A SUGGESTION OF NOISE EXPERIMENT FOR SHOWING BALLISTIC TRANSPORT

HAL Id: jpa-00221665

https://hal.archives-ouvertes.fr/jpa-00221665

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A SUGGESTION OF NOISE EXPERIMENT FOR SHOWING BALLISTIC TRANSPORT

P. Hesto, J. Vaissière, D. Gasquet, R. Castagné, J.-P. Nougier

To cite this version:

P. Hesto, J. Vaissière, D. Gasquet, R. Castagné, J.-P. Nougier. A SUGGESTION OF NOISE EX-

PERIMENT FOR SHOWING BALLISTIC TRANSPORT. Journal de Physique Colloques, 1981, 42

(C7), pp.C7-235-C7-241. �10.1051/jphyscol:1981728�. �jpa-00221665�

JOURNAL DE PHYSIQUE

Colloque Cl, supplément au n°10, Tome 42, octobre 1981 page C7-235

P. Hesto, J.C. Vaissiere , D. Gasquet , R. Castagne and J.P. Nougier

Institut d'Electronique Fondamentale, Laboratoire associé au C.N.B.S., LA 22 et Greco Microondes, Université Paris XI, 91405 Orsay, France

Université des Sciences et Techniques du Languedoc, Centre d'Etudes d'Electro- nique des Solides, Laboratoire associé au C.N.R.S., LA 21 et Greco Microondes, 34060 Montpellier Cedex, France

Résumé : La plupart des méthodes expérimentales tendant à mettre en évidence le régime quasi ballistique sont fondées sur la mesure de la survitesse.

Nous suggérons une expérience permettant de mesurer la variation de bruit.

Pour cela nous mettons en oeuvre une nouvelle méthode théorique de modélisa- tion du courant de bruit S\r dans la grille des FET, qui permet de montrer que Sig est relié à la température de bruit du canal actif sous la grille.

A SUGGESTION OF NOISE E X P E R I M E N T FOR SHOWING B A L L I S T I C T R A N S P O R T

Abstract : Most of the experimental methods intending to exhibit quasi bal- listic regimes involve the measurement of velocity overshoot. We suggest an experiment in which the variation of the noise would be measured. For this purpose we set up a new theoretical method for modeling the noise current S-j

in the gate of FETs, which shows that Sir is related to the noise tempe- rature of the active channel under the gate.

INTRODUCTION

In very short devices, the carriers, when flowing from one electrode to the other one, have no time to undergo enough scattering events so as to reach their stationary state velocity. The transport is then said to be ballistic or quasi ballistic. When the electric field is high, the acceleration during the free paths overcomes the deceleration during the few scattering events, so that the carriers are under velocity overshoot conditions. The d.c. characteristic of the device is then a steady regime in which every carrier is in non steady state. An important question is to estimate the effect of the velocity overshoot on the behaviour of the device, but a still more basic question is to get sure that this phenomenon actually occurs.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981728

JOURNAL DE PHYSIQUE

Plany authors t h e o r e t i c a l l y p r e d i c t e d b a l l i s t i c t r a n s p o r t i n many semiconduc- t o r s , by s o l v i n g t h e t r a n s i e n t Boltzmann equation. Flany experiments a r e now desi- gned t o b r i n g an experimental evidence o f t h i s phenomenon. Most o f them c o n s i s t i n evidencing v e l o c i t y overshoot. They use I - V c h a r a c t e r i s t i c s o f s h o r t devices [ 1 1

(although some problems may a r i s e u s i n g such a technique [ 2 I), microwave t e c h n i - ques [ 3 ] , o r o p t i c a l e x c i t a t i o n s [ 4 J [ 5 1.

