A SEMIPARTICULAR SIMULATION TO STUDY HIGH FREQUENCY BEHAVIOUR OF HOT CARRIERS IN IMPATT DEVICES

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A SEMIPARTICULAR SIMULATION TO STUDY HIGH FREQUENCY BEHAVIOUR OF HOT

CARRIERS IN IMPATT DEVICES

D. Lippens, E. Constant

To cite this version:

D. Lippens, E. Constant. A SEMIPARTICULAR SIMULATION TO STUDY HIGH FREQUENCY

BEHAVIOUR OF HOT CARRIERS IN IMPATT DEVICES. Journal de Physique Colloques, 1981,

42 (C7), pp.C7-207-C7-212. �10.1051/jphyscol:1981724�. �jpa-00221661�

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JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-207

Centre Hyper fréquence s et semi-conducteurs, Laboratoire associé au C.N.R.S.

n°287, Bâtiment P4, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France

RESUME : Les auteurs présentent une simulation semi-parti cul aire de la géné- ration et du transit des porteurs chauds dans les structures semi-conduc- trices unidimensionnelles soumises à une tension alternative superposée à la tension statique d'avalanche. Ce modèle est basé sur le formalisme proposé par HOCKNEY étendu aux cas :

- où l'ensemble des paramètres de transport sont fonction de l'énergie comme cela avait été fait dans le modèle de P Ô N E Q 7

- où une description microscopique de l'ionisation à l'aide des coefficients d'ionisation, également fonctions de l'énergie, est incluse.

Une telle simulation permet de tenir compte de tous les effets connus de dynamique non-stationnaire des porteurs chauds (survitesse, diffusion, re- laxation).

A SEMI PARTICULAR SIMULATION T O STUDY HIGH F R E Q U E N C Y B E H A V I O U R O F H O T C A R R I E R S IN IMPATT D E V I C E S

D. Lippens and E. Constant

ABSTRACT : A serai-particular simulation of generation and transit of hot car- riers in unidimensional semi-conductor structure, subjected to an ac voltage superposed on the breakdown dc voltage, is presented. The model is based upon the HOCKNEY formalism with two significant improvements :

- all the parameters of transport are assumed to be functions of the carriers energy as it was done in PONE model [l"]

- a microscopic description of the ionizing event by means of ionization coef- ficients, functions also of the energy, is included.

Therefore, the model takes into account all known nonstationaryhot electron dynamic effects(velocity-overshoot, diffusion, relaxation).

INTRODUCTION : Host of the published works about high frequencies properties of IflPATT diodes are extrapolations of simulations carried out at lower frequencies.

To take into account new phenomena which may occur at millimeter wave fre- quencies, it's necessary to solve the well known set of partial differential equa- tions that govern charge transport, including the relaxations equations [ 2 ] .

This macroscopic approach appears to be a powerfull tool for studying semi- conductors devices but fails when one try to incorporate in an easy manner the ef- fect of the "black space"! 3 ] and the overshoot velocity of the generated carriers.

In this paper, we present the first developpments of a semi-particular simu- lation of the generation and transit of hot carriers subjected to an ac-field surim- posed on a strong dc field.

In the first part, the main features of the model are discussed : the hot carriers dynamics and the simulation of the ionizing collision.

In part two, we present some results obtained for a PIN diode and an IMPATT diode and we show that such particle model could be successful for the

modeling of millemetric IHPATT diodes.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981724

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JOURNAL DE PHYSIQUE

F i e: Electronic mobility versus energy.

-

Fiq. 2 : Diffusion coefficient o f electrons versus enerqy.

-

F i g . 3 : Enerqy relaxation time o f electrons versus enerqv --

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I - SIMULATION OF W T CARRIERS IN IMPATT DEVICES 1-1- ELECTRONS AND HOLES TRANSPORT

Two p o s s i b i l i t i e s e x i s t f o r h i g h f r e q u e n c i e s IMPATT m o d e l i z a t i o n e i t h e r a b i p o l a r u n i d i m e n s i o n a l MONTE CARL0 s i m u l a t i o n w h i c h g i v e e x a c t s o l u t i o n s b u t does n o t a l l o w d e v i c e o p t i m i z a t i o n o r a s i m p l i f i e d model such as d i f f u s e model proposed i n 1974 b y HOCKNEY

[

4

1

The two main f e a t u r e s o f such s i m p l i f i e d procedure a r e :

-

t h e m o t i o n o f b o t h t y p e s o f p a r t i c l e s ( h o l e s and e 1 e c t r o n s ) I s s t u d i e d o n l y i n u n i - dimensional g e o m e t r i c a l space.

-

t h e s c a t t e r i n g mechanisms and e l e c t r i c f i e l d a c t i n g on t h e c a r r i e r s a r e t r e a t e d u s i n g m o b i l i t y d i f f u s i v i t y a p p r o x i m a t i o n s and phenomenological c o n s e r v a t i o n equations.

