INELASTIC ENERGY LOSSES AT ELECTRON TUNNELING THROUGH POTENTIAL BARRIERS

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INELASTIC ENERGY LOSSES AT ELECTRON TUNNELING THROUGH POTENTIAL BARRIERS

R. Bakhtizin, V. Valeyev, Yu. Kukharenko

To cite this version:

R. Bakhtizin, V. Valeyev, Yu. Kukharenko. INELASTIC ENERGY LOSSES AT ELECTRON TUN- NELING THROUGH POTENTIAL BARRIERS. Journal de Physique Colloques, 1987, 48 (C6), pp.C6-21-C6-26. �10.1051/jphyscol:1987604�. �jpa-00226807�

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INELASTIC ENERGY LOSSES AT ELECTRON TUNNELING THROUGH POTENTIAL BARRIERS

R.Z. Bakhtizin, V.G. Valeyev, Yu. A. Kukharenko

Department of Experimental Physics, Bashkir S t a t e University, 450074, Ufa, U.S.S.R.

Abstraot

.&

^{A n}attempt was made t o construct a simple theory of tunneling electron ineLastic losses due t o the excitation of e l e c h ron-hole pairs, surface plasmons a s well a s t o the excitation sf eleo- t r o n t r a n s i t i o n s i n adsorbed atoms and molecules.

Passing through the potentla1 barrier* f i e l d emitted e3.ectrons may bear energy losses due t o the exoitation of electron-hole pairs. auxb face plasmons a s well a s t o the excitation of electron t r a n s i t i o n s i n atoms and molecules adsorbed upon t h e metal surface. It m y r e s u l t in additional "smearingN and condition c e r t a i n p e c u l i a r i t i e s in the d i s t r i b u t i o n function f (

&

) of electrons tunneling 3.u t h e lowenem gy edge. The rnen%ioned prooessesl occur, being accompanied by t h e ele- ctrons tunneling from the s t a t e s below the bemP surface. However, the influence of this i n t e r f e r i n g contxibution may be ignored, provided the p o t e n t m b a r r i e r width grows rapidly alongeide the change in the difference of energies ( &p

E

)

.

The theoretical study of this group of phenomena has already been undertaken in / I

-

^{3 /}

.

^The

temperature Green function technique and l i n e a r response fonaalisan used in these papers necessitate r a t h e r complicated calculations and l i m i t s the range of application of the obtained r e s u l t s t o the e q b librium case. In the present paper we construct a simple theory of tunneling electron i n e l a s t i c losses f r e e of such limitations.

I n e l a s t i c losses of electrons tunneling f r o m the condensed system may be described by means of the tunneling Hamiltonian /3,4/ comple-

ted with terms allowing f o r the many-particle correlation. In i t s usual terms the Hamiltonian i s a s follows:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987604

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C6-22 JOURNAL DE _PHYSIQUE

A

"+

where

q6(3) , v6J)6(%')

a r e operators of eleckron destruction and creation a t the point with the spin 6

,

^& ^{( 3 ,}^*)i s the electron k i n e t i c energy operator,

3

( (

g')

is the tunneling probabUity amplitude determined by the b a r r i e r transparency factor.

i s the operator f i e l d a f f e c t i n g the a e c t r o n a s a combined action of a l l t h e electrons and ions i n the system,

9, (5) =z8 (2

- g j ( t ) ) i s the ion d i s t r i b u t i o n density at points R (t). 4

~ ~ n d r i n ~

t h e electron volume interaction, i t may be assumed that

3

an electron tunnel- ing through the b a r r i e r 3,s described by the single-particle Green function &3C,

3Cl,

s a t i s f y i n g the ~ c h r a d i n g e r equation

and the initial condition

g(t'3, ^{t ' ~ )}

⁼

^&it'- ^s),

3C

=

t ,

%,

spin index

6

i s omStted,

h=

1. Such an approach is used i n / 5 / f o r the description of i n e l a s t i c lowenergy electron r e f l e c - t i o n from s o l i d surface. Its shplLcLty i s conditioned by the possi- b i l i t y t o apply a simple diagram technique, elaborated in 1 6

-

^9/

f o r wave s c a t t e r i n g upon random uniformities, f o r the many-electron system. f r o b a b i l i t y

w

of electron i n e l a s t i c tunneling through the b a r r i e r i s found with t h e help of the mean square fluctuation of the Oreen function:

where

8@ = @

* G, G

=

( $ 7 ; averagiag i s performed f o r every st*

t e of the substance by means of t h e GLbbs density matrix, correspon- ding t o the specified temperature, jmpulses

'iJ

and

a

describe the electron s t a t e before and a f t e r t h e b a r r i e r i s passed, respectively, Ihe averaged Green function i s obtained by the summation of the bis- grams :

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teraction potential

<

^%s)G(3ct)>

9,. x

ar

stands for the triple correlation function

<G( x)$(

x')

U(x")>

etc. The pair Green function G12 =

<$,

/O G2>. where indices 1 and 2 denote a set of c e ordinates ( tt

f

⁾ ⁽^ti

$)

^and⁽^t2

ii;!

), ( ti Z; ), is described by the summation of the diagrams:

-

^4.

A

^A-

-

^* ⁺

Functions G and GI satisfy the Dyson and BefhhSolpeter equations respectively:

where

Z

^and

z,

are sums of 811 the single-particle compact dia- grams. Inelastic tunneling probability is expressed by means of the vertex

rI2:

( 8 )

its graphic representation will be as follows

where the arrowed line

-

denotes the averaged Green function G,

prd

is the vertex function of

r .

The following diagram, des- cribing successive processes of tunneling and inelastic losses gives the bulk of inelastic losses at tunneling:,

According to the definition of the tunneling transition probability amplitude TA (

& 1,

the averaged Green function must contain graphs

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C6-24 JOURNAL DE PHYSIQUE

f e a t u r i n g one broken l i n e onlyo

Provided t h e c r y s t a l s t r u c t u r e of t h e substance i s ignored, inelas- t i c tunneling pxobabilfty by (8

-

¹⁰⁾can be q u i t e acrcurately repre-.

sented as a product of e l a s t i c s u b b a r r i e r t r a n s i t i o n p r o b a b i l i t y

1

T(

3, 6

⁾

1 *

and p r o b a b i l i t y of e x c i t a t i o n of a many-electron system.

The constructed theory permits a r a t h e r simple c a l c u l a t i o n of 5nelas- t i c tunneling p r o b a b i l i t y accompanied by t h e e x c i t a t i o n of phonons in t h e system of molecules adsorbed upon t h e s o l i d surface, as well as by t h e e x c i t a t i o n of e l e c t r o n t r a n s i t i o n s in t h e adsorbates. For example, in case of moleculas oscJllations, t h e c o r r e l a t i o n f u n c t i o n of S n t e r a c t i o n p o t e n t i a l

<

^{U ~ X )}U(X')> is, according t o t h e f l u c t u - a t i o n - d i s s i p a t i o n theorem, expressed by means of t h e imaginary p a r t of t h e retarded phonon Green function, and in case of e l e c t r o n tran- s i t i o n s i n t h e adsorbate i t i s expressed by means of t h e imaginary p a r t of t h e r e t a r d e d Green f u n c t i o n of t h e l o n g i t u d i n a l e l e c t r i c f i e l d .

Consider i n e l a s t i c l o s s e s at f i e l d emission, conditioned by t h e s w f a c e plasma waves exoitation. We shall describe t h e electron-plasmon i n t e r a c t i o n i n t h e manner o r i g i n a l l y suggested in

/lo/ ,

where t h e method of c o l l e c t i v e v a r i a b l e s /11/ is generalized f o r t h e case of plasma being s i t u a t e d in t h e p e r i o d i c f i e l d of t h e c r y s t a l l a t t i c e . The r e s u l t f o r i n e l a s t i c component of t h e f i e l d emission e l e c t r o n energy d i s t r i b u t i o n f u n c t i o n J; (

