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SIMULATION OF CROSS-CORRELATION FUNCTIONS BY A SUSPECTED SOLITON
MECHANISM OF ALKALI SUBMONOLAYER FIELD EMISSION NOISE
C. Kleint
To cite this version:
C. Kleint. SIMULATION OF CROSS-CORRELATION FUNCTIONS BY A SUSPECTED SOLI-
TON MECHANISM OF ALKALI SUBMONOLAYER FIELD EMISSION NOISE. Journal de
Physique Colloques, 1989, 50 (C8), pp.C8-47-C8-52. �10.1051/jphyscol:1989809�. �jpa-00229907�
COLLOQUE DE PHYSIQUE
Colloque C8, supplement au n o 11, Tome 50, novembre 1989
SIMULATION OF CROSS-CORRELATION FUNCTIONS BY A SUSPECTED SOLITON MECHANISM OF ALKALI SUBMONOLAYER FIELD EMISSION NOISE
C. KLEINT
Sektion .Physik, Karl-Marx-Universitlt Leipzig, Lim6strasse 5, DDR-7010 Leipzig, D.R.G.
ABSTRACT
-
Cross-correlation (CC) measurements of alkali metal submonolayers adsorbed on the field emission tip show partly very high 'signal velocities' and negative CC functions. These features of the field emission flicker noise can hardly be explained by surface diffusion-
concentration fluctuations and therefore soliton or domain wall move~nents have been proposed a s a possible explanation. By proper use of a well known analytical function which describes the soliton propagation the cross-correlation function CCF for a two collector arrangement is simulated. The influence of soliton shape and velocity parameters and of the probe distance is studied and the resulting C C F sare presented. Curves with increasing correlation frorrc negative values at small delay times to the positive region are indeed obtainable which mimic very well experimental results. They are in fact an indication of possible soliton movements in adsorbed layers.
I. INTRODUCTION
The flicker -noise of the field emission current of adsorbate izovered emitters in contrast t o clean tips /1/ proved t o be strongly dependent on the adparticle mobility /2/.This implies an influence of the activation energy of surface diffusion which appears ,to be dependent on the surface structure including plane and directional peculiarities and on adsorbate binding.
While random walk surface diffusion of single adatoms is observed by well known FIM investigations the adparticle mobility at higher coverages has to be studied more indirect via the stochastic properties of the emission current. In different models developed s o far it is mostly assumed that surface diffusion induced concentration fluctuations /3,4/ which modulate the F E current or trap kinetics combined with surface diffusion / 5 / represent the origin of the field emission flicker noise (cf / 2 / for further references).
The interpretation of some observations, however, especially of the cross-correlation, by the usual diffusion mechanism seems to be difficult.
These are very high 'signal velocities' / & / (as compared to surface diffusion 'velocities') and negative parts of the cross-correlation function (CCF) at relatively low temperatures /7/ and large probe distances 8 . T o explain these anomalies a soliton or domain boundary movement mechanism was suggested /9/ and further specified /&,10/. Soliton processes in adsorbed alkali metal layers were already discussed by Naumovets 1 , and by Pokrovsky and coworkers / I 2 9 13/ an adlayer soliton model was developed.
The following simulation is intended t o show by a simple description of adlayer soliton or domain boundary movements their effect on hypothetical cross- correlation functions which might be compared with CC two collector measurements. While such movements are in fact assumed (see below) we use a well known analytical soliton solution of the sine-Gordon equation a s their representation. The influence -of different parameters can therefore easily be studied and we get a first insight into their possible effect on the cross- correlation function. In contrast to an adparticle concentration fluctuation model negative CCF parts are easily obtained and we end with a discussion about the implications and on possible improvements of this model.
11. SIMULATION OF DOMAIN WALL MOVEMENTS
Several approaches to soliton or domain wall movements have their origin in the Frenkel-Kontorova model /14/. We start with the nonlinear sine-Gordon
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989809
differential equation ( c p / l a / ) 6
'
9
- -
c=-
S"P + Q= s i n @ = 06t" & ;.: =
T h e kink motion of a dislocation line is described by its simplest one- soliton solution
@ ( x , t ) = 4 tan-' e>:p((x
-
v t ) / r ) ( 2 )According t o t h e simulation procedure for t h e kink dynamics applied by H o m e a / 1 5 / w e construct a similar soliton representation o r domain function Z starting w i t h t h e following difference function
w h e r e v is t h e velocity and
r
t h e kink width. T h e result is t h e movement w i t h t i m e of a r i d g e like soliton structure 1 Figs. 2-
7).Trying t o s i m u l a t e t h e movement of a periodic array of domain w a l l s and its effect w e construct a finite soliton comb structure by summation of spatially shifted ridges:
M
E P ( x t t ) = 4 G itan-% exp((x + b + mL
-
v t ) / r )m-O
-
tan-' exp( fx + mL-
v t ) / r ) )Referring t o Fig. 4 of Ref. / 1 0 / w e think of t h e alkali metal superstructure domains as being separated by dislocation lines w i t h missing akali adparticles. A domain w a l l movement a c r o s s t h e collector probe-region o n t h e surface t h e n implies a change in t h e effective w o r k function accompanied by a considerable ( F N equation induced) variation in electron emission. T h i s
leads t o fluctuations of t h e F E current about a t i m e averaged mean v a l u e 6 i ( t ) = ( 7
-
if*) ). W e simulate t h e change in s i g n of t h e fluctuation by subtracting a constant G ( a l l v a l u e s a r e g i v e n in arbitrary u n i t s ) w h i c h depends o n t h e s h a p e of t h e composed s o l iton group,W i t h increasing t i m e t h e soliton c o m b structure rrioves along t h e x direction.
