MOBILITY OF TWO DIMENSIONAL ELECTRON GAS IN A DELTA FET

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MOBILITY OF TWO DIMENSIONAL ELECTRON GAS IN A DELTA FET

P. Basu, K. Bhattacharyya

To cite this version:

P. Basu, K. Bhattacharyya. MOBILITY OF TWO DIMENSIONAL ELECTRON GAS IN A DELTA FET. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-251-C5-254. �10.1051/jphyscol:1987553�.

�jpa-00226757�

MOBILITY OF TWO DIMENSIONAL ELECTRON GAS IN A DELTA FET

P.K. BASU and K. BHATTACHARYYA

CAS i n Radio P h y s i c s and E l e c t r o n i c s , 92, Acharya P r a f u l l a Chandra Road, C a l c u t t a - 7 0 0 009, I n d i a

Abstract

The v a l u e s of m o b i l i t y of two d i m e n s i o n a l e l e c t r o n g a s i n a d e l t a doped FET a r e c a l c u l a t e d by s o l v i n g t h e Boltzmann e q u a t i o n by a n i t e r a t i v e t e c h n i q u e by c o n s i d e r i n g i m p u r i t y and p o l a r o p t i c phonon s c a t t e r i n g . I t i s concluded t h a t i n t h e r a n g e 77-300 K i m p u r i t y s c a t t e r i n g i s t h e most dominant p r o c e s s and t h e main c o n t r i b u t i o n comes from t h e l o w e s t subband.

1. I n t r o d u c t i o n

High E l e c t r o n M o b i l i t y T r a n s i s t o r s (HEMTs) made of GaAs and AlGaAs a r e b e l i e v e d t o b e t h e key e l e m e n t s f o r f u t u r e h i g h s p e e d d i g i t a l c i r c u i t s and computers [I]. An a l t e r n a t i v e f i e l d e f f e c t t r a n s i s t o r h a s r e c e n t l y been p r o p o s e d and f a b r i c a t e d [2-41 which o f f e r t h e same a d v a n t a g e s a s t h e HEMTs and i s composed e n t i r e l y of GaAs. I n t h e s t r u c t u r e , t h e i m p u r i t y d i s t r i b u t i o n t a k e s t h e form of D i r a c d e l t a f u n c t i o n and t h e FET h a s been named t h e b FET.

The m o b i l i t y b e h a v i o u r of two d i m e n s i o n a l e l e c t r o n g a s (2DEG) i n t h e c h a n n e l of t h e FET h a s been s t u d i e d [ 4 ] a s a f u n c t i o n of tem- p e r a t u r e and c a r r i e r d e n s i t y . The aim of t h e p r e s e n t work i s t o d e v e l o p t h e t h e o r y of m o b i l i t y and t o a s s e s s t h e r e l a t i v e s t r e n g t h s of v a r i o u s s c a t t e r i n g mechanisms. The e x p r e s s i o n f o r t h e r e l a x a t i o n t i m e and t h e method of c a l c u l a t i o n a r e g i v e n i n Sec.2. The r e s u l t s a r e p r e s e n t e d and compared w i t h t h e a v a i l a b l e d a t a i n Sec.3.

2. Theory

F o l l o w i n g S c h u b e r t e t a 1 [2-41, we have assumed t h a t t h e 2DEG i s c o n f i n e d i n a V shaped p o t e n t i a l w e l l i n which t h e wave f u n c t i o n s a r e s i n u s o i d a l . The subband e n e r g i e s En and t h e i r s p a t i a l e x t e n - s i o n s Ln a r e t h e n g i v e n by

where 6 i s t h e p e r m i t t i v i t y of t h e m a t e r i a l , ND i s t h e two dimen- s i o n a l d o p i n g d e n s i t y and t h e o t h e r symbols have t h e u s u a l meanings.

I n t h e p r e s e n t work, we s h a l l c o n s i d e r o n l y i m p u r i t y and p o l a r o p t i c phonon s c a t t e r i n g s , s i n c e d e f o r m a t i o n p o t e n t i a l a c o u s t i c and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987553

