LATERAL TRANSPORT IN SUPERLATTICES

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LATERAL TRANSPORT IN SUPERLATTICES

K. Hess

To cite this version:

K. Hess. LATERAL TRANSPORT IN SUPERLATTICES. Journal de Physique Colloques, 1981, 42

(C7), pp.C7-3-C7-17. �10.1051/jphyscol:1981701�. �jpa-00221637�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10_, Tome 42, oetobre 1981 page C7-3

LATERAL TRANSPORT IN SUPERLATTICES

K. Hess

Department of Electrical Engineering and the Coordinated Science Laboratory University of Illinois at Urbana-Champaign, Urhana, Illinois 61801, U.S.A.

Résumé. - Des résultats théoriques et expérimentaux sur le transport latéral dans des hétérojonctions sont présentés. On a montré que le changement des conditions aux li- mites (périodiques ou non) donne naissance à une série de nouveaux effets prometteurs d'applications aux composants.

Ces effets seront présentés en insistant sur l'analogie entre l'espace réel et l'es- pace des k.

Abstract. - Theoretical and experimental results are presented for lateral trans- port in layered heterojunction structures. It is shown that the variability of boundary conditions (periodic or nonperiodic) gives rise to a series of novel effects with high device potential. These effects will be presented stressing the real-space - k-space analogy.

1. Introduction. - When I heard in 1977 C. Hilsum's "Look Over The Shoulder" pre- sented at the meeting in Denton [1] , I did not realize that two years later I would almost experience another direct proof of his last quotation from Hegel [2] . In 1979 Hadis Morkoc, Ben Streetman and myself were led to the idea of real-space transfer in superlattices [3] , an effect which inter alia can be used to produce the real-space analogy of the Gunn effect in properly designed samples. We per- formed some preliminary calculations of this effect and submitted a manuscript [4]

which was characterized by the first referee as unimportant, incomplete, and mostly wrong. We would have stopped the work on this project, if we had not obtained con- firmation of our estimates by sophisticated Monte Carlo calculations performed by Glisson et al. [5] and calculations with the method of moments by Shichijo et al. [6]

as well as encouragement by discussions with J. Bardeen and H. Kroemer.

Meanwhile we also learned more about already well known variations of the real-space transfer effect. Among these variations are the electron emission from silicon into silicon dioxide which was discovered by Ning [7], the transfer of electrons to a floating gate in read only memories and the diffusion of hot elec- trons in graded gap semiconductors measured by Dargis et al. [8]. Superlattices and small quantum well heterostructure layers add many possibilities to these effects. In fact, a general correspondence of k-space and real-space transport effects can be established. It is the purpose of this paper to describe the lateral transport in superstructures and to illustrate this k-space-real-space correspondence on the basis of concrete examples.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981701

C7-4 JOURNAL DE PHYSIQUE

The major i m p e t u s t o t h e interest i n l a t e r a l t r a n s p o r t i n h e t e r o l a y e r s came from t h e e x p e r i m e n t a l v e r i f i c a t i o n o f m o d u l a t i o n doping by D i n g l e e t a l . [ 9 ] . I n s m a l l h e t e r o l a y e r s c h a r g e n e u t r a l i t y need n o t b e conserved ( i n a f i r s t approxima- t i o n ) and donors and e l e c t r o n s can b e s e p a r a t e d o v e r d i s t a n c e s much l a r g e r t h a n t h e e f f e c t i v e Bohr r a d i u s . For example, i f one dopes A 1 Ga As n e i g h b o r i n g t o GaAs

x 1-x

t h e e l e c t r o n s move t o t h e m a t e r i a l w i t h t h e lower band gap i f t h e band edge d i s c o n - t i n u i t y ( a f u n c t i o n of x ) is s u f f i c i e n t l y l a r g e . The e l e c t r o n s a r e t h e n s e p a r a t e d from t h e i r p a r e n t d o n o r s and e x p e r i e n c e much r e d u c e d i m p u r i t y s c a t t e r i n g . I n 1979 we r e a l i z e d [ 3 ] t h a t t h e i n v e r s e o f t h i s e f f e c t c a n happen i n a h i g h e l e c t r i c f i e l d p a r a l l e l t o t h e i n t e r f a c e ( n o t p e r p e n d i c u l a r , which was proposed by E s a k i i n a d i f f e r e n t c o n t e x t [ l o ] ) . E l e c t r o n s a r e a c c e l e r a t e d by h i g h f i e l d s and move up t h e (2 o r 3 d i m e n s i o n a l ) continuum o f s t a t e s i n t h e GaAs u n t i l t h e y r e a c h enough energy t o t r a n s f e r o u t . Then t h e y a r e p u l l e d back by t h e p o s i t i v e l y charged donors. It i s c l e a r t h a t t h e n t h e e l e c t r o n s e x p e r i e n c e s t r o n g i m p u r i t y s c a t t e r i n g and n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e c a n occur. These a r e t h e major i d e a s which w i l l be p r e - s e n t e d in d e t a i l . I n t h e second s e c t i o n I w i l l d e s c r i b e t h e t h e o r y o f t h e low f i e l d c o n d u c t i v i t y i n h e t e r o l a y e r s and t h e major d i f f e r e n c e s of t h e s c a t t e r i n g mechanisms t o b u l k m a t e r i a l . Then t h e t r a n s f e r mechanism and t h e g e n e r a l c o r r e -

spondence of k-space-real-space

+

e f f e c t s w i l l be d i s c u s s e d . T h i s s e c t i o n i s f o l l o w e d by e x p e r i m e n t a l r e s u l t s conf i n n i n g t h e r e a l - s p a c e t r a n s f e r e f f e c t a n d d e s c r i b i n g l a t e r a l t r a n s p o r t i n t h e Ohmic, warm e l e c t r o n and h o t e l e c t r o n range. F i n a l l y , i n t h e c o n c l u s i o n I w i l l p r e s e n t a n o u t l o o k on f u t u r e p r o s p e c t s i n t h i s a r e a .

