- Email: [email protected]
Interaction of tunneling electrons with collective modes of the electrodes
Solid State C o m m u n i c a t i o n s ,
Vol. 6, pp. 1 9 3 - 1 9 7 ,
1968.
Pergamon Press.
P r i n t e d in G r e a t B r i t a i n
I N T E R A C T I O N O F TUNNELING E L E C T R O N S WITH C O L L E C T I V E MODES O F THE E L E C T R O D E S * L . C . D a v i s ¢ and C . B . Duke D e p a r t m e n t of P h y s i c s and M a t e r i a l s R e s e a r c h L a b o r a t o r y , U n i v e r s i t y of I l l i n o i s , U r b a n a , I l l i n o i s , U . S . A . ( R e c e i v e d 2 J a n u a r y 1968)
The a l t e r a t i o n of the t u n n e l i n g c u r r e n t due to b u l k e l e c t r o n phonon c o u p l i n g in one of the e l e c t r o d e s i s c a l c u l a t e d u s i n g the t u n n e l i n g - H a m i l t o n i a n m e t h o d . L i n e s h a p e s a r e p r e s e n t e d for d e f o r m a t i o n - p o t e n t i a l c o u p l i n g to o p t i c a l phonons in p - t y p e s i l i c o n . The p r e d i c t e d l i n e s h a p e of d e I / d V 2 in a metal-oxide-semiconductor junction shows deviations from the o n e - e l e c t r o n b a c k g r o u n d which a r e s y m m e t r i c a l about z e r o b i a s in c o n t r a s t to a n t i s y m m e t r i c a l s t r u c t u r e p r e d i c t e d due to i n e l a s t i c phonon c r e a t i o n b y the t u n n e l i n g e l e c t r o n .
a r i s e s f r o m a b a n d of c a r r i e r s at e n e r g y ¢ = - eV of width eAV. The t u n n e l i n g p r o b a b i l i t y of t h e s e carriers is determined ~ by their associated v a l u e s of ~k. T h e r e f o r e f o r V -~ =L ~mo/e the p o l a r o n d i s p e r s i o n in the c a r r i e r s ' e n e r g y s p e c t r a c a u s e s l a r g e c h a n g e s in t h e i r t u n n e l i n g p r b b a b i l i t i e s and h e n c e in d ~ I / d V 2. A s i m i l a r m e c h a n i s m h a s b e e n i n v o k e d b y C o n l e y and M a h a n to e x p l a i n s t r u c t u r e at the o p t i c a l - p h o n o n e n e r g y in A u - G a A s c o n t a c t s , 9 u s i n g p o l a r e l e c t r o n - p h o n o n coupling. Our a n a l y s i s d i f f e r s f r o m t h e i r s in that (a) the b a r r i e r n e e d not b e i n s i d e t h e s e m i c o n d u c t o r ; (b) the f i n i t e l i f e t i m e of the c a r r i e r s f o r I ¢ [> ~,:o i s c o n s i d e r e d ; (c) the c o u p l i n g m e c h a n i s m i s that of d e f o r m a t i o n - p o t e n t i a l c o u p l i n g a s r e l e v a n t f o r h o m o p o l a r m a t e r i a l s l i k e St, Ge, Bi, and m e t a l s ; and (d) f o r d e f o r m a t i o n - p o t e n t i a l c o u p ling, the d e n s i t y - o f - s t a t e s r e n o r m a l i z a t i o n of the t u n n e l i n g c u r r e n t v a n i s h e s w h e r e a s it a l o n e c a n c a u s e the s t r u c t u r e in d e g e n e r a t e p o l a r m a t e r i a l s with s c r e e n e d p o l a r coupling. ~0
IN NORMAL m e t a l - i n s u l a t o r - m e t a l j u n c t i o n s , s t r u c t u r e in d 2 I / d V 2 r e c e n t l y h a s b e e n a s s o c i a t e d t e n t a t i v e l y with the phonon s p e c t r u m of the e l e c t r o d e s . 