My Magister

Modelling 3D of the straining by Avalanche In the structures in PN junction to GaAs Semi - Insulator Presenting Deep Centers - Traps. SIM3D Software. Mr.A.RESFA * . Pr Mrs Brahimi.R.Menezla. Pr Mrs Bourzig Y.Smahi. Laboratory of modelling and conception of the circuits electronic, department of electronics. University Djillali Liabès. BP89, Sidi Bel Abbes 22000. ALGERIA.

Summarized: In this work, we simulated, the behavior of the ionization phenomenon by impact " straining by avalanche " of the PN junctions, subject to an inverse polarization. While taking into account the trapping model in a stationary regime the P+N structures, PN+, and P+N+ using like material of basis the III-V compounds and mainly the GaAs semi - insulator in which the deep centers are in important densities. The survey retailed of the physical and electric behavior of the semiconductors and notably to really surround the influence of the deep center presence on the characteristic I(V) current - tension, that requires the secondary size calculation as the electrostatic electric, potential field, integrals of ionizations, the density of the states traps, the current of diffusion of the minority in the regions (1) and (3), the current of generation thermal region (2), and the current of flight in surface, so the tension of the straining..etc. We introduced these quantities thereafter in pretender SIM 3D, who is conceived for the survey of the components to weak geometry of conception permitting to determine in the volume of a structure, the distributions of potential and of free carrier densities according to a given polarization. The system have been solved by a coupled Newton, Newmann algorithm. The resolution method consist of the linearisation of the transport equations by the finite difference method. The actual version use a combined method involving at the same time Gummel' algorithm and Newton's algorithm, in order to enable a better convergence and consequently to improve the computation time in 3 dimensions (3D). This simulator gives the spatial distributions of the electrocstatic potentiel and density of carriers in the simulation domain under biais. I. INTRODUCTION We present in this work a program of threedimensional simulation, studying the phenomenon of ionization by impact " Effect Of avalanche " of the PN junctions to basis of GaAs, while taking into account the trapping model clean to the

semiconductors to relaxation. The GaAs made the object these last years of a particular interest because he proves to be a component presenting interesting potentialities in many domains (logical, oscillators, the development of the high range components, it is the objective of the state-of-the-art technology to high precision). The advantages offered by this component are, on the one hand, the absence of effect bound in the DX centers in the non dopey layers and, on the other hand the good uniformity of the tension doorstep that makes of him a candidate privileged for the conception of built-in circuits. To get the elevated tensions of straining close to the value of a plane junction, the components of power were and are achieved again in part in technology Mesa. However, this one remained reliable and conducted little to an elevated dismissal rate. It is therefore important, to remain competitive in the international context, to be able to use the Planar configuration, of setting in more attractive œuvre to manufacture the components of power. Another interest of this technology resides in the possibility to associate several devices on the same crystal; she/it permits to consider the realization of the circuits integrated of power more conveniently therefore. On the other hand, it presents the inconvenience to form cylindrical or spherical junction sides around the openings of masks breves that entail the existence from a region to elevated field and therefore a reduction important of holding in tension of the peripheries. This work has for objective, the determination of the characteristic I(VBDV) of the inverse current in a PN junction in GaAs basis, subject to an inverse polarization while specifying the parameters that influence the straining of the diodes. II.2. SYSTEMS OF BASIS EQUATIONS INTERVENING IN THE PHYSICAL MODEL

The systems of equations that translate the mechanism of transportation, in stationary regime, and that we will solve by numeric simulation, are presented like follows: Equation of Poisson

(

)

div ε. gradψ = e [ n − p + N A + nr − pt ] with :

nr = N R

τ pre . n + τ nre . plr τ pre. (n+nlr ) + τ nre. ( p+ plr )

(II.1)

(II.2)

τ nte . p + τ pre . nlt τ pte . (n+nlt ) + τ nte . ( p+ plt )

Jn = e.µ n.n.E + e.Dn. grad n

(II.6)

another atom pulls a new electron himself accéléré[7], etc... This phenomenon named effect of avalanche is generally unverifiable and conducted to a destruction of the junction. • When the junction is polarized in inverse by a tension of Vi module, the largueur of the zone of space load increases as Vi whereas tension to his boundary-marks augmente[2,4,9] as Vi. He results some that the electric field inside this zone increases. In the case of the junction abrupt dissymétrique p+n for example, the following expressions show that the field is maximum to the metallurgic junction, in x=0, and vary as :

Jp = e.µ p.p.E − e.Dp. grad p

(II.7)