C5-252 JOURNAL DE PHYSIQUE

p i e z o e l e c t r i c phonon s c a t t e r i n g s a r e u s u a l l y n e g l i g i b l e i n t h e r a n g e of t e m p e r a t u r e of i n t e r e s t [5-71. F o l l o w i n g t h e p r o c e d u r e o u t l i n e d i n [ 6 , 7 ] we o b t a i n from t h e Boltzmann e q u a t i o n t h e f o l l o w i n g e x p r e - s s i o n v a l i d f o r e a c h subband i n t h e a b s e n c e of i n t e r s u b b a n d

s c a t t e r i n g

where So(E) = ~ [ ( ~ ~ + l ) ( l - f ~ ( E - ~ ~ ~ ~ ) H1(y-) + No(l-fo(E+?iwo)> H1(y+)I, S,(E) = ~ [ ~ ~ ( l - f ~ ( E + f i w ~ ) ) fi+ H ~ ( Y + ) I ,

s,(E) = B(N~+I.) Cl-fo(~-fiw,)1

C-

2Tc 2%

H1(y) .=

So

So

- -

I n Eq.(3) t i n d e n o t e s t h e r e l a x a t i o n t i m e f o r i m p u r i t y s c a t t e r i n g i n t h e n t h subband t a k i n g o n l y t h e i n t r a s u b b a n d p r o c e s s e s i n t o a c c o u n t . The e x p r e s s i o n f o r rin may b e o b t a i n e d f o l l o w i n g t h e p r o c e d u r e o u t - l i n e d i n [ 8 ] and c o n s i d e r i n g t h e i m p u r i t y d i s t r i b u t i o n t o b e a 6 -

f u n c t i o n . One t h u s o b t a i n s

B i s t h e a n g l e between t h e i n i t i a l and f i n a l wave v e c t o r s . The form f a c t o r s used i n t h e above E q . ( 4 ) a r e g i v e n by

2 - a 1 / 2

~ ~ ( q , L ~ / 2 ) = ~ ~ [ 4 q ~ ~ + ( 8 n /qL1) (1-e ) I t P1 = ( q 2 L1 2 +4x 2 - 1

,

The s c r e e n i n g p a r a m e t e r q s n [ 8 ] f o r t h e n t h subband i s g i v e n by

Eq.(3) i s t h e n s o l v e d by a n i t e r a t i v e t e c h n i q u e d i s c u s s e d i n [5-71 t o o b t a i n t h e v a l u e s of @(E) and t h e n t h e m o b i l i t y i s o b t a i n e d from

Fig.2. V a r i a t i o n of m o b i l i t i e s w i t h 2D d e n s i t y a t 300 K. S u b s c r i p t s 0, 1 and EXP r e f e r t o t h e v a l u e s f o r ground-, f i r s t e x c i t e d sub- band and i n e x p e r i m e n t . i s t h e w e i g h t e d a v e r a g e o f po and pl.

v a l u e s of m o b i l i t y :- m* = 0.072 mo, Ks = 12.9, KoD = 11.1, %wo/kg = 416 K.

The c a l c u l a t e d v a l u e s of m o b i l i t y a t 77 K assuming a l l t h e c a r r i e r s ( e q u a l t o N ~ ) t o occupy t h e l o w e s t subband and t h e e x p e r i -

m e n t a l v a l u e s a r e shown i n Fig.1. F o r

t h e e n e r g i e s ' c a l c u i a - t e d from Eq. ( 1 ) and t h e s t a n d a r d e x p r e - s s i o n f o r 2DEG den- s i t y i n t e r m s of sub- band e n e r g i e s and Fermi l e v e l [8].

However, t o make t h e m o b i l i t y c a l c u l a t i o n s i m p l e r , we have assumed t h a t t h e ground subband con- t a i n s t h e e x a c t num- b e r no a s c a l c u l a t e d

-- 3.0 8

V)

5

(

u; 2.0

E 2

0

above, b u t t h e r e s t

t h e n e x t h i g h e r sub:

band. The c o r r e s p o n - d i n g v a l u e s of mobi- l i t y po and p1 and t h e o v e r a l l v a l u e

-

F o r h i g h e r v a l u e s o f

0 1 6 - 2 ' 6 c o n c e n t r a t i o n s , a few

n ( ~ 1 0 M ) subbands a r e occupied.

2 0 S i n c e t h e m o b i l i t y i s h i g h e r i n u p p e r sub- Fig.1. V a r i a t i o n of c a l c u l a t e d m o b i l i t y b a n d s , t h e o v e r a l l

f o r ground subband ( p,) and v a l u e of m o b i l i t y may e x p e r i m e n t a l v a l u e s ( kD) w i t h be h i g h e r , s o t h a t

agreement between 2D c a r r i e r d e n s i t y a t 77 K. e x p e r i m e n t a l d a t a and

t h e c a l c u l a t e d v a l u e s may be p o s s i b l e .