1-~ One m e c h a n i s m which could c a u s e such s t r u c t u r e i s the i n e l a s t i c i n t e r a c t i o n of t u n n e l i n g e l e c t r o n s with i m p u r i t i e s n e a r the electrode interface. ~ However, measurements of the l i n e s h a p e s f o r d 2 I / d V 2 in MOS j u n c t i o n s 5 s u g g e s t the o c c u r r e n c e of s t r u c t u r e at the o p t i c a l - p h o n o n e n e r g y , Ev = ~:o, which i s ( a p p r o x i m a t e l y ) s y m m e t r i c a l about z e r o b i a s . We d e s c r i b e a c a l c u l a t i o n in which the b u l k d e f o r m a t i o n p o t e n t i a l e l e c t r o n - o p t i c a l phonon coupling in p - t y p e s i l i c o n g i v e s r i s e to such s y m m e t r i c a l s t r u c t u r e in the w e a k - c o u p l i n g l i m i t . T h e r e f o r e our r e s u l t p e r m i t s e x p e r i m e n t a l d i s t i n c t i o n b e t w e e n the m e c h a n i s m s of bulk e l e c t r o n - o p t i c a l phonon coupling and i n e l a s t i c - p h o n o n e x c i t a t i o n 4 e which c a u s e s s t r u c t u r e in d 2 I / d V 2 a n t i s y m m e t r i c about z e r o b i a s . The p h y s i c a l o r i g i n of o u r r e s u l t is i l l u s t r a t e d by F i g . 1. At low t e m p e r a t u r e s , the inc r e m e n t a l c u r r e n t due to a change in b i a s AV
Although in t h i s l e t t e r we c o n s i d e r only e l e c t r o n - p h o n o n coupling, the m e c h a n i s m i s r e l e vant f o r the c o u p l i n g of the i n j e c t e d c a r r i e r s to any o t h e r e l e m e n t a r y e x c i t a t i o n of the b u l k e l e c trodes.
* T h i s r e s e a r c h was s u p p o r t e d in p a r t by the Advanced Research Projects Agency under C o n t r a c t SD-131.
The current density through a MOS junction, in which bulk electron-phonon coupling is considered only in the semiconductor is given by, 11
*National Science Foundation Post-Doctoral Fellow. 193
194
I N T E R A C T I O N O F TUNNELING E L E C T R O N S
/ METAL
Vol. 6, No. 4
/OXIDE
/
p-TYPE SEMICONDUCTOR
~~k-~"~__~__~O~L 15~0~F
i,~,~
__
Fo = 20meV Fe = .I meV t'i% = 65 meV
FIG. 1 S c h e m a t i c r e p r e s e n t a t i o n of a MOS tunnel j u n c t i o n and the p o l a r o n d i s p e r s i o n c u r v e E ( k ) - ~ - R e Z (E(k)) = 0. At z e r o t e m p e r a t u r e , the tunnel j u n c t i o n i n j e c t s a d d i t i o n a l e l e c t r o n s of e n e r g y E = - eV in a b a n d of width 5(eV) when the b i a s i s c h a n g e d b y an a m o u n t 5V. T h e p a r a m e t e r s u s e d to c a l c u l a t e the d i s p e r s i o n c u r v e (using e q u a t i o n s (3) in the text) a r e shown in the f i g u r e .
-F o J =2h. d ¢[f(¢)-f(e+eV)] _co
(la) Im G(~k, ¢)
t~ •
eF
d~ k
d2kll
,[
D (k,,,~k ) (2~) e
The s i n g l e - p a r t i c l e p r o p a g a t o r i s c a l c u l a t e d u s i n g t h e s e c o n d - o r d e r p r o p e r s e l f e n e r g y ~° a s s o c i a t e d with a c o n s t a n t d e f o r m a t i o n p o t e n t i a l e l e c t r o n - p h o n o % i n t e r a c t i o n ~3 of s t r e n g t h E~ ( ~ o / 2 p v , 2) ~ =- K. T a k i n g ~ n ( x ) t o b e r e a l f o r x > 0, we find, 1
F~(c) = 2e(-c+)K2 p ( - ~ + ) t a n -1 ( - O F / C + ) 2 f(e) = [1 + e x p ( c / k T ) ] -1
(lb)
In e q u a t i o n s 1 (a) and 1 (b) k , d e n o t e s the c o m ponent of e l e c t r o n m o m e n t u m p a r a l l e l to the junction, G(~k, c) i s the o n e - e l e c t r o n p r o p a g a t o r in the p r e s e n c e of the e l e c t r o n - p h o n o n i n t e r a c tion, u and D ( k , , ~k ) i s the b a r r i e r - p e n e t r a t i o n f a c t o r f o r the tunnel junction. T 8 ~ To s t u d y the p h o n o n - i n d u c e d l i n e s h a p e , we adopt the m o d e l f o r the b a r r i e r - p e n e t r a t i o n f a c t o r , ~
±
exp
- 3m*Eo
!
