The influence of a strong infrared radiation field on the conductance properties of doped semiconductors
TThe influence of a strong infrared radiation field on the conductance properties ofdoped semiconductors
I. F. Barna , , M. A. Pocsai and S. Varr´o ,
1) Wigner Research Centre for Physics of the Hungarian Academy of Sciences,Konkoly - Thege Mikl´os ´ut 29 - 33, 1121 Budapest, Hungary,2) ELI-HU Nonprofit Kft., Dugonics T´er 13, 6720 Szeged, Hungary (Dated: November 6, 2018)This work presents an analytic angular differential cross section formula for the electromagneticradiation field assisted electron scattering by impurities in semiconductors. These impurities areapproximated with various model potentials. The scattered electrons are described by the well-known Volkov wave function, which has been used describe strong laser field matter interaction formore than half a century, which exactly describes the interaction of the electron with the externaloscillating field. These calculations show that the electron conductance in a semiconductor could beenhanced by an order of magnitude if an infrared electromagnetic field is present with 10 < I < W/cm intensity. PACS numbers: 61.82.Fk,72.20.-i,72.20.Dp,32.80.Wr
I. INTRODUCTION
The key issue in understanding the electric conduction phenomena in semiconductors is the study of the corre-sponding scattering processes of electrons by impurities. One possible way to evaluate the rate of scattering transitionprobabilities is to solve the one-electron time-dependent Schr¨odinger equation up to the first Born approximation. Nu-merous models exist to approximate the electron-impurity interaction via a central potential of U ( r ) and the standarddescription can be found in textbooks [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 non-linearresponse of atoms, molecules and plasmas can be both investigated theoretically and experimentally [5]. These resultin many phenomena including high harmonic generations or plasma-based laser-electron acceleration. The originaltheory of potential scattering in external EM fields was developed about half a century ago and can be found invarious papers of [6–13]. Numerous studies on laser assisted electron collisions on atoms are also available [14]. Kanyaand Yamanouchi generalized the Kroll-Watson formula [15] for a single-cycle infrared pulse and applied it to time-dependent electron diffraction. There are only two studies where heavy particles e.g., protons, are scattered by nucleiin strong electromagnetic fields [16, 17]. Similar theoretical studies of solid states or semiconductors in such strongelectromagnetic fields are rare and recently, it has became possible to investigate the band-gap dynamics [18] and thestrong-field resonant dynamics [19] of semiconductors in the attosecond (as) time scale.This paper contains a self-contained overview of electron conduction calculations in a doped semiconductor; thetheory of laser assisted potential scattering and the numerical calculation of Lindhard dielectric function – all of whichare essential tools for the presented theoretical description.Finally, numerical calculations were performed for a model potential in infrared electromagnetic fields with in-tensities 10 < I < W/cm . The photon energy of such fields are below 1 eV which is comparable to somesemiconductor band gaps. It was shown by Kibis [20] that the backscattering of conduction electrons are suppressedby strong high-frequency electromagnetic field and this effect does not depend on the absolute form of the scatter-ing potential. Later, Morina et al. [21] calculated the transport properties of a two-dimensional electron gas whichinteracts 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 amagnitude by the presence of a strong infrared radiation field. This may open the way to a new forms of electronicgating. These kind of coherent infrared radiations will be soon available e.g., at the ELI-ALPS Research institute inSzeged, Hungary [22]. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov II. THEORYA. Electron scattering on impurities in semiconductors
