BAND-STRUCTURE DEPENDENT IMPACT IONIZATION IN SILICON AND GALLIUM ARSENIDE

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BAND-STRUCTURE DEPENDENT IMPACT IONIZATION IN SILICON AND GALLIUM

ARSENIDE

J. Tang, H. Shichijo, K. Hess, G. Iafrate

To cite this version:

J. Tang, H. Shichijo, K. Hess, G. Iafrate. BAND-STRUCTURE DEPENDENT IMPACT IONIZA-

TION IN SILICON AND GALLIUM ARSENIDE. Journal de Physique Colloques, 1981, 42 (C7),

pp.C7-63-C7-69. �10.1051/jphyscol:1981707�. �jpa-00221643�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, oatobve 1981 page C7-63

BAND-STRUCTURE DEPENDENT IMPACT IONIZATION IN SILICON AND GALLIUM ARSENIDE

J.Y. Tang, H. Shichijo, K. Hess and G.J. Iafrate

Coordinated Science Laboratory and Department of Electrical Engineering,Univer- sity of Illinois at Urbana-Champaign, Urbana, Illinois, U.S.A. 61801

*U.S. Army Electronics Technology and Device Laboratory, Ft. Monmouth, New Jersey U.S.A. 07703

Résumé. - Nous avons développé une simulation par la méthode de Monte Carlo pour du silicium et de l'AsGa en incluant une structure de bande réaliste. Les taux d'ioni- sation par impact et les vitesses de dérive en régime continu sous forts champs élec- triques (>100 kV/cm) ont été calculés à différentes températures.

Abstract. - We have performed a Monte Carlo simulation for GaAs and Si with the realistic band structure included. Steady state impact ionization rates and drift velocities under high electric fields (> 100 kV/cm) were calculated at various temperatures.

1. Introduction. - Impact ionization is an essential mechanism in the operation of semiconductor devices such as avalanche photo-diodes or transit time devices. The dependence of impact ionization on the crystallographic orientation has attracted substantial interest because of its relevance to noise and other phenomena in these devices. This dependence, however, is not shown by any of the theories as given by Wolff [1], Shockley [2], and Baraff [3], since none of them include a realistic band structure.

We have developed a complete theory for impact ionization and generally high field transport in semiconductors by combining a Monte Carlo simulation with the realistic band structure calculated by the empirical pseudopotential method. We do take into account scattering by all possible phonon types, the change in the density of states high in the band, the exact velocity v = j- V E(k) (no effective mass approximation), the collision broadening of the electronic states, and the tempera- ture effect.

Details of the model and the results for GaAs at 300 K can be found in two of our previous papers [4,5]. In the case of Si, the first two conduction bands were included. Besides the X - X scattering we also include the X - L scattering in Si.

In Section 2, we describe briefly our model and point out its differences from the commonly used model. The effects of the inclusions of the second band and the transition from X - L in Si are discussed in Section 4.

2. Theoretical Model. - The model for Monte Carlo simulation has two main ingredi- ents: (i) the band structure and (ii) the scattering rate. We describe briefly in the following the different features that have been included in our model and the advantages it has over other models.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981707

C 7 1 6 4 JOURNAL DE PHYSIQUE

2.1. Band S t r u c t u r e - I n s t e a d o f u s i n g a p a r a b o l i c band a p p r o x i m a t i o n , we i n c o r - p o r a t e d i n t o t h i s Monte C a r l o s i m u l a t i o n a r e a l i s t i c band s t r u c t u r e c a l c u l a t e d by t h e e m p i r i c a l p s e u d o p o t e n t i a l method [ 6 ] f o r b o t h GaAs and S i . The r e a l i s t i c band s t r u c t u r e e n a b l e s u s t o c a r r y o u t t h e s i m u l a t i o n up t o v e r y h i g h e l e c t r i c f i e l d s w i t h o u t worrying a b o u t t h e breakdown of a c o n s t a n t mass a p p r o x i m a t i o n which i s u s u a l l y adopted. The f i r s t two c o n d u c t i o n bands a r e i n c l u d e d f o r S i . Only t h e f i r s t c o n d u c t i o n band i s i n c l u d e d f o r GaAs b e c a u s e t h e second band l i e s s i g n i f i c a n t - l y h i g h e r up.

