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Field evaporation theory: A re-analysis of published field sensitivity data
498
Surface Science 114 (1982) 498-514 North-Holland Publishing Company
FIELD EVAPORATION THEORY: A REANALYSIS OF PUBLISHED FIELD SENSITIVITY DATA Richard G. FORBES, R.K. BISWAS and K. CHIBANE Department
of Physics, University of Aston, Costa Green, Birmingham 84 7ET, UK
Received 2 July 1981; accepted for publication
29 September
1981
Data concerning the field sensitivity of field evaporation flux were published for six refractory metals by Tsong in 1978. These data are converted into the form of partial energies, and used in a re-examination of the consistency of published field evaporation formalisms describing the escape step. The general conclusion is that the data are not compatible with the conventional simple image-hump formalisms, even when experimental error is taken into account, but are compatible with an analysis of charge exchange based on a simple parabolic-jug formalism.
1. Introduction Field evaporation has an established significance as the emission process in the atom-probe field-ion microscope [1,2] and related techniques, and a new importance as a possible emission mechanism in connection with liquid-metal field-ion sources [3,4]. Following recent work by Ernst [5,6] and by Kingham [7,8], it now seems virtually certain that for most metals field evaporation is a two-stage process. First a metal atom escapes as a singly-charged or possibly doubly-charged ion; subsequently, it may be post-ionized into higher charge states. Kingham’s demonstration, that theoretically post-ionisation is a likely process, provides a most plausible exljlanation of the high charge states sometimes observed amongst field-evaporation products [9], and serves to redirect theoretical attention towards the nature of the escape process. As is well known [lo], there are two commonly discussed alternatives for the escape process; the Mtiller-Schottky (or “image-hump”) mechanism [ 11,121 in which ionization precedes escape; or some form of charge-exchange process [ 131 in which ionization and escape occur together. These proposed mechanisms are incompatible alternatives, that lead to quantitatively different physical interpretations of basic field evaporation data [14], and it would be useful to decide firmly between them. Detailed comparison of the mechanisms has, however, been handicapped by two things. First, the Mtiller-Schottky mechanism is often discussed in terms of simple image-hump formalisms that are of dubious validity [ 15- 171(but do 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
R.G. Forbes et al. / Field ~uporafion
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499
apparently lead to reasonable predictions of evaporation field-see, for example, ref. [9]). Second, most mathematical analyses of the charge-exchange mechanism prior to 1978 contain a disabling mathematical flaw [ 14,181. Recently, Tsong [ 191 has reported some measurements on the field dependence of evaporation flux (“evaporation current”). The objectives of this note are to retabulate these in a form analogous to that used by Forbes [18,20], and to explore whether the resulting data are compatible with the existing fieldevaporation formalisms. The structure of the paper is as follows. In section 2 we set down some basic theory relating to the concept of “partial energies”. Section3 converts the experimental data into the form of partial energies_ Sections 4, 5 and 6 then discuss this data in the contexts of the published formalisms. Section 7 draws our conclusions together.
2. Background thmy It is first necessary to deal with some matters of notation and terminology. The amount of material evaporated per unit time from a field-ion emitter may be expressed in the form [21]: J = n&h,,
(1)
where nhr is the amount of material (or “count of atoms”) at high risk of evaporation, and k, is the fi&euaporation rate-constant for the atoms at ‘high risk” sites (which in practice are the kink sites). The rate-constant k,, is measured in s - ‘, The quantity nhr is regarded as having the dimension [amount-of-substance], but is measured in the units “atoms” or “layers” (these being regarded as non-S1 units of amount of substance). The quantity J is regarded as having the dimensions ~amount-of-subs~ce/time], is measured in layers/s (or atoms/s), and is here called the ~~~d-~uF~rutiun flux. J has previously been called “evaporation current” [20], this notionally being an abbreviation for “amount-of-substance current”. However, it now seems to us that the term “current” is better used to refer to the electric current associated with field evaporation, particularly in the context of liquid-metal field-ion sources. Thus we have adopted the term “flux” as a name for rate-like quantities having dimensions [amount-of-substance/time], and the name “field -evaporation flux” for J. In the literature, J is also called “evaporation rate”, “measured evaporation rate”, and “relative evaporation rate”, but we prefer not to use these names. Our system has been adopted in order to avoid ambiguity, to be consistent with the revised SI system introduced in 1971, and to keep k as having the traditional chemical meaning of “rate-constant”. Note that Tsong’s k is our J, and Tsong’s K is our k,,. The three parameters in eq. (1) are all functions of the field strength F at the
R.G. Forbes et al. / Field evuporutlon theov
500
high-risk sites. Consider some reference situation, in which F has the value F”, and let the corresponding values of these parameters be Jo, nL andkk. It follows that: ln( J/Jo)
= In( n,/n&)
+ ln( k,/kk).
(2)
Each of the terms in eq. (2) has value zero at F = F”, and it is convenient to Taylor-expand them about F”. If we introduce a variable g by: g= (F-
FO)/FO,
(3)
then the expansion of the rate-constant form: ln(&/W
= (C.c&Wg+t(I-(JkT)g*
term (for example) can be put in the +....
(4)
The (p,/kT), (p2/kT), etc., are the Taylor coefficients, which are written in this way because the quantities p,, cl2 are expected to be temperature independent if field evaporation is a thermally-activated process obeying the Arrhenius equation. By analogy with eq. (4), we may expand the other terms in a similar way, thus: ln(J/J’)
=(A,/kT)g++(X,/kT)g’+...,
ln(n,,/&)
= (v,/kT)g+i(v2/kT)g2
(5) + . .. .
(6)
Clearly we have: x, =jJ, + Y,, A, =jJ, + v*.
(7)
Existing theories of field evaporation are, of course, formalisms that deal with rate-constant field sensitivity rather than evaporation-flux field sensitivity. Each formalism provides an expression for the field-evaporation activation energy Q(F); expressions for the “partial energies” /.L, and cl2 can then be obtained as follows [l&20]. The Arrhenius equation may be written in the form: k,, = exp(M/kT),
(8)
where M= kTln{A}
- Q,
(9)
where {A} is the numerical value of the field evaporation pre-exponential expressed in s-‘. It follows that: ln(k,/kk)
=[M-M(F’)]/kT.
Taylor-expanding
A, (10)
M in terms of F, about F = F”, we obtain:
A4=M(F”)+(F-F’)+$l
= F
++(F-F’)‘$ F”
_ F-F’
+....
(11)
R. G. Forbes et ul. /
501
Field euuporation theory
Substituting from eq. (3) and then substituting into eq. (10) gives: ln(k,/k&)
= (F”g/kT)+$lpO
++[(F”)2g2\kT]$IF0
+....
(12)
Hence, by comparison with eq. (4): OdM
C1,=F
dFFO
N
-FOde
034
dF p’
p, = (F”)‘s
N - (F”12s FO
(13b)
FO’
The second step in each of these relationships assumes that the field dependence of the pre-exponential A may be neglected. Previous analyses, for example refs. [ 14,17,18], have taken this to be an adequate approximation, particularly at temperatures near 80 K and above, and we continue with this assumption. The arguments principally involved are as follows. First, the pre-exponential A involves a factor relating to the vibrational frequency of the bound neutral. This frequency .depends on the shape of the neutral potential well, which (in comparison with the activation energy) is unlikely to be sensitively affected by a small field variation. Second, the pre-exponential involves a factor that relates to the possibility of “ion tunneling” [ 161 and is in principle field-dependent for this reason. It has always been supposed that, at temperatures near 80 K and above, any field dependence due to this cause is insignificant. Neglect of posssible field dependence in the pre-exponential A is, in practice, an integral part of the formalism under discussion in later sections. The question .of whether this neglect is really justified is far from straightforward and would require detailed mathematical analysis that regretfully is beyond the scope of the present paper.
3. Experimental data 3.I. Conversion of data Tsong’s experimental results [ 191 are in the form of a plot of lg(J/JO) versus F/F’, where F” is the field at which the evaporation flux is 10 -2 layers/s. He seems to have treated this data by malting the (perfectly reasonable [22]) assumption that lg(nk/&) is effectively zero, and hence that: lg( J/Jo) = lg( k,/kk)
= a0 + a,( F/F’)
+ a2(F/Fo)‘.
(14)
He does not explicitly mention any assumption equivalent to taking lg(nk/&) zero, but he does state that a, and a2 are best-fit coefficients to the experimental data, so we shall assume that the correct logical identification between his
R.G. Forbes et ul. / Aeld evaporution theory
502
parameters and ours is: A, = ln(10) X kT- (u, + 2a,), A, =2ln(lO)XkT-a,.
For convenience in dealing with errors, it is useful to define working parameters (Y,,01~by: cy, =ln(lO)
X kT+a,,
@a)
a2 = In{ 10) X kT- a2.
(16b)
Tsong’s values of a, and a2, together with corresponding values of (Y, and (Ye are shown in table 1; T has been put equal to 77.3 K. Existing theories of field evaporation deal with rate-constant field sensitivity rather than evaporation-flux field sensitivity, so we require p1 and pcIzrather than A, and A,. We shah thus follow Tsong and other workers (for example, Brandon [15]) in assuming that there is no significant field dependence in the amount of material at high risk of evaporation. This implies that ZJ,and Yecan be neglected, and hence that experimental estimates of partial energies are given by: p,(exp) +(exp)
= ai + 2or,
(17a)
pz(exp) = X,(exp) = 2im,.
(17b)
Partial energies derived via eq. (17) are shown in table2, together with corresponding values of their ratio. For simplicity in later argument, we label the columns here and later in terms of c,, p2 and derived expressions. However, it should be remembered that strictly (except in the case of table 3) the columns show experimental estimates of X,, A, and derived expressions. For comparison with the tungsten estimates in table 2, we show in table 3 values of &, p; and & /r_1T,derived by Forbes and. Pate1 [23,24] from Tsong’s measurements [17] on the field sensitivity of evaporation rate-constant, in the
Table I Values of F”, a, and az, as given in ref. [19], together with working parameters q and a2 derived via eq. (16); the error in a, is not explicitly stated in ref. [19], but is assumed here to be * 5 in each case; for working purposes, the values and errors for a, and az are given to more places than are physically significant Species
F” (V/nm)
(I*
Mo(ll0)
46
Ru(ll~1) Hf(lOi0) W(l10) Ir(100) Pt(100)
42 40 55 52 48
715~5 20015 335*5 805*5 205r5 200*5
-310*10 -75220 - 135*20 -350* 10 - 80-‘20 -75’20
*I (evt
a2 (ev)
10.%7r+O.O77 3.068*0.077 5.138*0.077 12.347*0.077 3.i44*0.077 3.068-cO.077
-4.754*0.153 - 1.150~0.307 -2.071 -co.307 -5.368*0.153 - 1.227*0.307 - 1.150*0.307
503
R.G. Forbes et (11./ Field euuporution theory
Table 2 Experimental estimates of the partial energies c, and as, and their ratio, derived from the data in table I, using eq. (17); it has been assumed that the quantities Y, and Y*appearing in eq. (7) are negligibly small Species
CI (eV)
Mo(1 IO) Ru(l111) Hf( loio) W(110) Ir( 100) Pt(100)
1.46kO.4 0.77 -co.7 1.00*0.7 1.61kO.4 0.69kO.7 0.77kO.7
~2
WI
C2/CI
-6.5~2 -3.0*3 -4.2k3.5 -6.7*2 -3.6k3.5 -3.o-c3
-9.5120.3 -2.3OkO.6 -4.14-tO.6 - 10.74*0.3 -2.45kO.6 -2.3020.6
case of tungsten. (The suffix c on the partial energies indicates that these relate to an evaporation field defined by a unity-rate-constant criterion [20] rather than the evaporation-flux criterion used in table 2.) Bearing in mind that different criteria may lead to different field values being identified as the “evaporation field” [20], and that the value of p, will depend on the choice of “evaporation field”, we feel that the comparison of table 3 (based on rate-constant field sensitivity) with the tungsten estimates in table2 (based on flux field sensitivity) is entirely satisfactory, and goes some way towards confirming that Y, and Ye can be neglected in the context of low-temperature field evaporation. 3.2. Treatment of errors Some comment on our treatment of errors is necessary. In deriving the errors on the p’s from those on the u’s we have used simple linear formulae. For example, the standard deviation o(p,) is obtained from the formula, based on eq. (17a). 4%)
= 4%)
+ 2 4%).
(18)
Table 3 Experimental estimates of the partial energies pt and ~‘2. and their ratio, derived by Pate1 [23] from original data concerning rate-constant field sensitivity taken by Tsong [ 171; in the analysis of the fig. 9 data, certain deviant points have been excluded from the analysis on statistical grounds; the original data were obtained from pulse-type experiments on the field evaporation of tungsten adatoms from a tungsten substrate Data source
PC (ev)
PEG (eV
W, ref. [ 171,fig. 9 W, ref. [17], fig. 10
1.43-to.07 1.31*0.08
- 1222 -923
-8.5% 1 -7 23
504
R.G. Forbes et al. / Field evaporatron theory
This simple procedure will usually give an error estimate larger than any other reasonable procedure, and - as will become clear later - is adequate for the purposes of this paper. But in reality this procedure is not statistically correct, because a, and (Ye are not statistically independent variables. A better estimate of error would be obtained from: ~*(cL,) = e*(ai)
+ 4c2(02)
+ 4p12 e(ot)
4a2),
(19)
where p,* is the correlation coefficient between (Y,and 02. Information about the covariance of a, and a, is not available in ref. [19], and to obtain such information it would be necessary to carry out new regression calculations on the original data sets. New regression calculations should result in estimates of error lower than those given in table 2. However, because our objective is to test formalisms, it has not been found necessary to perform these calculations. A formalism that fails when errors are overestimated would fail when the errors were correctly estimated; on the other hand, if derived parameter values are “reasonable”, then the error estimate is of lesser significance. In deriving the error estimates in tables 4 to 6 simple methods have been used, and in consequence the errors are probably overestimated. For this reason, in some places parameter values are given to more significant figures than the stated error limits would seem to justify. Finally, we would point out that if the temperature of the emitter tip is in fact higher than the boiling point of liquid nitrogen (77.3 K), then there will be a small systematic error in some of the parameter values stated.
4. Compivlaon with theory: I. Tsong’s fomda In his paper, Tsong [19] has interpreted his coefficients a, and a2 in terms of an analysis of the charge-exchange mechanism that he gave some years earlier [ 171. There is, however, a mathematical flaw in this earlier analysis. This arises in the differentiation of the “pseudolinear” term in F, in the expression for activation energy. The correct expression [ 14,181 is:
&
( neFxP) = nexP + neF%
,
where e is the elementary (proton) charge, ne is the charge on the ion, and xp is the distance of the “crossing point” (of neutral and ionic standard potential curves) from the emitter’s electrical surface. Tsong has neglected the second term in eq. (20), apparently thinking it to be relatively small [17]. But in reality the second term is almost equal to the first, but opposite in sign. A corrected analysis [ 181, assuming the bound neutral to be moving in a parabolic potential well, with force-constant K, leads to the
R. G. Forbes et al. /
Field evaporation theory
505
result (see appendix):
(21)
-& (neFx*) = urx*/F,
where r is the distance of the crossing-point from the bottom of the well. This term is smaller than nex* by about an order of magnitude. The physical implication of this flaw is that Tsong’s derivation of values of “polarisability” and of “xc + A” from his tabulated regression data is meaningless. His values may be disregarded.
5. Comparison with theory: II. Simple image-hump formalisms 5.1. Data manipulation With the simple image-hump formalism, it may be shown by elementary analysis that, to second order in F, the activation energy Q(F) for the field evaporation of an atom bound as a neutral is given by: Q(F)=(A’+H,
-n~E)-(n3e3F/41rro)“2+~(co-~~)F2,
(22)
where H,, is the energy needed to form a n-fold ion, from the neutral, in remote field-free space, being given by the sum of the first n ionization energies; A0 is the binding energy of the neutral to the emitter surface, in the absence of the field; +E is the relevant local work-function of the emitter; and co is the electric constant. co is a parameter associated with polarisation of the neutral in an external field F (or, possibly, with partial charge transfer from evaporating atom to emitter [25]), and c, is the corresponding parameter for an n-fold ion. Note that, although these coefficients are sometimes called “polarisability”, neither coefficient is a polarisability in the sense of the ordinary text-book definition [26]. Using eq. (20) in eqs. (13a) and( 13b), we obtain: p, =~(n3e3F?/41n0)1’2-(~o p2 = -~(n3e3F0/4wco)1’2Pz/P,
=
-40
-c,,)Fo2,
(23)
(co -c,)Fo2,
(24)
+@,
(29
where 8 is a “polarisation correction factor” given by: B=6(c,
-c.)F02/[(n3e3F0/4mo)1’2-
By further manipulation
2(c, - c.)Fo2].
of eqs. (23) and (24) we obtain:
( n3e3F0/4wco)“* = $(p, - p2) =$(Y,, $(c, -c,,)Fo2
= +(p,
(26)
+2/~,)-
-(a2
(27) +4,/6).
(28)
506
R.G. Forbes et ui, / Field evupo~utron theory
Finally, by substituting eqs. (27) and (28) back into eq. (22), we obtain the consistency relationship, at field F”: K, -Q(F’)
=$r,
where, for notational
-142 +,
+q,
W-4
convenience, we have defined a new symbol K,, by:
K”=A”+H,-n+E.
(29b)
5.2. Detailed comparisons The simplest version of the image-hump formalism ignores the F2 (“polarisation”) terms in eqs. (22) to (25). If this were a reasonable approximation, then we should expect p2 /,u, ~5:- 3. Inspection of table 2, however, shows that experimental estimates (ignoring error limits) lie between -3 and -7. Because of the relatively wide error limits stated for pa/p,, it is only in the cases of MO and W that it is possible to say decisively that the in~~dual experimental results are incompatible with the simple no-polarisation formalism. This incompatibility can also be illustrated graphically. From earlier equations it follows that: lg(J/JO)
+(F”)
- Q(F)]/kTln(lO).
(30)
Hence, on using eq. (22), but ignoring the F2 term, we obtain: lg(J/‘J”)
=B[(F/F’)“*-
11,
(3la)
where B is a constant defined by B = ( ~3=3Fo/4~~o)“2/~~~(
10).
(3lb)
Fig. 1 shows Tsong’s results [ 191 for tungsten, replotted with a superimposed theoretical curve obtained by adjusting B to fit the observed slope at F/F0 = 1. The discrepancy is obvious and it also deserves note that the F” values derived from eq. (3lb), using the observed B, are much less than the observed field. (The derived values are: for n = 1, F” = ‘7V/nm; for n = 2, F” = 0.9 V/run: the observed value is 55 V/m.) When the observed field is used to plot a theoretical curve (assuming n = 2}, the discrepancy is even more obvious. If all the six ratio estimates are taken together as a group a clear result is also achieved. Because the simple formalism, without F2 terms, makes a prediction that is independent of species and of charge-state at escape, the six ratio values in table 2 can be taken as six independent estimates of a quantity whose value is known. From these estimates (ignoring the a-priori errors) we may deduce that p2/p,(exp) = -4.48 + 0.69. Using a Student t test, it is found that even at the 0.5% level there is a significant difference between this “experimental” mean and the %ue” mean of -0.5. Thus there is at least a 99.5% probability that these experimental results are not compatible with the simple formalism that neglects the F2 terms.
R.G. Forbes et al. / Field evaporation theory
501
The next simplest image-hump formalism, in which the F2 terms are included, as in eqs. (22) to (25), has been widely used in the literature. A test of its self-consistency may be based on eq. (29). Values of the 1.h.s. of this
g
(= F/F"-11
Fig. I. A comparison of Tsong’s (1978) experimental field sensitivity data [ 191for tungsten with the predictions of the simple image-hump (no F* terms) formalism. Two theoretical comparisons are shown: one in which the parameter F” in eq. (31b) is taken as equal to the observed field, with n = 2 assumed; the other in which the parameter F” is adjusted so that the theoretical curve has the same slope at the origin as does the experimental curve. (In making these comparisons it has been assumed that the quantity Y is effectively constant.)
equation, shown in table 4 for n = 1 to 3, are obtained by: (a) setting Q( F’) = 0.2 eV; (b) using the p values shown, which have been taken from refs. [27] and [28]; (c) using values of A0 and ionization energy I, taken from ref. [9]. