Possible Generation of Transient THz Electronic Drift Effects in a Semiconductor by a High Electric Field

(1)

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Possible Generation of Transient THz Electronic Drift Effects in a Semiconductor by a High Electric Field

V. Filip, C. Plavitu

To cite this version:

V. Filip, C. Plavitu. Possible Generation of Transient THz Electronic Drift Effects in a Semicon- ductor by a High Electric Field. Journal de Physique I, EDP Sciences, 1996, 6 (3), pp.403-412.

�10.1051/jp1:1996165�. �jpa-00247193�

(2)

J.

Phys.

I France 6

(1996)

403-412 MARCH 1996, PAGE 403

PossibJe Generation of Iransient THz Electronic Drift Effiects in

a

Semiconductor by

_a

High Electric Field

V.M.

Filip (*)

^and ^C.N. ^Plavitu

University

of Bucharest, Faculty of

Physics,

P-O- Box MG-il Bucharest

Magurele

76900, Romania

(Received

9 December 1994, revised ii

September

1995,

accepted

27 November

1995)

PACS.72.10.Bg

General formulation of transport theory PACS.72.20.Ht

High-field

and nonlmear effects

PACS.72.30.+q High-frequency

effects:

plasma effects)

Abstract.

Using

the Boltzmann equation formalism for _a

simple

semiconductor

model,

the

transient drift

velocity

of the electrons _is

computed

when _a

high

uniform electric field is

suddenly

turned _on.

Rapid

oscillations

superimposed

_on _an overshoot behaviour _are obtained. It is showed that this behaviour results from the combined action of the field and of the

electron-phonon

interaction.

Résumé. En utilisant le formalisme de l'équation de Boltzmann _pour _un modèle simple de

semiconducteur, _on _a calculé la vitesse de "drift" transitoire des électrons

quand

le

système

est

soumis à l'action

brusque

d'un

champ électrique

foi-t et uniforme. On _a obtenu des oscillations

rapides superposées

à

une évolution du type ~'overshoot" et _on _a démontré

qu'elles

résultent de l'action combinée du

champ

extérieur et des interactions

électron-phonon.

1. Introduction

It has

recently

become

possible

to fabricate small size electronic devices- The short distances involved lead _to

high

electric fields and the electronic

transport

^takes

place

_over _very short time scales.

Consequently,

the

study

of the transient effects became crucial for the

understanding

of the

operation

of such devices and in the effort to increase their

speed.

The first steps were

mainly theoretical,

based both _on Monte Carlo simulations and solutions of the Boltzmann

transport equations iii.

Some of these

investigations [2,

3]

predicted velocity

overshoot

phenomena

_as _an _important transient effect to be used in faster devices.

Recently, optical techniques

have been

developed

in order to achieve the

appropriate spatial

and

temporal

resolutions for the

experimental study

of the

problem [4-7].

These methods

opened

the way ^to direct

investigation

of the transient behaviour which _now _seems to be _more

complicated

^than ^that

theoretically predicted là, 6].

In the

present

paper, a

simple

theoretical method

(based

_on the Boltzmann

equation)

will be used in order to

explam

the overshoot of the drift

velocity.

We will also _prove the

possible

(*)

Author for

correspondence je-mail: fillum©lgnm.sfos.ro)

©

Les

Éditions

_de

Physique

1996

(3)

existence of _an

oscillatory shape

of the

peak

in the THz

domain,

which is influenced

by

the

types

^of

scattering

mechanisms of the electrons.

In

practice

^this

oscillatory

behaviour may lead to

electromagnetic pulses (which

in

fact,

have

already

been used in _some

investigations

of the transient

phenomena [6]),

and which could contain useful information about the electronic interactions.