!
- 2 ~ ( - e _ ) K ~ p ( - c _ ) [ t a n - ~ ( - C F / C _ ) ½ - ~ / 2 ] (3a) ! ± ± ! _ ~(¢_)K~p(¢_) ~ n [ ¢ F e + ¢ _ a ) / ( e F 2 _ c e ) ]
c+ = ¢ + c F ~ ~ o p ( e ) = (2~) - e ( 2 m * / ~ z ) 3 / 2
D ( k , , ~k ) = Do exPL
!
+ 8(¢+)Kep(e+) ~n[(¢F2 +¢+2)/(¢+2_ CF 2 ) ]
(3b) ~2 .
(3c)
(2)
so the one-electron lineshape is parameterized by E o .
T h i s r e s u l t i s an e x t e n s i o n of that of E n g e l s b e r g and S c h r i e f f e r ~4 to i n c l u d e the ek ~ d e p e n d e n c e of the o n e - e l e c t r o n d e n s i t y of s t a t e s (which is i m p o r t a n t in d e g e n e r a t e s i l i c o n b e c a u s e [(~o~O/CF)~ 1 ].
Vol. 6, No. 4
INTERACTION OF TUNNELING E L E C T R O N S I
I
Tiw o =
20--
64
I
r
I
I
195 r
I
6 5 . 5 meV
meV
~F
= 120
meV
Eo
= 750
meV
T
=
0 I
L
0 6 7 meV
% - 2 0 --
/i
-40 --~~
/.I
- 6 0 --
- l~
- 8 0 -II
il
. . . .
ro : 3mev r'o(l'iw o) = 1 5 m g V
-50
-70
meV
Fo (l'i~ o) = 12 meV
I {64meVl
90
-
F o = 22
I -30
J -I0
I I0 eV(rneV)
I
30
1 50
I 70
J
90
II0
FIG. 2 S t r o n g - c o u p l i n g l i n e s h a p e s o b t a i n e d u s i n g e q u a t i o n s (1) - (3) i n the text with the p a r a m e t e r s i n d i c a t e d in the f i g u r e . Fo (~mo) = K 2 p(CF) i s u s e d to p a r a m e t e r i z e the s t r e n g t h of the e l e c t r o n - p h o n o n coupling. The d a s h e d line gives a n e x c e l l e n t fit to the f o r w a r d - b i a s l i n e s h a p e i n r e f e r e n c e five. The v e r t i c a l s c a l e is a r b i t r a r y . A t e m p e r a t u r e of 4 . 2 ° K does not change the c u r v e s on the s c a l e of the f i g u r e , and t e m p e r a t u r e s KT >~ F~ s e r v e p r i m a r i l y to s m o o t h out the s h a r p peaks.