2. A n a l y t i c a l C o n s i d e r a t i o n s

2.1. C o n d u c t i v i t y in t h e Ohmic Range

The i d e a l i z e d modulation-doped s t r u c t u r e (e.g., AlxGal-xAs-GaAs) i s shown i n F i g u r e 1 a - c . Fig. l a r e p r e s e n t s t h e c o n d u c t i o n band edge f o r a n e g l i g i b l e e f f e c t of t h e s p a c e charge. The d i s t a n c e d i s t h e s p a c i n g i n t h e A 1 Gal-xAs l a y e r which

i x

i s f r e e o f i n t e n t i o n a l doping. z i s t h e d i s t a n c e o f t h e doping from t h e maximum of t h e s q u a r e of t h e e l e c t r o n e n v e l o p e wave-function f o r t h e l o w e s t quantum s t a t e E which i s above t h e GaAs c o n d u c t i o n band edge Ec. I n F i g . l b , I have made v i s i - b l e t h e e f f e c t of t h e s p a c e c h a r g e which c a u s e s some band bending. Note t h a t zo i s s t i l l a b o u t t h e same. I n F i g . l c t h e band bending i s s o s t r o n g t h a t two q u a s i - two-dimensional l a y e r s form, one a t e a c h i n t e r f a c e . z i s t h e n much s m a l l e r . A s we w i l l s e e , t h e s c a t t e r i n g r a t e f o r i o n i z e d i m p u r i t y s c a t t e r i n g i s t h e n s t r o n g l y i n c r e a s e d b e c a u s e i t depends s e n s i t i v e l y on t h e d i s t a n c e zo. T h i s d i s t a n c e c a u s e s t h e major r e d u c t i o n of i m p u r i t y s c a t t e r i n g ; a l l t h e o t h e r e f f e c t s s u c h a s two- d i m e n s i o n a l i t y have a s m a l l e r i n f l u e n c e on t h e s c a t t e r i n g rate and m o b i l i t y .

A r i g o r o u s t h e o r y which e x p l a i n s t h e o b s e r v e d d a t a h a s t o i n c l u d e m u l t i - subband conducPion and multisubband s c r e e n i n g o f remote and background i m p u r i t i e s . It must a l s o t a k e i n t o a c c o u n t i n t e r f a c e and b u l k p o l a r o p t i c a l phonons. A l s o , t h e continuum a p p r o x i m a t i o n f o r t h e b u i l t - i n f i e l d ( F i g . 1 ) i s r a t h e r poor. I n t h e f o l l o w i n g we u s e a s i m p l e model, which i g n o r e s t h e s e d e t a i l s . W e can e x p e c t t h a t

F i g . 1: Schematic diagram o f t h e c o n d u c t i o n band edge in modulation- doped GaAs-A1 Ga _{x 1-x} A s l a y e r s i n c l u d i n g

( a ) no band bending ( b ) moderate band bending ( c ) s t r o n g band bending

Note t h e change i n z o , which i s t h e d i s t a n c e o f t h e d o n o r s t o t h e maximum e l e c t r o n c h a r g e d e n s i t y .

t h i s model d e s c r i b e s t h e t r a n s p o r t i n modulation-doped s t r u c t u r e s w i t h i n a f a c t o r of two o r so. Our model i s p u r e l y two-dimensional w i t h r e s p e c t t o t h e e l e c t r o n s . I n r e a l i t y , however, more t h a n one subband w i l l be p o p u l a t e d and s c a t t e r i n g c h a n n e l s ( i n t e r s u b b a n d s c a t t e r i n g ) a r e opened. T h i s c a n i n c r e a s e t h e s c a t t e r i n g r a t e by a b o u t 1 1 3 o v e r t h e p u r e l y two-dimensional r a t e . A c a r e f u l d i s c u s s i o n of t h i s e f f e c t was g i v e n by P r i c e [ I l l .

The e l e c t r o n d i s t r i b u t i o n i s approximated by a 6 - l i k e s h e e t o f e l e c t r o n c h a r g e l o c a t e d a t z = 0 where z i s t h e d i r e c t i o n ~ e r ~ e n d i c u l a r t o t h e l a y e r s . The donor i m p u r i t i e s a r e homogeneously d i s t r i b u t e d a t z

>

^z According t o [12,13

1 ,

t h e

0 '

s c a t t e r i n g r a t e i s g i v e n by:

where

Here k i s t h e a b s o l u t e v a l u e of t h e e l e c t r o n i c wave v e c t o r , m

*

t h e e f f e c t i v e mass, N t h e remote i m p u r i t y d e n s i t y , E t h e r e l a t i v e d i e l e c t r i c c o n s t a n t of t h e semi-

R

c o n d u c t o r , E t h e d i e l e c t r i c c o n s t a n t o f f r e e s p a c e , fi is P l a n c k ' s c o n s t a n t d i v i d e d by 2 a , and S i s t h e two-dimensional s c r e e n i n g c o n s t a n t . I n t h e l i m i t of a h i g h e l e c t r o n d e n s i t y , n a t t h e i n t e r f a c e of t h e AlxGal-xAs/GaAs l a y e r s , S i s a con-

s'

s t a n t g i v e n by S = 2 / a g where ag i s t h e e f f e c t i v e Bohr r a d i u s i n t h e GaAs [ I 2

I .

C7-6 JOURNAL DE PHYSIQUE

I f t h e s c r e e n i n g i s s t r o n g , s c a t t e r i n g o c c u r s o n l y f o r s m a l l v a l u e s of 0(kz > I ) and s i n 0 can b e r e p l a c e d by 0 i n t h e exponent of e q u a t i o n ( 1 ) . Then

The r e s u l t h o l d s f o r background i m p u r i t i e s a l s o ( i . e . , z = 0 ) and s h o u l d a l l o w a n e s t i m a t e o f t h e i n f l u e n c e of i n t e r f a c e c h a r g e s (kz v e r y s m a l l ) . I n most p r a c t i c a l c a s e s k may b e r e p l a c e d by t h e Fermi wavevector k in e q u a t i o n s (1) and ( 2 ) . T h i s

F

can b e done b e c a u s e Fermi s t a t i s t i c s i s a p p r o p r i a t e a t low t e m p e r a t u r e s where i m p u r i t y s c a t t e r i n g i s i m p o r t a n t . For one subband we have

E q u a t i o n s (1) - ( 4 ) g i v e a rough d e s c r i p t i o n of i m p u r i t y s c a t t e r i n g i n modulation- doped s t r u c t u r e s . I f t h e t r u e z-dependence of t h e subband e n v e l o p e f u n c t i o n s i s t a k e n i n t o a c c o u n t , t h e formulae a r i s i n g a r e much more i n v o l v e d . The e s s e n c e of t h e t h e o r y , however, i s r e f l e c t e d by e q u a t i o n (3) i f z i s r e g a r d e d a s a n a d j u s t a b l e p a r a m e t e r

.

F i g . 2 : Comparison of t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s o f t h e m o b i l i t y of modulation-doped s t r u c t u r e p l o t t e d v s t e m p e r a t u r e a f t e r Drummond e t a l . [13

1

---- Theoret ica I

0

Experiment

50 100 150 200 250 300 Temperature (K)

^YS-83

An example o f t h e good f i t o f t h e o r y and e x p e r i m e n t a l d a t a i s shown i n F i g u r e 2 where t h e m o b i l i t y of a modulation-doped s t r u c t u r e is p l o t t e d v s t e m p e r a t u r e . Note, however, t h a t no s a t i s f a c t o r y e x p l a n a t i o n h a s been g i v e n up t o now f o r

5 2

e x t r e m e l y h i g h m o b i l i t i e s

(2

2 x 1 0 cm /Vs) a t low t e m p e r a t u r e s (10 K) a s t h e y a r e c u r r e n t l y r e a c h e d [14

I .

