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Submitted on 1 Jan 1981
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HOT CARRIER SPACE AND TIME DEPENDENT TRANSIENTS IN SHORT CHANNEL GALLIUM
ARSENIDE DEVICES
H. Grubin, G. Iafrate, D. Ferry
To cite this version:
H. Grubin, G. Iafrate, D. Ferry. HOT CARRIER SPACE AND TIME DEPENDENT TRANSIENTS
IN SHORT CHANNEL GALLIUM ARSENIDE DEVICES. Journal de Physique Colloques, 1981, 42
(C7), pp.C7-201-C7-206. �10.1051/jphyscol:1981723�. �jpa-00221660�
HOT CARRIER SPACE AND TIME DEPENDENT TRANSIENTS IN SHORT CHANNEL GALLIUM ARSENIDE DEVICES
55HrL. Grubin, G.J. Iafrate* and D.K. Ferry**
Scientific Research Associates, Inc. P.O. Box 498, Glastonbury, CT 0S0S3, U.S.A.
*U.S. Army Electronics Technology and Devices Laboratory, Fort Monmouth, New Jersey, U.S.A.
**Colorado State University, Fort Collins, Colorado, U.S.A.
Abstract - We examine transient carrier transport in gallium arsenide when the fields change temporally and spatially at a finite rate. For temporal changes and uniform fields the peak overshoot velocity is shown to be sensitive to bias rise times. For spatial changes the mean carrier velocity is extremely sensitive to gradients, and while overshoot can occur it is also below the peak uniform field value.
Introduction - Among the earliest papers to deal with transient carrier transport (TCT) and to include overshoot contributions were those of Butcher and Hearn and
2 3 Rees . These studies were followed by Ruch's , whose significance was highlighted
4
by Frey and coworkers in a long series of papers. These calculations established that on short time scales (of the order of an LO phonon intravalley scattering time for GaAs) the mean velocity of an ensemble of carriers could attain values of velocity substantially greater than their steady state values. These results have since generated a great deal of scientific and technological interest. However, while experimental data ' are consistent with the concept of velocity overshoot,
there are difficulties in directly linking its effect to the operating properties of room temperature devices, particularly FETs. Although this absence of linkage may be partly due to limitations of resolution we think part of the problem lies in the unfinished picture of TCT. At the center of this is the fact that virtually all discussions designed to isolate overshoot phenomena are uniform field calculations in which carriers respond to fields that suddenly change from one value to another.
The results of this calculation are useful in providing upper bounds for the peak transient velocity. But, they do not provide a realistic estimate of the true transient velocities within a device. The theoretical data for device design is thereby limited, and we require more representative device/circuit simulations to
KSupported by the U.S. Office of Naval Research
Résumé - Nous examinons i c i l e s phénomènes t r a n s i t o i r e s de t r a n s p o r t dans l ' a r s e n i u r e de gallium lorsque l e s champs v a r i e n t dans le temps e t l ' e s p a c e avec une v i t e s s e f i n i e . Pour des v a r i a t i o n s temporelles e t des champs u n i - formes ; l ' o n montre que le p i c de s u r v i t e s s e e s t s e n s i b l e au temps de montée de l a p o l a r i s a t i o n . Pour des v a r i a t i o n s dans l ' e s p a c e la v i t e s s e moyenne des porteurs e s t extrêmement sensible aux gradients e t quoique l ' o n puisse o b t e n i r une s u r v i t e s s e , c e l l e - c i e s t plus f a i b l e que c e l l e obtenue dans un champ uniforme.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981723
C7-202 JOURNAL DE PHYSIQUE
a c c o u n t f o r TCT and m o d i f i c a t i o n s a r i s i n g from f i e l d s t h a t change t e m p o r a r i l y and s p a t i a l l y a t a f i n i t e r a t e . Some s p a t i a l s t u d i e s have a l r e a d y a ~ p e a r e d , ~ - ' ~ b u t we a r e s h o r t of a consensus a s t o a phenomenology of t r a n s p o r t w i t h i n submicron d e v i c e s . The p u r p o s e of t h i s p a p e r i s t o i s o l a t e some of t h e d e t a i l s of TCT f o r g a l l i u m a r s e n i d e when f i e l d s change a t f i n i t e r a t e s and t o s u g g e s t t h e t y p e s of c a l c u l a t i o n s t h a t would b e u s e f u l f o r p r o v i d i n g a v i a b l e phenomenology. The r e s u l t s s h o u l d h e p a r t i c u l a r l y r e l e v a n t i n t h e d e s i g n o f such d e v i c e s a s a submicron f i e l d e f f e c t t r a n s i s t o r .
