M. Zarcone Istituto Nazionale per la Fisica della Materia and

1. High harmonics generation in plasmas and in semiconductors M. Zarcone Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica e Tecnologie Relative, Viale delle Scienze, 90128 Palermo, Italy e-mail :zarcone@unipa.it

2. harmonics generation in atoms Have been observed harmonics up 295th order of a radiation.

3. An electron initially in the ground state of an atom, exposed to an intense, low frequency, linearly polarized e.m. field 1) first tunnels through the barrier formed by the Coulomb and the laser field 2) then under the action of the laser field is accelerated and – can leave the nuclei (ionization) – or when the laser field changes sign can be driven back toward the core with higher kinetic energy giving rise to emission of high order harmonics

4. Harmonics generation in plasma and semiconductors • Plasma: case of anisotropic bi-maxwellian EDF • We study how the efficiency of the odd harmonics generation and their polarization depend on process parameters as: • i) the degree of effective temperatures anisotropy; • ii) the frequency and the intensity of the fundamental wave; • iii) the angle between the fundamental wave field direction and the symmetry axis of the electron distribution function. • Semiconductors: low doped n-type bulk semiconductors • Silicon • GaAs, InP

5. Electron-Ion Collision Induced Harmonic Generation in a Plasma with Maxwellian Distribution • the efficiency is lower • than in gases • no plateau • no cut-off Similar behavior found for semiconductors ! D. Persano Adorno, M. Zarcone and G. Ferrante Phys. Stat. Sol. C 238, 3 (2003). The intensity of the harmonics (2n + 1) for 4 different initial values of the parameter vE/vT(0) = 40 (squares); 20 (void circles); 10 (black circles); 4 (triangles). G. Ferrante, S.A. Uryupin, M. Zarcone, J. Opt. Soc. Am, B14, 1716,(1997)

6. Harmonics generation in plasma anisotropic bi-maxwellian EDF • Plasma: • Fully ionized • Two-component • Non relativistic The velocity distribution of the photoelectrons is given by anisotropic bi-Maxwellian EDF with the effective electron temperature along the field larger than that perpendicular to it:

7. Harmonics generation in a plasma with anisotropic bi-maxwellian distribution Such a plasma interacts with another high frequency wave, assumed in the form We consider also and the frequency and the wave vector are linked by the dispersion relation

8. Harmonics generation in a plasma with anisotropic bi-maxwellian distribution Tz and T are the electron effective temperatures along and perpendicularly to the EDF symmetry axis

9. Harmonic Generation The efficiency of HG of order n is given by To obtain the electric field of the n-th harmonic we have to solve the Maxwell equation where is the electron density current EDF in the presence of the high frequency field

10. For the EDF in the presence of a high frequency field we can write the following kinetic equation : the electron-ion collision integral in the Fokker-Planck form where (v) is the electron-ion collision frequency

11. If the frequency  largely exceeds both the plasma electron frequency and the effective frequency of electron collisions, in the first approximation it is possible to disregard the influence of the collisions on the quickly varying electron motion in the high-frequency field. In this approximation for the distribution function of electrons we have the equation the solution is given in the form where is the quiver velocity

12. In the next approximation we take into account the influence of the rare collisions on the high-frequency electron motion. For the correction To the distribution function due to collisions we have the equation

13. Harmonic Generation the current density generated by the high-frequency field. where the source of non linearity is given by the e-i correction to the time derivative of the current density, Taking into account, that in electron-ion collisions the number of particles is conserved we have Using a bi-maxwellian EDF

14. Harmonic Generation Using the bi-maxwellian for the the time derivative of the non linear current density With J2n+1 the Bessel function of order 2n+1.

15. Harmonic Generation The current density can be written as: The n-th component of the electric field is obtained as a solution of the Maxwell equation

16. Harmonic Generation we obtain the electric field of the n-th harmonics resulting from nonlinear inverse bremsstrahlung as: the field of the harmonic En, similarly to that of the fundamental field , has only two components and the efficiency of generation of the harmonic is characterized by the ratio with

17. Harmonic Generation g Intensity, d  anisotropy  is the angle between the field and the oZ axis

18. Harmonic Generation where In is the modified Bessel function of n-order

19. Efficiency of the Third Harmonic a is the angle between E and the anisotropic axis is the anisotropy degree

20. Efficiency of the Third Harmonic a is the angle between E and the anisotropic axis

21. Efficiency of the 5,7,9 Harmonic dashed continuous a is the angle between E and the anisotropic axis fifth (n=2), seventh (n=3) and ninth (n=4) harmonics

22. Efficiency of the 5,7,9 Harmonic dashed continuous a is the angle between E and the anisotropic axis fifth (n=2), seventh (n=3) and ninth (n=4) harmonics

23. Polarization of Harmonics Y is the angle between E and En  is the angle between the field and the oZ axis

24. Polarization of Harmonics Where the function G has the form: with

25. Polarization of the Third Harmonic a is the angle between E and the anisotropic axis is the anisotropy degree

26. Polarization of the Third Harmonic a is the angle between E and the anisotropic axis

27. Polarization of the 5,7,9 Harmonic dashed continuous a is the angle between E and the anisotropic axis fifth (n=2), seventh (n=3) and ninth (n=4) harmonics

28. Polarization of the 5,7,9 Harmonic dashed continuous a is the angle between E and the anisotropic axis fifth (n=2), seventh (n=3) and ninth (n=4) harmonics

29. Electron-Ion Collision Induced Harmonic Generation in a Plasma with a bi-maxwellian Distribution: Conclusions • We have shown how the harmonics generation efficiency and the harmonics polarization depend on the plasma and pump field parameters. • The reported results are expected to prove useful for optimization of the conditions able to yield generation of high order harmonics and for diagnosing the anisotropy of the EDF itself. • Though the results have been obtained for a plasma exhibiting a bi-Maxwellian EDF, they are of general character and open the avenue of the treatment of anisotropy effects in plasmas with more complicated initial EDF, which may result from different physical processes.