A short overview of the derivation, using first quantum mechanical and statistical physical principles, of the electronconductivity without the laser field is now presented. A much detailed derivation can be found in many theoreticalsolid 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 thefollowing form ϕ i ( r ) = A (2 π (cid:126) ) / exp (cid:18) i (cid:126) p i · r (cid:19) , ϕ f ( r ) = A (2 π (cid:126) ) / exp (cid:18) i (cid:126) p f · r (cid:19) . (1)The considered perturbation is simply due to the extra potential energy of the impurity U ( r ) and therefore thetransition rates can be evaluated as U fi = (cid:90) ϕ f ( r ) ∗ U ( r ) ϕ i ( r ) d r = 1 A (cid:90) U ( r ) e − i q · r d r . (2)This is the two-dimensional Fourier transform of the scattering potential and q = p i − p f is the momentum transferof the scattering electron, where p i and p f stand for initial a final electron momenta. The differential Born crosssection of the corresponding potential is given by dσ B d Ω = (cid:0) m π (cid:126) (cid:1) | U ( q ) | .The well-known total scattering cross section σ T for the elastic process can be calculated from the differentialscattering cross section via an angular integration, where the back scattered electrons gives significant contributionstherefore a [1 − cos ( θ )] factor appears σ T = 2 π (cid:90) π (cid:18) dσ B ( θ ) d Ω (cid:19) [1 − cos ( θ )] sin ( θ ) dθ. (3)The relaxation time or the τ single-particle life time against impurity scattering is defined in terms of the totalscattering cross section by multiplying with the number of impurities n imp /τ = σ T 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 derivationof Eq. (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 effectsof multiple scattering can be neglected. For silicon this value lies around 10 cm − , the degeneracy level is at 10 cm − . B. Electromagnetic field assisted potential scattering
The following section contains a detailed summarization of the non-relativistic quantum mechanical description ofthis system. The coherent infrared field is treated semi-classically via the minimal coupling. The IR beam is taken tobe linearly polarized and the dipole approximation is used. The non-relativistic description in dipole approximation isonly valid of the dimensionless intensity parameter (or the normalized vector potential) a = 8 . · − (cid:113) I ( Wcm ) λ ( µm )of the external field is smaller than unity. A laser wavelength of 3 µm correlates to a critical intensity of I = 1 . · W/cm , however much smaller laser intensities will be assumed and a moderate electron kinetic energy below one eVwill 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 µ minfrared (IR) electromagnetic field has a photon energy of 0.41 eV. Therefore, the ionization of the highest energybound valence electron would require a three photon absorption process in the perturbative regime. The probability ofabsorbing N photons depends on the laser intensity, I , as I N . The characteristic field strength in an atom are rather 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 theinitial and final scattered electron momenta, θ is the electron scattering angle, the EM pulse propagates parallel to the x axisand is linearly polarized in the x-z plane. The χ angle is needed for calculating the EM-electron momentum transfer. high and correspond to a laser intensity of 3 . · W/cm . In this work, it is considered that IR field intensities I = 10 W/cm , which is much lower than the atomic units.The following Schr¨odinger equation has to be solved in order to describe the non-relativistic scattering process ofan electron on an impurity by an external EM field (cid:20) m (cid:16) ˆp − ec A (cid:17) + U ( r ) (cid:21) Ψ = i (cid:126) ∂ Ψ ∂t , (6)where ˆp = − i (cid:126) ∂/∂ r is the momentum operator of the electron and U ( r ) represents the scattering potential of theimpurity atom, A ( t ) = A e cos ( ωt ) is the vector potential of the radiation field with unit polarization vector of e .Figure 1 shows the geometry of the scattering event. Without the scattering potential U ( r ) the particular solution of(6) can be immediately written down as non-relativistic Volkov states ϕ p ( r , t ) which exactly incorporate the interactionwith the EM field,The Volkov states are modulated de Broglie waves; parametrized by momenta p and form an orthonormal