USE OF AN STM TO DEFINE AND MEASURE AN OPERATIONAL TUNNELING TIME

HAL Id: jpa-00226820

https://hal.archives-ouvertes.fr/jpa-00226820

Submitted on 1 Jan 1987

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

USE OF AN STM TO DEFINE AND MEASURE AN OPERATIONAL TUNNELING TIME

P. Cutler, T. Feuchtwang, Z. Huang, T. Tsong, H. Nguyen, A. Lucas, T.

Sullivan

To cite this version:

P. Cutler, T. Feuchtwang, Z. Huang, T. Tsong, H. Nguyen, et al.. USE OF AN STM TO DEFINE

AND MEASURE AN OPERATIONAL TUNNELING TIME. Journal de Physique Colloques, 1987,

48 (C6), pp.C6-101-C6-106. �10.1051/jphyscol:1987617�. �jpa-00226820�

JOURNAL D E PHYSIQUE

Coffoque C6, suppl6ment a u nO1 1, Tome 48, novembre 1987

USE O F AN STM T O DEFINE AND MEASURE AN OPERATIONAL TUNNELING TIME

+* *(I)

P.H. C u t l e r , T.E. ~ e u c h t w a n ~ * , Z. Huang, T.T. Tsong, H. Nguyen ,

A.A. ~ u c a s + , + + ( ~ ) and T.E. Sullivan

(3)

Department of Physics, The Pennsylvania S t a t e University, University Park, Pennsylvania 16802, U.S.A.

(1) AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A. a n d

Department of Physics, t h e Pennsylvania S t a t e University, University Park, Pennsylvania 16802, U.S.A.

(2) IBM Almaden Research Center, San Jose, C A 95120, U.S.A. and Facult&

Universitaires Notre-Dame d e l a Paix, 5 5 0 0 0 Namur, Belgium (3)RCA Solid S t a t e Technology C e n t e r , Sumrnerville, NJ 08540, U.S.A.

A b s t r a c t

-

Recent measurements of d c c u r r e n t - v o l c a g e c h a r a c t e r i s t i c s of S c a n n i n g T u n n e l i n g Microscope j u n c t i o n s have c o n f i r m e d t h e i r e x p e c t e d h i g h r e c t i f i c a t i o n p r o p e r t y . W e have e x p l o i t e d t h i s p r o p e r t y t o ( i ) s t u d y r e c t i f i c a t i o n a t i n f r a - r e d f r e q u e n c i e s and t o ( i i ) a r r i v e a t a p r o c e d u r a l d e f i n i t i o n of a n e l e c t r o n t u n n e l i n g time. A l a s e r beam of l i n e a r l y p o l a r i z e d l i g h t i s f o c u s e d on a n STh1 j u n c t i o n and t h e r e s u l t i n g d c b i a s i n d u c e d a c r o s s t h e j u n c t i o n by t h e a l t e r n a t - i n g , a s y m m e t r i c a l t u n n e l c u r r e n t is d e t e c t e d . T h i s d c b i a s s h o u l d v a n i s h a t s u f f i c i e n t l y h i g h l a s e r f r e q u e n c y f o r f i x e d t i p - t o - s u r f a c e d i s t a n c e o r when w i t h - d r a w i n g t h e t i p away from t h e s u r f a c e f o r f i x e d l a s e r frequency. The a v e r a g e e l e c t r o n t u n n e l i n g t i m e i s t h e n t h e i n v e r s e c u t - o f f f r e q u e n c y o b s e r v e d i n s u c h a l a s e r r e c t i f i c a t i o n e x p e r i m e n t . R e s u l t s have been o b t a i n e d t h a t e x h i b i t r e c t i - f i c a t i o n f o r W-Si, W - N i , and W-Au STM j u n c t i o n s . The r e c t i f i c a t i o n e x p e r i m e n t w i t h a W-Si j u n c t i o n and a YAC l a s e r a t 1 . 0 6 m i c r o n s s h o w s s t r o n g r e s p o n s i v i t y and a n e a r l i n e a r d e c a y of r e c t i f i e d s i g n a l v e r s u s i n c r e a s i n g t i p - s u r f a c e d i s - t a n c e . The o b s e r v e d cut-off o c c u r s a r o u n d 25