The purpose o f t h i s paper i s t o suggest t h e p o s s i b i l i t y o f measuring t h e f l u c t u a t i o n s o f t h e v e l o c i t y i n s t e a d o f i t s average value, since under b a l l i s t i c behaviour, when no o r few c o l l i s i o n s occur, t h e f l u c t u a t i o n s o f t h e v e l o c i t y a r e expected t o diminish. The b a s i c i d e a i s then t o measure, i n t h e gate o f a f i e l d e f f e c t t r a n s i s t o r , t h e n o i s e c u r r e n t induced by t h e f l u c t u a t i o n s o f t h e v e l o c i t i e s o f t h e c a r r i e r s i n t h e channel : t h e gate behaves l i k e a probe and o n l y t h e f l u c - t u a t i o n s i n the a c t i v e channel, under t h e gate, a r e detected by t h i s probe, t h e

" p a r a s i t i c r e g i o n " between t h e source and t h e g a t e and between t h e gate and t h e d r a i n have no s i g n i f i c a n t e f f e c t . Thus, when t h e l e n g t h o f t h e t r a n s i s t o r i s redu- ced (submicron FET), t h e e l e c t r i c f i e l d i s h i g h underneath t h e gate, t h e c a r r i e r s a r e under quasi b a l l i s t i c c o n d i t i o n s i n t h e a c t i v e channel, so t h a t t h e n o i s e cur- r e n t o f t h e gate i s expected t o d i m i n i s h w i t h r e s p e c t t o t h e one which c o u l d be observed i n 1 ong FETs .

We s h a l l b r i e f l y r e c a l l ( s e c t i o n 2 ) t h e new method developped f o r g e t t i n g t h e impedance f i e l d , which i s p a r t i c u l a r l y w e l l s u i t e d f o r modelling t h e n o i s e o f FETs [ 7 1 ⁸ 1, then use t h i s method f o r g i v i n g a method t o determine t h e n o i s e induced i n t h e gate ( s e c t i o n 3 ) , and g i v e t h e r e s u l t s f o r a s i m p l i f i e d b u t r e a l i s - t i c model ( s e c t i o n 4).

2. DETERMINATION OF THE INPEDANCE FIELD

L e t us consider a FET (Fig. 1) where x i s along a d.c. c u r r e n t l i n e . A t abscisse x, t h e p o t e n t i a l i s V(x), and t h e c u r r e n t f l o w i n g through e q u i p o t e n t i a l surface V(x) i s I ( x ) . The d.c. c u r r e n t i s given by I ( x )

f [ ~ ( x ) ] . W r i t i n g V(x)

Vo(x) + bV(x) exp i w t and I ( x )

I o ( x ) + 6 I ( x ) exp i w t , t h e f i r s t order terms o f t h e conduction equations g i v e

A 4

where % i s a l i n e a r operator. L e t ( x , x l ) be t h e Green f u n c t i o n o f % , d e f i n e d as :

? z ( x , x ' ) = 6 ( x - x ' ) ( 2 )

where 6 ( x - x ' ) i s t h e D i r a c f u n c t i o n . Eqs. (1) and ( 2 ) g i v e :

O f course 2 ( x , x i ) may depend on a. By s e t t i n g x

L i n eq. ( 3 ) , one gets t h e v o l t a g e v a r i a t i o n &V(L) between t h e source and the d r a i n , produced by a c u r r e n t

6 1 ( x i ) i n t h e s l i c e dx' : t h e r a t i o Z ( x

L, x l ) i s t h e impedance f i e l d [7 ][ 8 1.

Source Gate Du*,

F i g u r e 1 : Example o f s t r u c t u r e o f a FET ( t h e x x+dx

f i g u r e shows a JG.FET, b u t a s i m i l a r represen-

t a t i o n c o u l d have been made f o r a MOS FET).

Figure 2 : Charge and current balance i n a s l i c e o f the channel.

3. NOISE CURRENT INDUCED I N THE GATE

Due t o t h e noise sources i n s i d e t h e channel, t h e p o t e n t i a l V(x,t) o f t h e e q u i p o t e n t i a l surface c r o s s i n g t h e c u r r e n t l i n e a t abscissa x f l u c t u a t e s around i t s average ( = d.c.) value Vo(x), thus producing a f l u c t u a t i o n o f the extension o f t h e space charge and hence o f t h e channel o f area A(x,t). The v a r i a t i o n o f t h e charge i n the space charge r e g i o n produces a v a r i a t i o n o f t h e c u r r e n t i n t h e gate, through t h e c a p a c i t i v e c o u p l i n g between t h e gate and the channel.