2.2. S c a t t e r i n g Rate -

( i ) GaAs: P o l a r o p t i c a l s c a t t e r i n g and e q u i v a l e n t and n o n e q u i v a l e n t i n t e r - v a l l e y s c a t t e r i n g were i n c l u d e d i n t h e model. We d i d n o t i n c l u d e i m p u r i t y s c a t t e r - i n g and a c o u s t i c phonon s c a t t e r i n g whose e f f e c t a r e smallcompared t o o t h e r s c a t t e r - i n g mechanisms. The r e a d e r s h o u l d r e f e r t o o u r p r e v i o u s p a p e r f o r more d e t a i l s and n u m e r i c a l p a r a m e t e r s [ 4 ] .

( i i ) S i : We f o l l o w e d c l o s e l y t h e work by C a n a l i e t al.17-91 i n t h e c a s e o f S i . F o r s i m p l i f i c a t i o n , t h e i n t r a v a l l e y a c o u s t i c phonon s c a t t e r i n g was c o n s i d e r e d e l a s t i c , which i s good f o r t h e h i g h f i e l d r e g i o n we were i n t e r e s t e d i n . The non- p a r a b o l i c i t y f a c t o r was a l s o i n c l u d e d 191. For X - X c o u p l i n g we c o n s i d e r e d s e v e r a l p o s s i b l e phonon t y p e s (3f and 3g). The d i f f e r e n c e between f and

s c a t t e r i n g was t a k e n i n t o a c c o u n t i n t h e s i m u l a t i o n p r o c e s s .

B e s i d e s t h e e q u i v a l e n t (X-X) i n t e r v a l l e y s c a t t e r i n g , we a l s o i n c l u d e d t h e n o n e q u i v a l e n t ( X - L ) i n t e r v a l l e y s c a t t e r i n g mechanism. The X - L c o u p l i n g c o n s t a n t s were assumed t o be t h e same f o r a l l f o u r p o s s i b l e phonons d e t e r m i n e d from t h e pho- non s p e c t r u m [ l o ] and were found t o b e 3 . 5 ~ 1 0 eV/cm by f i t t i n g t h e c a l c u l a t e d

r e s u l t s t o t h e e x p e r i m e n t a l r e s u l t s .

The l o n g i t u d i n a l and t r a n s v e r s e e f f e c t i v e masses of t h e L-valley were d e t e r - mined from t h e p s e u d o p o t e n t i a l band s t r u c t u r e t o b e mQ

1.59 m and mt

0.121 mo.

0

The L - v a l l e y s l i e a b o u t 1 e V above t h e X-valleys. The e f f e c t of t h e L - v a l l e y s i s i m p o r t a n t o n l y a t h i g h f i e l d s (> 100 kV/cm).

( i i i ) S c a t t e r i n g r a t e a t h i g h energy: The e f f e c t i v e mass d e n s i t y of s t a t e s

w i t h t h e p n p a r a b o l i c i t y f a c t o r i n c l u d e d was u s e d i n c a l c u l a t i n g t h e s c a t t e r i n g

r a t e s . I t i s n o t a p p r o p r i a t e t o simply e x t e n d t h e s e s c a t t e r i n g r a t e s t o h i g h e r

e n e r g i e s b e c a u s e t h e e f f e c t i v e mass a p p r o x i m a t i o n b r e a k s down a t h i g h e r e n e r g i e s

and t h e d e n s i t y of s t a t e s s t a r t s d e c r e a s i n g a t some c r i t i c a l p o i n t . The i n t e r -

v a l l e y s c a t t e r i n g r a t e , which i s p r o p o r t i o n a l t o t h e f i n a l d e n s i t y of s t a t e s ,

s h o u l d be m o d i f i e d a t h i g h e r e l e c t r o n energies.. T h e r e f o r e we have c a l c u l a t e d t h e

t o t a l d e n s i t y o f s t a t e s f o r one c o n d u c t i o n band f o r GaAs and two c o n d u c t i o n bands

f o r S i . The c r i t i c a l p o i n t s beyond which t h e d e n s i t y of s t a t e s d e c r e a s e s f o r

GaAs and S i were found t o be 1 . 8 eV and 2 e V , r e s p e c t i v e l y . The d e n s i t y of s t a t e s

d e c r e a s e s a l m o s t q u a d r a t i c a l l y above t h a t . We t h e r e f o r e have assumed a q u a d r a t i -

c a l d e c r e a s e o f t h e s c a t t e r i n g r a t e s above t h e c r i t i c a l p o i n t s .