2. The

Transport Equation

Let _us suppose that _a

sample

of

n-type nondegenerate

semiconductor is

subject

to an extemal uniform electric field. The

general

form of the transport

equation

^for ^electrons ^will be:

)

=

Éf Ùf, _Ii)

where

Éf

=

~~~ E

~~ (2)

is the field operator

describing

the action of the

applied

electric field E-

Ù

_is _the

_general

collision

operator

for the electrons

(which

for

nondegenerate

semiconductors _can be considered

as hnear and

homogeneous),

and

f(k,t)

is the

position mdependent

electronic distribution function for which the normalisation condition

~~ / _{f(k, t)dk}

=

1, (V t) (3)

4~

must hold [8]. ^The

integral

is taken _over the Brillouin _zone and

1

is the volume of the

primitive

cell of the lattice.

In order to

simplify

the discussion _we shall consider E _as

suddenly applied

at t ₌ 0 and constant thereafter- The initial form of the distribution function will then be the

equihbrium form, which,

for

nondegenerate semiconductors,

is well

approximated by

_[8]:

~2

fl

^3/2 ~2

~°~~~

2Vo ~

~~~

~~~~~~

~~~

where _a _is the _constant of the

(cubic)

lattice and

~ aÎÎÎ~ÎT

^~~~

Here _m _is the effective _mass of the electrons and

kB

_is the Boltzmann constant.

Since the Brillouin _zone has _an inversion

symmetry

^around ^its

centre,

^the ^function

f(k, t)

can

always

be written _as _a _sum of its

symmetric

and

antisymmetric

parts [9]:

f+lk,t)

=

lflk,t)

+

fl-k,t)1 16)

f-jk,t)

=

jfjk,t) fj-k,t)j. ii)

The drift

velocity

defined

by

_[10]

v~jt)

t

) jvjk)fjk,t)dk ^j8)

(4)

N°3 POSSIBLE GENERATION OF THz DRIFT EFFECTS 405

should then be

produced only by f-(k,t)

since the _group

velocity

of the electrons is

given by _[10]:

v(k)

=

~

E(k), (9)

and since the electronic _energy band _can be considered _as

isotropic:

E(k)

=

E(k). (10)

One then obtains:

vd(t)

=

j / ^~~~~~ _{f-(k, t)dk} ₍₁₁₎

4~ h ôk

which

implies

that _an

equation

for

f-(k, t)

is needed in order _to calculate the drift.

Writing (1)

for k and for

(-k), adding

and

subtracting

the obtained

equations,

_one arrives at the

following equivalent system:

~j/

=

É f+ Ù f- (12)

~~+

=

É f- Ù f+ (13)

In

obtaining this,

_one must use the fact that the action of

É

or

Ù

on a function _reverses and

respectively

_preserves its

symmetry.

This _is obvious for

É

_from _the _definition

(2)

and for it _can be demonstrated for every

scattering

mechanism of interest in

nondegenerate

semiconductors

[11].

Since the field is constant and since k and t are

independent

variables

[12],

^the ^time ^derivative

commutes with

É

_and

Ù _operators,

_and _thus _the

following equation

can be obtained from

(12)

and

(13):

~ÎÎ

"

f- Î ~~f+

_~~~~

As it will become dear from the _more detailed

analysis

_in the _next

section,

the contribution of the last term in

(14)

is very small. For

example,

if the interaction is

elastic,

it vanishes.

Therefore, neglecting

this term in

(14),

trie

remaining

contains

only f-,

and its form _is close to an

equation

for a

damped

_wave

propagating

in the

reciprocal

_space in the field direction.

This is dearer if _one _uses _in

(14)

the

projection

^k ^{of k in} ^this ^direction:

ÎÎ

=

llEl~ (ii _ô~lt

lis)

If the

apphed

electric field is

strong enough

^and if _some suitable

boundary

and initial conditions _are satisfied

by f-,

then _an

oscillatory

solution _is

possible

for

(15), leading

to _a similar

behaviour for the transient drift

velocity iii ).

The

boundary

condition _may be derived from

(3)

if _one takes into account the relation

/Ù f(k, t)dk

=

0, (V t), (16)

which expresses the fact that the

Ù

_operator

can

only

redistribute the conduction electrons among their bounded _set of _states. This statement _can be verified

directly

_on the

general

form of

[11].

(5)

By integrating

both members of

(1)

_over the Brillouin _zone, and

by taking advantage

of

(3)

and

(16),

and usmg the Gauss

Theorem,

_one finds that

/f(k, t)(E dE)

=

0, (V t), (17)

z

where 1 is the

boundary

of the Brillouin _zone and dE is its out~vard oriented surface element.

Since the

integral

for

f+(k, t)

vanishes _on the

symmetric boundary L,

it follows from

(17)

that

/f-(k, t)(E dE)

=

0, (V t), (18)

z

which represents ^the

boundary

condition for the function

f-.

