HOT ELECTRON NOISE PROPERTIES OF SEMICONDUCTORS IN THE NON-ZERO COLLISION DURATION REGIME

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HOT ELECTRON NOISE PROPERTIES OF SEMICONDUCTORS IN THE NON-ZERO

COLLISION DURATION REGIME

P. Das, D. Ferry, H. Grubin

To cite this version:

P. Das, D. Ferry, H. Grubin. HOT ELECTRON NOISE PROPERTIES OF SEMICONDUCTORS

IN THE NON-ZERO COLLISION DURATION REGIME. Journal de Physique Colloques, 1981, 42

(C7), pp.C7-227-C7-234. �10.1051/jphyscol:1981727�. �jpa-00221664�

H O T ELECTRON NOISE PROPERTIES OF SEMICONDUCTORS IN THE NON-ZERO COLLISION DURATION REGIME

P. Das

, D.K. Ferry* and H. Grubin**

Rensselaer Polytechnic Institute, Troy, NY 12181, U.S.A.

*Colorado State University, Fort Collins, CO 80523, U.S.A.

**Scientific Research Associates, Glastonbury, CT 06033, U.S.A.

Abstract.- To obtain the noise properties of devices under hot electron condi- tion, Langevin's equation is solved including the effects of energy and momen- tum relaxation time in the displaced maxwellian distribution approximation. The collision duration, which can be a significant fraction of energy or momentum relaxation time for very small structured devices, is also included. It is known that the noise power spectral density and thus the small signal diffusion co- efficient and velocity auto-correlation are related to the small signal micro- wave conductivity of semiconductors. Both the transverse and longitudinal com- ponents of these quantities are calculated including their dependence on the magnitude of collision duration. Numerical calculation using the constants of silicon has been performed and it shows the significant contribution of the non- zero collision duration.

I. INTRODUCTION

It is well-known that the microwave conductivity of semiconductors varies as a function of frequency and that this functional dependence becomes quite complicated when hot electron transport is included [1], This occurs because one must consider not only the momentum relaxation time of the carriers but also the energy relaxation time and any consequent differential repopu-

35

Partially supported by ONR, NSF and AFOSR

** On leave at the Electrical Engineering and Computer Science Department, University of California at San Diego, U.S.A.

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-227

Résumé.- Pour c a r a c t é r i s e r l e b r u i t de composants en régime d ' é l e c t r o n s chauds, on résoud l ' é q u a t i o n de Langevin en incluant l e s e f f e t s du temps de r e l a x a t i o n de l ' é n e r g i e e t du moment dans l'approximation d'une d i s t r i b u t i o n maxwellienne déplacée. On t i e n t aussi compte de l a durée de c o l l i s i o n qui peut ê t r e une fraction non négligeable du temps de r e l a x a t i o n de l ' é n e r g i e e t du moment. On s a i t que la densité s p e c t r a l e de puissance de b r u i t e t par s u i t e l e coefficient de diffusion (en p e t i t s signaux) e t la fonction d ' a u t o c o r r é l a t i o n de l a v i t e s s e sont r e l i é e s à l a conductivitê microondes des semiconducteurs (pour l e s p e t i t s signaux). Les composantes t r a n s v e r s a l e s e t longitudinales de ces grandeurs sont calculées en incluant leur dépendance de l a durée de l a c o l l i s i o n . Un calcul numérique u t i l i s a n t les constantes du s i l i c i u m a été développé e t montre l a contribution importante des durées de c o l l i s i o n non n u l l e s .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981727

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l a t i o n i n many v a l l e y systems [2-41. I n p a r t i c u l a r , enhanced c o n d u c t i v i t y a r i s e s a t h i g h frequencies due t o t h e process o f v e l o c i t y overshoot, I n small semiconductor devices, t h e time scale o f c a r r i e r t r a n s p o r t through t h e device, w i t h t h e expected h i g h f i e l d s present, i s such t h a t t h e device dynamics may w e l l be dominated by t h e t r a n s i e n t response c h a r a c t e r i s t i c s o f t h e

c a r r i e r v e l o c i t y and d i s t r i b u t i o n f u n c t i o n [5]. However, i f t h i s becomes t h e case, major m o d i f i c a t i o n s a r e r e q u i r e d t o t h e Boltzmann t r a n s p o r t equa- t i o n [6] and t o t h e c u r r e n t response equations w i t h i n t h e devices [7,8]. _The f a c t t h a t t h e r e l a x a t i o n times a r e varying, due t o t h e e v o l v i n g o f t h e average energy, on t h e same time scale a p p r o p r i a t e t o t h e v e l o c i t y response and t h a t a f i n i t e , non-zero c o l l i s i o n d u r a t i o n e x i s t s both l e a d t o a complicated, mu1 t i p l y - c o n v o l v e d form f o r t h e t r a n s p o r t balance equations [8]