Fig. 1 s h o w s t h i s a d v a n c e for a n assumed Z ( x , t ) w i t h M = 2 1 maxima. Starting a t t, = 100 (Fig. l a ) t h e group moves t o t h e right a n d a part of t h e
maxima subsequently passes t h e first collector region assumed t o be located a t x = 0
.
After 40 t i m e u n i t s fdt = 0.1) or a t te = 140 t h e group arrives a t t h e position s h o w n by Fig. Ib. T h e s h i f t of t h e group depends of c o u r s e o n t h e 'velocity' v which is t a k e n here t o be 0.7. W h i l e t h e 'domain width' represented by L in eq. 4 h a s been considered t o be L = 8 , t w o other e x a n ~ p l e s of t h e soliton group s h a p e a r e g i v e n in Figs. 2 . and 3 w h e r e L corresponds t o 6 a n d 12, respectively. W h i l e - t h e variation of a s h a p e parameter ( b r L y T ) a l s o changes t h e 'amplitude' of t h e domain function Z(x,t), t h e Z 7 s u s e d in t h e following cross-correlation s t u d i e s a r e adjusted by means of t h e parameters a a n d G t o provide approximately t h e s a m e amplitude. ( T h e number o f domains iM+.ll is chosen large enough t o e n s u r e f u l f i l m e n t of t h e condition described a t t h e e n d of section 1 1 1 . )o
0 -80 -40 0 40 80 0 -80 -40 0 40 80
D i s t a n c e x D i s t a n c e x
Fig. 1. The domain function representing the FE current of a domain array with 21 maxima i width L = 8 ) moving with velocity v = 0.7 at time te
(a) and 40 time units later ib).
10
N N
5 5
C C
0 0
-..
-4+r c.
U 0 U 0
C c
3 3
L L
c -S c
... ...
rn .3
E -10 E -10
0 0
0 -80 D i s t a n c e -40 x 0 40 80 13
'
-80 D i s t a n c e -40 x 0 40 805
Fig. 2. A domain function with Fig. 3. A domain function with shorter period ( L = 6 ) . larger period iL = 12).
-12
f
8 5 8 108
T i m e D e L a y T
Fig. 4 , Dependerice of the cross-correlat ion function on +:tie doroain widi:h, a
-
c." L = 6 , 8 and 12, respectively, for a collector distance of b = 10.I I I. CALCULATION OF THE CROSS-CORRELATION FUNCTLON
T o obtain the CCF patterns for the assumed domain functions we need the CCF definition which reads in an unnormalized form (cf Appendix A in Ref. / 2 / )
Cir) = lin (f/T)
f
Si(t)% S i ( t + ~ ) ~ dt ( 6 )-,-..a a
X and Y refer to the two probed regions from where the FE currents are collected. Their distance bpilts the second region Y at 3: = b with our assumption that X refers to x = 0. Both signals have to be multiplied for an increasing time delay T and the products obtained must be averaged for different starting times t and normalized by division with the observation time T.
The digitalisation of this procedure finally yields a computing program for the simulated CCF R(T) which atas written in PASCAL for r = 0
...
100.Because our domain functions are nearly periodic within the 'soliton group"
created by means of eq. 4 , we do not need an infinite T for averaging. By a suff icientlr large T = W.dt an almost asymptotic function R(r) is obtained provided the 'soliton group' does not leave the collector region in the time intel-va:l that w a s u s e d (t,
...
(t, + N.dt + Ime).dt)).Fig. 4 shows the simulated cross-correlation function R(T) for different domain widths L. Beginning with L = 6 (curve a ) we find a relatively short period in R ( r > which starts with a negative cross-correlation coefficient (CCC, R(r=B))) and I-eveals with increasing delay r a first minimum, a maximum a.r\d isnother rninirr~um. Curves b for L = 8 and c iL = 12) clearly demonstrate an increased period and also different CCCs. It is obvious from Fig. 4 that a further increased width can lead to completely positive or n e g a t i ~ e CCFs.
The variation of the velocity is simulated in Fig. 5 . Small v (curve a: v=
@,5 1 I:J: fZ.65) reveal long periods af R(r) (t: v=8.9; d: 0 . 9 5 ) . Because of a constant c ~ c s t f ec.i:or diskance and soliton share the CC:Cs are similar.
An interest in? dependence is investigated by variation of the col lector distance which 1.5 an essential part af the colrrplete CC function R(r, 1).
