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Current fluctuations from small regions of adsorbate covered field emitters
SURFACE
SCIENCE 38 (1973) 373-393 8 North-Holland
CURRENT
FLUCTUATIONS
ADSORBATE
Publishing Co.
FROM SMALL REGIONS OF
COVERED
FIELD EMITTERS
A METHOD FOR DETERMINING DIFFUSION ON SINGLE CRYSTAL PLANES
COEFFICIENTS
ROBERT GOMER The James Franck Institute and The Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, U.S.A.
Received 29 December
1972
A novel method is presented for determining surface diffusion coefficients of adsorbates on single crystal planes of field emitters in terms of the time correlation function of current fluctuations. The method is based on the fact that current fluctuations are related to adsorbate density fluctuations, whose time correlation is governed by relaxation times simply related to diffusion coefficients. A general formalism is presented, some idealized cases are worked out analytically, and the case of a circular aperture is treated in detail numerically. Switching between different adsorption states, in addition to but not affected by diffusion, is also included. The forms of the correlation function are shown to be complicated, with a l/t tail at large t, and cannot be represented by simple exponential decay.
1. Introduction The development of the field emission microscope by E. W. Miillerl) made possible the direct observation of diffusion processes on clean metal surfaces 2). In most investigations the emitter was shadowed with adsorbate from one side, and the spreading of this non-uniform layer observed as a function of time, temperature and coverage. While a great deal of semi-quantitative information has been obtained in this way2), the method suffers from the defect that diffusion observed on a given crystal plane also depends on transport over adjacent regions, so that the unequivocal correlation between diffusion constants and substrate structure is often difficult. In addition, visual observations are possible only on highly emitting regions. To some extent this can be overcome by the use of a small probe-hole in the screen of the tube, through which a small portion of the electron beam is allowed to pass. In this way emission from very small regions (5 10 A in linear dimension) even of high work function planes can be measured. The arrival of adsorbate is signalled by changes in current3). Nevertheless, even this refinement does not resolve the uncertainties pointed out above. For ad373
374
R. GOMER
sorbates visible in the field ion microscope direct observation on individual planes is possible, as shown by the experiments of Ehrlich4). Unfortunately even the introduction of low field image gases does not extend the class of visible adsorbates beyond metallic onesa), although it does of course permit the observation of metal atoms with lower ionization potentials. The reason for this is almost certainly the absence of appreciable local density of states just above the Fermi level for most covalently bonded adsorbates. For such adsorbates one is therefore restricted to field emission. During experiments on field emission from single planes carried out by Engel and the authors), we noticed that heating of emitters covered with equilibrated CO layers led to fluctuations in probe-hole current. It seemed plausible that these were caused by concentration fluctuations over the small region probed. Since such concentration fluctuations build up and decay by surface diffusion, their time correlation function must contain the local surface diffusion coefficient D through some characteristic decay time z,-r#$4D, where r,, corresponds to the dimension of the region probed. An explicit expression for the correlation function of current fluctuations thus provides a means of determining diffusion coefficients on single crystal planes. The fact that current fluctuations in field emission may be related to surface diffusion has also been noted by Kleint6), who attempted to adapt a model by Bess7) for l/f noise in semiconductors. This model treats switching between two adsorption states, mediated by surface diffusion. It does not consider at all the build-up and decay of concentration fluctuations over small regions, which is central to the present work, and thus addresses itself to quite a different situation both experimentally and theoretically. In the following, the correlation function for current fluctuations from small regions is derived on the basis of diffusion limited concentration fluctuations.
2. Relation between emission current and adsorbate density In the absence of adsorbate the current density of field emission j is given by the well-known Fowler-Nordheim relation, which we may write as lnj = 1nB - (6.8 x lo7 o/F) @,
(1)
where B is a field-insensitive term, F the field in V/cm, 4,, the work function of the clean surface in eV, and o an image correction term l). The presence of an adsorbate modifies emission by changing 4 and in some cases B. The change in 4 is due to the dipole moment associated with each ad-particle. This consists of a zero field component and one induced by the applied field. Since the latter is proportional to applied field, a simple analysiss) shows that the work function change resulting from polarization of the ad-
CURRENT FLUCTUATIONS
375
sorbate by the applied field appears as a field-independent term, i.e. seems to modify B. For present purposes, however, we are not interested in the variation of current with field, and the separation into intrinsic and induced dipoles need not be made. While the apparent changes in B observed in plots of In i/F’ versus l/P can largely be explained by polarization, it may also happen that an adsorbate produces anti-resonances near the Fermi levels) [in slightly different language, a marked decrease in local density of states at the Fermi levelis)]. Since in most cases virtually all emission comes from within 0.5 eV of the Fermi level, electronic effects outside this energy range cannot affect total current markedly. This phenomenon, first pointed out in connection with field emission by Duke and Alferieff3), can lead to a real change in B. Experimentally it is usually found that IogB is proportional to coverage, so that we may write quite generally for the total current i from a region area A In i = In A& + cIc - (6.8 x lO’/F) #+u, (2) where c1 is a constant (positive, negative or 0), c the density of ad-particles in the region and C#J the effective work function in A at field F and for a given ad-particle distribution. Writing $=$+Sss,
(3)
where $ is the time-averaged work function and SC$its (small) fluctuation, we see that @z:*(l +ts4/d)=&t$$+s#, (4) so that 61ni~Ini-l~=c~6n3-c2F~,
(5)
where &r=n-E, cz=- 1.02 x IO’ m$*/F, and Ini the log of the current when a=2 and $=d;, and c; =cJA; n, is the number of ad-particles in A. To complete the relation between current and adsorbate density we must express 4, or alternately 4 and 64, in terms of n. It will be useful to work in terms of the work function increment caused by adsorption A@=#-40.
(6)
For sufficiently large areas and for uniform adsorbate distributions A4 = 2xPn{A,
(7)
where P is the dipole moment of the ad-particles, defined in such a way that the plane of zero potential bisects it. Thus the contribution per ad-particle, W becomes, in this limit, 2nP/A, and we may think of this as the average over A of the potential contributed by an ad-particle. This suggests that we look for A+ as the sum over all particles i in the system (not merely in A) of the
376
R. GOMER
quantity
s
1 Pd w (XTd) = A A [Cx _ x’)’ +.-zF
dx’ ’
(8)
where d is the distance above the surface where the potential is to be evaluated and x denotes the position of the dipole (within or without A) with respect to an origin, conveniently chosen at the center of the probed area A. It is easily verified that for a circular area of radius rg and a dipole placed at its center (9) which, for d
$i
tan-’ [
-Y> _....- (1 -x)(1 ( z[(l - Y)” + (1 - x2 + z2)]’ > (1 - x1 (1 + Y> + tan-’ ( z[(l + Y)2 +-(1 - X)” + z21+> (1 + x1 (1 - Y> + tan-” ( z[(l - Y)’ + (1 + x>’ + Z2]%) (1 + x> (1 + Y> + tan-’ z[(l + x)’ + (1 + Y)” + zz-J+ ’ (
)I (10)
where all quantities are expressed in units of a and z=d/a. It can readily be shown that, for a dipole at (0, 0), the same limits result as in the circular case. Thus the approximation that a dipole makes its full contribution 2nP/A to the work function when it is within A and zero otherwise is a reasonably good one for z 4 I. 3. General expression for the correlation function of current fluctuations
We can now express 6 lni=lni-E in terms of sums over the particle positions in the entire system by combining eqs. (5) and (8):
CURRENT
FLUCTUATIONS
371
The symbol S(xi) has the value unity when the coordinates Xi of particle i lie in A and zero otherwise, and the bars denote time (or ensemble) averages. It follows that the time correlation functionf(r) of the quantity 6 lni is given by f (7) = 6 In i (0) 6 In i (7)
= Cc;T 6i(O)+ c2T K (O)llIc;C 6j(T)+ c2C wj(T)l j j - CT(c;6i+ c2W,)l’ 9
(12)
with W,(z) = W(x,, T), etc., since
CTLc;6i(O)+ c2K(O>ll[F [c; 6j(z) + cZwj(T)ll= IL?“I16i + ‘2K)l ‘9 (13) etc. The last term in eq. (12) is the square of the quantity
1 (c;
di +
c2Wi)
=
N
i
s”
[c;6(x)
+ c,w (x)]
tot
40t
$tot c; + c,l7
N
=
=
c;ri+
c2
A3,
(14)
>
where Nis the total number of ad-particles in the entire system, A,,, the total area. We next separate the first term in eq. (12) into a sum over i =j, and a sum over i #j : 4
Cc;
6i co)
+
c2K
(O)lCc;6i(z) + +
i;j
(r)l
c2W
Cc;
si to)
+
c2 wi
(O)lCc;6j(T)+
c2
wj
(r)l*
We now assume that the displacements of different particles are not timecorrelated at all, so that the ensemble average of the double sum becomes (N2 - N)
[Sg
[c;6 (x) +
C2
w
(x)1]2.
tot
AWIt
Since N is very large, N2 - N E N2, and thus the double sum is cancelled by the term evaluated in eq. (14). Hence, f (r) = 4
Cci6i
to)
+
c2W,
(O)lCcisi@)+
c2 wi
(T)l.