It i s a l s o known t h a t the n o i s e p r o p e r t i e s o f semiconductors a r e r e l a t e d t o t h e h o t e l e c t r o n microwave c o n d u c t i v i t y [9,10,11,121. For an example, t h e noise power spectrum i s p r o p o r t i o n a l t o t h e r e a l p a r t o f the microwave c o n d u c t i v i t y . As t h e n o i s e power spectrum i s r e l a t e d t o t h e v e l o c i t y a u t o - c o r r e l a t i o n through F o u r i e r t r a n s f o r m and a l s o t o d i f f u s i o n c o - e f f i c i e n t , a l l these q a n t i t i e s can be obtained from t h e knowledge o f microwave c o n d u c t i v i t y as a f u n c t i o n o f frequency. Recently t h i s microwave c o n d u c t i v i t y has been c a l c u l a t e d i n t h e non-zero c o l l i s i o n r e g i o n [13], a p p l i c a b l e f o r a s i n g l e v a l l e y semiconductor. The purpose o f t h e present paper i s t o extend t h e c a l c u l a t i o n s t o m u l t i v a l l e y case and r e l a t e them t o t h e n o i s e power spectrum, v e l o c i t y a u t o - c o r r e l a t i o n and d i f f u s i o n c o - e f f i c i e n t through t h e s o l u t i o n o f Langevin's equation.

11. MICROWAVE CONDUCTIVITY i ) P a r a l l e l polarizat.ion case

L e t us consider t h a t t h e semiconductor i s under t h e i n f l u e n c e o f a d.c.

b i a s e l e c t r i c f i e l d . Fo, i n a d d i t i o n t o the microwave f i e l d g i v e n by ~ ~ e ~ ~ ~ where F Fo < < 1 and w i s t h e angular frequency o f t h e microwave f i e l d . For t h e p a r a l l e l p o l a r i z a t i o n case we assume t h a t t h e d i r e c t i o n o f F1 and Fo 11 i s c o l l i n e a r . Using t h e f o r m u l a t i o n o u t l i n e d i n r e f . 13 i t i s easy t o show t h e f o l l o w i n g equations f o r the perturbed d r i f t v e l o c i t y vlejWt i n t h e presence o f a d o c . value v Before we w r i t e down t h e equations we note

0 '

t h a t t h e i n c l u s i o n o f a non-zero c o l l i s i o n time, rc, does n o t change t h e f i n a l e q u i l b r i u m value, vo. Also t h e o n l y change i n t h e momentum and energy balance equations imposed by

i s t o change t h e momentum r e l a x a t i o n time

by - ~ ~ ( l + j w ~ ~ ) and t h e energy r e l a x a t i o n time, re by $(l+jwTC).

Thus f o r a s i n g l e v a l l e y case, one o b t a i n s

mvlrmo mv T'T eF1

jwvl= -:+jwTc - -.-&L1 + - (momen tum

i + j w ~ ~ m balance),

l + j w - r c - 7 + j w . r , 3 0 1 1 0

m i s t h e e f f e c t i v e mass o f t h e c a r r i e r s , k i s t h e Boltzmann's constant, T i s t h e e f f e c t i v e temperature i n t h e displaced Maxwellian d i s t r i b u t i o n approximation and

The s u b s c r i p t

denotes t h e e q u i l i b r i u m value and t h e prime denotes t h e d i f f e r e n t i a t i o n w i t h r e s p e c t T taken a t T = T o . Both

and .re a r e e f f e c t i v e r e l a x a t i o n times which i n c l u d e the e f f e c t s o f a l l p o s s i b l e forms o f s c a t t e r - i n g mechanisms r e l e v a n t f o r t h e case (e.g. acoustic, o p t i c a l , i n t e r v a l l e y phonons, i m p u r i t y s c a t t e r i n g e t c . ) .

For t h e two-valley case, f o l l o w i n g r e f . 14, one o b t a i n s t h e f o l l o w i n g m o d i f i e d equations,

jwvli

- mVlirmoi -

l + j w ~ mvOriT1i + (momentum balance)

1 +jw~, "'i ( 3 )

( 4 ) (energy balance)

jmn il = - n i l r n i - n i o r ; i + n . . r n j + n r '

J

j o n j ( 5 )

nl1 + nZl

0. (number balance)

The s u b s c r i p t s i, j =1, 2 and denotes t h e various parameters f o r t h e two v a l l e y s ,

and

denotes t h e r e l a x a t i o n t i m e s associated w i t h t h e number balance and i n t e r v a l l e y phonon r e s p e c t i v e l y ,

The q u a n t i t i e s rij, rnj, T i j and d 1 a r e f u n c t i o n s o f T . only. k0 i s t h e

n j J

c h a r a c t e r i s t i c phonon temperature used i n equ. ( 4 ) o n l y f o r convenience.