%

t i p - s u r f a c e s e p a r a t i o n and is c o n s i s t e n t w i t h a n e l e c t r o n t r a v e r s a l t i m e of t h e b a r r i e r a t t h e t u n g s t e n Fermi v e l o c i t y . These r e s u l t s a r e i n a g r e e m e n t w i t h t u n n e l i n g t i m e s o b t a i n e d from n u m e r i c a l s o l u t i o n s of wave p a c k e t t r a n s m i s s i o n t h r o u g h model p o t e n t i a l b a r r i e r s .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987617

JOURNAL

DE

PHYSIQUE

Recent r . ~ e a s u r c t ~ ~ c r t t s l I j o f d c c u r r c r t c - v o l c a l ; c c l ~ ; t r a c t e r i s t i c s o i a S c ; t t t n i r ~ ~ l:utttirl- irrg E l i c r o s c u l ~ e ju t r c t i o t i ltave cotti irtiicd t l l c i r c x l ~ c c t c d I l i & I l r c c t i t i c a t i u t r p r o p e r t y .

\;a Liava previously projibsed e x p l o i t i r 8 r ; t h i s p r G p e r t y C b ( i ) s c d y r e c C i l i c ~ t i o t r i r t f r o r c d attd u 1 1 t i c : ~ l f r c q u e r i c i c s attd t o ( i i ) a r r i v e a t a p r u c c u u r a l d c f i t t i r i o t t ot' all c l c c c r o n t t r n n e l i r ~ ) ; t i n c [ 2 j . k'e e t r v i s a g e d f o c u s i r t g a l a s e r b c a ~ t o i l i t i e a r l y p u l a r i z ~ ' d L i g l r t o n a n Sl'fl j u r r c t i o r r arid d e t e c t i r i g tire d c b i a s itrduced a c r o s s t h e j u r r c t i o t i by t ~ r c a l t c r t r a t i r r g , a s y m r a e t r i c a l t u ~ r r r e l ~ u ~ r e t ~ t . Tire asyertiiacry clay ue of ~ c o r ; r c t r i c ; l l , ~ . ; r r c - r i a l a t r d / o r t l ~ c r v ~ a l o r i g i n . Tlre J c b i a s s h o u l d vatrisli aC s u f f i c i e n t l y I t i ~ h l a s e r f r e q u e n c y f o r f i x e d t i p - L O - s u r f a c e d i s t a r r c e o r when ~ i t h d r a w i r v ~ ; clre t i p away t r o u t11e s u r r ' a c e f o r f i x e d l a s e r f r e q u e n c y . Tlre a v e r a g e e l e c t r o n c u n n e l i r ~ y t i n e is thetr one- h a l f t h e i n v e r s e c u t - o f f f r e q u e n c y o b s e r v e d i n s u c h a l a s e r r e c c i f i c a c i o n e x p e r i u e n t . S i m i l a r c x p e r i n e n c s h a v e a l r e a d y b e e n c o n d u c t e d s u c c e s s f u l l y wit11 litecal-whisker d i o d e s a n d h a v e r e s u l t e d i n cite r e c e n t r e d e f i n i t i o r r o f t h e m e t e r [ l l j . The new o p p o r t u n i t y o f f e r e d by STEl i s t h e c o ~ i t r o l l a b i l i t y o f t h e j u n c t i o n and i:s n i c r o s c o p i c s i z e , e n t a i l i n g liiylr r r s p o n s i v i t y . I;e h a v e o b t a i n e d new r e s u l t s t i t a t e x h i b i t r e c t i f i c a t i o n f o r W-Si, U-Ni, a n d W-Au SSRl j u n c t i o r r s [ 4 ] . T h e o r e t i c a l c a l c u l a t i o n s o f t h e r e c t i f i - c a t i o n r a t i o and I ( V ) c u r v e s u s i n g a hypcrboloidal-tip-plaxiar-vase z1oZe1 of t h e STH j u n c t i o n s have b e e n d o n e and show g o o d agreetitent w i t h W-Au e x p e r i m e n t a l r e s u l t s .

l.[ore e x a c t t h r e e - d i m e n s i o n a l e l e c t r o n c u n r r e l i n g c a l c u l a t i o n s have a l s o b e e n d o n e u s i n g a s c a t t e r i n g t h e o r e t i c a p p r o a c h r e c e n t l y d e v e l o p e d by L u c a s , e t a l [ B ] . T h e s e r e s u l t s c a n be u s e d t o d e t e r r i n e G o r e a c c u r a t e l y t h e g e o m e t r i c a l and t h e m a l asyn- m e t r y o f t h e t u n n e l i n g c u r r e n t , p r e d i c t e d i n t h e o n e - d i m e n s i o n a l t u n n e l i n g c a l c u l a - t i o n s l 5 - 7 1 .

