HOT HOLES AND NOISE IN SPACE CHARGE LIMITED CURRENT FLOW IN P-TYPE SILICON

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HOT HOLES AND NOISE IN SPACE CHARGE LIMITED CURRENT FLOW IN P-TYPE SILICON

N. Sawaki, T. Ishikawa, M. Morohashi, T. Nishinaga

To cite this version:

N. Sawaki, T. Ishikawa, M. Morohashi, T. Nishinaga. HOT HOLES AND NOISE IN SPACE CHARGE LIMITED CURRENT FLOW IN P-TYPE SILICON. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-221-C7-226. �10.1051/jphyscol:1981726�. �jpa-00221663�

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JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-221

Résumé.- Le coefficient de diffusion et le bruit de porteurs chauds dans le silicium de type p sont étudiés à la température 77 - 300 K à l'aide de méthode de l'équation du moment. Ces ré- sultats théoriques fondés sur un modèle ayant une bande paraboli- que sont comparés aux résultats expérimentaux mentionnés dans la littérature.

HOT HOLES AND NOISE IN SPACE CHARGE LIMITED CURRENT FLOW IN p-TYPE SILICON

N. Sawaki, T. Ishikawa, M. Morohashi and T. Nishinaga

Nagoya University, Department of Electronics, Chikusa-ku, Nagoya 464, Japan

Abstract.- By using the moment equation method with a displaced Maxwellian, the diffusion coefficient and the noise of the hot holes in Si were investigated at 77 - 300 K. The results based on a parabolic energy band structure were compared with experimental results in literatures.

1. Introduction.- Though recent development of numerical methods such as Monte Carlo method has made the analytical methods less interest- ing in the investigation of hot carrier problems, the latter is still to be one of the powerful methods to study the nonlinear phenomena in semiconductor devices. The moment equation method, e.g., has enabled us to study the high frequency response as well as the steady state dc response of devices.

In this paper, by developing the moment equation method by

1) 2) Robson and by using the relaxation rate by Das and Ferry,' the mo-

bility and the diffusion coefficient of p-type Si are studied. The results are applied to the noise power density of a hole current flow in a SCL diode.

2. Method of Calculations.- According the the theory by Robson and Cheung and Hearn,' the motion of electrons or holes under an electric field is described by the following coupled equations;

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where use of the displaced Maxwellian with an electron temperature Tg

has been made, n the carrier density, v, the drift velocity, m the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981726

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C7-222 JOURNAL DE PHYSIQUE

e f f e c t i v e mass and i s t h e e l e c t r i c f i e l d . The r e l a x a t i o n r a t e of 2 1 momentum rm and t h a t of energy re have been given by D a s and F e r r y , where i n c l u d e d a r e t h e i n t e r a c t i o n s w i t h i o n i z e d i m p u r i t i e s , t h e a c o u s t i c phonons and t h e non-polar o p t i c a l phonons. B y l i n e a r i z i n g t h e v a r i a b l e s i n (1) around t h e s t e a d y s t a t e ,

5 ⁼ f o + f ₁eiYt ,

n + n e i w t n =

0 1 ^I

3 -f -f i w t

vd ⁼ v _do+ vle ,

t h e dc m o b i l i t y yo and t h e s m a l l s i g n a l frequency dependent m o b i l i t y and d i f f u s i o n c o e f f i c i e n t a r e o b t a i n e d f o r both t h e components p a r a l - l e l and p e r p e n d i c u l a r t o t h e d c f i e l d Eo. 3 Among them, t h o s e necessa- r y f o r t h e following c a l c u l a t i o n s a r e

where n = 'eo mo eo r 1 ^T

'mo ('eof ';oTeo)

For s i m p l i c i t y , we have assumed t h e p a r a b o l i c energy band.

By u t i l i z i n g t h e r e s u l t s ( 3 ) , t h e c u r r e n t v e r s u s v o l t a g e and t h e n o i s e v e r s u s v o l t a g e c h a r a c t e r i s t i c s of t h e h o l e c u r r e n t flow i n a SCL diode a r e i n v e s t i g a t e d w i t h t h e following formula 4

where A i s t h e sample a r e a and L t h e sample l e n g t h , na i s t h e thermal c a r r i e r d e n s i t y , EL i s t h e f i e l d s t r e n g t h a t t h e cathode and E i s t h e p e r m i t t i v i t y .

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3 . R e s u l t s and Discussions.- M a t e r i a l parameters used i n t h i s work a r e summarized i n Table 1. Because of t h e ambiguities of t h e assumption of t h e p a r a b o l i c energy band, t h e coupling c o n s t a n t f o r t h e a c o u s t i c and o p t i c a l i n t e r a c t i o n s were determined s o t h a t t h e low f i e l d mo- b i l i t y a t 77 K and 3 0 0 K match t h e experimental v a l u e s . The r a t h e r l a r g e v a l u e s of t h e c o n s t a n t s compared t o t h o s e i n l i t e r a t u r e s a r e c l e a r l y due t o t h e n e g l e c t . of t h e warping e f f e c t of t h e band.