andcomplete set ϕ p ( r , t ) = 1(2 π (cid:126) ) / exp (cid:20) i (cid:126) p · r − (cid:90) tt dt (cid:48) m (cid:16) p − ec A ( t (cid:48) ) (cid:17) (cid:21) , (7) (cid:90) d rϕ ∗ p ( r , t ) ϕ p (cid:48) ( r , t ) = δ ( p − p (cid:48) ) , (cid:90) d pϕ p ( r , t ) ϕ ∗ p ( r (cid:48) , t ) = δ ( r − r (cid:48) ) . (8)To solve the original problem of Eq. (6) the exact wave function are written as a superposition of an incomingVolkov state and a correction term, which vanishes at the beginning of the interaction (in the remote past t → −∞ ).The correction term can also be expressed in terms of the Volkov states as these form a complete set (see Eq. (8)),Ψ( r , t ) = ϕ p i ( r , t ) + (cid:90) d pa p ( t ) ϕ p ( r , t ) , a p ( t ) = 0 . (9)It is clear that the unknown expansion coefficients a p ( t ) describe the non-trivial transition symbolized as p i → p ,from a Volkov state of momentum p i to another Volkov state with momentum p . The projection of Ψ into someVolkov state ϕ p ( t ) results in (cid:90) d rϕ ∗ p ( r , t )Ψ( r , t ) = δ ( p − p i ) + a p ( t ) . (10)The insertion of Ψ of Eq. (9) into the complete Schr¨odinger equation (6) results in the following integro-differentialequation for the coefficients a p ( t ), i (cid:126) ˙ a p (cid:48) ( t ) = (cid:90) d rϕ ∗ p (cid:48) ( r , t (cid:48) ) U ( r ) ϕ p i ( r , t (cid:48) ) + (cid:90) d pa p ( t ) (cid:90) d rϕ ∗ p (cid:48) ( r , t (cid:48) ) U ( r ) ϕ p ( r , t (cid:48) ) , (11)where the scalar product was taken with ϕ p (cid:48) ( t ) on both sides of the resulting equation and the orthogonalityproperty of the Volkov sates was taken after all (see the first Eq. of (8)). The initial condition a p ( t ) = 0, alreadyshown in (8), means that the formal solution of (6) can be written as a p (cid:48) ( t ) = − i (cid:126) (cid:90) tt dt (cid:48) (cid:90) d rϕ ∗ p (cid:48) ( r , t (cid:48) ) U ( r ) ϕ p i ( r , t (cid:48) ) − i (cid:126) (cid:90) tt dt (cid:48) (cid:90) d pa p ( t (cid:48) ) (cid:90) d rϕ ∗ p (cid:48) ( r , t (cid:48) ) U ( r ) ϕ p ( r , t (cid:48) ) . (12)In the spirit of the iteration procedure used in scattering theory the ( k + 1) − th iterate of a p ( t ) is expresses by thek-th iterate on the right hand side in (12) as a ( k +1) p ( t ) = − i (cid:126) (cid:90) tt dt (cid:48) (cid:90) d rϕ ∗ p (cid:48) ( r , t (cid:48) ) U ( r ) ϕ p i ( r , t (cid:48) ) − i (cid:126) (cid:90) tt dt (cid:48) (cid:90) d pa ( k ) p ( t (cid:48) ) (cid:90) d rϕ ∗ p (cid:48) ( r , t (cid:48) ) U ( r ) ϕ p ( r , t (cid:48) ) . (13)In the first Born approximation, where the transition amplitude is linear in the scattering potential U ( r ) , thetransition amplitude has the form T fi = lim t →∞ lim t →−∞ a (1) p f ( t ) = − i (cid:126) (cid:90) ∞−∞ dt (cid:48) (cid:90) d rϕ ∗ p f ( r , t (cid:48) ) U ( r ) ϕ p i ( r , t (cid:48) ) . (14)The A term drops out from the transition matrix element (14), because it represents a uniform time-dependentphase. By taking the explicit form of the Volkov states (7) with the vector potential A ( t ) = e A cos ( ωt ) means that T fi becomes T fi = ∞ (cid:88) n = −∞ T ( n ) fi , T ( n ) fi = − πiδ (cid:32) p f − p i m + n (cid:126) ω (cid:33) J n ( z ) U ( q )(2 π (cid:126) ) , (15)Before time integration, the exponential expression can be expanded into a Fourier series with the help of the Jacobi-Anger formula [24]. This results in the next formula including the Bessel functions of the first kind e izsin ( ωt ) = ∞ (cid:88) n = −∞ J n ( z ) e inωt . (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 its absolute value is q = (cid:113) p i + p f − p i p f cos ( θ p i ,p f ). In thecase where electrons have 0.1 - 1 eV energy in the n = 0 channel (which means elastic scattering), the followingapproximation is valid q ≈ p i | sin ( θ/ | .In general, the Dirac delta function describes photon absorptions ( n <
