Heine, V.

English

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Theory of Surface Ion Neutralization 
Vol ker He ine 
Department of Physics 
and 
Institute for the Study of Metals 
The University of Chicago 
Chicago, Illinois 60637 
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March 1966 
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(NASA CR OR TMX OR AD NUMBER) (CATEGORI) 
https://ntrs.nasa.gov/search.jsp?R=19660023145 2020-03-16T21:06:15+00:00Z
2. 
THEORY OF SURFACE ION NEUTRALIZATION* 
by 
Vol ker He ine t 
ABSTRACT 
The formal theory of surface ion neutralization is given. 
[An electron from a filled state in the conduction band in a 
solid tunnels onto a rare gas ion on the surface t o  neutralize 
it, the energy drop trco being used t o  excite another electron 
from the band t o  a sufficiently high state that it can escape 
from the metal into a collector.] 
*a, the Coulomb interaction which is responsible for the pro-  
cess is effectively unscreened, but it is shown that the long 
range of the unscreened potential does not lead t o  any diver- 
gences which might be expected in calculating the emergent 
current. The ejected electrons originate in the first few 
atomic layers of the solid and some even outside it. The 
variations of the matrix element and of the transmission 
coefficient through the surface can explain some directional 
dependence observed in the measurements. 
Because of the large energy 
. 
3-  
I. INTRODUCTION 
Fig. 1 shows the  p o t e n t i a l  energy seen by an e l e c t r o n  when 
some free i o n  such as Ne+ i s  nea r  a m e t a l l i c  sur face .  
'one'  can tunnel  from a s t a t e  4, i n  t h e  conduction band of t h e  
metal t o  t h e  ground s ta te  E i n  t h e  ion, thus  n e u t r a l i z i n g  it .  
The excess energy 7ru = Ea - E i s  communicated v i a  the Coulomb 
g 
i n t e r a c t i o n  to e l e c t r o n  ' two '  i n  s t a t e  4, of t h e  conduction 
band, which i s  exc i t ed  t o  s t a t e  4, and may emerge from the  
Elec t ron  
g 
metal w i t h  k i n e t i c  energy Tee  
t h e  n e u t r a l i z a t i o n  process,  e s t a b l i s h e d  by Hagstrum and 
o t h e r s  through t h e i r  experimental i n v e s t i g a t i o n s .  For 
ins tance ,  t h e  r a t e  i s  c l e a r l y  propor t iona l ,  among o t h e r  th ings ,  
Such i s  t h e  general  p i c t u r e  of 
t o  the  d e n s i t y  of s t a t e s  n(E) a t  Ea and Eb, and has been used 3 
t o  ob ta in  a p i c t u r e  of n (E)  i n  t h e  conduction band. 
However, some questions have remained, i n  p a r t i c u l a r ,  how 
f a r  below t h e  su r face  t o  t h e  emergent e l e c t r o n s  o r i g i n a t e .  One 
would suppose that  t h e  Coulomb i n t e r a c t i o n  i s  e f f e c t i v e l y  un- 
screened because o f  t h e  large energy t r a n s f e r  tra. However, i f  
one then c a l c u l a t e d  w i t h  it the t o t a l  emergent cu r ren t  from a l l  
e l ec t rons ,  one would o b t a i n  a divergent  con t r ibu t ion  from pro- 
cesses  deep i n s i d e  t h e  metal, a r i s i n g  from t h e  long range 
na ture  of t h e  Coulomb i n t e r a c t i o n .  It t h e r e f o r e  seems worth- 
whi le  t o  s e t  down a formal theory.  Other ques t ions  concern 
. 
4. 
t h e  na ture  of t h e  matr ix  element involved i n  t h e  process 
d i r e c t i o n a l  v a r i a t i o n  of t h e  emergent cu r ren t .  
The ra te  of  t h e  n e u t r a l i z a t i o n  process i s  given by 
and t h e  
where Meab i s  the  matr ix  elemer_t f o r  a p a r t i c u l a r  t r a n s i t i o n  
S t r i c t l y  w e  should s u b t r a c t  t h e  corresporLdlng excl-iange c o n t r i -  
but ion,  which we sha l l  ignore s i n c e  it, does e o t  a f f e c t  t he  
2 
12 followir,g argument. Vint i s  t h e  Cc,Iozb i c t e r a c t i o n  e /r 
screened as  appropr ia te .  It i s  a v e r y  i o ~ p l i c a t e d  f-m-ction 
s i n c e  t h e  space of rl, r2 includes ?>e r e g i o r  or;tside the metal 
around t h e  gas ion, as  w e l l  a s  tihe i n + , e r i o r  o f  t h e  metal .  I n  
(1) t r a n s i t i o n s  from d i f f e r e n t  sets of band s t a t e s  a ,b  simply 
add, and it i s  t h e r e f o r e  convenient t o  c o m i d e r  those involving 
a p a r t i c u l a r  p a i r  d,, 4,. 
then  f i x e d  a t  
The e2ergy of t h e  emergent beam i s  
. 
5 -  
With 4 
in (2). We can treat 
fixed we may also perform in principle the r1 integration a 
as a perturbation in the one-electron Eamiltonian of electron 
'two'. At large r2 it is a Coulomb potential, appropriately 
screened. It is centered at the point of maximum overlap of 
and 4, which is somewhere near the center of the surface 
ry 
4, 
ion because 4 is so highly localized. g 
. 
6 .  
11. THE COULOMB INTERACTION 
The main problem is the long range of  v(r2), i.e., the 
divergence of v(q) at small q, because the screening is very 
small for large energy transfers tw. We can obtain the essen- 
tial physical picture by taking plane waves f o r  4h, 4,, and 
expressing v in terms of a free-electron screening constant 
- 
v 
U 
€ ( q , 4 :  
The matrix element is simply v(Q) where Q = k 
is assumed to be some fixed state and ke can lie on a sphere of 
radius ke determined by ( 3 ) .  The poirit is t h a t  Q can vary from 
- kb. Here k 
-4 a& u enb 
LL 
ke kb to ke + kb ' bu t  componerit.s with - do not con- 
tribute t o  the excitation of electrons at the high energies 
studied. The minimum value of iru for an electron to be observed 
in this type of experiment, is the work function 2. 
other electrons with lower ?Tu are produced but carmot escape 
Of course 
through the surface. But f o r  these also, Q cannot tend to zero 
unless )IC0 tends to zero as well, and in that limit the screening 
G (q,Lu) removes the l / q 2  divergence in v(q). 
7 .  
The conclusions remain v a l i d  when w e  t ake  Bloch waves 
4(kb,  nb; r2) and 4(xe, ne; r2) where k i s  now t h e  reduced 
9 * N k 
wave v e c t o r  and n t h e  band index. It i s  now p e r f e c t l y  poss ib l e  
t o  have k = kb and f o r  small q terms i n  ( 5 )  t o  con t r ibu te .  L e t  
& A  
us focus a t t e n t i o n  t h i s  time on t r a n s i t i o n s  from a l l  4, i n  a 
f i l l e d  band nb t o  a p a r t i c u l a r  f i n a l  s ta te  4, i n  t h e  h igher  
band ne. 
t o  
The number of emergent e l e c t r o n s  i n  4, i s  propor t iona l  
where 
M ( q )  must vanish by orthogonalit ,y f o r  q = 0 ,  and from keg k v t u r -  
rcI 0 
ba t ion  theory we have M ( q )  O( q ,  which k i l l s  o re  of t h e  f a c t o r s  
ry 
of q i n  (5 ) .  The o the r ,  when squared, disappears  w i t h  t h e  
volume element 9 = 2 d% 
F i n a l l y  i t  remains t o  cons ider  t h e  e f f e c t  i n  t h e  r e a l  
s i t u a t i o n  of te rmina t ing  the wave func t ion  4, a t  t h e  su r face  of  
t h e  metal, w i th  some t a i l  overlapping t h e  su r face  ion .  
e x i s t s  both i n s i d e  and outs ide  t h e  metal .  If we choose c y l i n d r i -  
4, 
8. 
c 
c a l  coordinates  4 , 8 ,a wi th  axis perpendicular  t o  t h e  sur- 
f ace ,  w e  s ee  t h a t  t he  e* 
s t r o n g  enough t o  give a divergence wi th  l/r i n  t h e  mat r ix  
element 
i n  t h e  volume element i s  not 
when we i n t e g r a t e  over  the volume j u s t  ou t s ide  t h e  sur face  of 
t h e  metal. 