F o r g e t t i n g t h i s c u r r e n t , one w r i t e s t h e conservation o f t h e c u r r e n t through a s l i c e o f t h e channel o f thickness dx (see F i g . 2) :

d i ( x , t )

a d Q ( x , t ) / a t where dQ(x,t)

q ND [Ao(x) - ~ ( x , t ) ] d x ( 4 )

Since A(x,t)

A [ ~ ( x , t ) ] , eq. ( 4 ) gives :

C7-238 JOURNAL DE PHYSIQUE

where the s u b s c r i p t s 0 and x mean the d.c. value a t p o i n t x. When s e t t i n g V(x,t) = Vo(x) + 6V(x) exp i w t , thus d i ( x , t ) = d i o ( x ) + 6 d i ( x ) exp i w t , one gets ( i n p r a c t i c e the leakage d.c. c u r r e n t d i o ( x ) can be neglected) :

The a.c. gate c u r r e n t i s then obtained by i n t e g r a t i n g eq. ( 6 ) over t h e t o t a l l e n g t h of t h e FET :

t h a t i s , t a k i n g i n t o account eq. ( 3 ) :

6 i G = - iwq ND (: 1; (dA/dV)o,x E ( x , x l ) 61 ( x ' ) dx d x l (8)

For g e t t i n g t h e s p e c t r a l d e n s i t y S o f t h e f l u c t u a t i o n o f t h e g a t e c u r r e n t i G

i n t h e bandwidth A f [ 8 1, one m u l t i p l i e s 6 i G by i t s complex conjugate 6 i E and : 6 i G . ^{6 i G} *

'iG ^Af , so t h a t eq. ( 8 ) g i v e s :

L L L L dA

si = u2q2n,2 1 \ I ^(m) ~ ( x , x l ) ( g i if?U,ut)~l(X~,U~)dX d x l du dul (9)

G o o o o 0

0

Now t h e n o i s e sources a t two d i f f e r e n t p o i n t s a r e uncorrelated, and t h e n o i s e source K ( x t ) i s d e f i n e d [ 8 ] as

s,(x1,u') = A ( x l ) K ( x l ) 6 ( x 1 - u ' ) (10)

I n t e g r a t i n g eq. (9) over u ' gives, t a k i n g i n t o account eq. (10) :

Eq. (11) can be w r i t t e n :

Comparing eq. (12) w i t h the usual one dimensional expressions f o r t h e n o i s e C 8 1,

h

i t can be seen t h a t Z ( x l ) i s a " t r a n s f e r f i e l d " , analogous t o t h e impedance f i e l d o f Shockley e t a!. [ 9 1 .

Remark : t h e cross c o r r e l a t i o n between t h e gate c u r r e n t and t h e d r a i n v o l t a g e

can be o b t a i n e d by m u l t i p l y i n g 6 i G , g i v e n by eq. (a), by 6VD obtained by *

s e t t i n g x

L in eq. ( 3 ) , which gives, taking into account eq. (10) :

S vD

- i q N ( ^($1 ~ ( x , x ) ~ ~ L , x ' ) K(xl) A ( x l ) dx dx' (14)

0

,x

Two important r e s u l t s can be derived from eqs. (12) and (13) :

a) The noise i s proportional t o w2, resulting on the capacitive coupling, so t h a t i t can be expected t o prevail a t high frequency, over the other noise sour- ces, which areconstant o r decreasing with increasing frequency. However t h i s i s very d i f f i c u l t t o achieve experimentally [ 10 1 due t o the high impedance of the gate c i r c u i t .

b) Outside the a c t i v e channel, the area i s almost constant, so t h a t

A

(dA/dV)o,x - 0. Thus the contribution t o Z ( x l ) , and hence t o the noise, i s mainly due t o the a c t i v e channel, where the e l e c t r i c f i e l d i s high. This will be even b e t t e r evidenced on the model below.