0) and emissions ( n > J n ( z ) is theBessel function with the argument depending on the parameters of the laser field, the intensity and the frequency z ≡ a qsin ( θ/ cos ( χ ) (cid:126) ω/c where a , q, χ are the dimensionless intensity parameter, the momentum transfer of the electronand the angle between the momentum transfer and the polarization direction of the EM field, respectively.The general differential cross section formula for the laser assisted collision with simultaneous nth-order photonabsorption and stimulated emission processes are dσ ( n ) d Ω = p f p i J n ( z ) dσ B d Ω . (17)The dσ B d Ω = (cid:0) m π (cid:126) (cid:1) | U ( q ) | is the usual Born cross section for the scattering on the potential U(r) alone, withoutthe external EM field. The expression Eq. (17) was calculated by several authors using different methods [6–13] andif the Born cross section is exactly known, Eq. (17) can be substituted in Eq. (3) and the single-particle lifetime canbe easily calculated. C. Scattering model potentials in semiconductors
Different kinds of analytic model potentials are available to model the electron scattering on impurities in a semi-conductor and four are presented within this paper. The simplest model is the ”box potential” which is well-knownfrom quantum mechanics textbooks. It is capable to mimic the two-dimensional impurity scattering provide by acylindrical-barrier of radius a . It can be used to describe a neutral impurity such as an Aluminum atom that hasdiffused from a barrier into a GaAs well [3].Mathematically, U ( r ) = U if r ≤ a and U ( r ) = 0 if r > a, where U is the depth of the square well potential in eVand a is the radius in nm. The two dimensional Fourier transformation of the potential gives the first-order Besselfunction of the form of U ( q ) = πa U J (2 qa ) qa . The infinite range Coulomb potential has an infinite total scatteringcross section in the first-order Born approximation. However, considering a maximal limiting impact parameter dueto electron screening in semiconductors the isotropic elastic cross section can be evaluated (3) and done by [25]. Thisis the Brooks-Herring(BH) model [26, 27] and is applicable to describe electron scattering on an ionized impurityatom U ( r ) = εe − r/λ D π(cid:15) (cid:15) r r , (18)where λ D = (cid:113) (cid:15) (cid:15) r k B Tq n is the Debye screening length, (cid:15) and (cid:15) r are the vacuum permitivity and the dielectric constantof the present media. To avoid confusion ε is used for the charge of the ionized impurity atom instead of q which isfixed for the momentum transfer of the electrons. The Debye screening is just the solution of the linearized Poisson-Boltzmann theory and is the simplest way to handle the problem. More general theory that describe screeningwould be Landau’s approach known as Fermi liquid theory where there is a quantitative account of electron-electroninteraction [36].The Fourier transform of this potential is as follows [2], U ( q ) = ε λ D (cid:15) (cid:15) r (1 + q λ D ) . (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 effectse.g., dielectric of Thomas-Fermi screening, electron-electron interaction [23]. The original BH can also be calculatedfrom a realistic electron concentration of the ionized impurity containing the Fermi-Dirac integral [28].There are two additional potentials which are widely used to model impurities in semiconductors. The first has beendeveloped to investigate the electron charged dislocation scattering in an impure electron gas. The derivation of theformula can be found in [1]. The radial potential has the form of U ( r ) = (cid:15) π(cid:15) r c K (cid:16) rλ D (cid:17) where K is the zeroth-ordermodified Hankel function, (cid:15), (cid:15) r , /c, λ D are the elementary charge, dielectric constant, linear charge density and theDebye screening length. This dislocation is a two dimensional interaction and has a cylindrical symmetry. The Fouriertransform of the potential is U ( q ) = (cid:15)λ (cid:15) r c (1+ q λ ) . Jena [29] used this interaction to evaluate the quantum and classicalscattering times due to charged dislocation in an impure electron gas. Half a century earlier, P¨od¨or [30] calculated ananalytic formula for relaxation time and investigated the electron mobility in plastically deformed germanium. Theseare remarkable similar to the three dimensional BH potential.The last most advanced model is dipole scattering in polarization induced two-dimensional electron gas. Thisconsiders the electrical field of a dipole above a plane [31]. In the following, the BH model will be analyzed in details.
