1 m. zarcone istituto nazionale per la fisica della materia and dipartimento di fisica e tecnologie...

1

M. Zarcone

Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica e Tecnologie Relative,

Viale delle Scienze, 90128 Palermo, Italye-mail :[email protected]

High harmonics generation in plasmas and in semiconductors

2

Have been observed harmonics up 295th order of a radiation.

harmonics generation in atoms

3

An electron initially in the ground state of an atom, exposed to an intense, low frequency, linearly polarized e.m. field1) first tunnels through the barrier formed by the Coulomb and the laser field2) 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 withhigher 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

i) Silicon

ii) GaAs, InP

5

Electron-Ion Collision Induced Harmonic Generation in a Plasma with Maxwellian Distribution

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)

• 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).

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Harmonics generation in plasma anisotropic bi-maxwellian EDF

Plasma:

• Fully ionized• Two-component • Non relativistic

2 2

( ) exp2 2 2 2

z

z z

mv mvNm mF v

T T T T

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

Such a plasma interacts with another high frequency wave, assumed in the form

cos( )E t kr ''''''''''''''''''''''''''''''''''''''''''

2 2 2 2L k c

the frequency and the wave vector are linked by the dispersion relation

( 0 )x zE E E ''''''''''''''

Harmonics generation in a plasma with anisotropic bi-maxwellian distribution

L We consider also and

8

Tz and T are the electron effective temperatures along and perpendicularly to the EDF symmetry axis

zT T

Harmonics generation in a plasma with anisotropic bi-maxwellian distribution

9

Harmonic Generation

The efficiency of HG of order n is given by

2

n nn

o

I E

I E

To obtain the electric field of the n-th harmonic we have to solve the Maxwell equation

2

2 2 2

1 4( )

E jE

c t c t

where j e dv v f

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

cos( )o

e ff E t St f

t m v

''''''''''''''

21( ) ( ) ( )

2 ij i ji j

fSt f v v v v

v v

4

2 3

4( )

Ze Nv

m v

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

00 cos( ) 0o

fef E t

t m v

''''''''''''''

0 ( , ) ( sin )Ef v t F v v t

the solution is given in the form

where E

eEv

m is the quiver velocity

12

In the next approximation we take into account the influence of therare collisions on the high-frequency electron motion.

For the correction

To the distribution function due to collisions we have the equation

0cos( ) ( )o

ef E t f St f

t m v

''''''''''''''

( , )f f v t

13

Harmonic Generationthe current density generated by the high-frequency field.

0 ( ) sinEj e dv v f f env t j

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 vt

2 ( )2 ij i j

i j

edv v v v v v F v

v v

'''''''''''''''''''''''''' ''

4Lj E j

t t

'''''''''''''''''''''''''' ''

Using a bi-maxwellian EDF

14

Harmonic Generation

Using the bi-maxwellian for the the time derivative of the non linear current density

32 12 2

0

1( ) ( )E E n E

n

qj eNv v dq J qvt q

2 2exp sin (2 1)( )2 2

zz

T Tq q n t kr

m m

With J2n+1 the Bessel function of order 2n+1.

15

Harmonic Generation

is obtained as a solution of the Maxwell equation

The n-th component of the electric field

2 2

2 2 2 2

1 4( ) n L

n n n

EE E j

c t c c t

''''''''''''''''''''''''''''''''''''''''''

( ) sin (2 1)( )n nE r t E n t kr ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''

The current density can be written as:

0

( )nn

j j r t

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

2 22( )( )n T

n

E va n

E

3 2 22 12 2

1( ) ( ) exp

( 1) 2 2z

E E n zn EL

T TeN qv v dq J q q qvE n n q m m

''''''''''''''

2 2 2( ) x za n a a with

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Harmonic Generation

2 22 2( )n T

n x y

E va a

E

),,,(, naa yx

Intensity, anisotropy

( 2 ) 3T zv T m T T T T

20 4z ET T mv T

is the angle between the field and the oZ axis

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Harmonic Generation

2 2 2( ) x za n a a

21 1

20 1

( ) 1 1( ) 2 2 ( 1) 1

x

z

a n dy y xdx

a n n n xy

2

13 22 13

( cos 1 sin )exp

1 ( )n n

x y xW I W I W

x

22

2 13

cos 1 sin1 ( )