Equations (3-5) can be-solved t o o b t a i n an a n a l y t i c a l expression f o r t h e

e f f e c t i v e microwave c o n d u c t i v i t y , ull. However, i t i s r a t h e r cumbersome

although s t r a i g h t f o r w a r d and w i l l n o t be given here. For numerical c a l c u l a -

t i o n s i t i s b e t t e r t o t a c k l e t h e equations d i r e c t l y . The above equations

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JOURNAL DE PHYSIQUE

can a l s o be extended i n a r a t h e r s t r a i g h t f o r w a r d manner f o r t h e m u l t i - v a l l e y case having more than two v a l l e y s .

For t h e s i n g l e v a l l e y case, however, one o b t a i n s

where uo

- e

mrmo

~ ( w )

( l + j u ~ ) ^'eo c

and

y(w) = 1 + - r; + 7 ³ j w ~ ~ ~ ( l + j w ~ ~ ) 'eo

i i ) Perpendicular p o l a r i z a t i o n case.

I f F, i s normal t o Fo then t h e p e r t u r b a t i o n i n e f f e c t i v e temperature i s zero f o r f i r s t order c a l c u l a t i o n s . Thus f o r t h i s case equ. (1) s i m p l i e s t o

From t h i s we i m e d i a t e l y o b t a i n

For t h e two v a l l e y case one o b t a i n s f o r each i n d i v i d u a l v a l l e y , an expression f o r m o b i l i t y given by equ. (8). The microwave c o n d u c t i v i t y f o r t h e two v a l l e y case i s given by

111. HOT ELECTRON NOISE PROPERTIES.

To o b t a i n h o t e l e c t r o n n o i s e p r o p e r t i e s one need t o s o l v e t h e Langevin's equation. For t h i s purpose one assumes t h a t a b i a s e l e c t r i c f i e l d i s

~ p p l i e d across t h e semiconductor i n a d d i t i o n t o t h e small f l u c t u a t i n g random e l e c t r i c f i e l d . Thus t h e d i f f e r e n c e between t h e problem solved i n t h e l a s t s e c t i o n and t h a t o f Langevin's equation i s t h a t one should consider F, as t h e s p e c t r a l component o f t h e f l u c t u a t i n g e l e c t r i c f i e l d . As t h e equations (1)-(6) a r e l i n e a r equations around a b i a s e l e c t r i c f i e l d , t h e s o l u t i o n o f t h e Langevin's equation i s e q u i v a l e n t t o t h a t discussed i n references [9 and 121. The n o i s e s p e c t r a l d e n s i t y f o r t h e e q u i v a l e n t n o i s e v o l t a g e a w )

f o r t h e s i n g l e v a l l e y case i s given by f o r t h e p a r a l l e l p o l a r i z a t i o n case

@(w) = A2kT (nee) ^{Re kill ( w )} (10)

case pll is replaced by the v , . Applying Weiner-Khintchin theorem one obtains the velocity auto-correlation, < v(t)v(t+~)>,to be given by

<v(t)v(t+r)>

r-'[vr(w)l (11)

where F-I denotes inverse Fourier transform and +(w) is the real part of the microwave mobility. As the diffusion co-efficient ~ ( w ) is related to the Fourier transform of velocity auto-correlation, one obtains its value from the extended Einstein relationship

R~D(U) = - kT LI

q r (12)

where Re denotes the real part.

Numerical results and discussion

The small signal microwave conductivity was calculated for Si for an applied d.c. field at 30KvCm- 1 . The scattering mechanisms and coupling constants are those used previously for Si [15-171. The real and imaginary part of p are defined as

and are plotted in fig, 1 and 2 for the parallel and perpendicular polarization cases respectively. The peaking at high frequencies is more pronounced in the presence of the non-zero collision duration, a result

expected from calculations of overshoot velocity-itself.

Fig. 1. Parallel Polarization: The real (a) and

imaginary (b) parts of the a.c. small signal mobility

for an applied d.c. field of 30 kV cm-1inSi. The

solid curve includes the effect of a finite, non-zero

collision duration, while the dashed curve ignores

this effect.

C7-232 JOURNAL DE PHYSIQUE

The peaking observed leads t o an i n t e r e s t i n g s h i f t o f t h e apparent plasma edge i n t h e semiconductor. I n f i g . 3 we p l o t t h e r e f l e c t i v i t y

( R I d e f i n e d below f o r t h e para1 l e l p o l a r i z a t i o n case.

where

and

i s t h e s t a t i c d i e l e c t r i c constant o f t h e semiconductor and c0 i s vacuum p e r m i t t i v i t y . Three curves a r e shown i n f i g . 3. The

F i g . 2 Perpendicular P o l a r i z a t i o n : The r e a l (a) and imaginary ( b ) p a r t s o f t h e a.c. small s i g n a l m o b i l i t y f o r an a p p l i e d d.c. f i e l d o f 30 kV cm-1 i n S i .