( i v ) S c a t t e r i n g p r o b a b i l i t y f o r impact i o n i z a t i o n : Impact i o n i z a t i o n c a n b e t r e a t e d a s a n a d d i t i o n a l s c a t t e r i n g mechanism. We assume a n i s o t r o p i c t h r e s h o l d e n e r g y f o r b o t h GaAs ( 2 . 0 eV) and S i ( 1 . 8 eV) [ 1 2 ] . The f o r m u l a f o r t h e p r o b a b i l i t y of impact i o n i z a t i o n above t h r e s h o l d c a n b e found i n a p a p e r by Keldysh [ I l l .

2.3. Temperature E f f e c t - We assumed t h a t t h e band s t r u c t u r e i s e s s e n t i a l l y u n a l t e r e d w i t h changes i n t e m p e r a t u r e e x c e p t t h a t t h e band gap i s m o d i f i e d . We f u r t h e r assumed t h a t t h e t h r e s h o l d e n e r g y o f impact i o n i z a t i o n c a n b e s c a l e d a c c o r d i n g t o t h e f o r m u l a

The f o r m u l a f o r t h e t e m p e r a t u r e dependence o f t h e band gap E c a n b e found i n [ 1 9 ] g

b o t h f o r S i and GaAs.

2 . 4 . C o l l i s i o n Broadening E f f e c t - A s t h e s c a t t e r i n g r a t e goes up t o 1014 p e r s e c o n d , t h e e n e r g y o f t h e e l e c t r o n i s u n c e r t a i n w i t h i n AE % f i / r a c c o r d i n g t o uncer- t a i n t y p r i n c i p l e . AE i s a b o u t 100 meV f o r I / T h i g h e r t h a n 1014. I n d e t e r m i n i n g t h e f i n a l s t a t e a f t e r e a c h s c a t t e r i n g , we c o n s i d e r e d t h e s t a t e s w i t h i n a r a n g e AE o f t h e t r u e f i n a l s t a t e e n e r g y a s p o s s i b l e c a n d i d a t e s . We t h e n c h o s e randomly one o f t h o s e c a n d i d a t e s a s t h e f i n a l s t a t e , k e e p i n g t h e a v e r a g e e n e r g y l o s s ( g a i n ) c o n s t a n t . The c o l l i s i o n b r o a d e n i n g e f f e c t s h o u l d b e t a k e n i n t o a c c o u n t a u t o m a t i c a l l y by s o d o i n g .

3. R e s u l t s

3.1. GaAs: F i g u r e 1 shows t h e e l e c t r o n i o n i z a t i o n r a t e a a t two t e m p e r a t u r e s . The r e s u l t f o r 300 K was shown i n a p r e v i o u s p a p e r [ 4 ] and t h e shaded r e g i o n i s t h e r a n g e covered by a v a i l a b l e e x p e r i m e n t a l r e s u l t s f o r 300 R. The e x p e r i m e n t a l r e s u l t f o r 77

i s n o t a v a i l a b l e a t t h i s t i m e . The lower t e m p e r a t u r e of 77

s h o u l d r e s u l t i n a h i g h e r impact i o n i z a t i o n r a t e a s shown i n t h e g r a p h . As we h a v e d i s - c u s s e d i n t h e p r e v i o u s p a p e r , o u r c a l c u l a t i o n s showed no o r i e n t a t i o n a l dependence f o r f i e l d s h i g h e r t h a n 300 kV/cm.

3.2. S i : We h a v e c a l c u l a t e d d r i f t v e l o c i t i e s i n t h e < I l l > d i r e c t i o n f o r h i g h e l e c t r i c f i e l d s and compared them t o t h e r e c e n t h i g h f i e l d e x p e r i m e n t a l r e s u l t s by Smith e t a l . [ 2 0 ] . As shown i n F i g u r e

o u r c a l c u l a t e d h i g h f i e l d r e s u l t s a r e i n good agreement w i t h t h e e x p e r i m e n t a l r e s u l t s . I n F i g u r e 3 we show t h e c a l c u l a - t e d r e s u l t s of t h e e l e c t r o n impact i o n i z a t i o n r a t e a a t two t e m p e r a t u r e s , 100 K and 300 K. F o r 300 K o u r r e s u l t s seem t o a g r e e b e t t e r w i t h Lee and Logan's r e s u l t s t h a n w i t h O v e r s t r a e t e n and De Nan's r e s u l t s . Again, l i k e i n GaAs, we d i d n o t f i n d any o r i e n t a t i o n a l dependent i o n i z a t i o n r a t e i n S i . As we mentioned i n S e c t i o n 2 of t h i s p a p e r , t h e f i r s t two c o n d u c t i o n bands were i n c l u d e d i n t h e model. We show a n i n t e r e s t i n g b u t i m p o r t a n t r e s u l t i n F i g u r e 4 which t e l l s u s t h e i m p o r t a n t r o l e p l a y e d by t h e second c o n d u c t i o n band i n t h e t r a n s p o r t of e l e c t r o n s i n S i . We