E0 = eNd Wn (I.1) with Wn = (

pt = NT

(II.3)

Équations de continuité :

1 .divJn = Ur − Ut e

(II.4)

1. divJp = − Ur − Ut e

(II.5)

with :

E = − gradψ

(II.62) ;

D=

µ.kT e

ε

What gives for Vi >> Vd

(II.1)

and

E0 = (2eNd/ ε )1/2

U nr = U pr = U r = U nt = U pt = Ut =

n.p − ni 2 τ pre. (n+nlr ) + τ nre. ( p+ plr )

n.p − ni2 τ pte. (n+nlt ) + τ nte. ( p+ plt )

2ε (Vd +Vi ) )1/2 (III.1) eNd

(II.8)

(II.9)

These are these equations that will be solved in a numeric way for structures in permanent working (independent of the time). II.2.3.A. NEUTRAL REGION OF THE DEVICE

In the neutral regions, one takes ρ =0. It implies the solution particular Ψ =cste in the resolution of the equation of LAPLACE ∆Ψ =0. II.2.3.B. DESERTED REGION

In the deserted completely regions, one has p=0 and n=0. The position of the limit of the zone destitute is gotten while forcing the potential to remain in the applied tension domain.

Vi

(III.2)

The Vi tension cannot increase indefinitely because a r limit exists to the value of the field electric E0 . Indeed, when the electric field increases the electric r r strength F =−eE0 exercised on the bound electrons increases. When this strength is superior to the strength of link of the valence electrons on the cores, these electrons are freed, the material becomes driver and the Vi tension doesn't increase anymore. In other words the maximum electric field that one can establish in a crystal semiconductor is the one that provokes the direct excitation of an electron bound of the valence strip toward a state free of the strip conduction, it is To Say the ionization of the material. In silicon this maximum field is the order of 106 V/cm, and of the order of 107 V/cm in the GaAs.

II.4.3.C. INSULATING

III.2. Straining by effect zéner (greatly dopey material)

In the insulators, while supposing that they don't have a load, the equation of LAPLACE gives: ∆Ψ(x, y, z) =0 (II.10) If the insulators have a density of load fix in volume, Q, one takes the equation of POISON:

It is now about a non destructive straining because the electric field is not sufficient To accelerate sufficiently the electrons pulled to the links. So that there is straining, it is sufficient that the internal field is superior to his value criticizes (Ecri) that is the order of 2.107 V/cm. In these conditions and for an abrupt junction.

∆Ψ(x, y, z) = −

Q ε

(II.11)

III. Behavior of the diode in the two senses III.1. Straining by effect of avalanche (material little dopey).

In inverse, he reigns an intense electric field to the level of the junction. This field can become sufficient to pull an electron then to a link to accelerate him and to procure him a sufficient energy so that this electron at the time of a shock with

III.3.LE Phenomenon of ionization by impact

The ionization by impact or by shock appears in a material for intense electric fields. Indeed, an electron that drifts in a strong under the effect of an electric field, win the energy under kinetic shape and transmits it progressively to crystal by the numerous shock slant that he/it does with the phonons of the network.

.This process assures the thermal dissipation of the potential energy lost by the electrons. However, if the electric field is sufficiently intense, certain electrons can, during one free flight, to acquire an energy as their impact on an atom of the crystalline network, succeeds to the rupture of a link and the creation of an electron - hole pair. .This process, illustrated on the III.1 face, can become cumulative and can drive to the phenomenon of avalanche. This effect being especially important than the gap of the material is small (the energy of doorstep of the ionization by shock is roughly of 3/2 Eg) [9], it is to the heart of our survey aiming to optimize the structure of the HEMT transistors on InP substratum).

III-1 face: Phenomenon of ionization by shock. III.3.3. CONDITION OF AVALANCHE

So that the electron can make a shock ionizing it is necessary that his kinetic energy acquired during the free course means is superior to EG. Q.EAV .L >EG. EAV =EG/Q.L. (III.3) EAV = amplitude of the necessary electric field for the avalanche. L = free course means. This EG: Height of the Strip Forbidden of the used semiconductor-B.I, value of the electric field is gotten for a tension applied on the metallurgic junction. . The condition of avalanche is given therefore by M ∞ the equal integral to the unit. The tension of straining is based on the determination of the ionization integral to a (1) dimension, his/her/its value is defined like being the inverse polarization tension for which this integral is equal to 1. TO 3D, it is the inverse tension that corresponds to a maximum ionization integral equals to 1 calculated in the different directions of the electric field lines. III. 4. IONIZATION BY IMPACTE