(15)
R. GOMER
378
It might be thought at first that the neglected term N
,*t [J$x.
(c;6
-I-
,,W,]2
At,t
is of some significance. However, eq. (14) shows that it is of the form ;
(c;Z +
c2
AT)“.
We shall see shortly that it is of order A/A,,,, relative to the term retained in eq. (15). The evaluation of the correlation function now becomes simplein principle. We write f(r)=N
J
%X_[C;~(X)+C~~(X)]“P(~/X’,~)~X’)[~;G(X’)+C~~(XI)]. tot AtSIt
(161
Here dx/Atot is the absolute probability that a given particle is in dx at t =O, and 9(x
1x’, 2) dx’ = d&
exp( - l%!!!),
the conditional (normalized) probabiIity that, if it was in dx at 1--O, it will be in dx’ at t =z. Expanding eq. (16), we have
with (19a) W’ (x) W’ (x’) -- 4~Dr exp( - Eyg)dx
dx’,
(19b)
where we have written W’= W/(2~~/~). As it stands, eq. (19) can only be integrated numerically. To show the structure of f(z) we now make the assumption that W’=6, i.e. is unity in A and zero outside, so that f(z) reduces to
CURRENT
The quadratures
FLUCTUATIONS
319
in eq. (20) can be carried out for a square area, A = 4a2, and
yield, for that case
(21) with
(22) fi is the average number
of ad-particles
in A and
z0 = a2/D.
(23)
We may write equivalently (24) For z/r,, 4 1, g (z) reduces to (25a) and for z/r0 % 1 g(r)-&[exp(-:)-I]22
(25b)
so that f (7) decays as l/r at large z. Eq. (22) or (25a) shows that (26) which corresponds to the mean square “strength” of a fluctuation, and is proportional to k as it should be. We are now able to verify that the term neglected in our derivation, (l/FA,,,) (Ec,+c,A$)~, with eq. (26). 4.
is of order
A/A,,,
relative
to
f (0) by comparison
f (7)from decay of density fluctuations
It is interesting that eqs. (20) and (22) can also be derived from a slightly different viewpoint, by using Onsager’s hypothesis that fluctuations decay according to macroscopic laws, i.e. by studying the decay of a fluctuation according to a macroscopic diffusion equation. We can do this by postulating a delta function fluctuation at (x,0), with x somewhere in A. At time r the concentration profile will be c(X.,r)=exp(--Is)&,
(27)
380
R. GOMER
and the total fraction of the original strength of the fluctuation left in A after time z is found by integrating with respect to x’ over A. To get the ensemble average we must now average over the initial position x as well and thus obtain, using the mean square strength of fluctuations, eq. (20) and the subsequent development. The equivalence of these two methods is no accident of course, but stems from the relation between D and the random walk assumption inherent in eq. (17). Finally, it should be pointed out that the averaging over the initial positions of delta functions, followed (or preceded) by integration of the concentration profiles over A, is equivalent to postulating a uniform, step-like concentration fluctuation over A and following its time development. (The arbitrariness of this fluctuation profile comes from the weighting function w’=6; a more realistic weighting function would lead to a less abrupt profile.) It is possible to treat this diffusion problem in the standard way and to obtain solutions of the form C
(I’, 7) = C
A
j
Jo
(Ujl.) exp (- UjzDr)
i
(28)
(for circular probe areas), where the J,‘s are Bessel functions of order 0, with roots Uj chosen to make J, (u~Y,,) =O. The Aj’s are the appropriate coefficients for the expansion of c (Y, 0) in the orthonormalized set of Bessel functions. The roots are given approximately by Uj
E
7c
(j - &)jr,,
(29)
so that each term in eq. (28) is multiplied by a time factor of the form exp ( - r/zj), with zj= r$ [x” (j - $)‘D]. Integration of c (Y, r) over A would then give g (r). The point to note is that only for very special assumptions of the initial fluctuation profile, namely c (r, 0) equal to the lowest order Bessel function would g (T) show an exponential decay. The statistically correct assumption about the fluctuation leads to the more complicated relation eq. (22); this can be expressed, however, in the form of eq. (28), or more correctly as an integral over A of eq. (28) (which leaves the z dependence intact), i.e. as a series of different exponential decays terms. It is interesting that, even for r& < 1, eq. (25b) does not go over into a single exponential decay term, as eq. (28) might suggest. The reason is that the random positions of the individual fluctuations over A assign too small a weight to the A, term to ever let it predominate, despite the fact that it decays more slowly than any other term. As we shall see, the fact that eqs. (21)-(25) have been derived for a square probe area does not alter these conclusions, since the long time limit of g (T) for a circular aperture will be shown to be of the same form a eq. (25b).
CURRENT
FLUCTUATIONS
381
5. Circular aperture The preceding discussion has concentrated on a square aperture and an assumed step function form for W, to show analytically the salient features off (2). We treat here the general case of a circular aperture, starting with W(x). Changing to polar coordinates gives 2n
W(P)
W’(p) = ~
z
2rrPllrr,2 = G
1
ss
p’ dp’ de (30)
o o (p’ + p’2 + z2 - 2pp’ cos 0)s’
where p=r/ro, z=d/r,,, with r. the radius of the probed over p’ and partial integration over 8 gives
region.
Integration
L
W’(p)=
1-z
(p’ + z2 -pcos@dG (31)
nos (P 2sin28+z2)(p2+z2+1-2pc0se)*’
These integrals were evaluated for a number of values of z on an IBM 360 computer. The results, as well as graphs of w’ (p)p are shown in figs. 1 and 2. It is seen that for small z, W’(p) approaches a step function 6 (p)= 1 for p=Gl,Oforp>l. The integrals appearing in eq. (19) expressed in polar coordinates are all of the form
&(T) =
s
zo
dp dp’ de de R(p) R’(p’) exp
z
-
p2 + pf2 -2PP’
COS(~-8') 3
TOIT
(32) z. = rg/4D,
Fig. 1.
Plots of W’(p) versus p for various values of .z=d/ro. Here p=r/ro, the radius of the circular probed area; z = 0 refers to W’(p) = 6(p).
(33)
with ~0
382
R. COMER
100
150
200
250
P
300
Plots of p W’(p) versus p for circular probed areas.
Fig. 2.
and p =r/ro, p’=rr/r,,, and R and R’ are either W’ or 6. The angular dependence appears only as the difference of 8 and 13’, and thus can depend only on cp= 8 - 0 ‘. Thus we may use the equivalent angle variables 8 - 0 ’ and 0 +8’, and integrate at once over the latter obtaining a factor of 27~. The integral over cp is of the form II 2
s
exp VPP
(G/T)
~0s
‘PIdv = 2nJo W (+)I
0 = 2nlo
where 1, is the Bessel function given by
(~o/T)l>
(34)
of order zero and imaginary
Jo (ix) = IO(x) =
Thus the integrals
VPP
argument
cc Qx)‘” ___ m=O W2. c
in eq. (19) reduce to double
ix,
(35)
integrals
over p and p’:
11 9 lb>
=
4 (70/r)
IO (~PP’T~/~) exp I-
(P’
+ P”>
ss
~o/~lPP’
dp dp’ 9
(364
00 mcc
92 (T> =
4 (rolr)
ss
W’ (P> W’ (P’> 10 (2PP’7o/T)
0 0 x exp [ - (P’ m
93 (r) = 4 (to/r)
+ P”)
z,/~]
PP’ dp
dp’ ,
Wb)
1
W’ (P> IO C~PP’~~/~) ew [-
ss o=o o’=O
(P’
+ P”)
zohl
x pp’ dp dp’ .