8') .

d y d e ' i s as follows ( f o r s i m p l i c i t y we consider only t h e oase of continuum and assume t h e plasmon damping t o be2all) :

where

jL( ⁶ ^' ⁾

denotes t h e f i e l d emission e l e c t r o n energy d i s t r i b u t i o n in t h e absence of i n e l a s t i c l o s s e s ; according t o t h e ordinary WKB method f o r t h e t r i a n g u l a r near-surface pot e n t i & b a r r i e r in one-di- mensional case we have:

Here

&

denotes t h e or~orgy of e l e c t r o n in metal, F i s t h e e l e c t r i c f i e L d s t r e n g t h n e a r t h e surface,

9

stands f o r t h e metal work fun- c t i o n , d = @/2- . n o ( & ) i s Ferrni f u n c t l o n and & i s Permi energy, ^%t 1.

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of t h e energy spectrum of t h e s u r f a c e plasma waves u), (

g,,) -

^Us

⁺

f A;

K,,

^f.a.

Consider t h e oase of low temperature, where t h e N e l a s t i c " d i s t r i b u - t i o n behaviour (1 2) i s mainly conditioned by t h e f i e l d P

a ,

s e e below), h e m e t h e f u a o t i o n (1 2) is a t i t s ( K T 4 s 4 m

maximum f o r

8

and decreases in t h e order of its magnitude f o r

( & - 8 ) -

^Fq

^.

^Iti s crlear from (11)-(13) t h a t t h o

4 G - z . - -

d i s t r i b u t i o n 3; (

6 ' )

is at i t s maximum f o r

&'

¹^{Us, and}

t h e width of i n e l a s t i c peak -r

gS.

The behaviour of (1 1 ) n e a r t h i s maximum may be e a s i l y asalyzed f o r two limiting cases:

It i s important t h a t f o r t h e f i n i t e values 09 lltuning-outl*

&

(

-

Us) t h e height of i n e l a s t i c peak (3;)- I* 2 in(

EF/ la)

i s logarithmioally dixergent t o r

as-

⁰⁴

b) ~j hence j;

( & ' I =

^je

( a ) @

t & , , ~ ' ) , w h e r e 4-

g q

(F) denotes t h e t o t a l f i e l d emission ourrent d e n s i t y ignoring i n e l a s t i c losses.

REFERENCES

/I/ Scalapino D.I., Marcus S.M., Phys. Rev. Lett.

2

(1967) 459.

/2,, Flood D o l e Phys. Rev. Lett., A 2 (1969) 1001 J. Chem~ P h ~ s .

22

(1970) 13550

/3/ Gadzuk J.W., Plummer E.W., Rev. Mod. Phys. 45, N 3, /4/ Duke C.B., Tunneling in Solids. Academic, New York, 1969.

/5/ Kuleshov V.Fo, Kukharenko Yu.A., Fridrikhov S.A. e t al. Electron spectroscopy and d i f f r a c t i o n under t h e i n v e s t i g a t i o n of t h e s o l i d s t surface. Moscow, Nauka, 1985, p a r t 1.

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C6-26 JOURNAL DE PHYSIQUE

/6/ Bouret R.G., Canado J. Phys. (1962) 782.

/7/ Patarsky VoI., Ger-t;sensteia M.E., J, Theor, Exper. Physo

44

(1963) 6760

/8/ Edvarda S.F. Philose Mago,

2

(1958) 1020.

/9/ Abrilroaov A.A., Gorlkov LOP., Dzyaloahinslcy IoE. Quantized f i e l d theory methods i n s t a t i s t i c a l physics. Moscow, Fimatgiz, 1962.

/lo/ Duke C.B., Laramore G.E., Phys, Revo

a

(1971) 3183.

/11/ Bohm D., Piaes D,, Physr Rev.

5

(1952) 1510

INELASTIC ENERGY LOSSES AT ELECTRON TUNNELING THROUGH POTENTIAL BARRIERS

HAL Id: jpa-00226807

https://hal.archives-ouvertes.fr/jpa-00226807