The r e c t i f i c a t i o n e x p e r i m e n t v i t h a W-Si j u n c t i o n a n d a YAG l a s e r a t 1.06 m i c r o n s shows s t r o n g r e s p o n s i v i t y a n d a n e a r l i n e a r d e c a y o f r e c t i f i e d s i g n a l v e r s u s i n c r e a s - i n g t i p - s u r f a c e d i s t a n c e . We t e n t a t i v e l y i n t e r p r e t t h i s r e s u l t a s d u e t o a competi- t i o n b e t v e e n e x p o n e n t i a l l y d e c r e a s i n g t u n n e l c u r r e n t s a n d e x p o n e n t i a l l y i n c r e a s i n g r e c t i f i c a t i o n r a t i o s a s s o c i a t e d w i t h e n h a n c e d g e o m e t r i c a l asymnecry when w i t h d r a w i n g t h e t i p f r o m t h e s u r f a c e . It i s u n d e r s t o o d t h a t t h e o p t i c a l and t h e r m a l e x c i t a t i o n by t h e l a s e r r a d i a t i o n may h a v e a s i g n i f i c a n t e f f e c t o n t h e t u n n e l i n g c u r r e n t . Con- s e q u e n t l y a n y t h e o r e t i c a l n o d e l t h a t i s i n v o k e d t o e x p l a i n and d i s t i n g u i s h b e t w e e n t h e ' n o r m a l ' e f f e c t o f b a r r i e r t h i c k n e s s a n d t h e i n t r i n s i c f r e q u e n c y d e p e n d e n t re-

s p o n s e o r t r a v e r s a l t i m e o n t h e t u n n e l i n g c u r r e n t s s h o u l d t a k e a c c o u n z of t h e photo- e l e c t r i c a n d t h e r m a l e f f e c t s . T h e o b s e r v e d c u t - o f f o c c u r s a r o u n d 2 5 A t i p - s u r f a c e s e p a r a t i o n and i s c o n s i s t e n t v i t h a n e l e c c r o n t r a v e r s a l t i n e t h r o u g h t h e b a r r i e r a t t h e t u n g s t e n F e r n i v e l o c i t y . T h e s e r e s u l t s a r e i n a g r e e m e n t w i t h t u n n e l i n g t i n e s o b t a i n e d f r o m n u m e r i c a l s o l u t i o n s of wave p a c k e t t r a n s m i s s i o n t h r o u g h model p o t e n t i a l b a r r i e r s and p r e s e n t e d i n S e c t i o n 111. T h e s e are compared w i t h o t h e r t h e o r e t i c a l a n d e x p e r i n e n t a l e s t i m a t e s o f t u n n e l i n g t i n e .

T h e p r o c e d u r e t o d e f i n e and m e a s u r e a n o p e r a t i o n a l t u n n e l i n g t i m e u s i n g a n S M i s o u t l i n e d i n S e c t i o n 11. I n a d d i t i o n , p r e l i m i n a r y e x p e r i m e n t a l r e s u ? t s a r e p r e s e n t e d a n d compared w i t h wave p a c k e t c a l c u l a t i o n s a n d o t h e r e s t i m a t e s of t u n n e l i n g t i m e .