(36~)
CURRENT
383
FLUCTUATIONS
For large z, I, z 1, and with the further assumption that W’ (p) = W’ (p’ ) = 6, we immediately obtain g (7) = (~~~~)[ 1 - exp (- Q/Z)]“,
zoiz < 1,
(37)
which is of the same form as eq. (25b) for a square aperture. It is interesting that in terms of D we thus obtain [using eqs. (23) and (33)] g (2) + (a2/7cD) z - i g (T) + (~~/4~) z- r
square of half-width a, circle of radius rO,
(38a) (3W
asz+ co. The integrals in eq. (36) were evaluated numerically on an IBM 360 computer for various values of z and z=r/r,. In those integrations with infinite upper limits, approximate upper limits pU were chosen as follows: The average work function increment .Q can be expressed as an integral of m W(r): &$=2nP~=2~
EW(r)rdr, s
(39)
0
so
that
m W’(p)pdp=l.
(40)
0 The upper limits pU were chosen so that P”
l-2
[ W’(p)pdp
.i
(41)
Since it is clear from eq. (8) that for large p, IV (p)oz pm3, it can readily be shown that condition (41) is equivalent to 2p,2 W’ (p,) < 0.01.
(42) The following values of pU were used in accordance with this criterion: For z=l.O, p,,=52.2; for 2=0.5, pU=25.3; for 2=0.2, p,=11.4; for z=O.l, pu=7.5. In order to evaluate the integrals at t=O, it is simplest to start with the definition off (0). (W e sh ow explicitly only the case for c1 = 0. The extension to finite c, is obvious.) f (0) =
cjC i
W (ri, 0) W (t-i, 0) W’ (r) W’ (r’) r dr dtl de’ x 6 (Cl -
d’) S (r - r’) dr’ (43)
384 so
R.GOMER
that (44)
For W’=6 that
(i.e. z=O), this obviously yields g, (0) = 1. It is also easily shown
(45) For z#O both g, (0) and g, (0) are less than unity, and g, (O) 0, dipoles beyond the probed area contribute to emission in it, thus effectively increasing its size and consequently decreasing the relative fluctuations. Since g, represents a cross term between gi and g2, it represents a smaller effective area than g2 and hence a larger Auctuation. The results of the computer calculations are shown in figs. 3 and 4 respectively as functions of t =T/T, for various values of Z. The case z=O corresponds to W’=& and applies when c2 27rP -%c,. The other curves in fig. 3 apply directly for the appropriate z when c2 2nP + c, . For intermediate cases ci and P would have to be known separately in order to combine the appropriate curves of figs. 3 and 4. In general this may be rather difhcult but, as pointed out, ci is in fact generally small. At large t, g*(t) and g,(t) + l/t. It is easy to see from eqs. (36) and (40) that this must be the case. At small t the curves do not behave like e-‘, except over very narrow ranges.
Fig. 3. s%(t) versus t for various of z=dJr~. The curve marked z=O corresponds to &(t). Here f = r/50, with to = dj4D. The quantities $1 and 92 are defined by eqs. (36a) and (36b), respectively.
CURRENT
0
Fig.
4.
gs(t)
385
FLUCTUATIONS
2
4
,6
8
IO
versus t for various values of z. The quantity g3 is defined by eq. (36c), all other notation as in fig. 3.
The dependence of g2 and g3 on z requires some discussion. For a fixed t,,, i.e. for a given rO, g2 and g3 vary most rapidly with time as z decreases, as we should expect. It must be remembered, however, that d is fixed at -5 & so that zxl/r,, and hence z2ccl/z,,. Thus, a better idea of the real time behavior off(z) can be gained by plotting g2 (t ) versus t/z2 = (40/25) z. Since an experiment would look at the relative change inf (7) as a function in figs. 5 and 6. of T, we show g2 (t)lg2 (0) and g&1 g3 (0) versus log,,(t/z2) This mode of presenting the results shows two interesting features. First, as zO decreases, so does the real time required for the decay of the correlation function, as we should expect. Second, and more important, the shape of the normalized curves for different z is almost identical. This is true both for g2 and g3 and for the comparison of g2 with g3. The only major differences are displacements
along the log(t/z2) I
I
axis. It now becomes I
an obvious
I
Fig. 5. Plots of gz(t)/gz(O) versus loglo(r/9). Also shown as z = 0 curve is gl(t) versus 1. Dashedcurve is e+ versus t. Solid points represent a hypothetical mixture: g(t) = 0.25 gl(t) + 0.25 gz(t) + 0.5 gz(t) for z = 0.5, plotted against loglo(r/0.25). It is seen that the curve is shifted to the left, as expected.
386
Fig. 6.
R. GOMER
Plots of 93(t)/gs(O) versus log&/z2) for various values of z. The curve z=O refers to plot of g1 (t) versus loglo t. Dashed is e+ versus loglot.
extension to compare the curves with the corresponding z=O case. We may do this, for instance, by comparing t for g2 (z, t)/gz (z, 0) =0.5 with t for g1 (t) =0.5. These ratios can be though of as increases in the effective values of ri. We show in fig. 7 the factors & (k=2 or 3) defined by
fi = r. (effective)/r,
= [t (g)/t (gr)]+ .
(46)
Both p2 and p3 vary linearly with increasing z, i.e. decreasing r,,. As expected, p3
it would
take detailed
knowledge
of all the parameters
involved
to
Fig. 7. Plots of 8~ = (t&l)+ and 83 = (t3/f1)* versus z. The quantities tl, 12,and t3 are defined as the times required for g1 (t), g2 (t)/gz (O), and gs (t)/g3 (0) respectively to become 0.5 ; /3r0is the eflective radius of the probed region.
CURRENT
decide
on the exact
contributions
FLUCTUATIONS
of each g. A hypothetical
387
mixture
is
shown in fig. 5. In practice, the importance of these considerations can be minimized by choosing a reasonably large r,, and hence small z, for which fi - 1 is small in any case.
6. Relation of rO to single plane parameters The last section has touched on the effective radius within which time correlated events contribute to the measured correlation function. For the idealized, infinite plane case this is of academic or computational interest only. For a real field emitter of radius 1000-5000 A planes of given crystallographic orientation are of 50-200 A in radius, and it is therefore important to see if the effective r0 can be made smaller than these dimensions, since we wish to measure D for single planes. It is clear from fig. 7 that this is so. Another criterion, somewhat more arbitrary, would be to pick the value of p, at which 2JP,p’W’(p’)dp’ had reached some fixed value, say 0.80 or 0.90. It should be pointed out, however, that the curvature of the emitter imposes an absolute limit on the effective size of the probed region, since the horizon distance r, for a point d 8, above the surface of a sphere or radius r, is r,,, = (2r,d)*.
(47)
For d = 5 A and rt = 1000 A, r,,,= 100 A. It may now be asked to what extent the curvature of the surface affects W’ and the subsequent calculations, which were carried out for a plane surface. For all cases considered the effect is very small, and hence we may assume the validity of the calculations for the curved emitter with confidence. 7. Extension to interactions between ad-particles We have assumed so far that there is no interaction between diffusing particles. In many systems, however, this is unlikely to be true. A detailed calculation taking interaction into account would be very difficult. However, it is known that even for biased random walks the form of eq. (17) is not changed, although the value of D will be affected. Thus our results will most probably remain valid, with D a function of average concentration. 8. Spectral density representation The results obtained here have been presented as correlation functions. It would be possible, of course, to represent them as spectral densities by taking their Fourier transforms. Since experimental correlation functions
388
R. GOMER
can be obtained directly, this step is omitted here. It is worth pointing out, however, that the power spectrum will not be fitted with any accuracy by functions of the form
s (0) = because the correlation nential.
7;
‘/(co2f g),
function is so poorly approximated
(48) by an expo-
9. Inclusions of ~~flop processes For certain adsorbates, for instance CO on bee transition metals, different adsorption states characterized by different dipole moments can exist on a given crystal planerr). It is possible that under rather special conditions a switching back and forth could occur, although in most cases the change from a given ad-state to another seems irreversible. It is fairly easy, with some restrictions, to extend the present calculation to include flip-flop. The treatment we shall use is a simple adaptation of one worked out by Machlupls) to treat current noise in semiconductors. Let state A be characterized by a dipole moment PA and let its mean life with respect to flipping to state B be rA. Let state B be similarly characterized by PB and zg. Letf,(z) andfa(z) be the probabilities that an ad-particle is in a state A and B respectively. Then
To proceed further we must now make two key assumptions: (1) we assume that the flipping of different ad-particles is wholly uncorrelated, and (2) we assume that an average diffusion coefficient D can be defined as b = D,f,
+ DsfB.