The v e h i c l e s f o r o u r d i s c u s s i o n a r e s o l u t i o n s t o t h e f i r s t t h r e e moments of t h e 1 3 Boltzmann t r a n s p o r t g q u a t i o n (MBTE), assuming a d i s p l a c e d Maxwellian d i s t r i b u t i o n
.
-
Within t h i s framework we a t t a c h no meaning t o b a l l i s t i c t r a n s p o r t , s i n c e i m p l i c i t i n o u r assumptions a r e s t r o n g e l e c t r o n - e l e c t r o n i n t e r a c t i o n s . We examine two extreme c a s e s : (1) uniform f i e l d s , where a l l s p a t i a l d e r i v a t i v e s approach z e r o (
- a
0) and'(2) t h e s t e a d y s t a t e nonuniform f i e l d s , where a l l time d e r i v a t i v e s a r e z e r o (-a t
= 0 ) . For b o t h c a s e s t h e d e v i c e s i m u l a t e d i s p a r t of a r e s i s t i v e c i r c u i t . The d e v i c e , a s r e p r e s e n t e d by MBTE, i s i n p a r a l l e l w i t h a g e o m e t r i c c a p a c i t a n c e , b o t h of which a r e i n s e r i e s w i t h a r e s i s t o r and a d c s o u r c e .Uniform F i e l d Temporal T r a n s i e n t s
-
For uniform e l e c t r i c f i e l d s a h o s t of c a l c u l a - t i o n s 3'4714 have been performed i n which a t y p i c a l e l e c t r o n w i t h z e r o mean i n i t i a l v e l o c i t y r e s p o n d s t o a n e l e c t r i c f i e l d of f i n i t e v a l u e . We i n t e r p r e t t h i s c a l c u l a - t i o n t o mean t h a t a c o l l e c t i o n o f c a r r i e r s i n t h e r m a l e q u i l i b r i u m w i t h t h e background l a t t i c e i s suddenly s u b j e c t e d t o a f i e l d whose v a l u e i n c r e a s e s from z e r o t o a f i n i t e v a l u e i n z e r o time ( o r n u m e r i c a l l y i n one t i m e s t e p ) . The r e s u l t s of t h i s c a l c u l a - t i o n a r e r e p r e s e n t e d i n F i g . l a , where t h e y a r e t a k e n a s h a v i n g t h e s i g n i f i c a n c e of a n upper bound. The remaining c u r v e s which were o b t a i n e d f o r a b i a s t u r n e d on a t a f i n i t e r a t e show a n e x p e c t e d r e d u c t i o n i n peak v e l o c i t y . The r e s u l t s a t dFo/dT = 4kv/cm/30ps (Fig. l b ) a r e n e a r t h o s e e x p e c t e d from a c a l c u l a t i o n i n which t h e s t e a d y s t a t e f i e l d dependent v e l o c i t y r e l a t i o n i s used.- 1 5 F i g . 1: Uniform v ( t ) c a l c u l a t i o n s w i t h
b i a s rise t i m e , Tb, a s a p a r a m e t e r . The 1 maximum b i a s f i e l d i s 21.5kv/cm. For ( a )
- t h e maximum f i e l d i s reached i n one time s t e p . For (b) Tb=160 p s . ( c ) Tb=16ps.
v0=3 . 7 ~ 1 0 ~ c m / s e c . To=O .32ps.