However, t o t a k e i n t o a c c o u n t t h e n o n s t a t i o n a r y dynamic e f f e c t s , t h e t r a n s p o r t p a r a - meters a r e assumed t o be i n s t a n t a n e o u s f u n c t i o n s o f t h e energy o f c a r r i e r s .

The s t o c h a s t i c m o t i o n o f c a r r i e r ( h o l e and e l e c t r o n ) c o n s i s t o f a s i n g l e d i s - placement due t o t h e e l e c t r i c f i e l d and a random w a l k w i t h a r o o t mean square r a d i a l displacement equal t o

a

(where D i s t h e d i f f u s i o n c o e f f i c i e n t , A t t h e t i m e s t e p ) .

The p o s i t i o n o f t h e e l e c t r o n o r h o l e i a t t i m e j. A t i s g i v e n by t h e r e l a t i o n :

k = n f o r e l e c t r o n s ; k = p f o r h o l e s

where u(ckiJ-l) and ~ ( ~ ~ ~ j - l ) a r e r e s p e c t i v e l y t h e m o b i l i t y and t h e d i f f u s i o n c o e f - f i c i e n t o f t h e p a r t i c l e i a t t i m e j-1

.

E i s t h e i n t e r n a l f i e l d .

The energy o f each p a r t i c l e i s c a l c u l a t e d u s i n g t h e energy r e l a x a t i o n t i m e

where s o i s t h e r m a l energy and T , t h e energy r e l a x a t i o n t i m e .

The f u n c t i o n s p k ( s )

,

D k ( € )

,

T ~ ( E ) can be determined u s i n g t h e r e s u l t s o b t a i n e d by MONTE CARL0 s i m u l a t i o n r ( i n s t e a d y s t a t e c o n d i t i o n .

An e x a m ~ l e o f t h e r e s u l t s we o b t a i n e d , a r e o i v e n i n t h e f i g u r e s 1

,

2

, 3

i n c l u d i n g e x t r a p o l a t i o n s f o r e n e r g i e s l o w e r t h a n E,

.

---

r There a r e o n l y few r e f e r e n c e s about t h e h o t h o l e dynamic i n Ga As. Consequently, f o r t h i s p r e l i m i n a r y c a l c u l a t i o n , we have assumed, f o r an energy s u p e r i o r t o 1 eV, a h o l e dynamic s i m i l a r t o t h o s e o f e l e c t r o n s . O b v i o u s l y , f o r l o w e r e n e r g i e s t h e dynamic p r o p e r t i e s a r e o b t a i n e d from l i t t e r a t u r e d a t a .

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JOURNAL DE PHYSIQUE

4 Electronic ionization coefficient versus energy.

F ~ G : v o l t a g e and p a r t i c l e c u r r e n t versus t i n e f o r a P+IX+ d i o d e .

-

auk

^%^t^6.3&/c/ro.'

r , 00-'R)

4

0

2E

^4.5 ⁹ ^{t ( p 4}

fi7.

6 V o l t a q e and o a r t i c l e . c u r r e n t f o r a P+NN+ d i o a e .

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1-2- THE IONIZING EVENT

The necessary i n f o r m a t i o n t o account t h e c a r r i e r s generation has been obtained from t h e knowledge o f t h e i o n i z a t i o n r a t e s an and a

.

These macroscopic c o e f f i c i e n t s a r e a p p l i e d i n d i v i d u a l l y f o r each p a r t i c l e . P

Usually, t h e c o e f f i c i e n t s an and a are considered as f u n c t i o n s o f t h e e l e c - t r i c f i e l d . However, a t m i l l i m e t e r wave frequencies, t h e energy, responsable o f the P i o n i z a t i o n process, does n o t f o l l o w i n s t a n t l y t h e v a r i a t i o n s o f t h e e l e c t r i c f i e l d and a g r e a t improvement can be obtained assuming t h a t these c o e f f i c i e n t s a r e i n s t a n - taneous f u n c t i o n s of t h e i n d i v i d u a l energy o f each c a r r i e r .

I n f i g u r e 4, we g i v e an example o f t h e v a r i a t i o n o f an versus energy which has been obtained from MONTE CARL0 s i m u l a t i o n c a r r i e d o u t i n u n i f o r m steady s t a t e c o n d i t i o n .

From t h e knowledge o f t h e energy o f each p a r t i c l e , and consequently o f t h e microscopic i o n i z a t i o n r a t e an(€)

,

i t ' s p o s s i b l e t o c a l c u l a t e t h e p r o b a b i l i t y f o r one c a r r i e r t o c r e a t e an e l e c t r o n - h o l e couple over i t s displacement Ax. P r a c t i c a l l y s u i t a b l e random t r i a l s can be achieved t o f i n d o u t w h e t h e r a n i o n i z i n g c o l l i s i o n has occured d u r i n g Ax.