The zone of load of space of a junction in inverse is browsed by some carriers responsible for the flight current. These carriers are accelerated very strongly by the reigning intense electric field in this region. If this field is raised sufficiently (enters 105 and 106 Vcm-1) the carriers acquire in the crossing of

the ZCE enough kinetic energy to transfer to an electron hired in a link of valence, electron of the BV an energy of ionization capable to make bring up it in the strip of conduction, creating a pair electron thus hole. There is Multiplication of the Carriers therefore. The following faces give a ballistic representation " 11 and the other in the energizing diagram 12 " of the phenomenon [9]. . This integral depends on rates of ionizations of the electrons and holes αn and αp, it is necessary to know the relation between the rate of ionization and "amplitude of the electric field". • Two parameters characterize the phenomenon of ionization by impact: a) The coefficients of ionization [10] αn(ζ) and αp(ζ) cm-1 who represents the number of pairs created by centimeter of course of an electron or a hole and that are quickly functions increasing with the electric field (in If, an" Si, αn ≈103cm-1 for ζ=2.105 V cm-1. b) The coefficient of multiplication [10]M that represent the report of the inverse current of the Ir junction in presence of informed multiplication of Iro flight in the absence of multiplication: M = Ir (IV.1) Iro One can join these two parameters thanks to the reasoning simple next one (that supposes the coefficients of ionization of the electrons and the equal holes, what is not actually): a carrier crossing a ZCE of W thickness creates N W

pairs:

N = ∫ α (ξ (x ))dx

(IV.2)

0

For every created pair, the electron and the hole are swept quickly by the electric field each in an opposite direction what comes back, according to an already made reasoning, to the crossing of part of leaves from it of the ZCE by an electron or a hole. So the even N pair created by the initial particle provoke N2 ionizations and so forth. He results some that: Ir =Iro(1+N+N2 + …) – Iro

1 1− N

(IV.3)

W

of or

1-

1 = N = α (ξ (x ))dx ∫0 M

(V.4)

There is avalanche when every carrier creates, create on his turn a pair electron hole, what drives to N = 1 and by following M = ∞. W

∫α(ξ (x ))dx =1 0

(IV.5)

IV.1.1. CALCULATION OF THE IONIZATION INTEGRALS

To characterize the phenomenon of avalanche [10], one defines the coefficients of ionization αn and αp and the coefficients of multiplication Mn. and Mp.

To determine the coefficients of multiplication in a junction subject p-n, a strong inverse polarization tension, we write the evolution of the densities of Jn current and Jp (Face:II.13)

IV.1.2. THE COEFFICIENTS OF IONIZATION

αn and αp The coefficients of ionization represent the number likely of ionizing collisions that an incidental carrier undergoes, on an unit of length of course. The analyses of the ionization mechanism [8,9,10] permit to express the coefficients αn and αp according to the local electric field and various physical parameters as the energy of the optic phonon, the energy of ionization, the free course means. Such expressions are often heavy and unusable. Thus, most authors use empiric formulas deducted of the experience. We will keep the formulas generally used to determine the coefficients of ionization bp αn= an exp- bn (IV.6) and αp= ap exp (IV.7) E E Where E is the electric field and an, bn, ap, bp of the constants. These expressions can be considered like simplifications of the theoretical laws. The values of the constants proposed by various authors [18,19,20] (chapter II). In our simulation, we worked with the coefficients -R .VAN OVERSTRAETENS [20] are given in the following picture: A.G.

C.A.LEE

[18]

R.VAN OVERSTRAE

CHYNOWETH

TEN [19] [20]

Ap(cm-1)

6,71x105

Bp(V/cm)

1,693x106

3,26x106

2.036x106

An(cm-1)

7,03x105

3,80x106

7,03x105

Bn(V/cm)

1,231x106

1,75x106

1,23x106

Table II.1.

2,25x107

II.13 face: Density of current in a junction p-n polarized in inverse. IV.1.4. COEFFICIENT OF MULTIPLICATION OF THE ELECTRONS

Mn=

1 (IV.A)  xp  1− ∫αn exp− ∫ (αn −αp )dx'dx xn  x  xp

IV.1.5.COEFFICIENT OF MULTIPLICATION OF THE HOLES

Mp =

1

(IV.B)

  1− ∫αp exp+ ∫ (αn −αp )dx'dx xn  xn  xp

x

These expressions are demonstrated in the theses of URGELL[21], LEGUERRE [23], LETURCQ [24] and BONNAUD [5], and the publication of MILLER [25,26]. IV.1.6. THE INTEGRALS OF IONIZATIONS

1,582x106

The integrals of In ionizations and Ip can be

The coefficients of ionization

IV.1.3. THE COEFFICIENTS OF MULTIPLICATION MN AND MP .

The coefficients of multiplication of the carriers represent the report of the resulting current of the avalanche phenomenon to the crossing of the ionization zone on the incidental current. Because of the difference between the coefficients year and ap, one defines MN AND MP two coefficients of multiplication for the electrons and for the holes.

calculated from the coefficients of multiplications and we have:

I

n

=1− 1

M

(IV.C) n

et

I

p

=1− 1

M

(IV.D) p

 xp  n exp n p dx ' ( ) = − − α α α  dx (IV.E) I n xn∫ ∫  x  xp

 x  = p exp α − ∫(αp −αn )dx'dx I p xn∫  xn  xp

(IV.F)

According to SIEVE OVERSTRAETEN and al[17][20], one calculates the integral of the electrons to determine the tension of straining of one junction pn+ and the integral of the holes for a junction p+n. The infinite multiplication coefficient is the criteria of straining that we will use for our structures p+n, from

where the integral of ionization of the holes will be taken equal to 1. Important note: Concerning the geometry of our structure is like next one; In our case, it is about one interface of a diode pn can come closer then as to have a cylindrical shape along a right side and a spherical to a corner of an oblong model. We didn't take account on the order of the junction depth, xj. of the ray of curvature xj=0 that is there equal zero in our case, it is about a plane Profile of the field electric " Strong multiplication " under inverse polarization.

diode V.1. SIMULATION 3D OF THE STRUCTURE: P+N, PN+ and P+N+. V.1.1. POTENTIAL ELECTROSTATIC AND ELECTRIC FIELD

The electric field generated by a tension of polarization presents an intensity and a specific direction in every point in the ZCE. However it is important to find all d ' first the potential electrostatic, the interest of the potential resides in the fact that the value of the intensity of the electric field in a point drifts directly of the variation of the potential. The electric field caracteristics, current-tension, and the carriers densities are derived by iterative method of the transport's equations. The regime of balance thermodynamic

V.1 face: Profile of the distribution of the potential to the thermodynamic balance of a P+N diode in GaAS (Plan XOY, Z=0.8µm) Survey under polarization

V.2 face: Distribution of the lines of field of a P+N diode. The variation of the potential electrostatic represented here over (V.2 face), after having solved the equation of applied Fish on this structure " three-dimensional resolution 3D ". 1) The electric field is a vector, characterized in every point of the domain by a vector E(x, of it, z) with a sense and an intensity. In a threedimensional reference mark, he/it is marked by his/its three components scalar Ex(xes, of it, z), Ey(x, there ,z), Ez (x, of it, z). 2) The équipotentielles is generally given by closed lines possibly closing endlessly se (V.2 face). They include the loads and are perpendicular to the lines of field. 3) The V.2 face represents the electrostatic équipotentielles of a P+N diode to GaAs semi - insulator, while using the program of simulation implémenter in our software SIM3D. Where: NA = 4.7.1019cm-3, ND = 4.32.101cm-3, NT=2.32.1012cm-2, thickness of film=250nm, l=350nm, Vpol_inv = 5volts, VBDV=21.61 Volt ,WZCE=251nm.

•

•

•

•

•

The ZCE due to the metallurgic contact of which, his width varies inversely proportional to the concentration of dopage of the layer integrated. This zone is therefore more important since it presents an intensity of considerable electric field is of the density of state of the centers traps condensed to the surface of the metallurgic diode. The spacing of the cristallites of GaAs between them, act strongly on the phenomenon of trapping of the free carriers by the states traps that are going to disrupt the electric field evolution.

V.1.2. INTEGRAL OF IONISATION/COURANT INVERSE/TENSION OF Straining

The different parameters put in game in the model, and the effect of some capable to influence on the phenomenon of avalanche of the PN junction in general, well on play some on the parameter of dopage of the two regions (1) and (3). For the P+N structures, the variation of the ionization integral according to the inverse polarization tension as well as the tension of straining deducted for this kind of structures is represented in the (V.5 face), the integral of ionization is the order of 9,80.10-1 for the holes and 9.17.10-1 for the electrons in the whole structure. . The density of state [0.45E15 - 3E15] is inversely proportional to the tension of straining that limited [17 - 21.61Volts], it is due to the thickness of the Movie of the substratum of GaAs that is [350nm](V.3 face).

V.5 face: Variation of the inverse current according to the tension of straining & the inverse tension of the P+N diode.

Face V.5A-B :Variation of the inverse current according to the tension of straining & the inverse tension of the P+N+ diode.

V.4 face: Variation of the inverse current according to the tension of straining of the P+N diode. V.6 face: Variation of the inverse current according to the tension of straining & the inverse tension of the PN+ diode.

. Concerning calculates it of the The integral Of ionization increases with the increase of the inverse tension. What we note the face (V.6 face), that the tension of straining that corresponds to tension for which the equal ionization integral to 1. . Holding in tension is limited between 13.59 Volts & 17.21 Volts of a PN junction achieved to basis of silicon polycristallin [27] and for the P+N diodes some GaAS is the order 14.02 Volts & 21.61Volts in our survey. . The introduction of traps or deep centers in a semiconductor drives to a complexity in the interpretation of the transportation mechanisms and simultaneously to a big diversity in the shape of the run - tension features of a component. . A center participating in a mechanism of recombination of the Shochley-Read[18 type] is characterized by four new parameters capable to vary one of the other independently: n1t (and p1t) (cm -3) that are functions of the position in energy of the level in the forbidden strip and (s-1)parameters. . Bound to the sections efficient of capture for the electrons and the holes and finally the Nt density (cm -3)du centers himself. . The results of simulation acquired our P+N structure to basis of GaAs, we permitted ourselves to pull the following findings: . The existence of the centers traps to an important influence on the determination of the semiconductor's nature, according to the density of the center traps (he is in this case of P type). . The structure doesn't present any real variations that according to the perpendicular direction to the plan of interface. . With the secondary size introduction in pretender SIM 3D, the analysis of the transportation mechanisms in our P+NS structures, PN+, and P+N+ to basis of GaAs, permitted us to arrive to the following observations: . Under applied polarization reverses, and since there is an injection important of holes in the critical region, the tendency to the global neutrality of the space load requires a growth of the free electron density (in relation to the thermodynamic balance). . This growth entails an increase of the generation current and because it is in sense opposite informed total inverse, a local increase of the electric field results from it. . The comparison between the pace of the electric field with the one found by bidimensional simulation, showed that the mechanisms of transportation through a N+ contact in presence of deep centers in the forbidden strip, doesn't obey the classic model of the P-N junction of Shockley type even though with the

presence of the phenomena of generation recominaison. . Such An evolution of the temperature, limit the performance of the junctions strongly especially in the materials that present some centers traps deep materials to large forbidden strip. The side the more weakly dopey is the one that determines, according to the value of Jf current of flight. He results some that, to minimize the currents of flight, the concentrations of dopes, to the two sides of the junction, must be maximized and must be (Na >Nd). . The parameters fixes( 1) or variables(2), some three structures are noted stables(1 previously) and instables(2) in a middle simulation fork. Then We see therefore, that when the dopage of the ND substratum, for the " structure(2) " decreases, the tension of straining increases, That will imply a lateral extension of the bigger space load zone, it will be inversely proportional in the reciprocal case structures (1). . Indeed, when the dopage increases, the lateral extension of the zone of space load is slightly decreasing for the two first structure, and narrower for the greatly dopey diodes and therefore, it won't especially be necessary! remote the range of simulation of every parameter to reach the optimum of holding in tension of straining. It serves to prevent the important gaps capable from emerging between some values what could compromise the calculations . As we said that the variation of the straining tension decreases when the doping foulness concentration increases, and the density of states NT traps becomes inversely proportional to the BREAK DOWN Voltage. . If, one to admit the reciprocal case, it doesn't mean that this decrease of the break down voltage to admit weak values of the density of states NT traps, he is limited enters of [2.03E15-1.59E16] for the first case and [1.28.1015 - 2.3216]cm-2 concerning the second case. The fall of the tension holding is of 21.61 volts until 15.55 volts is joined inversely proportional with the density of state traps NT. · The thickness of movie of the substratum of GaAs that is going to do a change of the resistance of the material in the region (02) zone of desertion of the load carriers, for it one to admit 250E-9 for the first case, 258 E-9 for the second case and 245 E-9, the minority carrier life span so their thermal speed acts very quickly under the electric field effect so the effect of ionization by impacte. · Our survey on these structures (P+N, PN+ and P+N+), being only structures of test for other ulterior studies to understand the phenomena of transportation of the semiconductors submitted to an inverse polarization really, so the phenomenon of ionization by impact and the thermal generation process, with

the hold in account of the geometric effects in these structures in presence of elevated deep centers that presumes a straining premature of this kind of diodes. · Finally, and as perspectives, this work will be an impulse bearer to give to others of freed, thereafter to the different structure simulation 3D to basis of other types of devices that remain neutral tracks to this stage. The improvement of the pretender SIM 3D version, permitted us to create a tool of the simulation in a several senses.

