responsible for building up the mathematical and physical model, reading the entire literature of the investigated field, and writing the submitted manuscript.
Mihály A. Pocsai (Ph.D. student, his research field is different) helped in building up the final Mathematica 11 notebook which performed the numerical calculations He helped to choose the proper integral 601 routine as well. He is also responsible for the final corrections of the manuscript.
Sándor Varró contributed to the physical interpretations of the formulae and he improved conceptual and technical details at all stages in compiling and writting the manuscript.
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Cite this article as: Imre Ferenc Barna, Mihály Anrás Pocsai, Sándor Varró, The influence of a strong infrared radiation field on the conductance properties of doped semiconductors, Eur. Phys. J. Appl. Phys. 84, 20101 (2018)
All Figures
Fig. 1
The geometry of the scattering process. The impurity atom is located in the center of the circle, p _{ i } and p _{ f } are the initial and final scattered electron momenta, θ is the electron scattering angle, the EM pulse propagates parallel to the xaxis and is linearly polarized in the z direction. The χ angle is needed for calculating the EMelectron momentum transfer. 

In the text 
Fig. 2
The averaged cross section ⟨σ _{ T }⟩ (Eq. (29)) as the function of the field intensity for a λ = 3 μm laser wavelength at room temperature. The upper curve corresponds to the frequencydependent Lindhard dielectric function. The lower curve corresponds to the ϵ _{ r } = 35 dielectric constant case. 

In the text 
Fig. 3
The intensity dependence of the ratio of the two models at λ = 3 μm. 

In the text 
Fig. 4
The averaged cross section ⟨σ _{ T }⟩ (Eq. (29)) as the function of the wavelengths for I = 10^{12} W/cm^{2} field intensity at room temperature. The upper curve corresponds to the frequencydependent Lindhard dielectric function. The lower curve corresponds to the ϵ _{ r } = 35 dielectric constant case. 

In the text 
Fig. 5
The wavelength dependence of the ratio of the two models at I = 10^{12} W/cm^{2} intensity. 

In the text 
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