- 0 ,
. - ~ . ~ f l - -
0
o 100 M O 3m roo sm 600
I l M L / T O ( C U I Y E O A N O t l
c i a l l y i n t r o d u c e a s p a t i a l dependence through t h e r e l a t i o n d = vx30 p s , t h e n f o r v = 2 x
l o 7
cmlsec, d = 6 microns, and a f i e l d i n c r e a s i n g by 4kv/cm over t h i s d i s -t a n c e t h e v e l o c i t y w i l l
not
e x h i b i t s i g n i f i c a n t overshoot. S i m i l a r f i e l d changes over a d i s t a n c e of 0.6 microns c a u s e nonequilibrium e f f e c t s t o appear b u t t h e peak v e l o c i t y shows only a 30% i n c r e a s e , much below t h a t of curve l a . The r e s u l t s a r e p a r t i c u l a r l y r e l e v a n t because t h e s e f i e l d and v e l o c i t y changes a r e what we may expect over a major f r a c t i o n of a one micron g a t e l e n g t h FET. More r a p i d f i e l d changes such a s t h a t over a s h o r t e r g a t e r e g i o n a r e l i k e l y t o i n c r e a s e overshoot but h e r e f i e l d and c a r r i e r temperature g r a d i e n t s w i l l make major c o n t r i b u t i o n s and i n some c a s e s1 5 v i r t u a l l y e l i m i n a t e t h e e f f e c t s of overshoot
.
Nonuniform F i e l d S p a t i a l T r a n s i e n t s - From uniform f i e l d c a l c u l a t i o n s we have become accustomed t o t h i n k i n g i n terms of v e l o c i t y v e r s u s time f o r t r a n s i e n t c a l c u l a t i o n s . I n t h e absence of r e a c t i v e c i r c u i t elements t h i s s c a l e s c u r r e n t v e r s u s time which, i n p r i n c i p l e , i s e x p e r i m e n t a l l y a c c e s s i b l e . For submicron d e v i c e s i n which s p a t i a l c o n t r i b u t i o n s a r e s i g n i f i c a n t , c u r r e n t no l o n g e r s c a l e s v e l o c i t y . V e l o c i t y , w h i l e now d i f f i c u l t t o measure, i s s t i l l u s e f u l f o r phenomenology. We d i s c u s s t h i s w i t h i n t h e bounds of a one-dimensional c a l c u l a t i o n i n which we c a l c u l a t e c u r r e n t v e r s u s t i n e : j ( t ) = j n ( x , t )
+
E,
w i t hThrough c u r r e n t c o n t i n u i t y , j ( t ) i s s p a t i a l l y independent.
One dimensional s o l u t i o n s t o t h e Boltzmann t r a n s p o r t e q u a t i o n have appeared i n t h e literature7-''. What we emphasize i s t h e micron s c a l e s p a c e chage d i s t r i b u t i o n w i t h i n s t e a d y s t a t e , a l t h o u g h we a l s o i n c l u d e t h e time e v o l u t i o n t o s t e a d y s t a t e . The
3- a 3
0 s o l u t i o n s r e p r e s e n t t h e kind of i n f o r m a t i o n we need f o r FET and o t h e rt
d e v i c e modeling.