The r e s u l t s f o r 300 K a r e shown i n Fig.2. Here, we have assumed t h a t t h e e l e c t r o n s occupy a s many a s s i x s u b b a n d s , a s p o i n t e d o u t i n [ 4 ] . The Fermi l e v e l and t h e d e n s i t i e s i n e a c h subband a r e t h e n

c a l c u l a t e d by u s i n q

-

't

Po

\

-

\

\

<

. . . - _

NEXP -.

_

-.-.

-

- -

i I I 1 I I 1 .

l o w e r d e n s i t y t h e t h e o r e t i c a l v a l u e s a r e found t o be

l a r g e r . However, t h e v a l u e s a r e r e d u c e d i f a c o u s t i c phonon s c a - t t e r i n g l i m i t e d mobi- l i t y i s i n c l u d e d . P o l a r o p t i c phonons do n o t p l a y a n y s i g - n i f i c a n t r o l e a t 77K.

C5-254 JOURNAL DE PHYSIQUE

a l o n w i t h t h e e x p e ~ i m e n t a l d a t a . We f i n d t h a t t h e m o b i l i t y l i m i t e d b y

~8

ood f o r t h e c a l c u l a t i o n o f t h e t o t a l m o b i l i t y . When t h e 2 D d e n s i t y

? o r ND) i s s m a l l , t h e a v e r a g e v a l u e i s c l o s e t o t h e e x p e r i m e n t a l v a l u e , s i n c e t h e a s s u m p t i o n t h a t o n l y two s u b b a n d s a r e o c c u p i e d i s q u i t e v a l i d . However, when t h e 2 D d e n s i t y i s l a r g e , t h e p r e s e n t v a l u e s of a r e f o u n d t o b e l o w e r t h a n t h e e x p e r i m e n t a l d a t a . I n t h i s r a n g e d u e i m p o r t a n c e t o t h e c o n t r i b u t i o n f r o m a l l t h e s i x sub- b a n d s m u s t b e g i v e n . S i n c e i s h i g h e r f o r h i g h e r s u b b a n d s and a l s o f o r l o w e r c o n c e n t r a t i o n , we may e x p e c t t h a t t h e a v e r a g e m o b i l i t y , i f p r o p e r l y w e i g h t e d , w i l l b e h i g h e r t h a n g i v e n i n F i g . 2 a n d h e n c e c l o s e r t o t h e e x p e r i m e n t a l d a t a .

We h a v e n o t c o n s i d e r e d i n t h e p r e s e n t work t h e r o l e of i n t e r s u b - band s c a t t e r i n g . A r o u g h c a l c u l a t i o n i n t h e l i n e of R i d l e y [ 9 ] shows t h a t t h i s p r o c e s s i n v o l v i n g LO p o l a r phonon i s n o t o f much i m p o r t a n c e . E x a c t c a l c u l a t i o n t h a t i n v o l v e s e x c e s s i v e n u m e r i c a l work, h a s b e e n p e r f o r m e d b y V i n t e r [lo], b y c o n s i d e r i n g LO phonon a n d a l a r g e number of s u b b a n d s i n c a s e of h e t e r o j u n c t i o n s . I n t h e p r e s e n t c a s e , a d d i t i o n a l i n t e r s u b b a n d s c a t t e r i n g w i l l b e c a u s e d b y more d o m i n a n t i m p u r i t y s c a t t e r i n g a n d c a l c u l a t i o n w i l l b e f u r t h e r

c o m p l i c a t e d d u e t o i n v o l v e d n a t u r e o f s c r e e n i n g [8].

I n c o n c l u s i o n , we h a v e o b t a i n e d t h e e x p r e s s i o n f o r r e l a x a t i o n t i m e f o r i m p u r i t y s c a t t e r i n g a n d f o r Boltzmann e q u a t i o n f o r i m p u r i t y and LO phonon s c a t t e r i n g s i n &-doped s t r u c t u r e s . The c a l c u l a t e d v a l u e s of m o b i l i t y a r e i n q u a l i t a t i v e a g r e e m e n t w i t h e x p e r i m e n t s .

The a u t h o r s a r e t h a n k f u l t o P r o f e s s o r B. R . Nag f o r s t i m u l a t i n g d i s c u s s i o n s a n d t o t h e I n d i a n S t a t i s t i c a l I n s t i t u t e , C a l c u t t a f o r c o m p u t a t i o n a l f a c i l i t i e s . The work h a s b e e n p a r t l y s u p p o r t e d b y t h e I n d i a n N a t i o n a l S c i e n c e Academy.

R e f e r e n c e s

[ l ] MIMURA T

.

^,

^,

.

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