p l o t t e d on t h e g r a p h b o t h t h e second band e f f e c t and t h e a v e r a g e energy of e l e c t r o n s

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 . From t h e p s e u d o p o t e n t i a l band s t r u c t u r e , t h e

C7-66 JOURNAL DE PHYSIQUE

minima of t h e second conduction band a r e l o c a t e d e x a c t l y a t X. They a r e about 0 . 1 eV above t h e minima of t h e f i r s t conduction band. The e l e c t r o n s can b e s c a t t e r e d t o t h e second band once t h e i r e n e r g i e s g e t above 0 . 1 eV. The c o r r e l a t i o n between t h e second band e f f e c t and average e l e c t r o n energy shows c l e a r l y i n t h e graph. The d e c r e a s e of t h e e f f e c t of t h e second band a f t e r 100 kV/cm i s probably because of t h e slower r e l a t i v e i n c r e a s e of t h e d e n s i t y of s t a t e s of t h e second band compared t o t h e f i r s t band.

Fig. 1: R e s u l t s of c a l c u l a t e d impact i o n i z a t i o n r a t e a f o r GaAs a t two d i f f e r e n t tempera- t u r e s , 77 K and 300 K , p l o t t e d a s a f u n c t i o n of i n v e r s e e l e c t r i c f i e l d . The shaded r e g i o n i s t h e range covered by a v a i l a b l e experimental r e s u l t s [15-181.

Erratum.- A correction needs t o be made t o figure ( I / , t h e e l e c t r o n i o n i z a t i o n r a t e u s . Z/E. The curve for T = 77 K corresponds t o the s c a l e on t h e r i g h t hand s i d e . Inad- v e r t e n t l y the l a b e l s on the r i g h t hand side were omitted. These were a factor o f t e n lower than those corresponding t o the l e f t hand s i d e

6.0

i n order t o avoid c l u t t e r i n the 1/E cmxV-l) ~ P - ~ ~ ~ V d i a g r a m . The t e x t , conclusions, and

o t h e r diagrams remain v a l i d .

-Conali et al. 1975 ---.Srn~th. Inoue and _TH. Frev 1980

1

o Tong and Hess

Fig. 2: Calculated d r i f t v e l o c i t y of S i i n < I l l >

d i r e c t i o n . The s o l i d l i n e and t h e dashed l i n e a r e t h e experimental r e s u l t s of Canali e t a l . and Smith e t a l . , r e s p e c t i v e l y .

I I I

1 I

10

100 ₁₀₀₀ I

Electr~c Field (kV/crn) LPIDOO

F i g . 3: C a l c u l a t e d impact i o n i z a - t i o n r a t e a a t two d i f f e r e n t t e m p e r a t u r e s i n S i p l o t t e d a s a f u n c t i o n of i n v e r s e e l e c t r i c f i e l d . For T

300 K , we c a l c u l a t e d t h e i o n i z a t i o n r a t e a i n t h r e e d i f f e r - e n t c r y s t a l l o g r a p h i c d i r e c t i o n s . Only <loo> d i r e c t i o n s were c a l c u l a - t e d f o r 100 K. The l i n e s a r e t h e e x p e r i m e n t a l r e s u l t s from d i f f e r e n t a u t h o r s 112-141.

F i g . 4: We p l o t t e d t h e second band e f f e c t and t h e a v e r a g e e n e r g y of e l e c t r o n s a s a f u n c t i o n o f e l e c t r i c f i e l d . The l e f t s c a l e c o r r e s p o n d s t o t h e upper c u r v e and t h e r i g h t s c a l e c o r r e s p o n d s t o t h e lower c u r v e .

0;

' "

' "

' " "

10 100 1000

Electric Field in KV/cm

4. D i s c u s s i o n

4.1. GaAs: The t r a n s p o r t p r o p e r t i e s of GaAs were d i s c u s s e d i n d e t a i l i n o u r

p r e v i o u s p a p e r s [ 4 , 5 ] . From t h e p r e s e n t 77 K r e s u l t f o r t h e impact i o n i z a t i o n r a t e

a , we s e e a g a i n t h a t no o r i e n t a t i o n a l dependence was found. From t h e d r i f t v e l o c i t y

r e s u l t s [ 4 ] ,

a p p e a r s t h a t m o d i f i c a t i o n of t h e c o u p l i n g c o n s t a n t s i s needed f o r a

r e a l i s t i c band s t r u c t u r e . A b e t t e r f i t t o t h e e x p e r i m e n t a l d a t a s h o u l d r e s u l t from

t h e m o d i f i c a t i o n of t h e c o u p l i n g c o n s t a n t s .

C7-68 JOURNAL DE PHYSIQUE

4.2. S i : A s can b e s e e n from F i g . 4 , t h e second band e f f e c t i s of s i g n i f i c a n c e t o t h e t r a n s p o r t of e l e c t r o n s i n S i . P r e v i o u s Monte C a r l o C a l c u l a t i o n s by t h e I t a l i a n group [7-91 produced a d r i f t v e l o c i t y c o n s i s t e n t l y h i g h e r t h a n t h e e x p e r i - mental v a l u e s . We b e l i e v e t h e r e a s o n f o r t h i s l i e s i n t h e n e g l e c t of t h e second band which enhances t h e s c a t t e r i n g r a t e of e l e c t r o n s f o r e n e r g i e s above 0 . 1 eV. The s c a t t e r i n g p e r c e n t a g e d u e t o X-L s c a t t e r i n g , a s s e e n from o u r s i m u l a t i o n , i s s i g n i - f i c a n t o n l y when t h e e l e c t r i c f i e l d i s above 100 kV/cm. The d r i f t v e l o c i t y f o r e l e c t r i c f i e l d s under 1 0 0 kV/cm i s b a s i c a l l y n o t i n f l u e n c e d by a d j u s t i n g t h e X-L c o u p l i n g c o n s t a n t . The impact i o n i z a t i o n r a t e is v e r y s e n s i t i v e t o t h i s c o u p l i n g c o n s t a n t b e c a u s e X - L s c a t t e r i n g i s s i g n i f i c a n t when t h e e l e c t r i c f i e l d i s above 200 kV/cm. For a n assumed i s o t r o p i c t h r e s h o l d e n e r g y i n S i o f 1 . 8 eV [ 1 2 ] , a c o u p l i n g c o n s t a n t of 3.5 x lo8 eV/cm i s o b t a i n e d by f i t t i n g t h e e x p e r i m e n t a l d a t a a t 333 kV/cm. I n a r e c e n t p a p e r by Thornber [ 2 1 ] a t h r e s h o l d e n e r g y of 3 . 6 eV was o b t a i n e d from f i t s t o t h e e x p e r i m e n t a l d a t a . (The p a p e r c o n t a i n s s i m p l i f y i n g a s s u m p t i o n s . ) We have t r i e d t o a d j u s t t h e v a l u e s of t h e X - L c o u p l i n g c o n s t a n t and t h e t h r e s h o l d e n e r g y o f i o n i z a t i o n t o s e e i f such a h i g h t h r e s h o l d i s p o s s i b l e . We found t h a t a t h r e s h o l d e n e r g y o f 2 . 4 eV i s needed t o f i t t h e e x p e r i m e n t a l d a t a a t 333 kV/cm i f we c o m p l e t e l y n e g l e c t t h e X - L c o u p l i n g e f f e c t . A t h r e s h o l d of 3.6 eV seems t h e r e f o r e t o o h i g h .

Acknowledgements. - The a u t h o r s wish t o t h a n k K.

Brennan f o r h e l p i n t h e compu- t e r work and G. E. S t i l l m a n f o r many v a l u a b l e d i s c u s s i o n s . The u s e of t h e computer f a c i l i t i e s o f t h e M a t e r i a l s Research L a b o r a t o r y , U n i v e r s i t y o f I l l i n o i s , i s a l s o g r a t e f u l l y acknowledged. T h i s work was s u p p o r t e d by t h e O f f i c e o f Naval R e s e a r c h , t h e Army Research O f f i c e , and t h e J o i n t S e r v i c e s E l e c t r o n i c s Program.

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