D. Generalized field assisted potential scattering in a media
The previouly outlined laser assisted potential scattering description with the listed potentials is not sufficientfor a realistic model to evaluate electron conduction in a solid at finite temperature. Thereforer two additionalimprovements are considered.Firstly, the scattering electrons now move in a media (doped semiconductor) instead of a vacuum, therefore the effectof the media, the dielectric response functions, has to be taken into account. Eq. (17) is modified and generalized andthe numerically evaluated Lindhard dielectric function [32] 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 matrixof the electron [33] that the total interaction potential in the frequency domain is equivalent to the Fourier transformof the interaction potential in vacuum multiplied by the Lindhard dielectric function. Therefore the final differentialcross section formula is dσ ( n ) d Ω = p f p (cid:16) m π (cid:126) (cid:17) J n ( z ) | U ( q , (cid:15) r [ k, ω ]) | . (20)The dielectric function now depends on the angular frequency of the external applied field, the coherent IR field, andthe wave vector of the scattering electron. The correct form of the interaction for the BH model is : U ( q, k, ω ) = ε λ D (cid:15) (cid:15) r ( k, ω )(1 + q λ D ) , (21)where ε in the numerator is the charge of the impurity. The next technical step in the model is to calculate theLindhard dielectric function. For a fermion gas with electronic density n at a finite temperature, T , the form can beexpressed in terms of real and imaginary part [34] ε r ( k, ω ) = ε r R ( k, ω ) + iε r I ( k, ω ) . (22)At finite temperatures, the dielectric function contains singular integrals of the Fermi function which can be eliminatedwith various mathematical transformations. According to [35], the following expressions have to be evaluated: ε r R ( k, ω ) = 1 + 14 πk F κ [ g t ( λ + = u + κ ) − g t ( λ − = u − κ )] , (23)and ε r I ( k, ω ) = t k F κ ln (cid:20) exp ( α ( t ) − λ − ) /t exp ( α ( t ) − λ ) /t (cid:21) . (24)Where the Fermi wave vector is k F = [3 π n ] / , the reduced temperature is t = T /T F , the Fermi energy is E F = k F / k B T F (where the electron mass and (cid:126) were set to unity). The reduced variables u and κ introduced byLindhard are defined as u = ωv F k , κ = k k F , (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 (cid:90) + ∞ x
11 + exp( x − α ( t ) t ) dx = 13 . (26)After determining the chemical potential, the function g t ( λ ) can be calculated via an integral where the usual singu-larity is successfully eliminated by a appropriate mathematical transformation g t ( λ ) = λ (cid:90) ∞ (cid:20) − A X exp( AX − B )1 + exp( AX − B ) (cid:21) × (cid:20) − X + 1 − X (cid:12)(cid:12)(cid:12)(cid:12) X + 1 X − (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) dX. (27)where A = λ/t and B = α ( t ) /t . Exhaustive technical details can be found in the original paper [35].At this point, the static (cid:15) ( q, ω →
0) and the long wavelength limit (cid:15) ( q → , ω ) of the Lindhard function can bereduced to analytic formulas [32, 36]. For the static limit (cid:15) ( q,
0) = 1 + κ q , the 3D screening wave number, κ , (3Dinverse screening length) is defined as κ = (cid:113) πe ε ∂n∂µ where n, µ, ε are the particle density N/L , chemical potential andcharge, respectively. However, in the long wavelength limit in 3D, (cid:15) (0 , ω ) = 1 − ω plasma ω where the angular frequencyof the plasma reads ω plasma = πe NεL m .At a finite temperature in a realistic semiconductor, the scattering electrons are not monochromatic and thus anaveraging over the distribution has to be evaluated (cid:104) G (cid:105) = e n e m ∗ n imp (cid:104) σ T (cid:105) . (28)This means that there is an additional numerical integration of the total cross section multiplied by the Fermi-Diracdistribution function f ( E ) (for non-degenerate electrons) times the density of states g ( E ) according to Shang [2], (cid:104) σ T (cid:105) = (cid:82) ∞ σ T ( E ) f ( E ) g ( E ) dE (cid:82) ∞ f ( E ) g ( E ) dE (29)with E ( k ) = (cid:126) k m being the energy of the electrons. In this representation, the integration can be traced back to anintegral over k . The numerical value of the density of state function is well-known for one, two or three dimensionalsolids. A two dimensional system g ( E ) D = m e π (cid:126) is independent of the electron energy. In three dimensions, thecurrent BH potential case, g ( E ) D = m / e √ π (cid:126) √ E . The mentioned charged dislocation potential is a two dimensionalmodel.If