(51)
The latter assumption is certainly valid when flipping times are short relative to diffusion times. When the opposite is true, i.e. if there are two populations each with its own diffusion coefficient but with no switching state A to B, we are in fact dealing with a different problem whose solution will be indicated also. It should also be pointed out that our model assumes that the switching times have nothing to do with diffusion. Kleint’se) and
389
CURRENTFLUCTUATIONS
Bess”)
model,
on the other hand,
assumes
that switching
times are deter-
mined by diffusion, since an ad-particle in state A can find a B site only by migrating there. To return to our model, we have by analogy with our previous results exp( - ixLL”2)
dx dr’]
(O)pA + fB to) PB][fA CT)pA + fB CT)‘RI .
’ [fA
The term in brackets in eq. (52), involvingf, andf,, is independent integral over coordinates and thus appears as a modulating factor previous results. To evaluate it we note that
(52) of the of the
fA =
zA/(TA +
TB),
(5%
fB =
%i(TA
d>
(53b)
+
and that BAA
~.f~(O).f~(t)
=JA
SAA(T),
fAB
= fA
=
sAB
(O)fB
b)
fA
(544
(T)
(54b)
3
where S,,(r) is the probability of an even number of flips in time z for a particle starting in state A at t =O, and SO on. It is shown by Machlupi2) that fAA
where l/z, =(1/z,)
=
(fA>’
+ (l/rg).
+
Jk.63
exp
(-
zizl)
)
(554
?
(55b)
It is easy to show that
.& = (A),> +
fA_h
exp
(-
z/zl)
~A”,,=~~A=J~~[~-~~P(-z~z~)I. With these results we obtain f
(7)
=
E(q2)
g2
(T)
[@ApA
forf(z),
+ .@B)~
assuming
+ exp(-
(55c)
ci (A) = c1 (B) =O,
T/Z~)fA.h(PA
-
p~)21y
(56) with SA, fB given by eqs. (53a) and (53b) and g2 (7) by eq. (19b), with z,, = r,f/4D and c1 =O. Eq. (56) has the form of the previous result, with an average dipole moment, P = (fAPA+fBPB) substituted for P, plus a term which multiplies g (7) by a decaying exponential. If PA =PB, or if r1 4 r, the previous results apply with P replaced by p, i.e. the exponential term drops out. The explanation when PA =PB is obvious. The situation for z1 or can be thought of in two ways. If the emphasis is on diffusion, i.e. if ZNZ~, it simply means that flip-flop is so rapid relative to diffusion that all flip-flop
R.GOMER
390
correlations have washed out in times of order ra, so that the mean fractional populations of A and B are maintained. On the other hand, ‘5x47, also means that, for r-rl, g (2) remains constant, so that pure flip-flop dominates the correlation function for r Q re. This can be expressed formally by excluding diffusion; in that case g(r) must be replaced by 2s: (W’(p))“p dp in eq. (56), with the result that the correlation function then consists of a constant term plus a decaying exponential. For r,%r-zO (but T~-T a) (i.e. the presence of states A and B but no flip-flop in times of order ra), eq. (55) gives the pure diffusion result, but with P2 replaced by p: (J”A”+ &J’B) +
pB
(f-B”
+
_6&
=
piz
+
‘;z
(57)
*
However, in this limit the assumption ii=fADA+fBDB breaks down because a molecule in state A remains in state A over times of the order of r,,, and hence the problem must be reformulated. If it is assumed that the probabilities SAB= SBA=O, a straightforward extension of eq. (15) then leads to
where gA and gB contain the diffusion coefficients D, and D, respectively. Finally, eq. (56) can be generalized to include both different dipole moments and Fowler-Nordheim pre-exponentials for states A and B. The result, whose derivation is completely straighforward, is f (r) = n
isi (7) t(.LcL
+ fB413)~ + exp (- riri) _&.&I(CL - 41?]
2rCc, 2 +
92
(
+ 93
U_tA~A
-2
b>
+
fBW
+
exp
(-
+d
fa.h
PA
-
1 q
CT> 2 (
[f
AdlAP.
+
_&B&PB
f
.L?
(4APB
+
Q21
4BPJl
>
(59) where c 1A= cl for state A, etc. 10. Correlation function in terms of field emitted current
We give for completeness some elementary transformations of the measurable forms of the correlation function. According to eq. (12), the quantity we have calculated is In [i (0)/t] In [i (7)/f] = ln i (0) In I
- (lnQ2,
(60)
with I defined as B exp (- 6.8 x 107~~~~~). If the fluctuations are small the
CURRENT
logarithm
can be expanded
391
FLUCTUATIONS
so that
f (7) = [i (0) i (T) -
1’]/1’.
(61)
Thus, at z=O we have
The mean square current fluctuation seems to be proportional to the square of the mean current. It must be remembered, however, that c2 is proportional to in 1. Thus the strict proportionality between the mean square current fluctuation and f2 should occur only when c1 predominates, i.e. when preexponential changes dominate over work function changes in adsorption. It may be worth pointing out that, when the logarithm ln(i/l) can be expanded, the entire treatment presented in this paper becomes valid for regions of arbitrary size without the implicit restriction that the (instanta-
neous) work function must be uniform over the probed region. For we may then obtain the total current fluctuation over the region of interest by summing over arbitrarily small area elements j the quantites: c;&z~+C,~+~ to obtain eq. (5) without implicit restrictions. 11. Estimate of fluctuation size Eq. (62) provides a convenient way to estimate the relative root mean square current fluctuation. For simplicity we assume that c1 =0, so that
g$ (0) = Ae3c2 A$,,
writing (64) (65)
with tI = I?/&,,,,, where CT,,,,,is the maximum adsorbate concentration GLX
and
the maximum C#Ichange, we have
Thus we see that the relative root mean square current fluctuation is inversely proportional to ro, directly proportional to the maximum work function increment, almost proportional to the square root of relative coverage
392
R. GOMER
(since 4” is almost 8 independent),
and inversely
root of maximum adsorbate concentration. fluctuation alone has the form
proportional
By contrast,
to the square
the concentration
(67) The difference between eqs. (66) and (67) results from the fact that the relative current fluctuation is proportional to the absolute density fluctuation, ii*, not the relative one, ii-*. Although values of coverage and A5, vary from adsorbate to adsorbate and from plane to plane, we may gain idea of (Ai’> “ii by picking A$ = 0.5 eV, c,= 10’s molecules/cm’, $ = 6 eV, and F=4 x 10’ V/cm. With these assumptions we obtain (Ai2)*/i= 5.5 (e*/r,)g*(O), with Y,, in A. Thus for r. = 50 A and t3 = 0.5 we obtain a 7% root mean square fluctuation. This indicates that a substantial effect should be observable under most conditions. In fact, for alkali metal or inert gas adsorption on tungsten where concentrations are much lower and dipole moments higher than those chosen in our example, the fluctuations may be so large that the logarithmic form of the current fluctuations in eq. (12) must be used.
12. Effect of finite resolution The resolution 6 of the field emission microscope is and varies with rf, rt being the tip radiuslys). Thus that an electron originating in the probed region but not be recorded, and there is also a probability that
of the order of -20 A there is a probability near its boundary will electrons originating
near but outside the boundary will be recorded. (The net result must, of course, be an average current equal to that obtained with infinitely sharp resolution.) Consequently, the effective radius of the probed region will appear larger by roughly +3. An exact calculation is not difficult in principle under the conditions of validity of eq. (60). In the simplest case where only g1 need be considered it amounts to a double convolution of the resolution function (for p and p’) with the previous result. Even this becomes a very time consuming computer calculation, and was therefore omitted. Acknowledgements It is a pleasure to thank Professor Walter Kohn for valuable help and discussions. The computer calculations were carried out by Mr. Robert Fleischaker, NSF Undergraduate Research Participant, Summer 1972, and by Mr. J. J. Chen. This research was supported by the Advanced Research
CURRENT
Projects Agency of the Department 15-71-C-0253, with the University Foundation
Grant
FLUCTUATIONS
393
of Defense under Contract No. DAHC of Michigan, and by National Science
GP-29388.
References 1) R. H. Good and E. W. Mtiller, in: Handbuch der Physik, Vol. 21, Ed. S. Fliigge (Springer, Berlin, 1956) p. 176. 2) R. Gomer, Field Emission and Field Ionization (Harvard Univ. Press, 1961). 3) T. Engel and R. Gomer, J. Chem. Phys. 50 (1969) 2428; 52 (1970) 1832. 4) G. Ehrlich and F. G. Hudda, J. Chem. Phys. 44 (1966) 1039. 5) R. Lewis and R. Gomer, Surface Sci. 26 (1971) 197. 6) Ch. Kleint, Surface Sci. 25 (1971) 394. 7) L. Bess, Phys. Rev. 103 (1956) 72. 8) L. D. Schmidt and R. Gomer, J. Chem. Phys. 42 (1965) 3573. 9) C. B. Duke and M. E. Alferieff, J. Chem. Phys. 46, (1967) 923. 10) D. Penn, R. Gomer and M. H. Cohen, Phys. Rev. B 5 (1972) 768. 11) C. Kohrt and R. Gomer, Surface Sci. 24 (1971) 77. 12) S. Machlup, J. Appl Phys. 25 (1954) 341.