11. OPERATIONAL TUNNELING TIME

T h e c o n c e p t o f a " t u n n e l i n g t i m e " h a s b e e n t h e f o c u s o f much i n t e r e s t and c o n t r o v e r s y s i n c e t h e e a r l y d a y o f quantum mechanics[9-171. c o n d o n ( 181 a n d ~ l a c ~ o l l [ 191 f i t s t s u g g e s t e d t h a t t r a n s m i s s i o n o f a p a r t i c l e t h r o u g h a c l a s s i c a l l y f o r b i d d e n r e g i o n i s c h a r a c t e r i z e d by a " . . . t i m e d u r i n g w h i c h t h e t u n n e l i n g p a r t i c l e is a c t u a l l y t r a v e r s - i n g t h e b a r r i e r . " [ l O ] A m e a n i n g f u l d e f i n i t i o n o f a t u n n e l i n g t i m e a p p r o p r i a t e t o a p a r t i c u l a r p h y s i c a l s y s t e m c a n b e debated[Y-151. F o r e x a m p l e , we h a v e p o i n t e d o u t e a r l i e r , i n t h e c o n t e x t o f a b s o l u t e l a s e r f r e q u e n c y m e a s u r e m e n t s by s u p e r h e t e r o d y n e d e t e c t i o n t e c h n i q u e s u s i n g t h e p o i n t - c o n t a c t o r m e t a l - w h i s k e r d i o d e , t h a t t h e q u e s - t i o n o f w h a t c o n s c i t u c e s t h e r e l e v a n t t u n n e l i n g t i a l e i n t h e d i o d e , and how i t r e - l a t e s t o t h e u l t i m a t e r e s p o n s i v i t y o f t h e c i r c u i t a r e i m p o r t a n t t h e o r e t i c a l , a s w e l l as c c c h n i c a l , u n r e s o l v e d i s s u e s ( 9 - 1 2 1 .

Wc. lravc r e c c r t r l y pruI,used[ Z j usirrg r h c r i o r r i i ~ r c a r and r c c t i l i ~ a r 1 0 1 1 ~ r r u p ~ r t i c ~ 01 a11 STEl j u ~ i c t i o o to nreasurc a s u i t a b l y d e f i l r c c d e l e c t r o n t u n i l e l i ~ l b o r t r a r r s i t t i u e througlr tlir j u n c t i o n . I n t h i s e x p e r i m e n t , a l i n e a r l y polarized l a s e r beau is focus- cd o n t o chc j u n c t i o n . Tlre e l e c t r i c f i e l d compo~ierrt alorrg t h e a x i s of t h e c i p b i a s e s tlie j u n c t i o n and ilrduces a c t u n t l e l i i i g a c r o s s t n e j u n c t r o r i a t tlre l a s e r i r e q u e 1 1 c y [ 2 l ~ ] . A s a r e s u l t of tlre r c c t i f y i i l g p r o p e r t i e s d e s c r i b e d e a r l i e r , a dc b i a s d e v e l o p s and a d c c u r r c t i t flows through t h e e x t e r n a l c i r c u i t a s i n a d c t c c t o r . Tllc s t r e n g t h of t h e r c c t i f i c d s i g n a l should however e x h i b i t a dependence on t h e l a s e r frequelicy because of tllc i n l i e r e n t non-zero r e s p o n s e t i m e of t h e j u n c t i o n . T h i s rcsporrse time car1 be i n t e r p r e t e d a s a n a v e r a g e " t r a v e r s a l tir.taW[9-111 f o r i n d i v i d u a l e l e c t r o n s t o c r o s s t h e vacuum b a r r j e r . I n p r a c t i c e , u o s t cunnelirrg phenonena a r e observed e i t h e r clrrough a s t a t i c o r q u a s i - s t a t i c c u r r e n t i n t e n s i t y which d o e s n o t contaizi any t h e a e p a n d e n t i n f o r m a t i o n e x p l i c i t l y . A r e c e n t example of t h i s i s t h e e l e g a n t e x p e r i n e n t by C u e r e t e t a1.[211 t o i n d i r e c t l y measure a t r a v e r s a l ticre 111 a h e t e r o s t r u c c u r e t u n n e l i n g o a r r i e r . The measurement i s done i n a s t a t i c node i n v o l v i n g t h e d e c r e a s e of tu11r;eling c u r r e n t c a u s e d by t h e e f f e c t i v e i n c r e a s e i n b a r r i e r heigiit a s a f u n c t i o n of u a g n e t f i e l d a p p l i e d t r a n s v e r s e t o t h e b a r r d e r . The t u n n e l i n g t i m e s o o t a i n e a a r c a b o u t . 1 p i c o s e c o n d f o r b a r r i e r w i u t h of '480A.