the number of donors are enhanced, the Fermi level will rise towards the conduction band. At some stage theapproximations will no longer hold because a larger proportion of the Fermi Dirac function overlaps with the bandedge. The approximations break down when the Fermi level is closer than 3 k B T to one of the band edges. This isapproximately 75 meV at room temperature. In this case, the semiconductor become degenerate and the Boltzmanndistribution function has to be applied, instead of the Fermi-Dirac fucntion. To the best of our knowledge, these twocompletions were never added to the general laser assisted potential scattering to model electron scattering in realisticsolid states.In practical calculations the upper limit of the integral can be cut at the Fermi energy which is about 1 eV at roomtemperature for semiconductors. Numerical values obtained from (29) with or without external electromagnetic fieldcan be directly compared in the future to experimental data. III. RESULTS
Doped silicon has been considered as a semiconductors with a Fermi energy of 1 eV and coherent IR electromagneticsources ( λ = 1 – 5 µ m) are the external field. According to the ELI-ALPS’ white book [37], a λ = 3 . µ m wavelengthmid-IR laser will operate,which will be similar to [38]. An intensity range of 10 < I < W/cm is used to avoidlaser damage of the silicon sample. Stuart et al. [39] and Tien et al. [40] published experimental results for radiationdamage of silica for λ = 1 µ m wavelength laser pulses with different pulse length and found that the threshold liesat 10 W/cm for 100 fs pulse duration. Unfortunately, no experimental measurement values for λ = 3 µ m couldbe found. However, there is an empirical power law dependence for damage threshold for silica glass at variouswavelength I th ( λ ) = 1 . · λ . · I th , where the wavelength and the intensities should be given in µ m and W/cm units [41]. This means that the threshold at 3 µm should be 2 . · W/cm .According to [23], 6 electron mobility versus electron concentration measurement were presented and compared tovarious BH models on a logarithmic-linear scale below 200 K giving discrepancy of a factor of 2 − E = k B T which is 0.025 eVat room temperature. The parameters of the BH potential are the following the screening range λ d = 30 nm, thedielectric constant (cid:15) r = 35 and the charge ε = 1.In the following, calculations for two distinct models are presented. The first system simply considers a semi-conductor media with a dielectric constant. The literature value is 35 for semiconductor Silicon. The second morerealistic model fully includes the frequency dependent Lindhard function Eq. (22). This model even includes theenergy dependence of the scattering electrons. The two field independent cross sections are σ T,(cid:15) =35 = 1 . and × × × × × I [ W / cm ] σ [ nm ] σ ε = ε ( ω , k ) σ ε = FIG. 2: The averaged cross section (cid:104) σ T (cid:105) Eq. (29) as the function of the field intensity for a λ = 3 µ m laser wavelength at roomtemperature. The upper curve is for the frequency dependent Lindhard dielectric function. The lower curve is for the (cid:15) r = 35dielectric constant case. σ T,(cid:15) ( ω,k ) = 39 . , respectively. The (cid:15) ( k ) dependence and averaging over the electron energy makes the ratio lessthan 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 µ m wavelength.The more realistic modelgives larger cross sections which means that there is a smaller electron conductance. Byconsidering the field independent cross section σ T,(cid:15) =35 = 1 . as a standard value gives the suppression of theelectron conductance by the external field by a factor of 15.Fig 3. presents the ratio of the two models as the function of the field intensity with values between 21 and 26. Atlarger 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 W/cm . The cross sections obtained from the model incorporating the Lindhard function are still largerthan the simpler model. The cross sections of both models decay at large intensities. Note, that for 1 µ m wavelength,the ratio of the original cross section (1.9 nm ) goes up to 310 nm a gain factor of 155. Calculations below 1 µ mwavelength were not performed because such fields may excite valence electrons into the conductance band and thatwould lie out of the scope of this elastic scattering model. Figure 5 shows the wavelength dependence of the ratio ofthese two models. Larger wavelength means smaller cross sections or larger conductance. These results are in fullyagreement with the general theory of [20]. The ratio between the two models