Fig. 3. The r e f l e c t i v i t y o f S i i n a h i g h e l e c t r i c

f i e l d . The d o t t e d curve assumes c o l d c a r r i e r s w i t h

m o b i l i t y

b u t a d i f f e r e n t chordal m o b i l i t y . The

s o l i d curve i n c l u d e s t h e e f f e c t o f t h e f i n i t e

c o l l i s i o n d u r a t i o n . A doping o f 1018 cm-3 i s

assumed.

curve assumes h o t c a r r i e r s w i t h t h e same uo b u t a d i f f e r e n t chordal m o b i l i t y . The s o l i d curve includes t h e effect o f t h e f i n i t e c o l l i s i o n d u r a t i o n . A doping of 1018 cm-3 i s assumed. I t i s observed t h a t t h e presence of h o t c a r r i e r s merely serves t o smooth the apparent plasma edge. However, f o r

f 0, t h e minimum s h i f t s s i g n i f i c a n t l y t o h i g h e r frequencies.

The v e l o c i t y a u t o - c o r r e l a t i o n s obtained by d i r e c t l y F o u r i e r t r a n s - forming +(w) i s shown i n f i g s . 4 and 5 f o r t h e two p o l a r i z a t i o n cases.

As expected, i t i s observed t h a t t h e i n c l u s i o n o f non zero c o l l i s i o n d u r a t i o n c o n t r i b h t e s s i g n i f i c a n t l y t o t h e negative swing o f t h e v e l o c i t y a u t o - c o r r e l a t i o n .

I n conclusion, t h e small s i g n a l microwave c o n d u c t i v i t y o f semi- conductors i n t h e h o t e l e c t r o n c o n d i t i o n has been obtained i n t h e non- zero c o l l i s i o n regime which i n t u r n has been used t o o b t a i n n o i s e p r o p e r t i e s .

Fig. 4. V e l o c i t y a u t o - c o r r e l a t i o n ~ 5. i ~ v e l o c i t y . auto-corre~ation the p a r a l l e l case. Units are f o r the perpendicular case. U n i t s

a r b i t r a r y . a r e a r b i t r a r y .

REFERENCES

[I] P. Das & D. K. Ferry, S o l i d S t a t e Electron, 19, 851 (1976).

[Z]' D. K - F e r r y & P. Das, S o l i d S t a t e Electron, 20, 355 (1977).

[3] H - D. Rees, IBM J. Res. Develop, 13, 537 (1969).

[4] H. L. Grubin, D.K. F e r r y & J. R. Barker, Proc. IEDM, p. 394 (1979).

[5] D. K. F e r r y & J. R. Barker, S o l i d S t a t e Electron, 23, 545 (1980).

C7-234 JOURNAL DE PHYSIQUE

C6l J. R. Barker & D. K. Ferry, S o l i d S t a t e Electron, 23, 519 (1980).

[71 D. K. F e r r y & J. R. Barker, Sol i d S t a t e Commun. 30, 361 (1979).

[8] D. K. F e r r y & J. R. Barker,J. Phys. Chem. S o l i d s ( i n press).

[9] A. Van Der Z i e l , "noise i n S o l i d S t a t e Devices", i n Advances i n E l e c t r o n i c s and E l e c t r o n Physics, Ed. L. Harton, Vol. 46, Academic Press, (1 978).

[ l o ] J. P. Nougier, D. Sodini, M. Rolland, D. Gasquet and Gasquet and

G . Lecoy, S o l i d S t a t e Electron, 21, 133 (1978).

[ll] A. Van Der Z i e l , S o l i d S t a t e E l e c t r o n i c s , Vol. 23, pp. 1035-1036 (1 980).

[12] C. W. Helstrom, "Markov Processes and a p p l i c a t i o n s " , i n Communcation Theory", Ed. A. V. Balakrishnar, McGraw H i l l , (1968).

[ I 3 1 P. Das, D. K. F e r r y and H. Grubin, S o l i d S t a t e Commun., t o be published.

1141 P. Das, Appl. Phys. L e t t e r s , Vol. 11, pp-386-388 (1967).

[ I 5 1 D. K. Ferry, Phys. Rev. 814, 1605, (1976).

[16] D. K. Ferry, S u r f . Sci., 57, 218 (1976).

[ I 7 1 D. K. Ferry, Phys. Rev. B. 14, 5364 (1976).