Fig. 6 pres~!nts an example where the curves a to d correspond to h = 10, 15, 28, and 2:fi~resrectively. The CCF shape does not change but the curve is shifting to the left (or to smaller r) with rising distance. Depending on the other parameters used different CCF shapes can be generated (Fig. 7.3-d).
V. DISCUSSION
The results presented are a first approach Co simulate cross-correlation functions originating frorrr domain bcandary or soliton movements in alkali adsar bate supel-structuress The domain functions chosen have a relatively small wicl*:!~ but it is believed that the main CCF features are already obtair?ucl. We cl-lose a relatively wide delay range which shows the periodic shape of the CCFs. In this way it is more convenient
60
follow the development of the curves with changing parameters. It is obvious that negative CCF parts appear if on the average positive Z values at probe region X met negative ones at region Y and vice versa.In reality we expect at least a distribution of the domain widths and
additional irregularities and therefore a broadening of the CCFs. This can be simulated with some more effort. It is this possibility (stati5tical variation of the domain width) which also justifies the use of eq. 2 with the described soliton group composition by eqs. 3-5: Though a truncated sine wave would show the essential CCF dependences given here a further e,,tension to this more realistic description would not be possible.Another interesting feature to study would be the influence of more complicated domain movements, for instance a reflection of the solitons at the plane baundaries /16/. There are various publications on soliton crossings and collisions /17,18/ which might be helpful, in setting up the domain functions. As to the general. question of fluctuation induced soliton movements we can only refer to the literature /19, 1ZY13/.
Fig. 5. CCF dependence on the soliton velocity, a
-
d: v = 0.50, 0.65, 0.80?and 0.95, respectively, for h = It?.
Fig. 6. CCF dependence on the collector distance, a
-
d:. a = 10, 15, 20, and 25, respectivelr, for a velocity of v = B . 6 5 and L = 8.- 1 2 . I
8 5 8 3.88
T i m e P a L a Y T
Fig. 7a-b: CCF dependence on collector distance, a and b: a = 5 and 8.5, r e s p e c t i v e l ~ , for v = 0.45 and L = 8.
-
t-'-
a8 58
-
T i n e De l a Y T
F i g . 7 c - d . CCF d e p e n d e n c e o n col l e c t o r d i s t a n c e , c a n d d : h = 10 a n d 20, r e s p e c t i v e l y , f o r v = 0 . 4 5 a n d L = 8.
F o r a s h o r t c o m p a r i s o n w i t h e x p e r i m e n t a l CCFs we t a k e f i r s t F i g . 5 o f Re?.
/ 2 0 / w i t h t h e t e m p e r a t u r e d e p e n d e n c e o f t h e a d s o r p t i o n s y s t e m W ( 1 1 2 ) K / 8 = 0 . 3 5 . I n d e e d , w i t h i n c r e a s i n g t e m p e r a t u r e a s h o r t e n i n g o f t h e p e r i o d a c c o r d i n g t o t h , e p r e s e n t F i g . 5 l v e l o c i t y d e p e n d e n c e ) is o b s e r v e d w h i c h f i n a l l y l e a d s t o n e g a t i v e v a l u e s a t h i g h e r d e l a y s r. I n a d d i t i o n t h e e > : p e r i m e n t s h o w s a n i n d i c a t i o n o f o s c i l l a t i o n e w i t h a s h o r t p e r i o d w h i c h m i g h t b e e x p l a i n e d by a s u p e r p o s i t i o n o f s m a l l a n d larger d o m a i n w i d t h s a c c o r d i n g t o F i g . 4.
L o o k i n g a t t h e d i s t a n c e d e p e n d e n c e o f t h e same s y s t e m g i v e n i n F i g . 2 o f R e f . /2@/ we f i n d a c o r r e s p o n d e n c e o f t h e f u n c t i o n s f o r 500 a n d 600 a w i t h , t h o s e o f F i g . '7c a n d 7 d , r e s p e c t i v e l y . F o r t h e W ( 1 1 0 ) s u r f a c e w i t h p o t a s s i u m we o b s e r v e some s i m i l a r i t 9 b e t w e e n F i g . 4 o f R e f . /8/ a n d t h e p r e s e n t F i g s . 7a a n d 7 b . An o p t i m u m f i t t o e x p e r i m e n t s , h o w e v e r , w a s n o t t r i e d .
T h e f a c t t h a t t h e s i m u l a t i o n e a s i l y r e v e a l s a l s o n e g a t i v e p a r t s o f t h e c r o s s - c o r r e ' a t i o n f u n c t i o n s s e e m s t o be a s t r o n g s u p p o r t o f t h e s u g g e s t i o n t h a t s o l i t o n o r d o m a i n b o u n d a r y m o v e m e n t s c a n e x p l a i n t h e CCF e x p e r i m e n t s i n a c e r t a i n t e m p e r a t u r e r a n g e .
T am much o b l i g e d t o F r i e d e m a n n K l e i n t f o r w r i t i n g t h e CC p r o g r a m a n d h e l p w i t h o t h e r s o f t w a r e a n d t h e m a n u s c r i p t .
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