The e x p e r i m e n t proposed by t h e a u t h o r s u s e s , by c o n t r a s t , a dynamical r e s p o n s e f o r n e a s u r i n g t u n n e l i n g t i n e s . I n t h i s method, a n a t u r a l t i u e s c a l e i s provided by t h e l a s e r which c o n s t i t u t e s a n i n t e g r a l p a r t of t h e measuring a p p a r a t u s . The l a s e r c a u s e s t h e t u n n e l i n g and s i m u l t a n e o u s l y h e l p s d e t e r n i n e t h e s p e e d of t h e t u n n e l i n g p r o c e s s . I f t h e l a s e r f r e q u e n c y i s t o o h i g h , t h e e l e c t r o n t r a n s f e r p r o c e s s i s uuch d i m i n i s h e d d u r i n g h a l f t h e p e r i o d o f t h e l a s e r f i e l d . Hence, o n e s h o u l d o b s e r v e a c u t - o f f i n t h e magnitude of t h e r e c t i f i e d s i g n a l e i t h e r i ) when t h e f r e q u e n c y i s i n c r e a s e d beyond a c r i t i c a l v a l u e f o r a c o n s t a n t t i p - b a s e d i s t a n c e s, o r i i ) vhen t h e gap w i d t h is i n c r e a s e d beyond a c h a r a c t e r i s t i c cut-off w i d t h sc, w h i l e keeping t h e l a s e r f r e q u e n c y c o n s t a n t .

We have u s e d t h e s e o p e r a t i o n a l p r o c e d u r e s

t o

d e f i n e and measure t h e a v e r a g e e l e c t r o n t u n n e l i n g time i n t h e SR1 j u n c t i o n a s s i m p l y b e i n g one-half t h e i n v e r s e c u t - o f f f r e q u e n c y o r TT

-

^1/2vc. The c o n s i s t e n c y of t h i s d e f i n i t i o n i s checked by v e r i f y i n g t h a t t h e r a t i o sc/TT is a p p r o x i m a t e l y e q u a l t o t h e Fermi v e l o c i t y of t h e e l e c t r o n s i n che e l e c t r o d e s , which i m p l i e s t h a t d u r i n g t u n n e l i n g , p a r t i c l e s a c t a s i f t h e y obey c l a s s i c a l dynamics and move b a l l i s t i c a l l y ( 9 - I l l .

Kuk and c o l l a b o r a t o r s used a U-Si STM j u n c c i o n t o measure r e c t i f i c a t i o n and d o t h e c o n s t a n t f r e q u e n c y v e r s i o n of t h e t u n n e l i n g t i m e e x p e r i m e n t . Using a YAG l a s e r , w i t h

1

= l.Q6um, t h e y o b t a i n e d t h e p r e l i n i n a r y d a t a i l l u s t r a t e d i n F i g . 1. The a r r o w a t s

=

12A d e n o t e s r h e o n 2 e t of d e c r e a s i n g t u n n e l i n g c u r r e n t . The c u t - o f f d i s t a n c e

sc

i s a p p r o x i m a t e l y 25A. Assuming TT = 1/2vc s:XP/<v>, where

vc

= c/X, we f i n d T T 7 i . 8 x l 0 - ~ 5 s and <v>

;

1 . 4 0 ~ 1 0 cm/sec. S i n c e t h e Fermi v e l o c i t y is about 108 cm/s f o r n o s t u e t a l s [ 2 2 ] , t h e r e i s c o n s i s t e n t agreement w i t h t h e d e f i n i t i o n of t h e t u n n e l - i n g ticle TT.

A new a p p a r a t u s f o r more a c c u r a t e measurements i s c u r r e n t l y b e i n g c o n s t r u c t e d which w i l l i n c l u d e a s m a l l rugged STI.1 mounted o n an o p t i c a l bench and embedded i n a con- c r e t e b l o c k t o a c h i e v e g r e a t e r s t a b i l i t y and c o n t r o l of t h e l a s e r beam.

We have a l s o done c a l c u l a t i o n s f o r wave p a c k e t t u n n e l i n g t h r o u g h r e c t a n g u l a r b a r r i e r s u s i n g t h e a l g o r i t h m of C o l d b e r g e t a l . [ l 6 ] . F o r e n e r g i e s up t o 1-2 eV below t h e peak o f t h e b a r r i e r , we o b t a i n w e l l d e f i n e d t r a n s m i t t e d p a c k e t s . F o r example, t h e s e v e n f r a m e s i n Fig. 2 a r e a time s e q u e n c e d e p i c t i n g t h e t r a n s o i s s i o n of a wave p a c k e t , of mean e n e r g y E = 10.5 eV i n c i d e n t on a r e c t a n g u l a r b a r r i e r of h e i g h t Vo = 11 eV. The t r a n s m i t t e d p a c k e t is c l e a r l y e v i d e n t i n t h e l a s t f o u r f r a m e s , a l t h o u g h a ( l i n e a r ) a m p l i f i c a t i o n of 2 x 1 0 ~ is n e c e s s a r y b e c a u s e of s m a l l t r a n s m i t t e d p r o b a b i l i t y d e n s i t y