The fact that ~ depends only on c i m p l i e s that the d e n s i t y of s t a t e s r e n o r m a l i z a t i o n f a c t o r , :0
Z(¢) = [ l + 5 R e
r ( { k , C ) / 5 ~ k ] -1 k=k(c)
is i d e n t i c a l l y u n i t y i n c o n t r a s t to its l o g a r i t h m i c d i v e r g e n c e for p o l a r coupling. ~0 T h e r e f o r e all of the s t r u c t u r e in I, d I / d V and d 2 I / d V ~ n e a r eV ~_ ± ~ o a r i s e s e i t h e r f r o m the l o c a t i o n of the m a x i m u m of I m G ( ~ k , - e V ) or f r o m the s u d d e n i n c r e a s e of Im ~(e) for I cl ~ ~ o . The e v a l u a t i o n of G = d I / d V can be r e d u c e d to a s i n g l e i n t e g r a l which is p e r f o r m e d n u m e r i c a l l y on the IBM 360 at the U n i v e r s i t y of I l l i n o i s . In the n u m e r i c a l c a l c u l a t i o n s the r e p l a c e m e n t 1
±
1
±
~n leFe_¢±e]~½~n[(¢Fe - elz)~+F ~/4¢F]
is m a d e i n e q u a t i o n (3a) with ~, s i m u l a t i n g the effects of the m o d u l a t i n g v o l t a g e i n the e x p e r i m e n t , 4 the i m p u r i t y and a c o u s t i c a l phonon l i f e t i m e at l cl ~_ ~ o , and the effects of d i s p e r s i o n in the phonon s p e c t r u m . 15 In Fig. 2 t y p i c a l l i n e s h a p e s i n d 2 I / d V e a r e shown for coupling suffic i e n t l y s t r o n g that the f o r w a r d b i a s peak b e c o m e s p o s i t i v e . W e a k e r coupling ~5 l e a d s to s y m m e t r i c l i n e s h a p e s at eV -~ ± ~wo. The deep dip in d ~ I / d V 2 at eV >~-~o is due to the e x p o n e n t i a l i n c r e a s e in D as ~k " ~ • Its m a g n i t u d e s u b s t a n t i a l l y r e d u c e d 16 by a l e s s r a p i d i n c r e a s e of D with ~ for ~k ~ ~ o . An inelastic-tunneling m e c h a n i s m ~ 6 would c a u s e a s i m p l e dip in d 2 I / d V ~ at eV ~ - ~ o and a peak at eV - + ~ o • T h e r e f o r e its c o n s e q u e n c e s c a n be e x p e r i m e n t a l l y d i s t i n g u i s h e d f r o m t h o s e of the b u l k e l e c t r o n phonon coupling. The I n - S i O e - S i data s i n d i c a t e s that s o m e i n e l a s t i c t u n n e l i n g o c c u r s at f o r w a r d b i a s for which the s p a c e - c h a r g e r e g i o n i n the
196
INTERACTION OF TUNNELING ELECTRONS
silicon is l a r g e r than at r e v e r s e bias.
Vol. 6, No. 4
lineshape provides a direct experimental measurement of the polaron dispersion relation.
If the f o r m of the b a r r i e r penetration factor can be determined independently, then in units described by this model the tunnel-injection
Acknowledgments - The authors are indebted to P r o f e s s o r J. Bardeen for discussions of this mechanism for tunneling and to V¢. D. Compton and E . L . Wolf for discussions of their data.
References 1.
McMILLAN W L. and ROWELL J . M . Ito be published).
Tunneling and the Strong Coupling Superconductor,
2.
LAMBE J. and JAKLEVIC R . C . ,
Solid State Commun. 5,
3.
ESAKI L . , CHANG L . L . , (to be published).
4.
DUKE C . B . ,
5.
WOLF E . L . , Evidence for Hole-optical Phonon Interaction in Degenerate Silicon in Tunneling Measurements, (to be published).
6.
KLEINMAN L . ,
7.
ITSKOVICH F . I . ,
8.
BENDANI~L D . J . and DUKE C . B . ,
9.
CONLEY J.W. and MAHAN G . D . ,
STILES P . J . ,
O'KANE D . F . and WISER N.,
SILVERSTEIN S.D. and BENNETT A . J . ,
Phys. Rev.
17 (1967).
Phys. Rev. Lett.
IBM Report RC 1900 19__2 315 (1967).
140, A637 (1965).
Sov. Phys. J E T P , 2__44, 202 (1967).
10. MAHAN G.D. and DUKE C . B . ,
Phys. Rev. Phys. Rev.
Phys. Rev.
160, 679 (1967). 161, 681 (1967).
149,
705 (1966).
11. DUKE C . B . , A Unified Theory of Z e r o - B i a s Anomalies and Energy Loss Mechanism in the B a r r i e r , General E l e c t r i c Report 67-C-305 and Proceedings of the International Summer School on Electron (to be published), equation" (34).
12. MURPHY E.L. and GOOD Jr. R.H., 13. HARRISON W.A.,
Phys. Rev.
Phys. Rev.
102, 1464 (1956).
104, 1281 (1956).
14. ENGELSBERG S. and SCHRIEFFER J . R . ,
Phys. Rev.
131, 993 (1963).
15. DAVIS L.C. and DUKE C.B., (to be published).
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d ' ~ _ T , I d J V "2-
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Vol. 6, pp. 1 9 3 - 1 9 7 ,
1968.
Pergamon Press.
P r i n t e d in G r e a t B r i t a i n
I N T E R A C T I O N O F TUNNELING E L E C T R O N S WITH C O L L E C T I V E MODES O F THE E L E C T R O D E S * L . C . D a v i s ¢ and C . B . Duke D e p a r t m e n t of P h y s i c s and M a t e r i a l s R e s e a r c h L a b o r a t o r y , U n i v e r s i t y of I l l i n o i s , U r b a n a , I l l i n o i s , U . S . A . ( R e c e i v e d 2 J a n u a r y 1968)
The a l t e r a t i o n of the t u n n e l i n g c u r r e n t due to b u l k e l e c t r o n phonon c o u p l i n g in one of the e l e c t r o d e s i s c a l c u l a t e d u s i n g the t u n n e l i n g - H a m i l t o n i a n m e t h o d . L i n e s h a p e s a r e p r e s e n t e d for d e f o r m a t i o n - p o t e n t i a l c o u p l i n g to o p t i c a l phonons in p - t y p e s i l i c o n . The p r e d i c t e d l i n e s h a p e of d e I / d V 2 in a metal-oxide-semiconductor junction shows deviations from the o n e - e l e c t r o n b a c k g r o u n d which a r e s y m m e t r i c a l about z e r o b i a s in c o n t r a s t to a n t i s y m m e t r i c a l s t r u c t u r e p r e d i c t e d due to i n e l a s t i c phonon c r e a t i o n b y the t u n n e l i n g e l e c t r o n .
a r i s e s f r o m a b a n d of c a r r i e r s at e n e r g y ¢ = - eV of width eAV. The t u n n e l i n g p r o b a b i l i t y of t h e s e carriers is determined ~ by their associated v a l u e s of ~k. T h e r e f o r e f o r V -~ =L ~mo/e the p o l a r o n d i s p e r s i o n in the c a r r i e r s ' e n e r g y s p e c t r a c a u s e s l a r g e c h a n g e s in t h e i r t u n n e l i n g p r b b a b i l i t i e s and h e n c e in d ~ I / d V 2. A s i m i l a r m e c h a n i s m h a s b e e n i n v o k e d b y C o n l e y and M a h a n to e x p l a i n s t r u c t u r e at the o p t i c a l - p h o n o n e n e r g y in A u - G a A s c o n t a c t s , 9 u s i n g p o l a r e l e c t r o n - p h o n o n coupling. Our a n a l y s i s d i f f e r s f r o m t h e i r s in that (a) the b a r r i e r n e e d not b e i n s i d e t h e s e m i c o n d u c t o r ; (b) the f i n i t e l i f e t i m e of the c a r r i e r s f o r I ¢ [> ~,:o i s c o n s i d e r e d ; (c) the c o u p l i n g m e c h a n i s m i s that of d e f o r m a t i o n - p o t e n t i a l c o u p l i n g a s r e l e v a n t f o r h o m o p o l a r m a t e r i a l s l i k e St, Ge, Bi, and m e t a l s ; and (d) f o r d e f o r m a t i o n - p o t e n t i a l c o u p ling, the d e n s i t y - o f - s t a t e s r e n o r m a l i z a t i o n of the t u n n e l i n g c u r r e n t v a n i s h e s w h e r e a s it a l o n e c a n c a u s e the s t r u c t u r e in d e g e n e r a t e p o l a r m a t e r i a l s with s c r e e n e d p o l a r coupling. ~0
IN NORMAL m e t a l - i n s u l a t o r - m e t a l j u n c t i o n s , s t r u c t u r e in d 2 I / d V 2 r e c e n t l y h a s b e e n a s s o c i a t e d t e n t a t i v e l y with the phonon s p e c t r u m of the e l e c t r o d e s . 