Transcript Document

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 :[email protected]
1
harmonics generation in atoms
Have been observed harmonics up 295th order of a radiation.
2
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
3
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
i)
Silicon
ii)
GaAs, InP
4
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)
5
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 biMaxwellian EDF with the effective electron temperature along the field larger
than that perpendicular to it:
Nm
F (v ) 
2 T
 mv2 mvz2 
m
exp  


2 Tz
 2T 2Tz 
6
Harmonics generation in a plasma with anisotropic
bi-maxwellian distribution
Such a plasma interacts with another high frequency wave,
assumed in the form
E  ( Ex  0 Ez )
E cos(t  kr )
We consider also
  L
and
 
the frequency and the wave vector are linked by the dispersion
relation
 2  L2  k 2c2 
7
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
Tz  T
8
Harmonic Generation
The efficiency of HG of order n is given by
I n En
n  
Io
E
2
To obtain the electric field of the n-th harmonic we have to solve
the Maxwell equation
1 2 E
4 j
 ( E )  2 2   2
c t
c t
where
j  e  dv  v f
is the electron density current
EDF in the presence of
the high frequency field
9
For the EDF in the presence of a high frequency field we
can write the following kinetic equation :

e
f
f  E cos(ot )  St  f 
t
m
v
1
 2
f
St ( f )   (v)
(v  ij  vi v j )

2
vi
v j
the electron-ion collision integral in the Fokker-Planck form
where (v) is the electron-ion
collision frequency
4 Ze 4 N 
 (v ) 
m2v3
10
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
f 0

e
f 0  E cos(ot )
0
t
m
v
the solution is given in the form
f0 (v, t )  F (v  vE sin t )
where
eE
vE 
m
is the quiver velocity
11
In the next approximation we take into account the influence of the
rare collisions on the high-frequency electron motion.
For the correction
 f   f (v, t )
To the distribution function due to collisions we have the equation

e

 f  E cos(ot )  f  St ( f 0 )
t
m
v
12
Harmonic Generation
the current density generated by the high-frequency field.
j  e  d v v ( f 0   f )  env E sin t   j
L


j
E  j
t
4
t
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
 j
 e St ( f )dv  v 
t
e
  2
 
  d vv   v 
F (v)
 v  ij  vi v j 


2
vi
v j
Using a bi-maxwellian EDF
13
Harmonic Generation
Using the bi-maxwellian for the the time derivative of the
non linear current density


1
q
 j  2 eNvE3 (vE ) d q 2  J 2n1 (qv E )
t

q n 0
 2 Tz
2 T 
exp  qz
 q
sin (2n  1)(t  kr )  

2m
2m 

With J2n+1 the Bessel function of order 2n+1.
14
Harmonic Generation
The current density can be written as:

 j    j n (r t )
n 0
The n-th component of the electric field
E n (r t )  E n sin (2n  1)(t  kr ) 
is obtained as a solution of the Maxwell equation
1  2 En L2
4 
 ( E n )  2
 2 En   2
 jn
2
c t
c
c t
15
Harmonic Generation
we obtain the electric field of the n-th harmonics resulting from
nonlinear inverse bremsstrahlung as:
1
eN 3
q
 2 Tz
2 T 
v  (vE ) dq 2 J 2n1 (qv E ) exp qz
 q

En 
2 E

 n(n  1) L
q
2m
2m 

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
En
 (vT )  2
n 

a (n      )

E
  
2
2
with
a2 (n)  ax2  az2 
16
Harmonic Generation
En
 (vT ) 
2
2


n 

a

a
x
y


E
  
2
2
ax, y  a(n,  ,  , )
  Intensity,   anisotropy
vT  T m T  (Tz  2T )3   T 
  Tz  T  0   mvE2  4T 
 is the angle between the field and the oZ axis
17
Harmonic Generation
a2 (n)  ax2  az2 
1
 a x ( n) 
1
dy




a
(
n
)
 z  2 2 n(n  1) 0 1  y 2

( x cos   y 1  x 2 sin  )
1   ( x  13 ) 
2
3 2

 y 1  x2 

dx


1  x  


1
exp  W I n W   I n1 W  
where In is the modified Bessel function of n-order
W

2
 x cos   y 1  x sin   

1   ( x  13 ) 
2
2
18
Efficiency of the Third Harmonic
vE2
  2  0.1, 0.4, 1, 3, 10
vT
 is the angle between E and the
anisotropic axis

Tz  T 9
  (Tz  10T )
T
4
is the anisotropy degree
T
1
2T  Tz 
3
19
Efficiency of the Third Harmonic
vE2
  2  0.4
vT
 is the angle between E and the
anisotropic axis
9
4
12
   (Tz  5T ),
7
3
   (Tz  2T ),
4
   (Tz  10T ),
20
Efficiency of the 5,7,9 Harmonic
 3
dashed
  10
continuous
 is the angle between E and the
anisotropic axis