% 0
0 8 -
: \ 0 6 -
5 0 4 -
0 2 -
0
I OMICRON
0 2 0 4 0
F i g . 2: Nonuniform f i e l d c u r r e n t v e r s u s time w i t h J o = N ev
0 0 '
5 0
C7-204 JOURNAL DE PHYSIQUE
I n examining s p a t i a l t r a n s i e n t s we become aware of s i g n i f i c a n t d i f f e r e n c e s from t h a t of t h e uniform f i e l d c a l c u l a t i o n s . I n t h e uniform f i e l d c a l c u l a t i o n s i t i s t h e f i e l d d r i v e n s t e a d y s t a t e e l e c t r o n temperature t h a t determines t h e d i s t r i b u t i o n of c a r r i e r s i n t h e c e n t r a l and s a t e l l i t e v a l l e y . For nonuniform f i e l d s where t h e y a r e s t r o n g g r a d i e n t s i n s p a c e charge, f i e l d , e t c . , t h e s e s i g n i f i c a n t l y modify t h e uniform f i e l d e l e c t r o n temperature dependence. Two d i s t i n c t s e t s of computations i l l u s t r a t e t h e above i d e a . I n t h e s e c a l c u l a t i o n s , two elements were chosen, each w i t h a doping l e v e l of 1 0 ~ ~ / c r n ~ . One element was 1 micron l o n g , t h e second, 0.5 microns. For each, a doping d e p r e s s i o n of 0.9 x 101'/cm3 was i n t r o d u c e d . For t h e 1 micron element i t began a t 0 . 1 microns downstream from t h e s o u r c e and continued a n o t h e r 0.2 microns.
For t h e 0.5 micron element t h e beginning was a t 0.05 microns and continued f o r 0.1 microns. Each n o n l i n e a r element was i n s e r i e s w i t h a r e s i s t o r and a dc source. The magnitude of t h e dc s o u r c e was chosen s o t h a t , should t h e f i e l d s be uniform, t h e average f i e l d a c r o s s each would b e e q u a l i n v a l u e . F i g u r e 2 shows c u r r e n t v e r s u s time f o r t h e elements. These p r o f i l e s show a remarkable s i m i l a r i t y t o t h e uniform v e l o c i t y - t i m e p r o f i l e s , t e n d i n g t o camouflage a r i c h space charge d i s t r i b u t i o n . The c a l c u l a t i o n s f o r b o t h c a s e s d i s p l a y e d v e l o c i t y overshoot d u r i n g t h e i n i t i a l t r a n s i e n t ; only p a r t of which can b e a t t r i b u t e d t o t h e temporal c o n t r i b u t i o n s . I n s t e a d y s t a t e , t h e s h o r t e r element e x h i b i t e d no s p a t i a l overshoot w h i l e t h e longer element d i d .
The d i f f e r e n c e s i n t h e s e r e s u l t s a r e a t t r i b u t e d t o q u a n t i t a t i v e d i f f e r e n c e s i n t h e s p a c e c h a r g e d i s t r i b u t i o n a s i l l u s t r a t e d i n F i g u r e s 3 and 4. By comparison, t h e s h o r t e r element h a s s t e e p e r g r a d i e n t s i n c a r r i e r d e n s i t y , h i g h e r f i e l d s , c a r r i e r temperature and enhanced t r a n s f e r a c r o s s t h e notched r e g i o n . The mean v e l o c i t y ( n o t shown) i s n e a r s a t u r a t i o n even though t h e f i e l d a c r o s s t h e n o t c h i s n o t h i g h enough f o r s a t u r a t i o n , a s determined by t h e uniform f i e l d c a l c u l a t i o n s . The q u e s t i o n of n e g a t i v e d i f f e r e n t i a l m o b i l i t y t h e r e f o r e a r i s e s and we examined i t by e x t r a c t i n g c o n t r i b u t i o n s from t h e d e r i v a t i v e of e l e c t r o n p r e s s u r e , a term which t e n d s t o behave a s a d i f f u s i o n c u r r e n t , from t h e t o t a l c u r r e n t . The remaining term i s f o r m a l l y t h e f a m i l i a r conduction c u r r e n t (some c o n t r i b u t i o n s remain from d e r i v a t i v e s of t h e mean k i n e t i c energy) and s o should i n d i c a t e t h e presence o r absence of NDM. With t h e e x c e p t i o n of extrema, NDM i s marginal. Across t h e r i g h t hand p a r t of t h e element where t h e f i e l d changes more g r a d u a l l y , t h e r e were t r a c e s of NDM, b u t i t was n o t
c l e a r t o what e x t e n t t h i s r e s u l t was model dependent. For comparison we r e f e r t o an e a r l i e r uniform f i e l d s t u d y where we concluded t h a t t r a n s f e r a c r o s s a 0.1 micron r e g i o n would n o t l i k e l y y i e l d NDM 16
.