[10] Viviane Boisson, Thèse de Doctorat du 23 Avril 1985 L’école Centrale De Lyon. ‘Étude de la Géométrie Optimale Des Périphéries Des Jonction Planar. [11] W.SHOCKLEY-Read. Czech –J.Phys.,vol.B11,p81-104 (1961)

REFERENCES :

[1] H.Mathieu, Physique des semiconducteurs et des composants électroniques, Masson, Paris, 1996. ISBN: 2-225-85124-7. [2] CHARLES KITTEL Physique de l’état Solide 7ième édition DUNOD., vol1, 7917 [3] R.Menezla, N.M.Rahmani,H.Sehil, N.Taleb, M.Le Helley, ‘Multigride methode for the resolution of coupled system of partiel equations’, Plovdiv (Bulgaria), Aug 13/17, 1995. [4] S.M.SZE Physics of Semiconductor Devices –John Wiley-2nd Edition Nez york (1981) [5] O.BONNAUD Thèse de Doctorat 3ième cycle –École Centrale de Lyon (28 septembre 1978) [6] Mui D-S-L-, Biswas D., Reed J., Demirel A-L-, Strite Set Morkog H-, Investigations of the S13N4/SilGaAs insulator-semiconductor interface with low interface trap density, Apl. Phys. Lett. 60 (1992) 2511. [7] T. P. Pearsall, "Impact ionization rates for electrons and holes in Ga0,47In0,53As", Applied Physics Letters, vol.36, N°3, 1st February 1980, pp 218-220 [8] R.VAN Overstraeten, H.De Man, ‘measurement of the ionisation rates in diffused Silicon PN junction ‘, SolidState Electronnics, N°13, pp 583-608,(1970) [9] A.Vapaille et R.Castagné, Dispositifs et circuits intégrés semiconducteurs « Physique et technologie», Dunod (1987).

[12] A.S.GROVE Physique et technologie des dispositifs à semiconducteurs Dunod – Paris (1971) [13] W.FULOP Solid-St. Electron., vol 1, p 39-43 (1967) [14] S.M.SZE, G. GIBBONS App. Phys. Letters, vol. 8. n°5, p 112-113 (1966) [15] M.S. ADLER. V.A.K. TEMPLE IEEE Trans. Electron. Devices, vol. ED-25, n°10, p 1266-1270 (1978) [16] B.J. BALIGA, J.K.GANDHI Solid-St- Electron., vol. 19, n°9, p739-744(1976). [17] F.CONTI, M.CONTI Solid-St. Electron., vol. 15, n°1, p 93-105 (1972). [18] A.G.CHYNOWETH Phys. Rev., vol. 109, n°5, 1537-1540 (1950) [19] C.A.LEE,

R.A. LOGAN, R,L. BATDORF, J.J.

KLEIMACK, W.WEIGAN Physm Rev., vol, 134, n° 3A, p 761-773 (1964) [20] R.VAN OVERSTRAETEN, H. De MAN Solid St. Electron., vol. 13, p 583-608 (1970) [21] D.M. TAYLOR, D.W. TONG J. Qppl. Phys., vol 56, n°6, p 1881-1883 (1984)

[22] J.J. URGELL Thèse d’État – Université de Toulouse (10 mai 1969) [23] J.R.LEGUERRE Thèse d’Etat –Université de Toulouse (7 juillet 1976) [24] P.LETURCQ Thèse d’État–Université de Toulouse(28 septembre 1969) Zhenhua Wang, Current-Mode Analog IC and Linearization Techniques in CMOS technologies, Hartung-Gore Verlag Konstanz. [25] S.L. MILLER Phys. Rev., vol. 99, n°4 p 1234-1241 (1955) [26] M.S. ADLER. V.A.K. TEMPLE, R.C. RUSTAY Solid-St. Electron., vol. 25, n°12, p 1179-1186 (1982). [27] M.AMRANI Modélisation Mono Et Bidimensionnelle des Caractéristiques C-V et I-V d’une jonction PN Realisée à base de silicium polycristallin Thèse de Docteur d’état, 1996-2000.