[ s e e f o r example, t h e c o r r e s p o n d i n g s e q u e n c e s i n R e f s . 14 and 161. The time i n c r e - ment f o r e a c h sequence i s 1x10-1s s. The t u n n e l i n g time is e s t i m a t e d t o be a b o u t

-

2 5 x 1 0 - ~ ~ s . T h i s is o b t a i n e d by p l o t t i n g t h e d i s p l a c e n r e n t s of i n c i d e n t and cransnrit- t e d wave p a c k e t s a s a f u n c t i o n of time. The t u n t l e l i n g time c a n t h e n be determined

C6-I04 JOURNAL

DE

PHYSIQUE

Fig. 1 . ExpcriaellC;ll p l o t o i t l ~ c l a s e r itrrluccd d c c u r r e l ~ c ila a t t r l ~ c t i o t ~ of s o p d i s t n ~ l c c s . Tllc l o s e r w ; ~ v t . l c ~ l g t h is

-

¹ . U ~ ~ I I I . Tllr arrow i ~ ~ d i c s c c s C l l r

o t l s r c of d e c r e a s i n g c u r r e u L . The v a l u e

0

.

^{of s,}

⁼

²⁵

i.

0

1

1 I I

1

0 10 2 0 3 0

C A P O l S T A N C E A

F i g . 2. Sequence d e p i c t i n g t h e t r a n s - m i s s i o n of a wave packec of e n e r g y E-lOeV through a r e c t s n g ~ l a r ~ b a r r i a r of heighc Vo=12eV G I w i d t h ~ d=SA. The t r a n s n i t t e d packec emerging i n f r a ~ e 4 i s a m p l i f i e d by 2x104. The time i n t e r v a l between e a c h frame is 1x10'15s. The tun- n e l i n g t i m e is e s t i m a t e d t o be 0 . 2 5 ~ 1 0 ~ 5 ~ .

t h e y i n t e r c e p t t h e d i s p l a c e m e n t a x i s a t t h e p o s i t i o n s c o r r e s p o n d i n g t o t h e f r o n t

-

and

l a c e d back v a l u e s i d e of of T T t h e is l i s t e d b a r r i e r . i n The t h e c a l c u - t a b l e

-h

^,"

^{- -}

^,-

^{, ".-} :h!L

^I

^-

^s-

^--

w i t h o t h e r t h e o r e t i c a l and n u m e r i c a l estimates of t u n n e l i n g t i n e s . Most of t h e e s t i m a t e d v a l u e s of t u n n e l i n g t i n e a r e r o u g h l y c o n s i s t e n t w i t h e a c h o t h e r and o n t h e o r d e r of 10'l5s. However, B u t t i k e r and ~ a n d a u e r [ l O ] and ~ t e v e n s [ l l ] t h e o r e t i c a l l y p r e d i c t a n a p p r o x i m a t e tun- ,_

n e l i n g t i m e which i s p r o p o r t i o n a l t o d ,

t h e w i d t h of t h e b a r r i e r . T h i s c o n t r a -

:I , ^{I1 1}

d i c c s H a r t n a n ' s e x p r e s s f o r

r,

which is

-- - - -- -- ^{-.- ,.- - .-- --}

i n d e p e n d e n t of d[17]. We have d o n e c a l - c u l a t i o n s of wave p a c k e t t u n n e l i n g a s a f u n c t i o n of d f o r f i x e d b a r r i e r h e i g h t m

Vo and mean e n e r g y E of t h e p a c k e t . The r e s u l t s a r e g i v e n i n F i g u r e 3, which a r e