2.2. The Warm E l e c t r o n Range

Warm e l e c t r o n e f f e c t s i n b u l k GaAs having low e l e c t r o n d e n s i t i e s c a n be des- c r i b e d p r o p e r l y o n l y by i n c l u d i n g t h e non-Maxwellian (non-Fermi-type) n a t u r e o f t h e d i s t r i b u t i o n f u n c t i o n . The m o b i l i t y i n b u l k GaAs w i t h a h i g h donor c o n c e n t r a t i o n ( h i g h e l e c t r o n d e n s i t i e s ) i s v e r y low and t h e r e f o r e warm e l e c t r o n e f f e c t s (below t h e Gunn t h r e s h o l d ) a r e n o t u s u a l l y observed. S e l e c t i v e l y doped (Al,Ga)As/GaAs h e t e r o - s t r u c t u r e s combine t h e u n i q u e f e a t u r e o f h i g h m o b i l i t i e s a s s o c i a t e d w i t h l a r g e e l e c - t r o n c o n c e n t r a t i o n s . Warm e l e c t r o n e f f e c t s i n t h e s e s t r u c t u r e s s h o u l d t h e r e f o r e be t r e a t e d w i t h a model i n c l u d i n g 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 . The s i m p l e s t way o f i n c l u d i n g e l e c t r o n - e l e c t r o n s c a t t e r i n g i s t h e assumption of a F e r m i - d i s t r i b u - t i o n a t e l e v a t e d e l e c t r o n t e m p e r a t u r e T f o r t h e s p h e r i c a l symmetrical p a r t o f t h e d i s t r i b u t i o n f u n c t i o n .

The c a l c u l a t i o n i s t h e n s t r a i g h t f o r w a r d u s i n g t h e two-dimensional model f o r t h e m o b i l i t y a s d e s c r i b e d above. To c a l c u l a t e t h e c a r r i e r t e m p e r a t u r e T we can u s e t h e power b a l a n c e e q u a t i o n a s d e r i v e d b e f o r e [ 1 5 ] . (At low t e m p e r a t u r e s we have t o i n c l u d e t h e power l o s s t o a c o u s t i c modes [ 1 6 1 . ) E x p e r i m e n t a l c u r v e s f o r a v e r y h i g h m o b i l i t y (y = 2 x

l o 5

cm /Va a t 1 0 K) modulation doped s t r u c t u r e s [ 1 7 ] a r e shown i n 2 F i g u r e 3 . A t h e o r e t i c a l c u r v e i s shown f o r comparison.

Fig. 3: E x p e r i m e n t a l (0) and t h e o r e t i c a l (---) r e s u l t s f o r t h e n o r m a l i z e d m o b i l i t y o f

modulation- doped s t r u c t u r e s a t moderate e l e c t r i c f i e l d s . The f u l l l i n e s a r e o n l y g u i d e s f o r t h e e y e t o t h e e x p e r i m e n t a l r e s u l t s .

A f t e r Drummond e t a l . [ I 7 1

.

I n comparing e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s , two a s p e c t s a r e a p p a r e n t . A t v e r y low f i e l d s t h e t h e o r y p r e d i c t s a d e v i a t i o n from Ohm's law, which i s s t e e p e r t h a n t h e e x p e r i m e n t a l l y o b s e r v e d d e v i a t i o n . T h i s d i s c r e p a n c y can b e a c c o u n t e d f o r

C7-8 JOURNAL DE PHYSIQUE

o n l y by a s s i g n i n g a r e l a t i v e l y l a r g e e n e r g y l o s s r a t e t o i n t e r a c t i o n s w i t h phonons.

I n o u r model a d e f o r m a t i o n p o t e n t i a l o f 10 eV which i s r a t h e r l a r g e was assumed.

T h e r e f o r e , a n e x p l a n a t i o n seems t o b e p o s s i b l e o n l y by a d d i t i o n a l phonon s c a t t e r - i n g p r o c e s s e s , s u c h a s p i e z o e l e c t r i c s c a t t e r i n g and multiphonon e m i s s i o n .

At h i g h e r e l e c t r i c f i e l d s , t h e r a t e of d e c l i n e i n t h e e l e c t r o n m o b i l i t y d e c r e a s e s , which may b e a n i n d i c a t i o n of a lower e n e r g y l o s s r a t e t h a n what was used in o u r model. T h i s t y p e o f d e v i a t i o n between t h e o r y and experiment i s t y p i c a l f o r d e v i a t i o n s o f t h e d i s t r i b u t i o n f u n c t i o n s from t h e Fermi-shape

,

which we have assumed. Around 300 V/cm a s l i g h t n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e o c c u r s which c o u l d b e connected w i t h subband r e p o p u l a t i o n and t h e sudden o n s e t of (two-dimension- a l ) p o l a r o p t i c a l s c a t t e r i n g [18

1 .

2 . 3 . The Hot E l e c t r o n Range

The e f f e c t s d i s c u s s e d s o f a r a r e n o t e n t i r e l y d i f f e r e n t from e f f e c t s observed in MOS-transistors where t h e e l e c t r o n s a l s o a r e quasi-two-dimensional. k t v e r y h i g h e l e c t r i c f i e l d s (2 3000 V/cm), however, new e f f e c t s o c c u r i f t h e band edge d i s c o n - t i n u i t y BE i s s m a l l . S i n c e AE i s a f u n c t i o n o f t h e A 1 c o n t e n t , i . e . , of x , i t can

C

b e a d j u s t e d by v a r y i n g x . For x Q 0.23 t h e band edge d i s c o n t i n u i t y AE i s s m a l l e r t h a n t h e e n e r g y s e p a r a t i o n of t h e s a t e l l i t e minima. I n o t h e r words, t h e e l e c t r o n s can move o u t of t h e GaAs b e f o r e t h e Gunn e f f e c t o c c u r s and i n t h i s way c a u s e a r e a l - s p a c e n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e i f t h e m o b i l i t y ( a t h i g h f i e l d s ) i n t h e AlxGal-xAs i s much lower t h a n t h e m o b i l i t y ( a t h i g h f i e l d s ) i n t h e GaAs. T h i s r e - q u i r e s , o f c o u r s e , heavy doping i n t h e A1 Ga A s . I n o r d e r t o d e p l e t e t h e

x 1-x

A 1 Gal-xAs a t low f i e l d s c o m p l e t e l y i t may b e n e c e s s a r y t o compensate t h e m a t e r i a l .