1-~ One m e c h a n i s m which could c a u s e such s t r u c t u r e i s the i n e l a s t i c i n t e r a c t i o n of t u n n e l i n g e l e c t r o n s with i m p u r i t i e s n e a r the electrode interface. ~ However, measurements of the l i n e s h a p e s f o r d 2 I / d V 2 in MOS j u n c t i o n s 5 s u g g e s t the o c c u r r e n c e of s t r u c t u r e at the o p t i c a l - p h o n o n e n e r g y , Ev = ~:o, which i s ( a p p r o x i m a t e l y ) s y m m e t r i c a l about z e r o b i a s . We d e s c r i b e a c a l c u l a t i o n in which the b u l k d e f o r m a t i o n p o t e n t i a l e l e c t r o n - o p t i c a l phonon coupling in p - t y p e s i l i c o n g i v e s r i s e to such s y m m e t r i c a l s t r u c t u r e in the w e a k - c o u p l i n g l i m i t . T h e r e f o r e our r e s u l t p e r m i t s e x p e r i m e n t a l d i s t i n c t i o n b e t w e e n the m e c h a n i s m s of bulk e l e c t r o n - o p t i c a l phonon coupling and i n e l a s t i c - p h o n o n e x c i t a t i o n 4 e which c a u s e s s t r u c t u r e in d 2 I / d V 2 a n t i s y m m e t r i c about z e r o b i a s . The p h y s i c a l o r i g i n of o u r r e s u l t is i l l u s t r a t e d by F i g . 1. At low t e m p e r a t u r e s , the inc r e m e n t a l c u r r e n t due to a change in b i a s AV
Although in t h i s l e t t e r we c o n s i d e r only e l e c t r o n - p h o n o n coupling, the m e c h a n i s m i s r e l e vant f o r the c o u p l i n g of the i n j e c t e d c a r r i e r s to any o t h e r e l e m e n t a r y e x c i t a t i o n of the b u l k e l e c trodes.
* T h i s r e s e a r c h was s u p p o r t e d in p a r t by the Advanced Research Projects Agency under C o n t r a c t SD-131.
The current density through a MOS junction, in which bulk electron-phonon coupling is considered only in the semiconductor is given by, 11
*National Science Foundation Post-Doctoral Fellow. 193
194
I N T E R A C T I O N O F TUNNELING E L E C T R O N S
/ METAL
Vol. 6, No. 4
/OXIDE
/
p-TYPE SEMICONDUCTOR
~~k-~"~__~__~O~L 15~0~F
i,~,~
__
Fo = 20meV Fe = .I meV t'i% = 65 meV
FIG. 1 S c h e m a t i c r e p r e s e n t a t i o n of a MOS tunnel j u n c t i o n and the p o l a r o n d i s p e r s i o n c u r v e E ( k ) - ~ - R e Z (E(k)) = 0. At z e r o t e m p e r a t u r e , the tunnel j u n c t i o n i n j e c t s a d d i t i o n a l e l e c t r o n s of e n e r g y E = - eV in a b a n d of width 5(eV) when the b i a s i s c h a n g e d b y an a m o u n t 5V. T h e p a r a m e t e r s u s e d to c a l c u l a t e the d i s p e r s i o n c u r v e (using e q u a t i o n s (3) in the text) a r e shown in the f i g u r e .
-F o J =2h. d ¢[f(¢)-f(e+eV)] _co
(la) Im G(~k, ¢)
t~ •
eF
d~ k
d2kll
,[
D (k,,,~k ) (2~) e
The s i n g l e - p a r t i c l e p r o p a g a t o r i s c a l c u l a t e d u s i n g t h e s e c o n d - o r d e r p r o p e r s e l f e n e r g y ~° a s s o c i a t e d with a c o n s t a n t d e f o r m a t i o n p o t e n t i a l e l e c t r o n - p h o n o % i n t e r a c t i o n ~3 of s t r e n g t h E~ ( ~ o / 2 p v , 2) ~ =- K. T a k i n g ~ n ( x ) t o b e r e a l f o r x > 0, we find, 1
F~(c) = 2e(-c+)K2 p ( - ~ + ) t a n -1 ( - O F / C + ) 2 f(e) = [1 + e x p ( c / k T ) ] -1
(lb)
In e q u a t i o n s 1 (a) and 1 (b) k , d e n o t e s the c o m ponent of e l e c t r o n m o m e n t u m p a r a l l e l to the junction, G(~k, c) i s the o n e - e l e c t r o n p r o p a g a t o r in the p r e s e n c e of the e l e c t r o n - p h o n o n i n t e r a c tion, u and D ( k , , ~k ) i s the b a r r i e r - p e n e t r a t i o n f a c t o r f o r the tunnel junction. T 8 ~ To s t u d y the p h o n o n - i n d u c e d l i n e s h a p e , we adopt the m o d e l f o r the b a r r i e r - p e n e t r a t i o n f a c t o r , ~
±
exp
- 3m*Eo
!