As for the initial conditions _one may use trie

original

form

(4)

of the distribution and obtain

f~jk,t

=

o)

= o.

j19)

Then. from

(12)

it follows that

~j~ ^(k,t

⁼

⁰⁾

⁼

^Éfo(k). ⁽²⁰⁾

3. The Solution of the

Equation (14)

and the Drift

Velocity

of the Electrons

In order to obtain numerical

results,

one must use certain

approximations. First,

_we shall suppose a

simple parabolic

band structure for the electrons

:

E(k)

/~2

=

-k~. (21)

This will avoid unnecessary

complications.

We then notice the axial

symmetry

^of our

problem

around the field direction and _use _a

Legendre polynomials expansion

for the _cos à

dependence

of

f(k, t) (à being

the

angle

between k and

E).

From this _we shall retain

only

the first _two terms~ which is _a

frequently adopted

approximation [10,11,

_13]

f(k, t)

E£

çJo(k, t)

_{+ çJi}

(k, t)

_cos à

(22)

It follows

by (6)

and

(7),

that

f+(k, t)

E£

çJo(k, t) (23)

and

f-(k, t)

à

çJi(k, t)

cosô

(24)

As for trie collision

operator Ù,

_we shall consider the electron-acoustical

phonon

and electron-

polar optical phonon

to be trie dominant interactions and write

Ù

as a sum of

Ùac

and

Ùpo

[10,11,13]:

Ùa~

f

=

~~~~

~

Ea~k f(k, t) /

sm

ô'dô' / _{f(k, à',} _çJ', _t)dçJ' ₍₂₅₎

~21

4~

~ ~~

~~°~ ÎÎÎ° Î _' ^~~~~~

~~

k' (~~~~'~

~~

~

~~'

k2

~Î ^~~

~~~~

_~°~^~~ ^~

Î _~k(É~~~~'~

_~~^~

~~'

k2

~k( ^~~

~~~~

(6)

The

symbols

are defined _as follows:

°(X) j i _(~?)

~~

~~~ là) ⁾

^~

~~~~

~° Î Ù'

~~~~

Ea~

=

j)Î ₍₃₀₎

[e[a

h pu~

Epo

₌

~~(~°°

(£ic~

£ic~), (31)

where

El

is trie deformation

potential constant,

_p is trie

crystal density,

_us and _~oo _are the

mean sound

velocity

and the _mean

optical phonons angular frequency,

and _sc~, _s~ _are the

high frequency

and static dielectric

permitivities-

^In

(25)

and

(26)

the

nondegeneracy

of the electronic

system

was used.

Furthermore,

in

(25)

the electron acoustical

phonon

interaction

was considered _as elastic.

Using (23)

and

(24)

in

(25)

_one _can condude that

Ùa~f+

= 0

(32)

Ùa~f-

=

~(~~ Ea~kçJi(k,t)

_cos à

(33)

4~ h

In order to

obtain,

_in _a

simple form,

the action of

Ùpo

on

f+,

the functions _were

expanded

in _power series of

k/ko

for k <

ko,

and of

ko/k

for k >

ko,

and

only

the terms up ^to the first order _were retained:

Ùpo f+

E£

)~ Epo

[nov7o

(k, t) (1

+

no)v7o(ko, t)]8(ko k) (34)

ko Ùpo f-

E£

)~ Epo ()° ^8(ko ^k)

⁺ ^~°

( _~~~8(k ^o)j

v7i

(k, t)

_cos à

(35)

o

Using (24), (32-35)

in the

equation (14),

and

making

the

substitution, çJ(k,t)

=

VlçJi(k,t), ₍₃₆₎

we obtain

~

Çl +

2@E~(k)

^~

()E) (kçJ'l'

^~ +

çJA~(k)

₌

0, (37)

5 4

where

E~(k)

=

j _(())~

Ea~k

+

2Epo ~~)°8(ko ^k)

⁺ ^~°

(~~~8(k ^o))j

,

(38)

~ o

A~(k)

=

())~ )°Epo (())~ ^Ea~k

+

)°Epoj ^8(ko ^k), ⁽³⁹⁾

o ^~ o

and

by

and çJ' we mean the time and k derivatives

respectively.

The last term in

(37) originates

from the term

ÉÙ f+

of

(14)

where

çJ[

was substituted from

(12),

if _we _use

(23)

and

(24).