X

I n t h e v e r y f i r s t p u b l i c a t i o n on t h i s e f f e c t we assumed t h a t a r a t h e r wide l a y e r o f A 1 GalvxAs i s n e c e s s a r y f o r a h i g h p e a k . t o - v a l l e y r a t i o s o t h a t t h e e l e c t r o n r e a l l y

X

would " g e t l o s t " i n t h e p o o r l y c o n d u c t i n g A 1 Ga A s . The e x p e r i m e n t s showed, x 1-x

however, t h a t t h e a c t u a l t h i c k n e s s of t h e A 1 Gal-xAs i s r a t h e r u n e s s e n t i a l p o s s i b l y X

b e c a u s e t h e p o s i t i v e d o n o r s a t t r a c t t h e e l e c t r o n s s t r o n g l y enough t o keep them i n even s m a l l ( - 1 5 0 0

2

i n o u r e x p e r i m e n t s ) l a y e r s p r o v i d e d t h e e l e c t r i c f i e l d i s h i g h enough.

The d e t a i l s of t h e r e a l - s p a c e t r a n s f e r mechanism a r e f a i r l y c o m p l i c a t e d and cannot b e d e r i v e d a n a l y t i c a l l y

.

N e v e r t h e l e s s , one can g i v e e x p l i c i t d e s c r i p t i o n s f o r t h e most i m p o r t a n t f e a t u r e s . T h e r e f o r e , I w i l l t r e a t t h e s p e e d , t h e c r i t i c a l f i e l d , t h e s c a t t e r i n g mechanisms and t h e t r a n s f e r i t s e l f below a n d t h e n p r e s e n t t h e r e s u l t s o f Monte C a r l o c a l c u l a t i o n s performed by G l i s s o n e t a l . 151

.

To c a l c u l a t e t h e s w i t c h i n g speed we o b s e r v e t h a t t h e p o t e n t i a l w e l l s i n F i g . 1 a r e s i m i l a r t o t h e s t e p - l i k e w e l l s i n c h a r g e coupled d e v i c e s , where e l e c t r o n s move by d i f f u s i o n from one g a t e t o a n o t h e r . Using t h i s a n a l o g y , we o b t a i n t h e t i m e t which t h e e l e c t r o n s need t o f a l l back t o t h e GaAs l a y e r a f t e r s w i t c h i n g o f f t h e

" h e a t i n g " f i e l d [ 4 ] :

2 2

t

"

4 L 2 / s D (5)

where L2 i s t h e t h i c k n e s s of t h e AlxGal-xAs l a y e r , and D i s t h e d i f f u s i o n c o n s t a n t i n t h e A 1 Gal-xAs. T h i s formula i s v a l i d o n l y a s l o n g a s t h e d i f f u s i o n c o n c e p t a p p l i e s and t h e mean f r e e p a t h f o r phonon e m i s s i o n X i s s m a l l e r t h a n L1, t h e

ph

t h i c k n e s s of t h e GaAs l a y e r . I f X i s l o n g e r t h a n L1, t h e p r o b a b i l i t y of a n e l e c - ph

t r o n b e i n g c a p t u r e d i n t h e w e l l i s reduced by L1/Xph. Thus we have

For t y p i c a l v a l u e s such a s Ll = L2 = 400

8,

^{h <} cm, and D = 1 cm 2

Is,

we

-11 ph

o b t a i n t s < 1 . 6 2 ~ 1 0 s, which i s a n a t t r a c t i v e l y s h o r t t i m e f o r a v a r i e t y o f a p p l i c a t i o n s . Of c o u r s e , one must add t h e t i m e r e q u i r e d f o r h e a t i n g and c o o l i n g o f t h e e l e c t r o n s , about 5 x 10-l2 s , t o t s . I n d e r i v i n g Eq. ( 6 ) , it was assumed t h a t t h e r e i s n o p o t e n t i a l b a r r i e r caused by t h e i o n i z e d d o n o r s i n t h e AlxGal-xAs. The p o t e n t i a l b a r r i e r c r e a t e d by such donors i s comparable t o kTL/e f o r an A 1 Ga A s

x 1-x l a y e r width of 600

8,

a n e t donor doping c o n c e n t r a t i o n of 1016 ~ m - ~ , and TL = 300 K.

Of c o u r s e , t h e continuum approximation i s poor and t h e A1 Ga A s h a s t o be s t r o n g l y x 1-x

compensated t o a c h i e v e low m o b i l i t y .

I f t h e l a y e r s a r e v e r y t h i n and t h e p o t e n t i a l b a r r i e r c r e a t e d by t h e d o n o r s is h i g h , t h e d i f f u s i o n c o n c e p t b r e a k s down and t h e back t r a n s f e r w i l l be " t h e r m i o n i c emission" l i m i t e d . The p r i n c i p l e s o f t h e t h e r m i o n i c e m i s s i o n c u r r e n t a r e w e l l known and we can immediately o b t a i n t h e b a c k - t r a n s f e r t i m e [ 4

1

:

where A i s t h e Richardson c o n s t a n t , m

*

i s t h e e f f e c t i v e mass, m i s t h e f r e e - e l e c t - t r o n mass, $ i s t h e p o t e n t i a l c r e a t e d by t h e d o n o r s i n t h e A1 x Ga 1-x A s , and N c i s t h e d e n s i t y of s t a t e s i n AlxGal-xAs. For e@/kTL 5 2 , t i s a b o u t 10-I* s , and t h e r e f o r e t h e t r a n s f e r speed i s a l s o d e t e r m i n e d by t h e h e a t i n g and c o o l i n g t i m e and t h e t i m e needed t o r e p l e n i s h e l e c t r o n s w i t h h i g h enough k i n e t i c e n e r g y t o overcome t h e b a r r i e r . Both t i m e c o n s t a n t s a r e d e t e r m i n e d by c o l l i s i o n r a t e s and a r e about 5 x 1 0 - l 2 s. A c o r r e c t i o n f a c t o r o f L / A must b e i n t r o d u c e d f o r A > Ll.

1 ph ph

The t r a n s f e r o u t o f t h e GaAs i s a l s o dominated by t h e t i m e c o n s t a n t i n Eq. (7).

We o n l y have t o r e p l a c e @ by t h e band e d g e d i s c o n t i n u i t y and t h e l a t t i c e t e m p e r a t u r e T by t h e a c t u a l t e m p e r a t u r e o f t h e c a r r i e r s T which i s much l a r g e r t h a n TL i n h i g h

L

f i e l d s . Even i f T can be d e f i n e d ( h i g h e l e c t r o n - e l e c t r o n s c a t t e r i n g r a t e ) t h e a c t u a l c a l c u l a t i o n i s s t i l l n o t s t r a i g h t f o r w a r d b e c a u s e o f t h e p r e s e n c e of t h e i n t e r f a c e and e l e c t r o n i c h e a t c o n d u c t i o n e f f e c t s [ 6 1 . A l s o , w e have t o a c c o u n t f o r t h e two-dimensional n a t u r e o f t h e e l e c t r o n s a t low f i e l d s . Roughly s p e a k i n g , however, we can e x p e c t t h a t T i s h i g h enough f o r t h e t r a n s f e r a t a b o u t t h e same

( o r a l i t t l e s m a l l e r ) c r i t i c a l f i e l d which c a u s e s t h e Gunn e f f e c t , s i n c e p o l a r o p t i - c a l s c a t t e r i n g e x h i b i t s a s t e e p i n c r e a s e of T a t t h i s f i e l d ( a s l i g h t l y h i g h e r

C7-I0 JOURNAL DE PHYSIQUE

f i e l d f o r two-dimensional s y s t e m s [12]). We a r e now l e f t w i t h t h e d e s c r i p t i o n of t h e t r a n s f e r mechanism i t s e l f . I n a s i n g l e - l a y e r s t r u c t u r e we have t o c a l c u l a t e t h e quantum m e c h a n i c a l t r a n s m i s s i o n c o e f f i c i e n t 1191 which i s z e r o i f t h e e l e c t r o n e n e r g y i s below AE and a p p r o a c h e s o n e a t e n e r g i e s s u b s t a n t i a l l y h i g h e r t h a n AEc.

A t 1.25 AE i t i s a b o u t 0.6 [ I 9 1 i f t h e Bloch-nature o f t h e wave f u n c t i o n i s t a k e n i n t o a c c o u n t . I n a s u p e r l a t t i c e t h e e l e c t r o n d o e s n o t need t o r e a c h AE t o propa- g a t e ; it n e e d s o n l y be e x c i t e d t o t h e c o n d u c t i o n band among t h e minibands. A Monte Carlo s i m u l a t i o n which i n c l u d e s most o f t h e e f f e c t s d i s c u s s e d above ( e x c e p t t h e two- d i m e n s i o n a l e f f e c t s [ 2 0 ] ) was performed and t h e r e s u l t i n g c u r r e n t - v o l t a g e c h a r a c t e r - i z a t i o n i s shown in F i g . 4 , which a l s o shows t h e r e a l s p a c e t r a j e c t o r y o f a n e l e c - t r o n in t h e i n s e t . The f i g u r e shows c l e a r l y t h a t t h e c u r v e s can be t a i l o r e d by t h e doping c o n c e n t r a t i o n s u b s t a n t i a l l y . D e t a i l s a b o u t t h e dependences on x , AEc, e t c . can b e found i n r e f . 1 5 1 .

I I I I I I

+- cn

x.0.17

0

2 4

6 8

Field (kV/cm)

Lp-16B1

Fig. 4 : C u r r e n t - f i e l d c h a r a c t e r i s t i c s f o r t h e double-hetero- j u n c t i o n s t r u c t u r e shown i n t h e i n s e t w i t h AlxGal_,As d e n s i t i e s of 1017 cm-3 (p = 4000 cm /Vs) and 1 0 2 0 m - 3

(u

= 50 cm2/vs).

The t h i c k n e s s of t h e AlGaAs l a y e r i s 4000

2

and t h e GaAs i s 400

8

t h i c k and h a s 11 = 8000 cm2/vs i n b o t h c a s e s . The i n s e t shows a Monte C a r l o s i m u l a t i o n of t h e p a t h o f a n e l e c t r o n when a h i g h e l e c t r i c f i e l d i s a p p l i e d p a r a l l e l t o t h e l a y e r i n t e r f a c e s . I have d e s c r i b e d t h e n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e by r e a l - s p a c e t r a n s f e r in g r e a t e r d e t a i l b e c a u s e i t h a s meanwhile been v e r i f i e d e x p e r i m e n t a l l y . A c t u a l l y t h e e f f e c t i s o n l y o n e of many e f f e c t s which can be d e r i v e d from k-space-real-space

+

a n a l o g i e s . The i n t e r v a l l e y n o i s e in Gunn d e v i c e s h a s a r e a l s p a c e a n a l o g y which

can b e i m p o r t a n t i n c h a r g e coupled d e v i c e s [ Z l ] . The i n t e r v a l l e y d i f f u s i o n ( s p r e a d of c h a r g e p a c k e t b e c a u s e of d i f f e r e n t e f f e c t i v e masses i n d i f f e r e n t v a l l e y s ) h a s a r e a l - s p a c e a n a l o g y i n s u p e r l a t t i c e s a t v e r y h i g h f i e l d s ( s p r e a d o f c h a r g e p a c k e t because of d i f f e r e n t m o b i l i t i e s i n d i f f e r e n t l a y e r s ) . There i s a l s o a r e a l - s p a c e a n a l o g y t o t h e s p l i t band impact i o n i z a t i o n r e s o n a n c e which I would l i k e t o o u t l i n e

i n more d e t a i l b e c a u s e of i t s p o s s i b l e d e v i c e a p p l i c a t i o n .

The s i g n a l / n o i s e r a t i o of a v a l a n c h e p h o t o d i o d e s (APD) i s i n f l u e n c e d s i g n i f i - c a n t l y by t h e s t a t i s t i c s of t h e g a i n p r o c e s s and b e c a u s e of f e e d b a c k e f f e c t s more n o i s e i s g e n e r a t e d when b o t h e l e c t r o n s and h o l e s produce secondary p a i r s . The l o w e s t e x c e s s n o i s e i s a c h i e v e d in APDs i f t h e r a t i o o f 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) t o t h e i o n i z a t i o n r a t e of h o l e s ( B ) i s e i t h e r i n f i n i t e ( e l e c t r o n m u l t i p l i c a t i o n o n l y ) o r z e r o ( h o l e m u l t i p l i c a t i o n o n l y ) . A p o s s i b i l i t y of a c h i e v i n g t h i s i s o f f e r e d by t h e s p e c i f i c band s t r u c t u r e o f GaxAll-xSb.