W e  no te  i n c i d e n t a l l y  t h a t  s i n c e  t h e  c e n t e r  of v ( r )  l i e s  
nea r  t h e  sur face  ion,  an appreciable  f r a c t i o n  of (8)  may come 
U 
from the  reg ion  between the  metal su r face  and t h e  ion .  Thus, 
some of the  emergent e l ec t rons  may be regarded as generated 
ou t s ide  the  metal  proper,  f o r  t h e  complete emergent wave 
4 
packet o r i g i n a t i n g  from a p a r t i c u l a r  4, may be w r i t t e n  ( t r e a t i n g  
t h e  4, as f ree  waves f o r  s i m p l i c i t y )  
where i n  c l a s s i c a l  terms v ( r )  4, ( r2) i s  the source of t h e  wave. 
A s  regards  t h e  i n s i d e  o f  the metal ,  w e  have a l ready  shown tha t  
t h e  long range part of v,  corresponding t o  low q, does no t  con- 
t r i b u t e  anything s u b s t a n t i a l .  Only t h e  high q components con- 
t r i b u t e  e f f e c t i v e l y ,  and the emergent e l e c t r o n s  o r i g i n a t i n g  i n  
t h e  f irst  few atomic l aye r s .  
9 .  
111. INTENSITY VARIATIONS 
Two orientational effects arise from the surface. In the 
free-electron model the total number of electrons excited to k 
is independent of its direction. However, states with k per- 
pendicular to the surface have the largest transmission coeffi- 
6Re 
#& 
cient through it. 
outside the metal is largest when kb is perpendicular to the 
surface, and the extra contribution to the matrix element from 
Further, the magnitude of the tail of 6, 
Y 
outside the metal makes the largest contribution to the current 
when k is also perpendicular to the surface, for then q is 
&e 
smallest, v ( q )  largest. Both effects give an orientational 
dependence relative to the direction of the surface as 
observed, 2y5 the former probably being dominant. 
With a real band structure there can also be directional 
effects wihh respect to the crystal axes. 
energy Eb contribute to the emergent current in a particular 
state +e, giving a factor n(Eb) in the current if all ob are 
weighted equally. However, in the nearly-free-electron 
pseudopotential model, the 4, with dominant wave vector kb 
parallel to, or otherwise closest to, the strongest component 
A l l  states 4, of 
clr 
ke of will have the largest v(q) factor. Thus, the cor- 
cy 
2,5 - rectly weighted density of' states, n(Eb) say, depends somewhat 
on the final state 6,. 
10. 
The d-bands in transitional metals do not contribute as 
strongly, relative to the plane wave bands, as in optical 
excitation. 2 y 5  Firstly, the amplitude of the tail of +a over- 
lapping the ion is smaller, 2y5  and secondly, the matrix 
element ( 7 )  does not have the enhancement provided in the 
optical case by the momentum operator, i.e., by differentiating 
once. 
ACKNOWLEDGEMENT 
I am greatly indebted to Dr. H. D. Hagstrwn for formulating 
and putting the questions which prompted the present paper in 
reply. The work was largely carried c ? i _ z t  during 8 visit tr! the 
Bell Telephone Laboratories. 
L * 
4 
~ 
12. 
. 
REFERENCES 
* 
Supported in part by NASA. 
t Permanent address: Cavendish Laboratory, Free School Lane, 
Cambridge, England 
See for example: H. D. Hagstrum, Phys. Rev. 96, 336 (1954), 
- 122, 83 (1961); H. D. Hagstrum, Y. Takeishi and D. D. 
Pretzer, Phys. Rev. m, ~ 5 2 6  (1965). 
H. D. Hagstrum, to be published. 
H. D. Hagstrum and G. E. Becker, to be published. 
L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955), 
2nd edition, p. 165. 
H. D. Hagstrum, private communication. 
FIGURE CAPTION 
Fig. 1. Surface i o n  n e u t r a l i z a t i o n  process.  
E, 
t A 
a 
Fig. I 


Theory of surface ion neutralization

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