SCIENCE 38 (1973) 373-393 8 North-Holland
CURRENT
FLUCTUATIONS
ADSORBATE
Publishing Co.
FROM SMALL REGIONS OF
COVERED
FIELD EMITTERS
A METHOD FOR DETERMINING DIFFUSION ON SINGLE CRYSTAL PLANES
COEFFICIENTS
ROBERT GOMER The James Franck Institute and The Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, U.S.A.
Received 29 December
1972
A novel method is presented for determining surface diffusion coefficients of adsorbates on single crystal planes of field emitters in terms of the time correlation function of current fluctuations. The method is based on the fact that current fluctuations are related to adsorbate density fluctuations, whose time correlation is governed by relaxation times simply related to diffusion coefficients. A general formalism is presented, some idealized cases are worked out analytically, and the case of a circular aperture is treated in detail numerically. Switching between different adsorption states, in addition to but not affected by diffusion, is also included. The forms of the correlation function are shown to be complicated, with a l/t tail at large t, and cannot be represented by simple exponential decay.
1. Introduction The development of the field emission microscope by E. W. Miillerl) made possible the direct observation of diffusion processes on clean metal surfaces 2). In most investigations the emitter was shadowed with adsorbate from one side, and the spreading of this non-uniform layer observed as a function of time, temperature and coverage. While a great deal of semi-quantitative information has been obtained in this way2), the method suffers from the defect that diffusion observed on a given crystal plane also depends on transport over adjacent regions, so that the unequivocal correlation between diffusion constants and substrate structure is often difficult. In addition, visual observations are possible only on highly emitting regions. To some extent this can be overcome by the use of a small probe-hole in the screen of the tube, through which a small portion of the electron beam is allowed to pass. In this way emission from very small regions (5 10 A in linear dimension) even of high work function planes can be measured. The arrival of adsorbate is signalled by changes in current3). Nevertheless, even this refinement does not resolve the uncertainties pointed out above. For ad373
374
R. GOMER
sorbates visible in the field ion microscope direct observation on individual planes is possible, as shown by the experiments of Ehrlich4). Unfortunately even the introduction of low field image gases does not extend the class of visible adsorbates beyond metallic onesa), although it does of course permit the observation of metal atoms with lower ionization potentials. The reason for this is almost certainly the absence of appreciable local density of states just above the Fermi level for most covalently bonded adsorbates. For such adsorbates one is therefore restricted to field emission. During experiments on field emission from single planes carried out by Engel and the authors), we noticed that heating of emitters covered with equilibrated CO layers led to fluctuations in probe-hole current. It seemed plausible that these were caused by concentration fluctuations over the small region probed. Since such concentration fluctuations build up and decay by surface diffusion, their time correlation function must contain the local surface diffusion coefficient D through some characteristic decay time z,-r#$4D, where r,, corresponds to the dimension of the region probed. An explicit expression for the correlation function of current fluctuations thus provides a means of determining diffusion coefficients on single crystal planes. The fact that current fluctuations in field emission may be related to surface diffusion has also been noted by Kleint6), who attempted to adapt a model by Bess7) for l/f noise in semiconductors. This model treats switching between two adsorption states, mediated by surface diffusion. It does not consider at all the build-up and decay of concentration fluctuations over small regions, which is central to the present work, and thus addresses itself to quite a different situation both experimentally and theoretically. In the following, the correlation function for current fluctuations from small regions is derived on the basis of diffusion limited concentration fluctuations.
2. Relation between emission current and adsorbate density In the absence of adsorbate the current density of field emission j is given by the well-known Fowler-Nordheim relation, which we may write as lnj = 1nB - (6.8 x lo7 o/F) @,
(1)
where B is a field-insensitive term, F the field in V/cm, 4,, the work function of the clean surface in eV, and o an image correction term l). The presence of an adsorbate modifies emission by changing 4 and in some cases B. The change in 4 is due to the dipole moment associated with each ad-particle. This consists of a zero field component and one induced by the applied field. Since the latter is proportional to applied field, a simple analysiss) shows that the work function change resulting from polarization of the ad-
CURRENT FLUCTUATIONS
375
sorbate by the applied field appears as a field-independent term, i.e. seems to modify B. For present purposes, however, we are not interested in the variation of current with field, and the separation into intrinsic and induced dipoles need not be made. While the apparent changes in B observed in plots of In i/F’ versus l/P can largely be explained by polarization, it may also happen that an adsorbate produces anti-resonances near the Fermi levels) [in slightly different language, a marked decrease in local density of states at the Fermi levelis)]. Since in most cases virtually all emission comes from within 0.5 eV of the Fermi level, electronic effects outside this energy range cannot affect total current markedly. This phenomenon, first pointed out in connection with field emission by Duke and Alferieff3), can lead to a real change in B. Experimentally it is usually found that IogB is proportional to coverage, so that we may write quite generally for the total current i from a region area A In i = In A& + cIc - (6.8 x lO’/F) #+u, (2) where c1 is a constant (positive, negative or 0), c the density of ad-particles in the region and C#J the effective work function in A at field F and for a given ad-particle distribution. Writing $=$+Sss,
(3)
where $ is the time-averaged work function and SC$its (small) fluctuation, we see that @z:*(l +ts4/d)=&t$$+s#, (4) so that 61ni~Ini-l~=c~6n3-c2F~,
(5)
where &r=n-E, cz=- 1.02 x IO’ m$*/F, and Ini the log of the current when a=2 and $=d;, and c; =cJA; n, is the number of ad-particles in A. To complete the relation between current and adsorbate density we must express 4, or alternately 4 and 64, in terms of n. It will be useful to work in terms of the work function increment caused by adsorption A@=#-40.
(6)
For sufficiently large areas and for uniform adsorbate distributions A4 = 2xPn{A,
(7)
where P is the dipole moment of the ad-particles, defined in such a way that the plane of zero potential bisects it. Thus the contribution per ad-particle, W becomes, in this limit, 2nP/A, and we may think of this as the average over A of the potential contributed by an ad-particle. This suggests that we look for A+ as the sum over all particles i in the system (not merely in A) of the
376
R. GOMER
quantity
s
1 Pd w (XTd) = A A [Cx _ x’)’ +.-zF
dx’ ’
(8)
where d is the distance above the surface where the potential is to be evaluated and x denotes the position of the dipole (within or without A) with respect to an origin, conveniently chosen at the center of the probed area A. It is easily verified that for a circular area of radius rg and a dipole placed at its center (9) which, for d
$i
tan-’ [
-Y> _....- (1 -x)(1 ( z[(l - Y)” + (1 - x2 + z2)]’ > (1 - x1 (1 + Y> + tan-’ ( z[(l + Y)2 +-(1 - X)” + z21+> (1 + x1 (1 - Y> + tan-” ( z[(l - Y)’ + (1 + x>’ + Z2]%) (1 + x> (1 + Y> + tan-’ z[(l + x)’ + (1 + Y)” + zz-J+ ’ (
)I (10)
where all quantities are expressed in units of a and z=d/a. It can readily be shown that, for a dipole at (0, 0), the same limits result as in the circular case. Thus the approximation that a dipole makes its full contribution 2nP/A to the work function when it is within A and zero otherwise is a reasonably good one for z 4 I. 3. General expression for the correlation function of current fluctuations
We can now express 6 lni=lni-E in terms of sums over the particle positions in the entire system by combining eqs. (5) and (8):
CURRENT
FLUCTUATIONS
371
The symbol S(xi) has the value unity when the coordinates Xi of particle i lie in A and zero otherwise, and the bars denote time (or ensemble) averages. It follows that the time correlation functionf(r) of the quantity 6 lni is given by f (7) = 6 In i (0) 6 In i (7)
= Cc;T 6i(O)+ c2T K (O)llIc;C 6j(T)+ c2C wj(T)l j j - CT(c;6i+ c2W,)l’ 9
(12)
with W,(z) = W(x,, T), etc., since
CTLc;6i(O)+ c2K(O>ll[F [c; 6j(z) + cZwj(T)ll= IL?“I16i + ‘2K)l ‘9 (13) etc. The last term in eq. (12) is the square of the quantity
1 (c;
di +
c2Wi)
=
N
i
s”
[c;6(x)
+ c,w (x)]
tot
40t
$tot c; + c,l7
N
=
=
c;ri+
c2
A3,
(14)
>
where Nis the total number of ad-particles in the entire system, A,,, the total area. We next separate the first term in eq. (12) into a sum over i =j, and a sum over i #j : 4
Cc;
6i co)
+
c2K
(O)lCc;6i(z) + +
i;j
(r)l
c2W
Cc;
si to)
+
c2 wi
(O)lCc;6j(T)+
c2
wj
(r)l*
We now assume that the displacements of different particles are not timecorrelated at all, so that the ensemble average of the double sum becomes (N2 - N)
[Sg
[c;6 (x) +
C2
w
(x)1]2.
tot
AWIt
Since N is very large, N2 - N E N2, and thus the double sum is cancelled by the term evaluated in eq. (14). Hence, f (r) = 4
Cci6i
to)
+
c2W,
(O)lCcisi@)+
c2 wi
(T)l.
(15)
R. GOMER
378
It might be thought at first that the neglected term N
,*t [J$x.
(c;6
-I-
,,W,]2
At,t
is of some significance. However, eq. (14) shows that it is of the form ;
(c;Z +
c2
AT)“.
We shall see shortly that it is of order A/A,,,, relative to the term retained in eq. (15). The evaluation of the correlation function now becomes simplein principle. We write f(r)=N
J
%X_[C;~(X)+C~~(X)]“P(~/X’,~)~X’)[~;G(X’)+C~~(XI)]. tot AtSIt
(161
Here dx/Atot is the absolute probability that a given particle is in dx at t =O, and 9(x
1x’, 2) dx’ = d&
exp( - l%!!!),
the conditional (normalized) probabiIity that, if it was in dx at 1--O, it will be in dx’ at t =z. Expanding eq. (16), we have
with (19a) W’ (x) W’ (x’) -- 4~Dr exp( - Eyg)dx
dx’,
(19b)
where we have written W’= W/(2~~/~). As it stands, eq. (19) can only be integrated numerically. To show the structure of f(z) we now make the assumption that W’=6, i.e. is unity in A and zero outside, so that f(z) reduces to
CURRENT
The quadratures
FLUCTUATIONS
319
in eq. (20) can be carried out for a square area, A = 4a2, and
yield, for that case
(21) with
(22) fi is the average number
of ad-particles
in A and
z0 = a2/D.
(23)
We may write equivalently (24) For z/r,, 4 1, g (z) reduces to (25a) and for z/r0 % 1 g(r)-&[exp(-:)-I]22
(25b)
so that f (7) decays as l/r at large z. Eq. (22) or (25a) shows that (26) which corresponds to the mean square “strength” of a fluctuation, and is proportional to k as it should be. We are now able to verify that the term neglected in our derivation, (l/FA,,,) (Ec,+c,A$)~, with eq. (26). 4.
is of order
A/A,,,
relative
to
f (0) by comparison
f (7)from decay of density fluctuations
It is interesting that eqs. (20) and (22) can also be derived from a slightly different viewpoint, by using Onsager’s hypothesis that fluctuations decay according to macroscopic laws, i.e. by studying the decay of a fluctuation according to a macroscopic diffusion equation. We can do this by postulating a delta function fluctuation at (x,0), with x somewhere in A. At time r the concentration profile will be c(X.,r)=exp(--Is)&,
(27)
380
R. GOMER
and the total fraction of the original strength of the fluctuation left in A after time z is found by integrating with respect to x’ over A. To get the ensemble average we must now average over the initial position x as well and thus obtain, using the mean square strength of fluctuations, eq. (20) and the subsequent development. The equivalence of these two methods is no accident of course, but stems from the relation between D and the random walk assumption inherent in eq. (17). Finally, it should be pointed out that the averaging over the initial positions of delta functions, followed (or preceded) by integration of the concentration profiles over A, is equivalent to postulating a uniform, step-like concentration fluctuation over A and following its time development. (The arbitrariness of this fluctuation profile comes from the weighting function w’=6; a more realistic weighting function would lead to a less abrupt profile.) It is possible to treat this diffusion problem in the standard way and to obtain solutions of the form C
(I’, 7) = C
A
j
Jo
(Ujl.) exp (- UjzDr)
i
(28)
(for circular probe areas), where the J,‘s are Bessel functions of order 0, with roots Uj chosen to make J, (u~Y,,) =O. The Aj’s are the appropriate coefficients for the expansion of c (Y, 0) in the orthonormalized set of Bessel functions. The roots are given approximately by Uj
E
7c
(j - &)jr,,
(29)
so that each term in eq. (28) is multiplied by a time factor of the form exp ( - r/zj), with zj= r$ [x” (j - $)‘D]. Integration of c (Y, r) over A would then give g (r). The point to note is that only for very special assumptions of the initial fluctuation profile, namely c (r, 0) equal to the lowest order Bessel function would g (T) show an exponential decay. The statistically correct assumption about the fluctuation leads to the more complicated relation eq. (22); this can be expressed, however, in the form of eq. (28), or more correctly as an integral over A of eq. (28) (which leaves the z dependence intact), i.e. as a series of different exponential decays terms. It is interesting that, even for r& < 1, eq. (25b) does not go over into a single exponential decay term, as eq. (28) might suggest. The reason is that the random positions of the individual fluctuations over A assign too small a weight to the A, term to ever let it predominate, despite the fact that it decays more slowly than any other term. As we shall see, the fact that eqs. (21)-(25) have been derived for a square probe area does not alter these conclusions, since the long time limit of g (T) for a circular aperture will be shown to be of the same form a eq. (25b).
CURRENT
FLUCTUATIONS
381
5. Circular aperture The preceding discussion has concentrated on a square aperture and an assumed step function form for W, to show analytically the salient features off (2). We treat here the general case of a circular aperture, starting with W(x). Changing to polar coordinates gives 2n
W(P)
W’(p) = ~
z
2rrPllrr,2 = G
1
ss
p’ dp’ de (30)
o o (p’ + p’2 + z2 - 2pp’ cos 0)s’
where p=r/ro, z=d/r,,, with r. the radius of the probed over p’ and partial integration over 8 gives
region.
Integration
L
W’(p)=
1-z
(p’ + z2 -pcos@dG (31)
nos (P 2sin28+z2)(p2+z2+1-2pc0se)*’
These integrals were evaluated for a number of values of z on an IBM 360 computer. The results, as well as graphs of w’ (p)p are shown in figs. 1 and 2. It is seen that for small z, W’(p) approaches a step function 6 (p)= 1 for p=Gl,Oforp>l. The integrals appearing in eq. (19) expressed in polar coordinates are all of the form
&(T) =
s
zo
dp dp’ de de R(p) R’(p’) exp
z
-
p2 + pf2 -2PP’
COS(~-8') 3
TOIT
(32) z. = rg/4D,
Fig. 1.
Plots of W’(p) versus p for various values of .z=d/ro. Here p=r/ro, the radius of the circular probed area; z = 0 refers to W’(p) = 6(p).
(33)
with ~0
382
R. COMER
100
150
200
250
P
300
Plots of p W’(p) versus p for circular probed areas.
Fig. 2.
and p =r/ro, p’=rr/r,,, and R and R’ are either W’ or 6. The angular dependence appears only as the difference of 8 and 13’, and thus can depend only on cp= 8 - 0 ‘. Thus we may use the equivalent angle variables 8 - 0 ’ and 0 +8’, and integrate at once over the latter obtaining a factor of 27~. The integral over cp is of the form II 2
s
exp VPP
(G/T)
~0s
‘PIdv = 2nJo W (+)I
0 = 2nlo
where 1, is the Bessel function given by
(~o/T)l>
(34)
of order zero and imaginary
Jo (ix) = IO(x) =
Thus the integrals
VPP
argument
cc Qx)‘” ___ m=O W2. c
in eq. (19) reduce to double
ix,
(35)
integrals
over p and p’:
11 9 lb>
=
4 (70/r)
IO (~PP’T~/~) exp I-
(P’
+ P”>
ss
~o/~lPP’
dp dp’ 9
(364
00 mcc
92 (T> =
4 (rolr)
ss
W’ (P> W’ (P’> 10 (2PP’7o/T)
0 0 x exp [ - (P’ m
93 (r) = 4 (to/r)
+ P”)
z,/~]
PP’ dp
dp’ ,
Wb)
1
W’ (P> IO C~PP’~~/~) ew [-
ss o=o o’=O
(P’
+ P”)
zohl
x pp’ dp dp’ .
(36~)
CURRENT
383
FLUCTUATIONS
For large z, I, z 1, and with the further assumption that W’ (p) = W’ (p’ ) = 6, we immediately obtain g (7) = (~~~~)[ 1 - exp (- Q/Z)]“,
zoiz < 1,
(37)
which is of the same form as eq. (25b) for a square aperture. It is interesting that in terms of D we thus obtain [using eqs. (23) and (33)] g (2) + (a2/7cD) z - i g (T) + (~~/4~) z- r
square of half-width a, circle of radius rO,
(38a) (3W
asz+ co. The integrals in eq. (36) were evaluated numerically on an IBM 360 computer for various values of z and z=r/r,. In those integrations with infinite upper limits, approximate upper limits pU were chosen as follows: The average work function increment .Q can be expressed as an integral of m W(r): &$=2nP~=2~
EW(r)rdr, s
(39)
0
so
that
m W’(p)pdp=l.
(40)
0 The upper limits pU were chosen so that P”
l-2
[ W’(p)pdp
.i
(41)
Since it is clear from eq. (8) that for large p, IV (p)oz pm3, it can readily be shown that condition (41) is equivalent to 2p,2 W’ (p,) < 0.01.
(42) The following values of pU were used in accordance with this criterion: For z=l.O, p,,=52.2; for 2=0.5, pU=25.3; for 2=0.2, p,=11.4; for z=O.l, pu=7.5. In order to evaluate the integrals at t=O, it is simplest to start with the definition off (0). (W e sh ow explicitly only the case for c1 = 0. The extension to finite c, is obvious.) f (0) =
cjC i
W (ri, 0) W (t-i, 0) W’ (r) W’ (r’) r dr dtl de’ x 6 (Cl -
d’) S (r - r’) dr’ (43)
384 so
R.GOMER
that (44)
For W’=6 that
(i.e. z=O), this obviously yields g, (0) = 1. It is also easily shown
(45) For z#O both g, (0) and g, (0) are less than unity, and g, (O)
Fig. 3. s%(t) versus t for various of z=dJr~. The curve marked z=O corresponds to &(t). Here f = r/50, with to = dj4D. The quantities $1 and 92 are defined by eqs. (36a) and (36b), respectively.
CURRENT
0
Fig.
4.
gs(t)
385
FLUCTUATIONS
2
4
,6
8
IO
versus t for various values of z. The quantity g3 is defined by eq. (36c), all other notation as in fig. 3.
The dependence of g2 and g3 on z requires some discussion. For a fixed t,,, i.e. for a given rO, g2 and g3 vary most rapidly with time as z decreases, as we should expect. It must be remembered, however, that d is fixed at -5 & so that zxl/r,, and hence z2ccl/z,,. Thus, a better idea of the real time behavior off(z) can be gained by plotting g2 (t ) versus t/z2 = (40/25) z. Since an experiment would look at the relative change inf (7) as a function in figs. 5 and 6. of T, we show g2 (t)lg2 (0) and g&1 g3 (0) versus log,,(t/z2) This mode of presenting the results shows two interesting features. First, as zO decreases, so does the real time required for the decay of the correlation function, as we should expect. Second, and more important, the shape of the normalized curves for different z is almost identical. This is true both for g2 and g3 and for the comparison of g2 with g3. The only major differences are displacements
along the log(t/z2) I
I
axis. It now becomes I
an obvious
I
Fig. 5. Plots of gz(t)/gz(O) versus loglo(r/9). Also shown as z = 0 curve is gl(t) versus 1. Dashedcurve is e+ versus t. Solid points represent a hypothetical mixture: g(t) = 0.25 gl(t) + 0.25 gz(t) + 0.5 gz(t) for z = 0.5, plotted against loglo(r/0.25). It is seen that the curve is shifted to the left, as expected.
386
Fig. 6.
R. GOMER
Plots of 93(t)/gs(O) versus log&/z2) for various values of z. The curve z=O refers to plot of g1 (t) versus loglo t. Dashed is e+ versus loglot.
extension to compare the curves with the corresponding z=O case. We may do this, for instance, by comparing t for g2 (z, t)/gz (z, 0) =0.5 with t for g1 (t) =0.5. These ratios can be though of as increases in the effective values of ri. We show in fig. 7 the factors & (k=2 or 3) defined by
fi = r. (effective)/r,
= [t (g)/t (gr)]+ .
(46)
Both p2 and p3 vary linearly with increasing z, i.e. decreasing r,,. As expected, p3
it would
take detailed
knowledge
of all the parameters
involved
to
Fig. 7. Plots of 8~ = (t&l)+ and 83 = (t3/f1)* versus z. The quantities tl, 12,and t3 are defined as the times required for g1 (t), g2 (t)/gz (O), and gs (t)/g3 (0) respectively to become 0.5 ; /3r0is the eflective radius of the probed region.
CURRENT
decide
on the exact
contributions
FLUCTUATIONS
of each g. A hypothetical
387
mixture
is
shown in fig. 5. In practice, the importance of these considerations can be minimized by choosing a reasonably large r,, and hence small z, for which fi - 1 is small in any case.
6. Relation of rO to single plane parameters The last section has touched on the effective radius within which time correlated events contribute to the measured correlation function. For the idealized, infinite plane case this is of academic or computational interest only. For a real field emitter of radius 1000-5000 A planes of given crystallographic orientation are of 50-200 A in radius, and it is therefore important to see if the effective r0 can be made smaller than these dimensions, since we wish to measure D for single planes. It is clear from fig. 7 that this is so. Another criterion, somewhat more arbitrary, would be to pick the value of p, at which 2JP,p’W’(p’)dp’ had reached some fixed value, say 0.80 or 0.90. It should be pointed out, however, that the curvature of the emitter imposes an absolute limit on the effective size of the probed region, since the horizon distance r, for a point d 8, above the surface of a sphere or radius r, is r,,, = (2r,d)*.
(47)
For d = 5 A and rt = 1000 A, r,,,= 100 A. It may now be asked to what extent the curvature of the surface affects W’ and the subsequent calculations, which were carried out for a plane surface. For all cases considered the effect is very small, and hence we may assume the validity of the calculations for the curved emitter with confidence. 7. Extension to interactions between ad-particles We have assumed so far that there is no interaction between diffusing particles. In many systems, however, this is unlikely to be true. A detailed calculation taking interaction into account would be very difficult. However, it is known that even for biased random walks the form of eq. (17) is not changed, although the value of D will be affected. Thus our results will most probably remain valid, with D a function of average concentration. 8. Spectral density representation The results obtained here have been presented as correlation functions. It would be possible, of course, to represent them as spectral densities by taking their Fourier transforms. Since experimental correlation functions
388
R. GOMER
can be obtained directly, this step is omitted here. It is worth pointing out, however, that the power spectrum will not be fitted with any accuracy by functions of the form
s (0) = because the correlation nential.
7;
‘/(co2f g),
function is so poorly approximated
(48) by an expo-
9. Inclusions of ~~flop processes For certain adsorbates, for instance CO on bee transition metals, different adsorption states characterized by different dipole moments can exist on a given crystal planerr). It is possible that under rather special conditions a switching back and forth could occur, although in most cases the change from a given ad-state to another seems irreversible. It is fairly easy, with some restrictions, to extend the present calculation to include flip-flop. The treatment we shall use is a simple adaptation of one worked out by Machlupls) to treat current noise in semiconductors. Let state A be characterized by a dipole moment PA and let its mean life with respect to flipping to state B be rA. Let state B be similarly characterized by PB and zg. Letf,(z) andfa(z) be the probabilities that an ad-particle is in a state A and B respectively. Then
To proceed further we must now make two key assumptions: (1) we assume that the flipping of different ad-particles is wholly uncorrelated, and (2) we assume that an average diffusion coefficient D can be defined as b = D,f,
+ DsfB.
(51)
The latter assumption is certainly valid when flipping times are short relative to diffusion times. When the opposite is true, i.e. if there are two populations each with its own diffusion coefficient but with no switching state A to B, we are in fact dealing with a different problem whose solution will be indicated also. It should also be pointed out that our model assumes that the switching times have nothing to do with diffusion. Kleint’se) and
389
CURRENTFLUCTUATIONS
Bess”)
model,
on the other hand,
assumes
that switching
times are deter-
mined by diffusion, since an ad-particle in state A can find a B site only by migrating there. To return to our model, we have by analogy with our previous results exp( - ixLL”2)
dx dr’]
(O)pA + fB to) PB][fA CT)pA + fB CT)‘RI .
’ [fA
The term in brackets in eq. (52), involvingf, andf,, is independent integral over coordinates and thus appears as a modulating factor previous results. To evaluate it we note that
(52) of the of the
fA =
zA/(TA +
TB),
(5%
fB =
%i(TA
d>
(53b)
+
and that BAA
~.f~(O).f~(t)
=JA
SAA(T),
fAB
= fA
=
sAB
(O)fB
b)
fA
(544
(T)
(54b)
3
where S,,(r) is the probability of an even number of flips in time z for a particle starting in state A at t =O, and SO on. It is shown by Machlupi2) that fAA
where l/z, =(1/z,)
=
(fA>’
+ (l/rg).
+
Jk.63
exp
(-
zizl)
)
(554
?
(55b)
It is easy to show that
.& = (A),> +
fA_h
exp
(-
z/zl)
~A”,,=~~A=J~~[~-~~P(-z~z~)I. With these results we obtain f
(7)
=
E(q2)
g2
(T)
[@ApA
forf(z),
+ .@B)~
assuming
+ exp(-
(55c)
ci (A) = c1 (B) =O,
T/Z~)fA.h(PA
-
p~)21y
(56) with SA, fB given by eqs. (53a) and (53b) and g2 (7) by eq. (19b), with z,, = r,f/4D and c1 =O. Eq. (56) has the form of the previous result, with an average dipole moment, P = (fAPA+fBPB) substituted for P, plus a term which multiplies g (7) by a decaying exponential. If PA =PB, or if r1 4 r, the previous results apply with P replaced by p, i.e. the exponential term drops out. The explanation when PA =PB is obvious. The situation for z1 or can be thought of in two ways. If the emphasis is on diffusion, i.e. if ZNZ~, it simply means that flip-flop is so rapid relative to diffusion that all flip-flop
R.GOMER
390
correlations have washed out in times of order ra, so that the mean fractional populations of A and B are maintained. On the other hand, ‘5x47, also means that, for r-rl, g (2) remains constant, so that pure flip-flop dominates the correlation function for r Q re. This can be expressed formally by excluding diffusion; in that case g(r) must be replaced by 2s: (W’(p))“p dp in eq. (56), with the result that the correlation function then consists of a constant term plus a decaying exponential. For r,%r-zO (but T~-T a) (i.e. the presence of states A and B but no flip-flop in times of order ra), eq. (55) gives the pure diffusion result, but with P2 replaced by p: (J”A”+ &J’B) +
pB
(f-B”
+
_6&
=
piz
+
‘;z
(57)
*
However, in this limit the assumption ii=fADA+fBDB breaks down because a molecule in state A remains in state A over times of the order of r,,, and hence the problem must be reformulated. If it is assumed that the probabilities SAB= SBA=O, a straightforward extension of eq. (15) then leads to
where gA and gB contain the diffusion coefficients D, and D, respectively. Finally, eq. (56) can be generalized to include both different dipole moments and Fowler-Nordheim pre-exponentials for states A and B. The result, whose derivation is completely straighforward, is f (r) = n
isi (7) t(.LcL
+ fB413)~ + exp (- riri) _&.&I(CL - 41?]
2rCc, 2 +
92
(
+ 93
U_tA~A
-2
b>
+
fBW
+
exp
(-
+d
fa.h
PA
-
1 q
CT> 2 (
[f
AdlAP.
+
_&B&PB
f
.L?
(4APB
+
Q21
4BPJl
>
(59) where c 1A= cl for state A, etc. 10. Correlation function in terms of field emitted current
We give for completeness some elementary transformations of the measurable forms of the correlation function. According to eq. (12), the quantity we have calculated is In [i (0)/t] In [i (7)/f] = ln i (0) In I
- (lnQ2,
(60)
with I defined as B exp (- 6.8 x 107~~~~~). If the fluctuations are small the
CURRENT
logarithm
can be expanded
391
FLUCTUATIONS
so that
f (7) = [i (0) i (T) -
1’]/1’.
(61)
Thus, at z=O we have
The mean square current fluctuation seems to be proportional to the square of the mean current. It must be remembered, however, that c2 is proportional to in 1. Thus the strict proportionality between the mean square current fluctuation and f2 should occur only when c1 predominates, i.e. when preexponential changes dominate over work function changes in adsorption. It may be worth pointing out that, when the logarithm ln(i/l) can be expanded, the entire treatment presented in this paper becomes valid for regions of arbitrary size without the implicit restriction that the (instanta-
neous) work function must be uniform over the probed region. For we may then obtain the total current fluctuation over the region of interest by summing over arbitrarily small area elements j the quantites: c;&z~+C,~+~ to obtain eq. (5) without implicit restrictions. 11. Estimate of fluctuation size Eq. (62) provides a convenient way to estimate the relative root mean square current fluctuation. For simplicity we assume that c1 =0, so that
g$ (0) = Ae3c2 A$,,
writing (64) (65)
with tI = I?/&,,,,, where CT,,,,,is the maximum adsorbate concentration GLX
and
the maximum C#Ichange, we have
Thus we see that the relative root mean square current fluctuation is inversely proportional to ro, directly proportional to the maximum work function increment, almost proportional to the square root of relative coverage
392
R. GOMER
(since 4” is almost 8 independent),
and inversely
root of maximum adsorbate concentration. fluctuation alone has the form
proportional
By contrast,
to the square
the concentration
(67) The difference between eqs. (66) and (67) results from the fact that the relative current fluctuation is proportional to the absolute density fluctuation, ii*, not the relative one, ii-*. Although values of coverage and A5, vary from adsorbate to adsorbate and from plane to plane, we may gain idea of (Ai’> “ii by picking A$ = 0.5 eV, c,= 10’s molecules/cm’, $ = 6 eV, and F=4 x 10’ V/cm. With these assumptions we obtain (Ai2)*/i= 5.5 (e*/r,)g*(O), with Y,, in A. Thus for r. = 50 A and t3 = 0.5 we obtain a 7% root mean square fluctuation. This indicates that a substantial effect should be observable under most conditions. In fact, for alkali metal or inert gas adsorption on tungsten where concentrations are much lower and dipole moments higher than those chosen in our example, the fluctuations may be so large that the logarithmic form of the current fluctuations in eq. (12) must be used.
12. Effect of finite resolution The resolution 6 of the field emission microscope is and varies with rf, rt being the tip radiuslys). Thus that an electron originating in the probed region but not be recorded, and there is also a probability that
of the order of -20 A there is a probability near its boundary will electrons originating
near but outside the boundary will be recorded. (The net result must, of course, be an average current equal to that obtained with infinitely sharp resolution.) Consequently, the effective radius of the probed region will appear larger by roughly +3. An exact calculation is not difficult in principle under the conditions of validity of eq. (60). In the simplest case where only g1 need be considered it amounts to a double convolution of the resolution function (for p and p’) with the previous result. Even this becomes a very time consuming computer calculation, and was therefore omitted. Acknowledgements It is a pleasure to thank Professor Walter Kohn for valuable help and discussions. The computer calculations were carried out by Mr. Robert Fleischaker, NSF Undergraduate Research Participant, Summer 1972, and by Mr. J. J. Chen. This research was supported by the Advanced Research
CURRENT
Projects Agency of the Department 15-71-C-0253, with the University Foundation
Grant
FLUCTUATIONS
393
of Defense under Contract No. DAHC of Michigan, and by National Science
GP-29388.
References 1) R. H. Good and E. W. Mtiller, in: Handbuch der Physik, Vol. 21, Ed. S. Fliigge (Springer, Berlin, 1956) p. 176. 2) R. Gomer, Field Emission and Field Ionization (Harvard Univ. Press, 1961). 3) T. Engel and R. Gomer, J. Chem. Phys. 50 (1969) 2428; 52 (1970) 1832. 4) G. Ehrlich and F. G. Hudda, J. Chem. Phys. 44 (1966) 1039. 5) R. Lewis and R. Gomer, Surface Sci. 26 (1971) 197. 6) Ch. Kleint, Surface Sci. 25 (1971) 394. 7) L. Bess, Phys. Rev. 103 (1956) 72. 8) L. D. Schmidt and R. Gomer, J. Chem. Phys. 42 (1965) 3573. 9) C. B. Duke and M. E. Alferieff, J. Chem. Phys. 46, (1967) 923. 10) D. Penn, R. Gomer and M. H. Cohen, Phys. Rev. B 5 (1972) 768. 11) C. Kohrt and R. Gomer, Surface Sci. 24 (1971) 77. 12) S. Machlup, J. Appl Phys. 25 (1954) 341.