W x y xx

where In is the modified Bessel function of n-order

19

Efficiency of the Third Harmonic

is the angle between E and the anisotropic axis

is the anisotropy degree

zTTT 23

1

9 ( 10 )

4z

z

T TT T

T

2

20.1, 0.4, 1, 3, 10E

T

v

v

20

Efficiency of the Third Harmonic


2

20.4E

T

v

v

9 ( 10 ),

412

( 5 ),73

( 2 ),4

z

z

z

T T

T T

T T

21

Efficiency of the 5,7,9 Harmonic


3

10

dashed

continuous

9 ( 10 ),

4 zT T

fifth (n=2), seventh (n=3) and ninth (n=4) harmonics

22

Efficiency of the 5,7,9 Harmonic


3

dashed

continuous( 10 )zT T


( 2 )zT T

23

is the angle between E and En

Polarization of Harmonics

is the angle between the field and the oZ axis

24

Polarization of Harmonics

( ) arccos arccos ( )n

n

EEn G n

E E

''''''''''''''''''''''''''''

''''''''''''''''''''''''''''

2 21 1

3 220 1 2 13

1 ( cos 1 sin )( )

2 2 ( 1) 1 1 ( )

dy x y xG n dx

an n y x

1exp n nW I W I W

Where the function G has the form:

22

2 13

cos 1 sin1 ( )

W x y xx

with

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is the anisotropy degree

zTTT 23

1

9 ( 10 )

4z

z

T TT T

T

2

20.4, 1, 3, 10E

T

v

v

Polarization of the Third Harmonic

26


2

20.4E

T

v

v

9 ( 10 ),

412

( 5 ),73

( 2 ),4

z

z

z

T T

T T

T T

Polarization of the Third Harmonic

27


3

10

dashed

continuous

9 ( 10 ),

4 zT T


Polarization of the 5,7,9 Harmonic

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3

dashed

continuous( 10 )zT T


( 2 )zT T

Polarization of the 5,7,9 Harmonic

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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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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.

Harmonics generation in bulk semiconductors

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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.

High-order harmonic emission

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The propagation of an electromagnetic wave along a given direction z in a medium is described by the Maxwell equation

t

P

t

E

cz

Eo 2

2

2

2

22

2 1

is the polarization of the free electron gas in terms of the linear and nonlinear susceptibilities.

EEEP o .....)( 2321 where

t

PEvnej

)(

The source of the nonlinearity is the current density

The model

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The efficiency of HG or of WM at frequency , normalized to the fundamental one is given by::

21

22

v

v

E

E

I

I

oo

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 with n=1,3,5.....

34

ENERGY BAND STRUCTURE

35

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

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SCATTERING MECHANISMS IN OUR MODEL

Si

INTRAVALLEY

Acoustic Phonon Scattering [elastic and isotropic]Ionized Impurity Scattering [elastic and anisotropic; Brooks-Herring approximation]

INTERVALLEY

Longitudinal Optical Phonon Scattering [g-type inelastic and isotropic]Transverse Optical Phonon Scattering [f-type inelastic and isotropic]

GaAs and InP

INTRAVALLEY

Acoustic Phonon Scattering [elastic and isotropic]

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

Non Polar Optical Phonon Scattering [inelastic and isotropic]

(Equivalent)(Equivalent and non equivalent)

37-319o m 10n ,K 80T GHz, 200

Si InPHarmonics Generation

E

38

Harmonics GenerationSi InP

-319o m 10n ,K 80T GHz, 200

n

39

Harmonics Generation

InP

Minimum of the efficency is shifting to higher field intensity with the increasing of the field frequency !

n

40

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)

Harmonics Generation

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))

41E

Si Static Characteristic

saturation

Non-linearity

42E

InP Static Characteristic

Gunn Effect

saturation

Polar phonon emission

43

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

CONCLUSIONS

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 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.

W j E ''''''''''''' '

1 m. zarcone istituto nazionale per la fisica della materia and dipartimento di fisica e tecnologie...

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