9
 (Tz  10T ),
4
fifth (n=2), seventh (n=3) and
ninth (n=4) harmonics
21
Efficiency of the 5,7,9 Harmonic
(Tz  2T )
(Tz  10T )
dashed
continuous
 is the angle between E and the
anisotropic axis
 3
fifth (n=2), seventh (n=3) and
ninth (n=4) harmonics
22
Polarization of Harmonics
Y is the angle between E
and En
 is the angle between the
field and the oZ axis
23
Polarization of Harmonics
 EE 
n

  arccos G (n      )  
Y (n      )  arccos
 E En 


Where the function G has the form:
1
1
dy
G(n      ) 
2 2 an(n  1) 0 1  y 2

1
1
dx
( x cos   y 1  x 2 sin  )2
1   ( x  13 ) 
2
3 2

 exp  W  I n W   I n 1 W  
with
W

2
 x cos   y 1  x sin   

1   ( x  13 ) 
2
2
24
Polarization of the Third Harmonic
vE2
  2  0.4, 1, 3, 10
vT
 is the angle between E and the
anisotropic axis

Tz  T 9
  (Tz  10T )
T
4
is the anisotropy degree
T
1
2T  Tz 
3
25
Polarization of the Third Harmonic
vE2
  2  0.4
vT
 is the angle between E and the
anisotropic axis
9
4
12
   (Tz  5T ),
7
3
   (Tz  2T ),
4
   (Tz  10T ),
26
Polarization of the 5,7,9 Harmonic
 3
dashed
  10
continuous
 is the angle between E and the
anisotropic axis

9
 (Tz  10T ),
4
fifth (n=2), seventh (n=3) and
ninth (n=4) harmonics
27
Polarization of the 5,7,9 Harmonic
(Tz  2T )
(Tz  10T )
dashed
continuous
 is the angle between E and the
anisotropic axis
 3
fifth (n=2), seventh (n=3) and
ninth (n=4) harmonics
28
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 biMaxwellian 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.
29
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.
30
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.
31
The model
The propagation of an electromagnetic wave along a given
direction z in a medium is described by the Maxwell equation


2
E 1 E
2 P
 2 2  o 2
2
z c t
t
2
where
P   o ( 1   2 E   3 E 2  .....)E
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
P
j  nev( E ) 
t
32
The efficiency of HG or of WM at frequency , normalized to
the fundamental one is given by:
I
E
 

Io
Eo
2
v2
 2
v1
Where v 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 n 1 with n=1,3,5.....
33
ENERGY BAND STRUCTURE
34
The band structure of Silicon shows two kinds of minima. The
absolute minimum is represented by six equivalent ellipsoidal
valleys (X valleys) along the <100> directions at about 0.85 %
of the Brillouin zone. The other minima are situated at the limit
of the Brillouin zone along the <111> 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
35
SCATTERING MECHANISMS IN OUR MODEL
GaAs and InP
Si
INTRAVALLEY
INTRAVALLEY
(Equivalent and non equivalent)
(Equivalent)
Acoustic Phonon Scattering [elastic and isotropic]
Ionized Impurity Scattering [elastic and
anisotropic; Brooks-Herring approximation]
Piezoelecric Acoustic Scattering [elastic and
isotropic]
Acoustic Phonon Scattering [elastic and
isotropic]
Ionized Impurity Scattering [elastic and
anisotropic; Brooks-Herring approximation]
Optical Phonon Scattering [inelastic e anisotropic]
Non Polar Optical Phonon Scattering [inelastic
and isotropic; effective only in L valleys]
INTERVALLEY
Non Polar Optical Phonon Scattering [inelastic and
isotropic]
INTERVALLEY
Longitudinal Optical Phonon Scattering [g-type
inelastic and isotropic]
Transverse Optical Phonon Scattering [f-type
inelastic and isotropic]
36
Harmonics Generation
Si
InP
E
  200GHz, T  80 o K , n  1019 m-3
37
Harmonics Generation
Si
InP
n
  200GHz, T  80 o K , n  1019 m-3
38
Harmonics Generation
InP
n
Minimum of the
efficency is shifting to
higher field intensity with
the increasing of the field
frequency !
39
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))
40
Si Static Characteristic
saturation
Non-linearity
E
41
InP Static Characteristic
Gunn Effect
Polar phonon
emission
saturation
E
42
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
43
The work per unit time performed by the external electric field on the free
electron is given by
W  j  E
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.
44