(7)

The

boundary

and the initial conditions for _çJ _can be deduced from

(18), (19)

and

(20).

Substitution of

(24)

in

(18), assuming

^that the Brillouin _zone _can be

approximated by

_a

sphere

of radius

2~la, gives

çJ

~~,

_t

=

0, IV t). (40)

a

We also have from

(36)

that

çJ(0, t)

₌

0,

_(V

t), (41)

if _we notice that _çJi _is finite _at k ₌ 0

(in

fact it vanishes

by (7)

and

(24) _).

Then

by (19), (24)

and

(36)

_one _can deduce

çJ(k,

t

=

0)

₌

0, (42)

and

by (20); (24)

and

(36)

it follows that

#jk,t

=

o)

₌

'~j~ô~(j~~ ^j43)

The structure of the

equation (37)

and the

boundary

conditions

(40)

and

(41)

_suggest _a

solution

expanded

in a series of Bessel functions of

3/2

index

[14]

m

V~(k,t)

_"

~j Cn(t)J3/2(Ànk), (44)

n=1

where

a

(45)

~" 2~~"'

~on

being

_an

increasing

infinite sequence of

positive

numbers

generated by

the

equation

J3/2(~on)

= 0.

(46)

These numbers _can be well

approximated by (n +1/2)~,

for

high

_n values.

Using (44)

in

(47)

and the

orthogonality property

of the

J3/2(Ànk)

^functions

[14],

one obtains

~dÎ

ô~w

+

Ann'

_C"' ~

-i

f _2Enn'Cn'

+

~°~

~(~on

+ L°"

:~

~

~(~a~

+ ~d~

~ n =

1, 2,.

,

(47)

where

Enn,

₌

(Î)

^~

^~~~~ E~(k)kJ3/2(Ànk)J3/2(Àn,k)dk, (48)

2~

o

Ann> ₌

~

()~ /~~~~A~(k)kJ~/2(Ànk)J~/2(Àn,k)dk, ⁽⁴⁹⁾

3 _e

o

and

~~

ÎÎÎ~'

_~~~~

It _can _now be

verified, by

direct numerical

computations, that,

for fields

greater

^thon kV

/cm,

the elements Ann> are

negligible quantities

_in

(47).

This proves the assertion made

in Section 2

regarding

the weak contribution of the last term of

(14).

(8)

Evaluations of

(48)

show that the

diagonal

elements _are

greater (by

about _one order of

magnitude)

than the

nondiagonal

_ones. This

suggests

an iterative

approach

to the

coupled equations (47).

However _we shall content

ourselves,in

the present paper, with the zeroth order

approximation,1-e-

_we shall consider the matrix

Enn,

_as

diagonal.

So,

if _we define

En

₌ ^~~

~a

^~

~(~dn

+

~d[~ )Enn, (51)

ôn

=

~~~~En, (52)

and

~"

~Î _Î'

~~~~

we obtain from

(47)

the

following independent damped

oscillator

type equations:

én

+

lé( ~d(r()cn

₊

26nén

= 0 _n

=

1, 2,..

,

(54)

whose solutions may be

easily

constructed in order to meet the initial conditions

(42)

and

(43).

~5/2 ~2

cn(t)

=

-~LdÎ/~

lLdn ⁺

Ldi~)

exP

~-j) _exPl-ont)

~

1n~~"

~~~'

~~~~

By

_means of the relations

(11), (24), (36), (44)