I n t h i s m a t e r i a l i t happens a t a c e r t a i n composition t h a t t h e s p l i t o f f v a l e n c e band i s s e p a r a t e d from t h e t o p o f t h e v a l e n c e band j u s t by t h e amount o f t h e e n e r g y gap. T h e r e f o r e h o l e s a t t h e maximum of t h e s p l i t b a n d a r e a b l e t o impact i o n i z e . I n s u f f i c i e n t l y h i g h e l e c t r i c f i e l d s h o l e s w i l l p o p u l a t e t h e s p l i t b a n d b e c a u s e of i n t e r b a n d phonon s c a t t e r i n g . The e l e c t r o n s which a r e a b l e t o impact i o n i z e , however, do n o t have a n y s t a t e s n e a r k = 0 and t h e r e f o r e need a much h i g h e r e n e r g y t o impact i o n i z e t h a n t h e h o l e s as d i c t a t e d by b o t h t h e c o n s e r v a t i o n o f e n e r g y and momentum. The r e a l - s p a c e a n a l o g y t o t h i s e f f e c t i s c a u s e d by t h e

asymmetric band edge d i s c o n t i n u i t y f o r e l e c t r o n s AE and f o r h o l e s AEv. Consider a n e l e c t r o n a t a n e n e r g y of 1 . 5 eV i n A l A s e l e c t r o n s t r a n s f e r r i n g t o GaAs i n a l a y e r e d s t r u c t u r e . A s soon a s t h e e l e c t r o n h a s t r a n s f e r r e d it f i n d s i t s e l f

-

2 . 3 eV above t h e GaAs c o n d u c t i o n band edge and can t h e r e f o r e impact i o n i z e ( t h e t h r e s h o l d i s around 2 eV) w h i l e a h o l e w i t h s i m i l a r h i s t o r y would b e o n l y

-

1.65 eV below t h e t o p of t h e GaAs v a l e n c e band and c o u l d n o t c o n t r i b u t e t o impact i o n i z a t i o n . Device s t r u c t u r e s t o e x p l o i t t h i s e f f e c t have been proposed 1221.

Let me f i n a l l y emphasize t h a t t h e v a r i e t y of a c h i e v a b l e e f f e c t s by r e a l - s p a c e t r a n s f e r can be s u b s t a n t i a l l y i n c r e a s e d i f t h e l a y e r s can b e h e a t e d s e p a r a t e l y [ 3 ] . Switching and s t o r a g e between t h e l a y e r s w i t h t i m e c o n s t a n t s r e a c h i n g from h o u r s t o 10-'I s e c s h o u l d b e p o s s i b l e ( s e e Eq. ( 7 ) ) . T h i s makes t h e e f f e c t a t t r a c t i v e f o r d i g i t a l a p p l i c a t i o n s .

3 . E x p e r i m e n t a l R e s u l t s

Some e x p e r i m e n t a l r e s u l t s f o r t h e Ohmic and f o r t h e warm e l e c t r o n r a n g e have a l r e a d y been p r e s e n t e d i n t h e p r e v i o u s c h a p t e r . More d e t a i l e d a c c o u n t s can be found i n r e f s . 113,171.

R e s u l t s f o r t h e h o t e l e c t r o n r a n g e a r e shobm Ln F i g . 5. These r e s u l t s a r e r e p r e s e n t a t i v e f o r measurements on many w a f e r s o f d i f f e r e n t geometry and doping p r o f i l e s . The m a t e r i a l p a r a m e t e r s a n d dimensions of t h e samples which were p r e - p a r e d by l l o l e c u l a r Beam E p i t a x y a r e a s f o l l o w s . The doping d e n s i t y in t h e

JOURNAL DE PHYSIQUE

1

I

¹

I

I

I

^I

~ = 0 . 1 7 N, ~ m - ~

h

- ^AE=0.2eV

V)

.-

C

t 3

2 e

p

t

s

₊

-

C

!?? - Experimental

5 --- Theory (Monte Carlo)

0

0 2 4 6 8

Field (kV/cm)

LP-1977

Fig. 5 : E x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s f o r t h e c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of a m o d u l a t i o n doped (ND = 1017 cm-3 i n t h e

A l O , ¹ 7Ga0. e3As) s t r u c t u r e . Note t h e v e r y weak n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e i n b o t h t h e o r y and experiment. ( A f t e r Keever e t a l . [ 2 3 ] ) AlxGaldxAs l a y e r s was N

=

10

D ^{l 7} cm-3 and t h e GaAs was n o t i n t e n t i o n a l l y doped. The m o b i l i t y i n t h e GaAs l a y e r s was enhanced o v e r t h e b u l k v a l u e f o r e q u i v a l e n t doping and was t y p i c a l l y 2 x

lo4

cm / V s a t 77 K. 2 The m o b i l i t y in t h e A 1 Gal_As l a y e r s was around 1000 cm / V s between 300 and 77 K. 2 The doped A 1 Gal-xAs _X l a y e r was 1000

8

t h i c k i n a l l c a s e s , whereas t h e GaAs l a y e r s v a r i e d i n t h i c k n e s s from 400

8

t o - 1 . 0 um. I n some of t h e samples t h e GaAs l a y e r was sandwiched between t h e doped and a second (undoped) AlxGa A s l a y e r . As mentioned b e f o r e , we t h i n k t h a t t h e

1-x

a c t u a l w i d t h of t h e GaAs l a y e r i s r e l a t i v e l y u n i m p o r t a n t b e c a u s e o f t h e p u l l i n g f o r c e o f t h e d o n o r s , s i n c e t h e e l e c t r o n s in t h e G a A s w i l l a l w a y s b e w i t h i n 1000 of t h e doped AlxGal-xAs l a y e r . Although x was 0.17 f o r most of t h e d a t a r e p o r t e d h e r e , we have made s i m i l a r measurements w i t h x i n c r e a s e d t o 0.25. Au-Ge c o n t a c t s were e v a p o r a t e d on t o p o f t h e l a y e r ( t o p l a y e r AlxGa 1-x As) and a l l o y e d by h e a t i n g a t a r a t e of 400°c/min i n f l o w i n g H2 t o a f i n a l t e m p e r a t u r e o f 450'~. C o n t a c t s formed i n t h i s way proved t o b e ohmic i n most c a s e s .