!
- 2 ~ ( - e _ ) K ~ p ( - c _ ) [ t a n - ~ ( - C F / C _ ) ½ - ~ / 2 ] (3a) ! ± ± ! _ ~(¢_)K~p(¢_) ~ n [ ¢ F e + ¢ _ a ) / ( e F 2 _ c e ) ]
c+ = ¢ + c F ~ ~ o p ( e ) = (2~) - e ( 2 m * / ~ z ) 3 / 2
D ( k , , ~k ) = Do exPL
!
+ 8(¢+)Kep(e+) ~n[(¢F2 +¢+2)/(¢+2_ CF 2 ) ]
(3b) ~2 .
(3c)
(2)
so the one-electron lineshape is parameterized by E o .
T h i s r e s u l t i s an e x t e n s i o n of that of E n g e l s b e r g and S c h r i e f f e r ~4 to i n c l u d e the ek ~ d e p e n d e n c e of the o n e - e l e c t r o n d e n s i t y of s t a t e s (which is i m p o r t a n t in d e g e n e r a t e s i l i c o n b e c a u s e [(~o~O/CF)~ 1 ].
Vol. 6, No. 4
INTERACTION OF TUNNELING E L E C T R O N S I
I
Tiw o =
20--
64
I
r
I
I
195 r
I
6 5 . 5 meV
meV
~F
= 120
meV
Eo
= 750
meV
T
=
0 I
L
0 6 7 meV
% - 2 0 --
/i
-40 --~~
/.I
- 6 0 --
- l~
- 8 0 -II
il
. . . .
ro : 3mev r'o(l'iw o) = 1 5 m g V
-50
-70
meV
Fo (l'i~ o) = 12 meV
I {64meVl
90
-
F o = 22
I -30
J -I0
I I0 eV(rneV)
I
30
1 50
I 70
J
90
II0
FIG. 2 S t r o n g - c o u p l i n g l i n e s h a p e s o b t a i n e d u s i n g e q u a t i o n s (1) - (3) i n the text with the p a r a m e t e r s i n d i c a t e d in the f i g u r e . Fo (~mo) = K 2 p(CF) i s u s e d to p a r a m e t e r i z e the s t r e n g t h of the e l e c t r o n - p h o n o n coupling. The d a s h e d line gives a n e x c e l l e n t fit to the f o r w a r d - b i a s l i n e s h a p e i n r e f e r e n c e five. The v e r t i c a l s c a l e is a r b i t r a r y . A t e m p e r a t u r e of 4 . 2 ° K does not change the c u r v e s on the s c a l e of the f i g u r e , and t e m p e r a t u r e s KT >~ F~ s e r v e p r i m a r i l y to s m o o t h out the s h a r p peaks.