and

(55),

the electron drift

velocity along

the field _direction becomes:

~~~~~

ÎÎÎ _~~~~ ^Î~

_~"~~

x

[exp(rn~dEt) exp( -rn~dEt)] (56)

2rn

We notice first that the result

(56) gives

real

values,

_even if _some of the _rn numbers _are

imagmary. Then,

the _presence of the factor

exp(-~d(/(4fl))

_ensures the absolute convergence of the series. We shall denote

by

N the number of the

significant

terms of

(56),

which is _a

decreasing

"function" of

temperature.

^As an

example,

for Insb at T

= 77

K,

N _must be close

to 400 to _ensure _a

good precision.

Computations

of

(51)

and

(48)

showed that the _sequence

En

has _a

complicated n-dependence,

but this is in

quite

a narrow interval around _a certain _mean value. It follo~&rs

that,

for values of E that _are not too

high,

the first few _ternis _in

(56) (their

number

depends

_on

E,

_as _one _can

see from

(53)

have _an

overdamped

evolution and the others _are

"oscillatorally damped"-

It is also evident from

(52)

that the time extent of the transient response

(56)

is controlled

by

the

scattering

mechanisms

through

the factors

exp(-ont).

4. Discussion of the Results

The transient drift

velocity

_given

by (56)

_was

computed

for Insb

using

data from the literature

[10,13].

We _are

only

interested in the

general problem

of the transient response, ^not in _a

(9)

8

T=77 K

_

~ E=20 kV/cm

Î

~

4

CL E=10 kV/cm

>~ 2

0

O.O 0.2 0.4 0.6 0.8 1-Ù

t

[PS)

Fig.

l. Transient drift velocity of electrons in Insb calculated at T ₌ 77 K for two values of the field

strength.

particular

semiconductor and for this _reason _we have not included _some detailed features such

as

nonparabohcity

and other

complications

of the band structure.

The main results _are

presented

in

Figure 1,

where the time variation of the drift

velocity

is shown for two values of the external field

strength,

at 77 Ii- A

high frequency

oscillation

superimposed

_on _a

general

overshoot

shape

_may be observed in both _cases and its

frequency

increases with the electric field

applied.

From the formal

point

of

view,

this behaviour _can be

explained

_as follows. The response

is

expressed

_as _a

superposition

of _a certain number of

significant

time

dependent

terms

(N).

This number _is determined

by

the initial conditions and

by

the

temperature,

as was shown _in Section 3. There _are

generally

three types ^of _response _among ^these terms: the

overdamped

kind

(mainly

the first

kind),

the

damped slowly oscillating

kind

(for

intermediate _n values in

(56) ),

and the

damped high frequency oscillating

kind

(for

_n values closed to

N).

The first two

types

^determine ^the ^overall

shape

of the transient response and the last _one is

responsible

for the

high frequency

oscillations. As _can be _seen from

(53),

the

greater

the

applied field,

the

higher

the

frequencies

of these last terms allowed in the

significant

set.

From _a

physical point

of _view, such oscillations could _appear to be the result of the impact of the external field _on the electronic

system.

^This

produces,

first of

all,

_a

crowding

of the

electrons into states of

high

_wave vector

along

the field direction. The

scattenng

^from ^such

states _is _more

effective, deflecting

them out of the stream, without

important

relative variations of the energy. This results m a

drop

of the drift

velocity.

If the external field is

high enough,

it will rebuild the

streaming

motion before the end of the transient response

(whose

overall time extent is controlled

mainly by

the

collisions)

and _new deflections become

possible.

These

phenomena

could

repeat

_many times before

stationary

drift is

reached,

when _a _compromise

between the field and the

scattering

is established.

The

simple

result obtained

previously

indicates that the

scattering

mechanisms have _a crucial role in the transient response.

Indeed,

if _one computes

ud(t)

from

(56) by setting

the collisions

zero

(Ea~

=

Epo

=

0),

then the _curve

la)

of

Figure

2 is

obtained,

_i e. a constant acceleration

drift

(as

is

intuitively dear)-

In

addition,

if _one smoothes out the collision

operators ii

_e. if

they

_are

supposed

_as

indepen-

dent of

k,