4. EXAMPLE

This theory i s applied i n the present section t o a simplified but s t i l l r e a l i s t i c model of FETs (JG.FETs - MOS FETs), valid up t o the pinch off voltage of the s t a t i c c h a r a c t e r i s t i c s . We suppose t h a t the charge accumulation in the channel i s negligible, t h a t i s n = N D (indeed n l i e s in the range 0.9 N D - 1.1 N D up t o the pinch off voltage [ 11 1). The velocity-field r e l a t i o n will be taken, i n Si channel FETs, as : v(E)

- wo ^{E / ( 1} - E/Ec). The minus signs r e s u l t from the con- ventions we adopt : I i s s e t positive, although E i s negative since V D > O f o r a n type channel. The drain current i s then given by

I

- q ND wo E A / ( l - E/Ec) where E = - ^{dV/dx (14)}

and t h e function E ( x , x l ) i s 1 ⁷ 1, where the subscript 0 means the d.c. value and H(x - x ' ) i s the s t e p function :

2 ( x , x t )

- - H(x - x ' ) Eq. (15) carried into eq. (13) gives

Now - Eo(x) dx

dVo(x) and (dA/dV),,, dVo(x) = dA(x), so t h a t f i n a l l y :

A

(16)

This expression carried i n t o eq. (12) gives the noise current in the gate :

JOURNAL

PHYSIQUE

The noise can be expressed u s i n g t h e n o i s e temperature T n ( x l ) i n s t e a d of t h e noise source K ( x ' ), since i t was very r e c e n t l y shown [ 12 1 [ 8 t h a t b o t h were r e l a t e d though :

K ( x l )

- 4 kB T n ( x l ) Re [ 6 j / 6 ~ 1 (18)

where kB i s t h e Boltzmann constant, j the c u r r e n t d e n s i t y and the minus s i g n f o l l o w s from t h e convention o f signs adopted f o r E and I. Eq. (14), c a r r i e d i n t o

Eq. (19) c l e a r l y shows t h a t the c o n t r i b u t i o n o f t h e regions o u t s i d e the a c t i v e channel i s n e g l i g i b l e s i n c e then A ( x l ) = A ( L ) . The noise c u r r e n t i n t h e gate i s then d i r e c t l y r e l a t e d t o t h e n o i s e temperature i n the channel, thus c o n f i r m i n g t h a t t h e gate p l a y s the r o l e o f a probe useful f o r t e s t i n g t h e noise underneath.

Remark : t h e cross c o r r e l a t i o n between iG and VD can a l s o be achieved : eq. ( 1 5 ) c a r r i e d i n t o eq. (14) g i v e s :

iwq N

D d A

vD

- 7 ^Eo(L) [ d x K ( x r ) A ( x z ) [ - ] ^{d ( Eo(x)}

0

0,x

Carrying eqs. (14) and (18) i n t o (20), one gets : 2 2

4 i w q ND kB Eo(L)

S -

2 ! 1 T n ( x 1 ) A ( x L ) [ A ( ~ ) - A ( x l ) ] d x ' (21)

i~ v~

T h i s expression i s very s i m i l a r t o eq. (19), except t h a t t h e i n t e g r a l i n - volves A(L) - A ( x l ) i n s t e a d o f [A(L) - A ( x ' ) 1 2 , which means t h a t the e f f e c t s of t h e non a c t i v e regions, o u t s i d e t h e channel, i s l e s s damped f o r Si v than f o r

G D

Si G , which i s normal s i n c e VD takes those r e g i o n s i n t o account.

5. CONCLUSION

We presented i n t h i s paper a new method f o r determining t h e noise c u r r e n t S . i n t h e gate so as the cross c o r r e l a t i o h S i G vD. Q u i t e simple formulas were ob-

'G

t a i n e d on r e a l i s t i c models, and can be used f o r s i m u l a t i n g t h e noise o f a c t u a l de-

vices and comparying i t w i t h experimental r e s u l t s . I t was shown t h a t t h e gdte o f a

FET behaves l i k e a probe, so t h a t t h e n o i s e c u r r e n t depends on t h e n o i s e tempera-

t u r e o f t h e a c t i v e channel.

