M. Zarcone

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_at_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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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Harmonic Generation
Using the bi-maxwellian for the the time
derivative of the non linear current density
With J2n1 the Bessel function of order 2n1.
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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
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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
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Harmonic Generation
g ? Intensity, d ? anisotropy
? is the angle between the field and the oZ axis
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Harmonic Generation
where In is the modified Bessel function of
n-order
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Efficiency of the Third Harmonic
a is the angle between E and the anisotropic axis
is the anisotropy degree
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Efficiency of the Third Harmonic
a is the angle between E and the anisotropic axis
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Efficiency of the 5,7,9 Harmonic
dashed
continuous
a is the angle between E and the anisotropic axis
fifth (n2), seventh (n3) and ninth (n4)
harmonics
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Efficiency of the 5,7,9 Harmonic
dashed
continuous
a is the angle between E and the anisotropic axis
fifth (n2), seventh (n3) and ninth (n4)
harmonics
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Polarization of Harmonics
Y is the angle between E and En
? is the angle between the field and the oZ axis
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Polarization of Harmonics
Where the function G has the form
with
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Polarization of the Third Harmonic
a is the angle between E and the anisotropic axis
is the anisotropy degree
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Polarization of the Third Harmonic
a is the angle between E and the anisotropic axis
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Polarization of the 5,7,9 Harmonic
dashed
continuous
a is the angle between E and the anisotropic axis
fifth (n2), seventh (n3) and ninth (n4)
harmonics
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Polarization of the 5,7,9 Harmonic
dashed
continuous
a is the angle between E and the anisotropic axis
fifth (n2), seventh (n3) and ninth (n4)
harmonics
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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.

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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 devices in 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
  due only to the motion of free carriers in the
  presence of the electric field of the incident
  wave.

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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
  with different effective mass
• the inelastic character of some scattering
  mechanisms.

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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
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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 wn w1 with
  n1,3,5.....

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ENERGY BAND STRUCTURE
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The band structure of Silicon shows two kinds of
minima. The absolute minimum is represented by
six equivalent ellipsoidal valleys (X valleys)
along the lt100gt directions at about 0.85 of
the Brillouin zone. The other minima are situated
at the limit of the Brillouin zone along the
lt111gt directions (L valleys). In our simulation
the conduction band of Si is represented by six
equivalent X valleys. Since the energy gap
between X and L valley is large (1.05eV), for
the employed electric field and frequency, the
electrons do not reach sufficient kinetic
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
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SCATTERING MECHANISMS IN OUR MODEL
(Equivalent)
(Equivalent and non equivalent)
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Harmonics Generation
Si
InP
E
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Harmonics Generation
Si
InP
n
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Harmonics Generation
InP
Minimum of the efficency is shifting to higher
field intensity with the increasing of the field
frequency !
n
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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))
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Si Static Characteristic
saturation
Non-linearity
E
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InP Static Characteristic
Gunn Effect
Polar phonon emission
saturation
E
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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

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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 Te will 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.