The d i s t a n c e between t h e c o n t a c t s was 0.065 cm and t h e w i d t h o f t h e samples was 0 . 1 cm. R e c e n t l y we have a l s o u s e d v e r y d i f f e r e n t g e o m e t r i e s . Samples w i t h a c o n t a c t d i s t a n c e of

-

0.01 cm and a w i d t h of 0 . 1 cm and a l s o samples i n w i t h t h e i n v e r s e l e n g t h t o w i d t h r a t i o d i d n o t show s i g n i f i c a n t l y d i f f e r e n t e f f e c t s . The

measurements have been performed u s i n g s t a n d a r d s h o r t c u r r e n t p u l s e t e c h n i q u e s [ 2 3 ] . I n a l l samples u s e d up t o now we have n o t o b s e r v e d Gunn o s c i l l a t i o n s . The r e a s o n f o r t h i s i s n o t e n t i r e l y u n d e r s t o o d . However, we can p o i n t o u t t h e f o l l o w i n g d i s - t i n c t d i f f e r e n c e s between r e a l - s p a c e t r a n s f e r and t h e RWH mechanism. There i s no l o c a l m i c r o s c o p i c n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e in t h e

Al

x Ga 1-x A s a t t h e e l e c t r i c f i e l d s c o n s i d e r e d . Accumulation (and t h e r e f o r e d i p o l e ) domains cannot b e formed b e c a u s e a c c u m u l a t i n g e l e c t r o n s would b e e m i t t e d o u t o f t h e GaAs. The t o t a l l a c k of any k i n d o f i n s t a b i l i t y , however, d o e s n o t n e c e s s a r i l y f o l l o w from t h e s e arguments and might b e connected w i t h f i x e d i n t e r f a c e i n h o m o g e n e i t i e s a t which t h e e l e c t r o n s s p i l l o u t f i r s t and w i t h t h e f a c t t h a t t h e l a y e r s a r e e x t r e m e l y t h i n and t h e s e n s i - t i v i t y of t h e Gunn i n s t a b i l i t y t o t h e d i m e n s i o n a l i t y [ 2 4 ]

.

^{F i g . 6} shows r e s u l t s o b t a i n e d f o r samples which were made w i t h a s i n g l e l a y e r of doped A 1 GaldxAs on t o p

X

Field (kV/cm)

LP-LsaO

Fig. 6: C u r r e n t v o l t a g e c h a r a c t e r i s t i c of modulation doped s t r u c t u r e b e f o r e and a f t e r changing t h e s u r f a c e c o n d i t i o n s and l a y e r w i d t h by e t c h i n g .

of n o t i n t e n t i o n a l l y doped GaAs. The s u r f a c e c o n d i t i o n s and t h e A1 GalmxAs t h i c k -

X

n e s s was changed i n t h e s e samples by subsequent e t c h i n g . It was o b s e r v e d t h a t t h e Ohmic c o n d u c t i v i t y h a r d l y changed w h i l e d r a s t i c changes a p p e a r e d i n t h e h i g h f i e l d conductance. T h i s i s a v e r y s t r o n g argument f o r t h e a c t u a l t r a n s f e r of t h e e l e c - t r o n s i n t o t h e A1 Gal-xAs. G e n e r a l l y , t h e v a r i e t y of changes o f t h e c u r r e n t - v o l t a g e

X

c h a r a c t e r i s t i c which i s o b s e r v e d i n samples w i t h d i f f e r e n t doping makes i t a p p e a r t h a t new e f f e c t s o c c u r i n t h e s e s t r u c t u r e s and t h a t t h e n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e s and c u r r e n t s a t u r a t i o n s observed a r e n o t merely caused by t h e Gunn e f f e c t i n t h e GaAs l a y e r . A more d i r e c t proof o f t h e r e a L s p a c e t r a n s f e r e f f e c t can

C7-14 JOURNAL DE PHYSIQUE

p o s s i b l y b e a c h i e v e d by picosecond s p e c t r o s c o p y a n d / o r t i m e of f l i g h t measurements.

I n t h e e x p e r i m e n t s d e s c r i b e d above i n h o m o g e n e i t i e s of t h e e l e c t r i c f i e l d w i l l mask t h e t r u e m i c r o s c o p i c c h a r a c t e r i s t i c i n t h e r a n g e o f n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e . E s t i m a t e s o f t h i s e f f e c t a r e d i f f i c u l t b e c a u s e t h e continuum a p p r o x i - mation f o r t h e i m p u r i t y c h a r g e and t h e band bending is n o t good. (The s p a c i n g of t h e i m p u r i t i e s i s o f t h e o r d e r of t h e l a y e r w i d t h . ) The form of t h e c u r r e n t v o l t a g e c h a r a c t e r i s t i c a l s o depends s e n s i t i v e l y on whether f r e e c h a r g e c a r r i e r s a r e l e f t in i n t h e AlxGalPxAs o r i f t h e AlxGa As is e n t i r e l y d e p l e t e d . T h i s i s shown i n

1-x

Fig. 7. The c u r v e which shows t h e s a t u r a t i o n was measured on a sample which had no o r v e r y few e l e c t r o n s i n t h e A1 Ga As w h i l e t h e second c u r v e (almost ohmic) was

x 1-x

we*, ,,-' Slope 2 I

x.0.15 ,*'

I

ZI L

e

+ e2-

- ₁₀₀ -

0 1 2 3 4 5

Field (kV/crnl LP-1935

F i g . 7: C u r r e n t v o l t a g e c h a r a c t e r i s t i c o f modulation

-

doped s t r u c t u r e under d i f f e r e n t c o n d i t i o n s of A I D . 15Ga0. 85As d e p l e t i o n . ( A f t e r Keever et a l . [ 2 5 ] )

t a k e n from a sample w i t h e l e c t r o n s l e f t i n t h e AlxGa A s l a y e r . I n t h i s c a s e t h e 1-x

o n l y a p p e a r a n c e of t h e r e a l s p a c e t r a n s f e r i s g i v e n by t h e s l i g h t change i n s l o p e a t h i g h e l e c t r i c f i e l d s ( s e e i n s e t of F i g , 7 ) . I n a few samples w e have observed a n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e w i t h e x t r e m e l y h i g h p e a k - t o - v a l l e y r a t i o . I n t h e s e samples t h e m o b i l i t y i n t h e AlxGal-xA~ was p r o b a b l y v e r y low w h i l e t h e GaAs m o b i l i t y was s t i l l v e r y h i g h . A r e p r e s e n t a t i v e r e s u l t i s g i v e n i n F i g . 8. More d e t a i l e d d i s c u s s i o n s of some of t h e e f f e c t s mentioned can b e found in r e f s . 123,251.

F i g . 8: E x p e r i m e n t a l c u r r e n t - v o l t a g e c h a r a c t e r i s t i c o f modulation-doped s t r u c t u r e w i t h e x t r e m e l y h i g h peak- t o - v a l l e y r a t i o . Only two samples e x h i b i t e d such l a r g e e f f e c t s p o s s i b l y b e c a u s e of a v e r y low m o b i l i t y i n t h e A10.17Ga0.83As ( h i g h compen- s a t io n )

.

4 . Conclusions

It h a s been shown t h a t l a t e r a l t r a n s p o r t i n s u p e r l a t t i c e s o f f e r s new f e a t u r e s i n t h e Ohmic r a n g e , t h e warm e l e c t r o n r a n g e , and e s p e c i a l l y a t h i g h e l e c t r i c f i e l d s when t h e a v e r a g e energy of t h e e l e c t r o n i s comparable t o t h e band edge d i s c o n t i n - u i t y . I n t h e Ohmic r a n g e , h i g h m o b i l i t i e s can be a c h i e v e d by modulation doping.

The e f f e c t s a r e e s p e c i a l l y d r a m a t i c a t low t e m p e r a t u r e s . I n t h e r a n g e of warm e l e c t r o n s , s t r o n g e r d e v i a t i o n s from Ohm's law o c c u r a s compared t o b u l k m a t e r i a l . Also t h e e l e c t r o n d e n s i t i e s a r e e x t r e m e l y h i g h , making t h e 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 i m p o r t a n t . I n t h e h o t e l e c t r o n r a n g e t h e e l e c t r o n s l e a v e t h e i r h o s t l a y e r

(GaAs) and r e u n i t e e i t h e r w i t h t h e i r p a r e n t d o n o r s o r p o p u l a t e a n e i g h b o r i n g c o l d l a y e r ( i f no e l e c t r i c f i e l d i s a p p l i e d t o t h i s l a y e r ) which a c t s a s a s i n g l e g i a n t e l e c t r o n t r a p [ 3 ] . The s w i t c h i n g and b a c k - t r a n s f e r t i m e c o n s t a n t s depend exponen- t i a l l y on t h e composition x and on t h e doping, a s can b e s e e n from Eq. ( 7 ) . There- f o r e , t i m e c o n s t a n t s r a n g i n g from 1 0 -12 s e c t o many s e c o n d s and even y e a r s ( a t low t e m p e r a t u r e s ) can be a d j u s t e d a t w i l l . The e f f e c t s h o u l d be a t t r a c t i v e f o r s w i t c h - i n g a p p l i c a t i o n s of v a r i o u s k i n d s . F u r t h e r m o r e , I have shown t h a t a g e n e r a l c o r r e - spondence e x i s t s between t r a n s p o r t e f f e c t s i n k-space and r e a l - s p a c e . -f We have demonstrated t h e u s e of t h i s c o r r e s p o n d e n c e i n problems of n o i s e [ 2 1 ] and impact i o n i z a t i o n [ 2 2 ] e t c . , i n a d d i t i o n t o t h e u s e f o r Gunn-like r e a l - s p a c e t r a n s f e r d e v i c e s and s w i t c h i n g a p p l i c a t i o n s . A l l t h e s e e f f e c t s a r e consequences of t h e c o m p l i c a t e d boundary c o n d i t i o n s imposed ( b e t h e y p e r i o d i c o r n o t ) . It i s e x a c t l y t h e l a r g e v a r i a b i l i t y o f t h e boundary c o n d i t i o n s which make l a y e r e d h e t e r o j u n c t i c n s t r u c t u r e s s o a t t r a c t i v e f o r t r a n s p o r t a s d e s c r i b e d h e r e a s w e l l a s f o r o p t o e l e c - t r o n i c s [ 2 6

I .

C7- 16 JOURNAL DE PHYSIQUE

Acknowledgements. - I t i s a p l e a s u r e t o t h a n k my c o l l e a g u e s N. Holonyak, J r . , 9 . M o r k o ~ , G. E. S t i l l m a n , and B. G. S t r e e t 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 work was s u p p o r t e d by t h e O f f i c e of Naval Research, 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 E1ectronic.s Program.

R e f e r e n c e s

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5 (1978).

2. "Experience and h i s t o r y t e a c h t h a t p e o p l e and governments n e v e r have l e a r n t a n y t h i n g f r o m h i s t o r y . "

3. K. Hess, B. G. S t r e e t m a n , and H. Morkoq, "Negative r e s i s t a n c e h e t e r o j u n c t i o n d e v i c e s , " U. S. P a t e n t No. 4,257,055, March 1 7 , 1981.

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5445 (1980).

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2,

817 (1980).

T . Ning, S o l i d - S t a t e E l e c t r o n .

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273 (1978).

A. D a r g i s , S. Z h i l y o n i s , A. M a t u l i o n i s , I. Parshelyumas, Y u r i s K. P o z h e l a , A. Poshkus, E. S h i m u l i t y e , L i e t u v o s F i z i k o s R i n k i n y s

17,

Nos. 3 and 4 (1977), V i l n i u s ( i n R u s s i a n ) .

The e f f e c t was proposed by L. E s a k i and R. Tsu, IBM Research Report No.

RC-2418 (1969) and found e x p e r i m e n t a l l y by R. D i n g l e , 3. L. StBrmer, A. C.

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665 (1978).

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P. J . P r i c e , J. Vac. S c i . and T e c h n o l . , t o b e p u b l i s h e d and Annals of P h y s i c s , t o b e p u b l i s h e d .

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2,

484 (1979).

T. J. Drummond, H. M o r k o ~ , I<. H e s s , and A. Y. Cho, J. Appl. Phys., t o be p u b l i s h e d .

H a d i s Morkoq

,

p r i v a t e communication.

K. Hess, N. Holonyak, J r . , W. D. L a i d i g , B. A. Vojak, J . J . Coleman, and P. D.

Dapkus, S o l i d S t a t e Communications

34,

749 (1980).

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z,

1 9 1 (1978).

T. J. Drummond, M. Keever, W. Kopp, H. Morkoc, K. Hess, A. Y. Cho, and B. G.

S t r e e t m a n , E l e c t r o n i c s L e t t e r s , t o b e p u b l i s h e d .

If. Keever, H. M o r k o ~ , T. J. Drummond and K. Hess, t o b e p u b l i s h e d . G. C. Osbourn and D. L. Smith, Phys. Rev. B

19,

2124 (1979).

Two-dimensional e f f e c t s have been t r e a t e d by H. S h i c h i j o , T h e o r e t i c a l S t u d i e s of High F i e l d T r a n s p o r t i n 111-V Semiconductors, Ph.D. T h e s i s , U n i v e r s i t y of I l l i n o i s , C o o r d i n a t e d S c i e n c e L a b o r a t o r y Report No. R-893 (1980).

J. Y. Tang and K. Hess, IEEE T r a n s a c t i o n s o n E l e c t r o n Devices

E ,

285 (1981).

R. Chin, N. Holonyak, J r . , G. E. S t i l l m a n , J. Y. Tang, and K. Hess, E l e c t r o n i c s L e t t e r s

16,

467 (1980).

M. Keever, H. S h i c h i j o , K. Hess, S. B a n e r j e e , L. Witkowski, H. Morkoc, and B.

G. S t r e e t m a n , Appl. Phys. L e t t .

38,

36 (1981).

H. Kroemer, p r i v a t e communication.

25. M. Keever, T. Drummond, H. Morkoq, K. Hess, B. G. Streetman, and M. Ludowise,

J. Appl. Phys., to be published.

26.

K.

Hess and N. Holonyak, Jr., Physics Today, October (1980).