this leads _to _a constant

En

_sequence and

finally

to _a result

presented

_as the _curve

16)

(10)

12

~

^~

(Cl

~

~ 4

~

E=10 kV/cm ^~~~

° T=77 K

Ù-Ù 0.2 0.4 0.6 0.8

t

(ps)

Fig.

2. Transient drift

velocity

of electrons _m Insb calculated for E

= 10

kV/cm

and T

= 77 K for

three

particular

_cases:

lai

collisionless

(the

values of the

velocity

_on the _curve should be

multiplied by ten); (b)

smoothed collision operator;

ici

relative enhancement of

electron-polar optical phonon

interactions.

4

T=150 K

3 E=20 kV/cm

jf

E

~

2

= E=10 kV/cm

>~ l

0

O.O 0.2 0.4 0.6 0.8

t

(PS)

Fig.

3. Transient drift

velocity

of electrons in Insb calculated at T

= lso K for the _same field

strengths

_as those for

Figure

1.

of

Figure

2. Here

(56)

_was calculated with

En equal

to its _mean

value,

for the

example depicted

in

Figure

1- A

slight

overshoot _may be

observed,

but _no oscillation _appears. This

suggests

that the

scattering might operate

a selection among the electrons

allowing only

_some k values to be

substantially

^deflected. When the collisions _are smoothed out

they

cannot

distinguish

among the k states and _are

equally

effective for all of

them,

and from the _very

beginning.

This leads to a

rapid

balance with the external field and _so to a

stationary

drift.

From _a formal

point

of _view, the action of the collisions

corresponds

to _a selection

operated by

the

En

_sequence _among the terms of

(56)

it influences both the

damping

and the

amplitude

of each term- The absence of such _a selection leads to

nonoscillatory superpositions

of the terms

(11)

as in the _cases

la)

and

(b)

of

Figure

2.

In order to examine the relative

importance

of the collision operators ⁱⁿ ^the ^transient re-

sponse, the drift

velocity

_was calculated for

Ea~

diminished

by

_a factor of three and

Epo

increased

by

_a factor of

fifty (the electron-polar optical phonon coupling

is

relatively

weak in Insb

[10]).

The enhancement of the transient oscillations

(Fig. 2c)

shows that the electron-

acoustical

phonon coupling

has _an inhibitive action _on the electronic deflections because of its almost fluid-like

damping

effect.

Finally

the transient drift _was calculated for _a

higher temperature (Fig. 3).

As _one would

expect,

given that the

electron-phonon

interactions _are _more

important

^and ^their

damping

action is _more

pronounced,

the overshoots become smaller and the oscillations weaker.

In

conclusion,

the

simple

model used shows that when _a

high

electric field _is

suddenly apphed

on a

semiconductor,

THz transient oscillations could appear as a combined result of the impact of the field _on the electronic system ^and of the selective

scattering

_processes. The effect should be enhanced at low temperatures.

References

[ii

Constant

E-,

in Hot-Electron

Transport

in

Semiconductors, Topics

in

Applied Physics,

Vol.

58,

^L-

Reggiani

Ed.

(Springer, Berlin, 1985)

_p. 227.

[2] Ruch

J-G-,

IEEE Trans. Electron Devices ED-14

(1972)

652.

[3]

Maloney

T.J. and

Frey J.,

J.

Appi. Phys.

48

(1977)

781.

[4]

Meyer Ii.,

Pessot

M.,

Mourou

G.,

Grodin R. and Chamoun

S., Appi- Phys.

^Lett. 53

(1988)

2254.

[5] Sha

W.,

Norris

T.B.,

Schaff W.J. and

Meyer K-E-, Phys.

Reu. Lett. 67

(1991)

2553.

[6] Son

J.,

Sha

W.,

Kim

J.,

Notas

T.B.,

Whitaker J-F- and Nlourou

G-A-, Appi. Phys.

Lett.

63

(1993)

923.

[7] Grann

E-D-,

Sheih

S-J-,

Tsen

K-T-, Sankey O.F., Giinçer S-E-, Ferry D.K.,

Salvador A., Botcharev A. and Morkoc

H., Phys.

Reu. B

51(1995)

1631.

[8]

Seeger K.,

Semiconductor

Physics, Spnnger

Senes _in Solid-State

Sciences,

liol. 40

(Spnnger, Berlin, 1985).

[9] Carelman

T.,

Problèmes

Mathématiques

dans la Théorie

Cinétique

des Gaz

(Almqvist

&

Wiksells

Boktryckeri AB, Upsala, 1957).

[10]

Reggiani L.,

_in Hot-Electron

Transport

in

Semiconductors, Topics

_in

Applied Physics,

Vol.

58,

L.

Reggiani

Ed.

(Springer, Berlin, 1985) p.7.

[Il] Haug

A., Theoretical Sohd State

Physics,

Vol. 2

(Pergamon Press, Oxford, 1972).

[12]

1(rieger

J-B- and Iafrate

G-J-, Phys.

Reu. B 35

(1987)

9644.

[13] ^Conwell

E.àI.,

Sand State

Physics, Suppl-

9

(1967).

[14] Nikiforov A. and 0uvarov

V., Éléments

_{de la} _Théorie _des _Fonctions