The fact that ~ depends only on c i m p l i e s that the d e n s i t y of s t a t e s r e n o r m a l i z a t i o n f a c t o r , :0
Z(¢) = [ l + 5 R e
r ( { k , C ) / 5 ~ k ] -1 k=k(c)
is i d e n t i c a l l y u n i t y i n c o n t r a s t to its l o g a r i t h m i c d i v e r g e n c e for p o l a r coupling. ~0 T h e r e f o r e all of the s t r u c t u r e in I, d I / d V and d 2 I / d V ~ n e a r eV ~_ ± ~ o a r i s e s e i t h e r f r o m the l o c a t i o n of the m a x i m u m of I m G ( ~ k , - e V ) or f r o m the s u d d e n i n c r e a s e of Im ~(e) for I cl ~ ~ o . The e v a l u a t i o n of G = d I / d V can be r e d u c e d to a s i n g l e i n t e g r a l which is p e r f o r m e d n u m e r i c a l l y on the IBM 360 at the U n i v e r s i t y of I l l i n o i s . In the n u m e r i c a l c a l c u l a t i o n s the r e p l a c e m e n t 1
±
1
±
~n leFe_¢±e]~½~n[(¢Fe - elz)~+F ~/4¢F]
is m a d e i n e q u a t i o n (3a) with ~, s i m u l a t i n g the effects of the m o d u l a t i n g v o l t a g e i n the e x p e r i m e n t , 4 the i m p u r i t y and a c o u s t i c a l phonon l i f e t i m e at l cl ~_ ~ o , and the effects of d i s p e r s i o n in the phonon s p e c t r u m . 15 In Fig. 2 t y p i c a l l i n e s h a p e s i n d 2 I / d V e a r e shown for coupling suffic i e n t l y s t r o n g that the f o r w a r d b i a s peak b e c o m e s p o s i t i v e . W e a k e r coupling ~5 l e a d s to s y m m e t r i c l i n e s h a p e s at eV -~ ± ~wo. The deep dip in d ~ I / d V 2 at eV >~-~o is due to the e x p o n e n t i a l i n c r e a s e in D as ~k " ~ • Its m a g n i t u d e s u b s t a n t i a l l y r e d u c e d 16 by a l e s s r a p i d i n c r e a s e of D with ~ for ~k ~ ~ o . An inelastic-tunneling m e c h a n i s m ~ 6 would c a u s e a s i m p l e dip in d 2 I / d V ~ at eV ~ - ~ o and a peak at eV - + ~ o • T h e r e f o r e its c o n s e q u e n c e s c a n be e x p e r i m e n t a l l y d i s t i n g u i s h e d f r o m t h o s e of the b u l k e l e c t r o n phonon coupling. The I n - S i O e - S i data s i n d i c a t e s that s o m e i n e l a s t i c t u n n e l i n g o c c u r s at f o r w a r d b i a s for which the s p a c e - c h a r g e r e g i o n i n the
196
INTERACTION OF TUNNELING ELECTRONS
silicon is l a r g e r than at r e v e r s e bias.
Vol. 6, No. 4
lineshape provides a direct experimental measurement of the polaron dispersion relation.
If the f o r m of the b a r r i e r penetration factor can be determined independently, then in units described by this model the tunnel-injection
Acknowledgments - The authors are indebted to P r o f e s s o r J. Bardeen for discussions of this mechanism for tunneling and to V¢. D. Compton and E . L . Wolf for discussions of their data.
References 1.
McMILLAN W L. and ROWELL J . M . Ito be published).
Tunneling and the Strong Coupling Superconductor,
2.
LAMBE J. and JAKLEVIC R . C . ,
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ESAKI L . , CHANG L . L . , (to be published).
4.
DUKE C . B . ,
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WOLF E . L . , Evidence for Hole-optical Phonon Interaction in Degenerate Silicon in Tunneling Measurements, (to be published).
6.
KLEINMAN L . ,
7.
ITSKOVICH F . I . ,
8.
BENDANI~L D . J . and DUKE C . B . ,
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CONLEY J.W. and MAHAN G . D . ,
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O'KANE D . F . and WISER N.,
SILVERSTEIN S.D. and BENNETT A . J . ,
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17 (1967).
Phys. Rev. Lett.
IBM Report RC 1900 19__2 315 (1967).
140, A637 (1965).
Sov. Phys. J E T P , 2__44, 202 (1967).
10. MAHAN G.D. and DUKE C . B . ,
Phys. Rev. Phys. Rev.
Phys. Rev.
160, 679 (1967). 161, 681 (1967).
149,
705 (1966).
11. DUKE C . B . , A Unified Theory of Z e r o - B i a s Anomalies and Energy Loss Mechanism in the B a r r i e r , General E l e c t r i c Report 67-C-305 and Proceedings of the International Summer School on Electron (to be published), equation" (34).
12. MURPHY E.L. and GOOD Jr. R.H., 13. HARRISON W.A.,
Phys. Rev.
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102, 1464 (1956).
104, 1281 (1956).
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Phys. Rev.
131, 993 (1963).
15. DAVIS L.C. and DUKE C.B., (to be published).
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