30. Harmonics generation in bulk semiconductors • The investigation of non-linear processes involving bulk semiconductors interacting with intense F.I. radiation is of interest: • to explore the possibility to build a frequency converter of coherent radiation in the terahertz frequency domain • to understand the dynamics of the conducting electrons in semiconductors in the presence of an alternate field • to study the electric noise properties in semiconductor devicesin the presence of an alternate field The F.I. frequencies are below the absorption threshold and the linear and non-linear transport properties of doped semiconductors are dueonly to the motion of free carriersin the presence of the electric field of the incident wave.

31. High-order harmonic emission • Low-doped semiconductors (Si, GaAs, InP), show an high efficiency in the generation of high harmonic in the presence of an intense a.c. electric field having frequency in the Far Infrared Region (F.I.).   • Several mechanisms contribute to the nonlinearity of the velocity-field relationship: • the nonparabolicity of the energy bands; • the electron transfer between energy valleys withdifferent effective mass; • the inelastic character of some scattering mechanisms.

32. The model The propagation of an electromagnetic wave along a given direction z in a medium is described by the Maxwell equation where is the polarization of the free electron gas in terms of the linear and nonlinear susceptibilities. The source of the nonlinearity is the current density

33. The efficiency of HG or of WM at frequency w, normalized to the fundamental one is given by: Where vw is the Fourier transform of the electron drift velocity. the time dependent drift velocity of the electrons is obtained from a Monte Carlo simulation using the standard algorithm including alternating fields • We find peaks in the efficiency spectra: • For Harmonic Generation when w=nw1 with n=1,3,5.....

34. ENERGY BAND STRUCTURE

35. The band structure of Silicon shows two kinds of minima. The absoluteminimum is represented by six equivalent ellipsoidal valleys (X valleys)along the <100> directions at about 0.85% of the Brillouin zone. Theother minima are situated at the limit of the Brillouin zone along the <111> directions (L valleys).In our simulation the conduction band of Siis represented by six equivalent X valleys. Since the energy gap between X and L valley is large (1.05eV), for theemployed electric field and frequency, the electrons do not reach sufficientkinetic energies for these transitions. In our simulation the conduction bands of GaAs and InP are represented by the Gamma valley, by four equivalent L-valleys and by three equivalent X-valleys. The energy gap between X and L valley is (0.3eV for GaAs and 0.85eV for InP) and transition between non equivalent bands must be included

36. Si GaAs and InP INTRAVALLEY INTRAVALLEY AcousticPhononScattering [elastic and isotropic] AcousticPhononScattering [elastic and isotropic] Ionized Impurity Scattering [elastic and anisotropic; Brooks-Herring approximation] Ionized Impurity Scattering [elastic and anisotropic; Brooks-Herring approximation] Piezoelecric Acoustic Scattering [elastic and isotropic] Optical Phonon Scattering [inelastic e anisotropic] Non Polar Optical Phonon Scattering [inelastic and isotropic; effective only in L valleys] INTERVALLEY INTERVALLEY Longitudinal Optical Phonon Scattering[g-type inelastic and isotropic] Non Polar Optical Phonon Scattering [inelastic and isotropic] Transverse Optical Phonon Scattering[f-typeinelastic and isotropic] SCATTERING MECHANISMS IN OUR MODEL (Equivalent and non equivalent) (Equivalent)

37. Harmonics Generation Si InP E

38. Harmonics Generation Si InP n

39. Harmonics Generation InP Minimum of the efficency is shifting to higher field intensity with the increasing of the field frequency ! n

40. Harmonics Generation • High efficiency (10 -2 for the 3rd harmonic) • Saturation of the efficiency for high fields • Presence of a minimum in the efficiency vs field intensity (for polar semiconductor) EXPERIMENTS: Experiments on Si have shown conversion effciencies of 0.1% (Urban M., Nieswand Ch., Siegrist M.R., and Keilmann F., J. Appl. Phys. 77, 981 (1995))

41. Si Static Characteristic saturation Non-linearity E

42. InP Static Characteristic Gunn Effect Polar phonon emission saturation E

43. CONCLUSIONS • In general the efficiency of high harmonics is relatively high, at least as compared with similar processes in media like plasmas. • The efficiency strongly depend on the semiconductor type and on the field intenity • The efficiency strongly depend on the relative importance of the different scattering mechanisms However the same scattering mechanisms (except for the intervalley transitions) are responsible for the harmonics generation in both cases, Plasma and Semiconductors

44. The work per unit time performed by the external electric field on the free electron is given by Since the velocity v and the current density j oscillate at the frequency  of the electric field E, the work W and consequently the electron temperature Tewill oscillate at frequency 2 and the total collision frequency (Te) will be modulated also at frequency 2. Then we expect that, the free electron drift velocity will acquire, because of the collisions, a component oscillating at frequency 3that will give rise to the third harmonic generation. Iteratively, at higher order we will get all the odd harmonics.