The s t a t e o f t h e i n i t i a l and c r e a t e d c a r r i e r s have been a r b i t r a r i l y chosen a t a n o t too low value E = 0.6 eV i n o r d e r t o l i m i t t h e overshoot v e l o c i t y . I f necessary, o t h e r e n e r g e t i c r e p a r t i t i o n c o u l d be considered i n t h e model w i t h o u t main com- p l i c a t i o n .

I 1

-

APPLICATION OF THE METHOD TO THE MODELING OF UNI-DIMENSIONAL DEVICE I n t h i s s e c t i o n , we present some r e s u l t s o f t h e method a p p l i e d t o a P'IN' diode and t o a P'NN' diode u n i f o r m e l y dopped (ND = 6 10 16 ) .

The c o n t a c t s a r e assumed b l o c k i n g . The i n t e r n a l f i e l d i s c a l c u l a t e d from t h e POISSON'S e q u a t i o n t a k i n g i n t o account i m p u r i t i e s , holes and e l e c t r o n s concentra- t i o n s .

The study i s c a r r i e d o u t when t h e diode i s subjected t o an ac v o l t a g e and biased by a dc c u r r e n t obtained by means o f a dc feedback.

11-1- P+IN+ DIODE

The f i r s t r e s u l t has been obtained f o r a 0.2 micron, w i d t h diode, sub- m i t t e d t o a sinusoSda1 s i g n a l a t t h e frequency o f ' l l O

GHz,

w i t h a modulation deepth o f 28 %.

In

t h e f i g u r e 5, we g i v e the e v o l u t i o n versus time r e s p e c t i v e l y o f t h e v o l - tage accross the avalanche r e g i o n ( a ) and t h e p a r t i c l e c u r r e n t .

This i l l u s t r a t e s c l e a r l y t h e generation process: when t h e v o l t a g e exceeds breakdown, t h e number o f c a r r i e r s grows u n t i l t h e v o l t a g e swing below t h e avalanche v o l t a g e t a k i n g o f f the process.

The phase s h i f t c l o s e t o 90° between t h e v o l t a g e and t h e c u r r e n t , i n h e r e n t

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C7-212 JOURNAL DE PHYSIQUE

i n t h e avalanche process, may be observed.

11-2- P+NN+ DIODE

I n t h e f i g u r e 6 t h e r e s u l t s obtained f o r a 0.5 micron diode (ND =

16 3

6 10 /cm ) , i n t h e same c o n d i t i o n o f frequency a r e presented.

One can note t h a t , due t o t h e t r a n s i t o f c a r r i e r s , t h e d u r a t i o n o f t h e p a r - t i c l e c u r r e n t i s l o n g e r than i n t h e PIN case. The mean e f f i c i e n c y i s o f t h e o r d e r o f 4 %.

F i n a l l y , l e t us mention t h a t t h i s s i m u l a t i o n performed w i t h 1000 t o 2000 p a r - t i c l e s takes about 3 minuts f o r a p e r i o d o f 9 1 0 - l 2 s and a step time o f 5 10-l4 s.

CONLUSION :\A new method o f s i m u l a t i o n o f h i g h frequency behaviour o f h o t c a r r i e r s i n I m p a t t devices, u s i n g a s e m i - p a r t i c u l a r model., have been presented.

We have shown, t h a t such a method appears t o be successful t o describe t h e avalanche and t r a n s i t process and should be a promising t o o l t o study t h e funda- mental frequency l i m i t a t i o n o f I m p a t t diodes.

REFERENCES :

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J.F. P ~ N E , R. CASTAGWE, J.P. COURAT, C. ARNODO

2D P a r t i c l e modeling o f submicron GaAs FET near Pinch o f f , accepted i n IEEE E l e c t r o n i c Devices

[21 H.J. KAFKA, K. HESS

A c a r r i e r temperature model s i m u l a t i o n o f a d o u b l e - d r i f t I m p a t t diode IEE Transactions on e l e c t r o n devices Vol.ED-28 No 7 J u l y 1981

[31 Y. OKUTO, C.R. CROWEL

I o n i z a t i o n c o e f f i c i e n t s i n semi-conductors : a non l o c a l i z e d p r o p e r t y Physical Review

B y

Vol. 10

,

number 10 (1974)

[ 4 ] R.W. HOCKNEY, R.A. WARRINER, M. REISER

Two dimensional p a r t i c l e models i n semi-conductor device a n a l y s i s E l e c t r o n i c s l e t t e r s

,

Vol. 10, no 24

,

?p. 484-485

[51

PI.

SHUR

Two dimensional p a r t i c l e models i n semi-conductor device a n a l y s i s E l e c t r o n i c s l e t t e r s , Vol

.

10, no 24, pp. 484-485