Figure I shows t h e d r i f t v e l o c i t y a s a f u n c t i o n of t h e e l e c t r i c f i e l d s t r e n g t h a t s e v e r a l temperatures. The agreement of t h e r e s u l t s w i t h t h e experimental ones a r e e x c e l l e n t .

Figure 2 shows t h e f i e l d dependence o f t h e d i f f u s i o n c o e f f i c i e n t , t h e agreement of which with t h e experimental ones a r e n o t s o good a s i n t h e c a s e o f t h e m o b i l i t y . The d e v i a t i o n i s enhanced i n t h e high f i e l d r e g i o n , b u t p r e s e n t r e s u l t g i v e s a c o r r e c t value a t t h e low f i e l d l i m i t where t h e

E i n s t e i n r e l a t i o n i s h e l d i n t h e theory. A s i m i l a r c a l c u l a t i o n f o r e l e c t r o n s 5

showed an e x c e l l e n t agreement with t h e Monte Carlo r e s u l t s by C a n a l i e t ~ 3 1 . ~ ) f o r a p a r a b o l i c band ⁽ a = 0 i n t h e i r work ) . Therefore, t h e d e v i a t i o n s shown i n Fig.2 i s thought due t o t h e n e g l e c t of t h e warping e f f e c t a s w e l l a s t h e non- p a r a b o l i c i t y . Figure 3

Table 1. C o n s t a n t s o f p-Si Q u a n t i t y Value

P * ^2.33 ^g/cm³

* O t t a v i a n i e t a l . Ref. [ 9 ]

** Runyan. Ref.[lO]

*** P r e s e n t work.

l o 5 1 ¹ ¹l l i l l l I 1 111111 ^I l l l l l l i ¹ l l l l L ~

1 0 l o 2 l o 3 lo4 lo5

ELECTRIC FIELD ( V/cm ) F i g . 1. The d r i f t v e l o c i t y a s a f u n c t i o n of

e l e c t r i c f i e l d s t r e n g t h . The s o l i d l i n e s a r e p r e s e n t r e s u l t s and ( 0 ) and ( A ) a r e e x p e r i m e n t a l p o i n t s . Ref.[6, 91.

100

n Si-Holes

- _Theory

- p r e s e n t work

l o 2 ¹⁰ ¹⁰

ELECTRIC FIELD (V/cm)

Fig.2. The d i f f u s i o n c o e f f i c i e n t a s a func- t i o n o f e l e c t r i c f i e l d s t r e n g t h . _{( 0 )} and ( A ) a r e e x p e r i m e n t a l p o i n t s [ 6 , 111.

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C7-224 JOURNAL DE PHYSIQUE

r e p o r t s t h e n o i s e t e m p e r a t u r e d e f i n e d by

= eD,(O)/u, ( 0 ) ,

a l o n g w i t h t h e e x p e r i m e n t a l and Monte C a r l o r e s u l t s by Nava e t a l . 6 I t i s n o t a b l e t h a t i n s p i t e o f t h e l a r g e d i s c r e p a n c i e s i n t h e d i f f u - s i o n c o e f f i c i e n t a s i n Fig.2 t h e agreement o f t h e n o i s e t e m p e r a t u r e i s e x c e l l e n t .

By u s i n g t h e f o r m u l a ( 4 ) , t h e e l e c t r i c a l p r o p e r t i e s o f a SCL d i o d e was i n v e s t i g a t e d . The c a l c u l a t i o n s were s t r a i g h t forward. I n o r d e r

t o compare t h e r e s u l t s w i t h e x p e r i m e n t a l o n e s , t h e sample dimensions were s e t a t t h e a r e a A = 0 . 0 1 cm-2 , t h e sample l e n g t h L = 40 um.

F i g u r e 4 shows t h e I - V c h a r a c t e r i s t i c s , o m i t t e d ( p u r e SCL c u r r e n t ) . The c u r - v a t u r e i s s i m i l a r t o t h a t o b t a i n e d by Nougier e t a l ? ) , b u t t h e magnitude o f t h e c u r r e n t i s a b o u t f o u r t i m e s l a r g e r

Si-Holes Experiment

0 Nava e t a 1 (1979) Theory

1 I . ' l , I 5 . ^{! I}^{! . .}^{I ,} ⁴ . ¹ ^I, ⁶ ⁸ ^{1 ' .}

l o 2 10 10

ELECTRIC FIELD (V/ cm) F i g . 3 . The n o i s e t e m p e r a t u r e o f p-type

S i as a f u n c t i o n of t h e e l e c t r i c f i e l d s t r e n g t h .

where t h e ohmic c u r r e n t i s

Fig.4. I - V c h a r a c t e r i s t i c s of a SCL d i o d e o f p-Si. The ohmic c u r r e n t a t t h e low v o l t a g e i s o m i t t e d .

Fig.5. The w h i t e n o i s e power d e n s i t y of a SCL d i o d e of p-Si. See t e x t . L e f t ( Theory 300 K, Exp. 300 K ) , R i g h t ( Theory 150 K , Exp. 145 K ) .