In submicron FETs, t h e c a r r i e r s undergo ball i s t i c t r a n s p o r t i n t h e a c t i v e channel, so t h a t one would expect t h e diminution o f t h e noise s i n c e l e s s s c a t t e - r i n g events occur. This e f f e c t could l i k e l y be evidenced attemperature low enough f o r t h e Fermi energy l e v e l be about t h e impurity energy l e v e l : f o r example i n n-Si a t 77 K , p a r t of t h e donors a r e ionized, and t h e e l e c t r i c f i e l d enhances impurity i o n i z a t i o n through impact i o n i z a t i o n o r Poole Frenkel e f f e c t , thus producing a high n o i s e temperature [ 13 ] , and one might t h i n k t h a t i t s diminution, produced by quasi b a l l i s t i c t r a n s p o r t , could be observable.

REFERENCES

L.F. EASTMAN, Experimental s t u d i e s of b a l l i s t i c e l e c t r o n t r a n s p o r t i n semiconductors, Journal de Physique, Proc. 3rd I n t . Conf. on Hot C a r r i e r s in Semiconductors, Montpell i e r ( F r a n c e ) , 7-10 J u l y 1981.

J.R. BARKER, D . K . FERRY and H . L . GRUBIN, IEEE El. Dev. L e t t . EDL 1, 209(1980) S . J . A L L E N , D.C. TSUI, F. DEROSA, K.K. THORNBER and B . A . WILSON, D i r e c t measurement of v e l o c i t y overshoot by hot e l e c t r o n , submillimeter wave con- d u c t i v i t y i n Si i n v e r s i o n l a y e r s , Journal de Physique, Proc. 3rd I n t . Conf.

on Hot C a r r i e r s i n Semiconductors, PZontpellier (France), 7-10 J u l y 1981.

S. LAVAL, C. BRU, R. CASTAGNE and C. ARNODO, Gallium Arsenide and Related Compounds 1980, Ed. H.W. T h i m , I n s t i t u t e of Physics, Conf. S e r i e s no 56, London, p. 171, 1980.

C.V. SHANK, R . L . FORK and B.I. GREENE, Picosecond nonequilibrium c a r r i e r t r a n s p o r t i n GaAs, Journal de Physique, Proc. 3rd I n t . Conf. on Hot Carriers i n Semiconductors, Montpellier ( F r a n c e ) , 7-10 J u l y 1981.

J . SHAH, i n v e s t i g a t i o n o f hot e l e c t r o n r e l a x a t i o n with picosecond l a s e r p u l s e s , Journal de Physique, Proc. 3rd I n t . Conf. on Hot C a r r i e r s i n Semi- conductors, Montpellier (France), 7-10 J u l y 1981.

J.P. NOUGIER, J.C. VAISSIERE and D. GASQUET, Proc. 6 t h I n t . Conf. on Noise i n Physical Systems (Washington, USA, 6-10 April 1981), t o be published.

J.P. NOUGIER, Modeling t h e noise of devices working under hot c a r r i e r con- d i t i o n s , i n v i t e d paper, Proc. 6th I n t . Conf. on Noise i n Physical Systems

(Washington, USA, 6-10 April 1981), t o be published.

W. SHOCKLEY, J.A. COPELAND and R.P. JANES, i n Quantum Theory o f Atoms, Mo- l e c u l e s and t h e S o l i d S t a t e , P. Lowdin e d . , Acad. Press, New York, 1966.

G. LECOY, D. RIGAUD and D. SODINI, S o l i d S t a t e Electron. 17, 11 (1974).

D. SODINI, ThPse de Doctorat Ps Sciences, Montpellier ( 1 9 7 9 ) ( a v a i l a b l e on r e q u e s t )

J.P. NOUGIER, J.C. VAISSIERE, D. GASQUET and A. FIOATADID, Noise sources of hot c a r r i e r s i n space charge regime, J . Appl. Phys., t o be published.

A. VAN DER ZIEL, R. JINDAL, S.K. KIM, H . PARK and J.P. NOUGIER, S o l . S t a t e

Electron. 22, 177 (1979).