still lie at a factor of 23 to 25.Multiplying with the remaining constants of e n e m e n imp where e is the charge of the electron, m e is the effective massof an electron in a semiconductor is about 0 . × . · − kg and the number of the impurities per cm lies between10 − therefore the obtained final conductance values would lie between 10 − − Si/cm [42]. This is a verybroad range of conductance, hence not reporting exact numerical conductance values. However, these ratio of theconductances with or without strong external IR fields can vary by more than a magnitude.A doped semiconductor has a complex nature and the physical value of the resistivity change in an external IRfield can of course only be investigated in a real physical experiment but these calculation shows that it would be aninteresting project.These models only include elastic scattering processes, without any photon absorption or emission. The inclusionof one photon absorption only requires changing the zeroth order Bessel function to the first order term, otherwisethe process and the way of calculation are the same. Note, that the corresponding cross sections or the probabilitiesof a first order process is at least one order of magnitude lower than elastic one.These presented calculations cannot include additional effects coming from the complex nature of a real solid statelike, valence dielectric screening, band-structure details, electron-electron scattering, non-linear screening, multipleelectron scattering and impurity dressing – all of which are mentioned in the review of [23].The two-fold numerical integrations of (29) for various laser parameters were evaluated with Wolfram Mathematica[Copyright 1988 − × × × × × I [ W / cm ] σ ε = ε ( ω , k ) / σ ε = FIG. 3: The intensity dependence of the ratio of the two models at λ = 3 µ m. λ [ μ m ] σ [ nm ] σ ε = ε ( ω , k ) σ ε = FIG. 4: The averaged cross section (cid:104) σ T (cid:105) Eq. (29) as the function of the wavelengths for I = 10 W/cm field intensity atroom temperature. The upper curve is for the frequency dependent Lindhard dielectric function. The lower curve is for the (cid:15) r = 35 dielectric constant case. λ [ μ m ] σ ε = ε ( ω , k ) / σ ε = FIG. 5: The wavelength dependence of the ratio of the two models at I = 10 W/cm intensity. IV. SUMMARY
A formalism based on an interrelated model to calculate electron conductance in a doped semiconductor in strongexternal IR fields with intensities between 10 − W/cm with a wavelength range of 1 −− µ m. The mathematicaldescription of multi-photon processes was coupled to the well-established potential scattering model based on the first-Born approximation. An application of this general formalism in the present paper has been the modification of thescattering elastic cross sections in the elastic channel.In solid sate physics, the elastic scattering of electrons on impurities modeled by the BH potential can model theelectric conductance up to a factor of 2 to 5, a reliable background [23]. Electron scattering has been treated in aperturbative manner whilst the influence of the external strong radiative IR field (which can cause photon absorption)has been treated non-perturbatively. The solid state 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 theinclusion of dielectric constant, which models the semiconductor. The second improvement is that the final electronenergy distribution above the Fermi function at room temperature is averaged. These two improvements give at leasta factor of 15 suppression in the final conductance. It has been demonstrated that due to the joint interaction ofthe conduction electrons with the impurity scattering potential and the laser field that there could be a considerablechange in the conduction as was expected at the beginning of this studies. These theoretical results have inspiredour experimental colleagues in ELI-ALPS to perform measurements on silicon samples at 3 µ m wavelength. Workis in progress to build up a setup to measure the prognosticated change in the conductivity. If the change of theconductivity lies in the same order of time scale as the duration of the mid-IR laser pulse (tenth of femtoseconds) thanonly pump-probe type measurements can be applied to measure the change of the optical properties of the sample[43]. If the experiments verify these theoretical predictions than it may be possible to start speculating about possiblephysical application of the phenomena, like a quick gating, a quick moldulator or even a mid-IR light intensity sensor. V. ACKNOWLEDGMENTS
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