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t h a n t h e i r e x p e r i m e n t a l r e s u l t s , and s l i g h t l y s m a l l e r t h a n t h e r e s u l t s by G i s o l f and ~ i j l s t r a ? ) F i g u r e 5 r e p o r t s t h e n o i s e power d e n s i t y a t

two t e m p e r a t u r e s . The c u r v e ( a ) i s t h e p r e s e n t r e s u l t w i t h h o t c a r - r i e r e f f e c t and ( c ) i s t h a t f o r t h e r m a l c a r r i e r s w i t h o u t h o t c a r r i e r e f f e c t . The c u r v e ( b ) i s t h e r e s u l t s by r e t a i n i n g t h e h o t c a r r i e r e f f e c t w i t h a c o u s t i c s c a t t e r i n g s ? ) I n t h e c a l c u l a t i o n of t h e n o i s e , t h e r e s i s t i v i t y of t h e sample was s e t a t 100R-cm a t 300 K a s i n t h e c a s e o f t h e e x p e r i m e n t by G i s o l f and Z i j l s t r a , whose r e s u l t s a r e shown by d o t s i n t h e f i g u r e s . The agreement o f t h e e x p e r i m e n t a l p o i n t s w i t h t h e c u r v e s ( a ) i s good and shows t h a t t h e h o t c a r r i e r e f - f e c t i s e s s e n t i a l i n t h e n o i s e power o f a SCL d i o d e . However, t h e f a i r agreement would g i v e o n e a c o n f u s i o n b e c a u s e t h e r e i s a l a r g e d i s c r e p a n c y i n t h e b u l k d i f f u s i o n c o e f f i c i e n t a s shown i n F i g . 2 .

According t o t h e s t u d y o f t h e warped band o f S i by Nava e t a l . 6 ,

t h e warping and t h e n o n p a r a b o l i c i t y would s u b s t a n t i a l l y r e d u c e 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 h o l e s , e s p e c i a l l y a t a h i g h f i e l d , which i s a l s o t h e c a s e f o r e l e c t r o n s ? ) T h e r e f o r e , t h e i n c l u s i o n o f t h e e f f e c t w i l l make t h e s l i g h t d e v i a t i o n s i n t h e n o i s e power d e n s i t y much

l a r g e r . I n f a c t Reggiani e t a l ? ) c o u l d n o t f i n d a good agreement i n t h e i r Monte C a r l o c a l c u l a t i o n s o f t h e n o i s e t e m p e r a t u r e s o f SCL c u r - r e n t , even though t h e c a l c u l a t e d b u l k d i f f u s i o n c o e f f i c i e n t showed a n e x c e l l e n t agreement w i t h e x p e r i m e n t a l ones by Time-of-Flight.

4 . Conclusions.- B y u s i n g t h e moment e q u a t i o n method w i t h d i s p l a c e d Maxwellian d i s t r i b u t i o n f u n c t i o n , t h e n o i s e o f h o t h o l e s i n S i was i n v e s t i g a t e d . Though agreement o f t h e r e s u l t s f o r a SCL d i o d e w i t h t h e e x p e r i m e n t a l d a t a i n l i t e r a t u r e s were e x c e l l e n t , t h e a b s o l u t e v a l u e o f t h e c u r r e n t o f t h e SCL d i o d e and t h e i n c o n s i s t e n c y i n t h e d i f f u s i o n c o e f f i c i e n t i n b u l k S i and t h e c o r r e s p o n d i n g n o i s e o f SCL d i o d e have been remained unsolved.

Acknowledgement.- The n u m e r i c a l c a l c u l a t i o n s were performed a t t h e Computation C e n t e r a t Nagoya U n i v e r s i t y .

R e f e r e n c e s

1) R.E. Robson : Phys. Rev. L e t t . 31 (1973) 825.

2) P . Das and D.K. F e r r y : S o l i d S t , E l e c t r o n . 19 (1976) 851.

3) P.S. Cheung and C . J . Hearn : J. Phys. C : ~ o l i d S t . Phys. 5 (1972) 1563.

4 ) A . G i s o l f and R . J . J . Z i j l s t r a : J. Appl. Phys. 47 (1976) 2727.

5 ) C. C a n a l i e t a l . : Appl. Phys. L e t t . 27 (1975) m 8 . 6) F. Nava e t a l . : J . Appl. Phys. 50 ^(1-9) 922.

7) J.P. Nougier e t a l . : " Noise i n P h y s i c a l Systems " Ed. D. Wolf, S p r i n g e r 1978, pp.110.

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C7-226 JOURNAL DE PHYSIQUE

8) L. Reggiani : Proc. Intern. Conf. Physics of Semiconductors, Kyoto 1980, pp.317.

9) G. Ottaviani et al. : Phys. Rev. B g (1975) 3318.

10) W.R. Runyan : " Silicon Semiconductor Technology " McGraw Hill 1965.

11) T.W. Sigmon and J.F. Gibbons : Appl. Phys. Lett. 15 (1969) 320.