"

p l o t s of TT v e r s u s d f o r G a u s s i a n pack-

F i g u r e 4 , we have p l o t t e d B u t t i k e r ' s

!IAiA

e t s of two d i f f e r e n t h a l f - w i d t h s 0 - I n

-

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

- - - - ^{.- .,, ,.- -- .-- --}

t T [ 1 9 ] - The agreement b o t h a t low b a r r i e r t h i c k n e s s and f o r i n c r e a s i n g l y opaque b a r r i e r s i s q u i t e good. We have a l s o computed t h e dependence of T T a s a f u n c t i o n of che k i n e t i c e n e r g y of t h e wave p a c k e t and f i n d t h a t i t i s i n a g r e e -

turlit w i t h t h e p r e d i c t i o n s of B u c t i k e r _,- and ~ n n d a u e r ( 1 O i and S t e v e n s ( l l j , b u t n o t ttartman[ 171.

'l'r\ULI: I . L'UI~IPAI(ISON OF TIIEDI~ETICAL AND EXI'EKIPIENTAL TUNNELlNC; TINES

K = (k;-k2]1/2 where ko = [2n,V0/ 2 ] 1 / 2 a n d k = 12!1iE/ 2 ] 1 / 2 .

SCliidUPY 14 J I4I;kIERLCAL 15 8.5 7 3

(Uave p a c k e t )

BUTTIi;ER and LAl;DLtiEX[ 101

IIUANG NUHEKICtU 8 12

e t a l . (Wave P a c k e t )

P

.

^CUEPET EXPEKINENTAL

e t a 1 . [ 2 1 ] (Hecero- 430 40 meV 33 meV 0.6~10-13s

s t r u c t u r e )

KUK e t a l . EXPERIPIENTAL 2 5 i ( STPI)

F i g . 3. P l o t of t h e t u n n e l i n g t i m e K T a s a f u n c c i o n of d. The r e s u l t s d r e o b c a i n e d from t h e n u m e r i c a l s o l u c i o n o f wave p a c k e t t r a n s m i s s i o n t h r o u g h a r e c t a n g u l a r b a r r i e r of h e i g h V o = l l e v . The mean e n e r g y o f t h e p a c k e t i s i0eV.

The t y o c u r v e s c o r r e s p o n d t o U =l6A (. ) and 8 A ( A ) .

F i g . 4. P l o c ,of B u c c i k e r ' s e x a c t e x p r e s s i o n f o r t h e t r a v e r s a l time TT a c r o s s a r e c t a n g u l a r b a r r i e r a s a f u n c t i o n s of b a r r i e r w i d t h d[Eq. 2.20a i n Ref. [ 9 ]

1.

The b a r r i e r h e i g h c Vo i s 10.5eV and t h e e n e r g y o f che cunnel- i n g p a r t i c l e is 10.0eV.

C6-106 JOURNAL DE PHYSIQUE

The r e s u l t s of t h e STM t u n n e l i n g t i m e e x p e r i m e n t , a l t h o u g h p r e l i m i n a r y , and t h o s e of t h e wave p a c k e t c a l c u l a t i o n s l e a d t o t h e f o l l o w i n g t e n t a t i v e c o n c l u s i o n s :

1 ) There i s a d e l a y t i m e i n v o l v e d i n quantum mechanical t u n n e l i n g , which i s of f i n i t e d u r a t i o n and 10-15s f o r o r d i n a r y b a r r i e r dimensions.

2) The t h e o r i e s of B u t t i k e r and Landauer and Stevens f o r t r a v e r s a l time i n b a r r i e r s a r e i n good agreement q u a l i t a t i v e l y and q u a n t i t a t i v e l y w i t h numeri c a l s o l u t i o n s of wave p a c k e t dynamics and t u n n e l i n g .

More d e t a i l e d d i s c u s s i o n s on t h e e x p e r i m e n t a l r e s u l t s and t h e o r e t i c a l a n a l y s i s of t h e t u n n e l i n g t i m e measurements w i l l be r e p o r t e d elsewhere.

One of u s (N.H) w i s h e s t o s i n c e r e l y thank D r . Young Kuk and M r . P a u l Silverman f o r t h e i r guidance and a s s i s t a n c e d u r i n g t h e e x p e r i m e n t a l phase of t h s work. Two of u s (PHC and ZH) would l i k e t o acknowledge h e l p f u l c o n v e r s a t i o n s w i t h P r o f e s s o r s E. Kazec and R. H. Good.

REFERENCES

+, *

T h i s work h a s been s u p p o r t e d i n p a r t by t h e O f f i c e of Naval Research, A r l i n g t o n , V i r g i n i a , C o n t r q c t No. N00014-86K-0160, t h e NATO Research Grant Program No. 1902(.+), and IBM, Belgium

J. A. S t r o s c i o , R. M. F e e n s t r a and A. P. F e i n , Phys. Rev. L e t t . 5 7 , 2579 (1986).

P. H. C u t l e r , T. E. Feuchtwang, T. T. Tsong, H. Nguyen and A. A. Lucas, Phys.

Rev. B: Rapid Commun,

35,

7774 (1987).

K. M. Evenson, "Frequency Measurements from t h e Microwave t o t h e V i s i b l e , t h e Speed of L i g h t and t h e R e d e f i n i t i o n of t h e Meter," i n Quantum Metrolopy and Fundamental P h y s i c a l C o n s t a n t s , ed. by P. H. C u t l e r and A. A. Lucas, NATO AS1 S e r i e s B-98, (Plenum P r e s s , New York, 1983).

A. A. Lucas, P. H. C u t l e r , T. E. Feuchtwang, T. T. Tsong, Y. Kuk, H. Nguyen, P. H. Silverman and T. E. S u l l i v a n , t o be p u b l i s h e d , J . Vac. S c i . Technol.

A. A. Lucas, A. Moussiaux, M. Schmeits and P. H. C u t l e r , Commun. Phys.

2,

169 (1977).

N. M. Miskovsky, P. H. C u t l e r , T. E. Feuchtwang and A. A. Lucas, I n t e r n . J.

I n f r a r e d and M i l l i m e t e r Waves

2,

739 (1981).

H. Q. Nguyen, P. H. C u t l e r , T. E. Feuchtwang, N. Miskovsky and A. A. Lucas, Surf. S c i .

160

(1985).

A. A. Lucas, H. Morawitz, G. R. Henry, J. P. Vigneron, Ph. Lambin, P. H. C u t l e r and T. E. Feuchtwang, B u l l . Am. Phys. Soc. 32, 648 ( 1 9 8 7 ) , Abs I07 and t o be p u b l i s h e d , Phys. Rev. B.

A. A. Lucas and P. H. C u t l e r , S o l i d S t a t e Comm. l3, 361 (1973).

T. E. S u l l i v a n , P. H. C u t l e r and A. A. Lucas, S u r f . S c i . 54, 561 (1976); 62, 455 (1977).

N. M. Miskovsky, S. J. Shepherd, P. H. C u t l e r , T. E. S u l l i v a n and A. A. Lucas, Appl. Phys. L e t t .

3,

560 (1979);

37,

189 (1980).

S e e , f o r example, F i g s . 2 and 3 i n T. E. Feuchtwang, P. H. C u t l e r , N. M.

Miskovsky and A. A. Lucas i n Quantum Metrology and Fundamental P h y s i c a l C o n s t a n t s , ed. by P. H. C u t l e r and A. A. Lucas, NATO AS1 S e r i e s B-98 (Plenum, New York, 1983).

M. B u t t i k e r , Phys. Rev.

E,

6178 (1983).

M. B u t t i k e r and R. Landauer, Phys. Rev. L e t t . 49, 1739 (1982).

K. W. S t e v e n s , Eur. J . Phys. 1, 98 (1980).

T. E. S u l l i v a n , A. A. Lucas ayd P. H. C u t l e r , Appl. Phys.

14,

289 (1977).

J. W. Gadzuk and A. A. Lucas, Phys. Rev.

g ,

4770 (1973).

P. Schnupp, T h i n S o l i d Films

2,

177 (1968).

K. K. Thornber, T. McGill and C. A. Mead, J. Appl. Phys. 38, 2384 (1967).

V. Goldberg, H. M. Schey and 3 . L. Schwartz, Am. J. Phys. 3 5 , 177 (1967).

T. E. Hartman, J. Appl. Phys.

33,

3427 (1962).

E. U. Condon, Rev. Mod. Phys.

3,

43 (1931).

L. A. MacColl, Phys. REV. 40, 621 (1932).

P. G u e r e t , A. B a r a t o f f and E. Marclay, Europhys. L e t t . 3 , 367 (1987).

G. Burns, S o l i d S t a t e P h y s i c s , (Academic P r e s s , N.Y., 1985), T a b l e 9.4.