For t h e l o n g e r element we d i s p l a y t h e mean c a r r i e r v e l o c i t y v ( x , t ) = j ( x , t ) / n e n ( x , t ) , which i s a more g e n e r a l e x p r e s s i o n than t h e uniform f i e l d v e l o c i t y and i n c l u d e s d i f f u s i o n i m p l i c i t l y . For t h i s element point-by-point correspondence between v ( x , t ) and F ( x , t ) d i s p l a y s s p a t i a l overshoot. However, t h e s e e x c e s s v a l u e s of v e l o c i t y a l s o i n c l u d e s i g n i f i c a n t c o n t r i b u t i o n s from g r a d i e n t s (See a l s o Ref. 1 5 ) . Indeed, i f we were t o d i s c u s s a c a r r i e r v e l o c i t y through t h e f a m i l i a r phenomenological
f i e l d , ( c ) mobile c a r r i e r d e n s i t y (d) background d e n s i t y , ( i n s e t ) c e n t r a l v a l l e y d e n s i t y .
Fig. 4: S p a t i a l d i s t r i b u t i o n s : ( a ) mean c a r r i e r v e l o c i t y , (b) i n t e r n a l f i e l d , ( c ) mobile c a r r i e r d e n s i t y , ( i n s e t ) c e n t r a l v a l l e y d e n s i t y .
C7-206 JOURNAL DE PHYSIQUE
s t e a d y s t a t e ( s s ) c o n d u c t i o n p l u s d i f f u s i o n r e l a t i o n v = vss -
12
n -aT an, t h e n n e a r a d e p r e s s i o n i n n , v s h o u l d b e g r e a t e r t h a n vss.Conclusion
-
What a r e t h e s e r e s u l t s t e a c h i n g . On one l e v e l we can i n t e r p r e t n u m e r i c a l r e s u l t s , a s above. On a second l e v e l we can i n f e r some g e n e r a l p a t t e r n s . Here, i f we go back t o t h e l a t e s i x t i e s and t h e i n t e r p r e t a t i o n s of t r a n s f e r r e d e l e c t r o n d e v i c e b e h a v i o r , we r e c a l l t h e c o n c l u s i o n t h a t s i g n i f i c a n t , l o c a l l y c o n f i n e d , s p a t i a l non- u n i f o r m i t i e s c o n t r o l d e v i c e b e h a v i o r . The above r e s u l t s a r e c o n s i s t e n t w i t h t h e s e e a r l i e r NDM s t u d i e s 1 7 . For h e r e , t h e s h o r t e r element s u s t a i n e d a l a r g e r f i e l d g r a d i e n t n e a r one o f t h e b o u n d a r i e s , enhanced e l e c t r o n t r a n s f e r , s a t u r a t i o n i n c a r r i e r v e l o c i t y , and a low n e t c u r r e n t , w h i l e i n t h e l o n g e r element t h e f i e l d n e a r one of t h e b o u n d a r i e s i s lower, t h e r e i s a more e q u i t a b l e s h a r i n g of p o t e n t i a l between t h e r e s i s t o r and t h e n o n l i n e a r element and t h e c u r r e n t i s h i g h e r . ( I n sub- micron d e v i c e s we may e x p e c t some s p a c e c h a r g e p r o p a g a t i o n . For t h e l o n g e r element d i s c u s s e d h e r e t h e r e was s t i l l some r e s i d u a l time dependence b u t i t was n o t examined.) The r e s u l t s a l s o r e a c h t h a t t h e achievement of h i g h s p e e d s i n p r a c t i c a l d e v i c e s w i l l n e c e s s a r i l y r e q u i r e t h e i n c o r p o r a t i o n of s p a t i a l t r a n s i e n t s i n t o d e v i c e modeling.R e f e r e n c e s
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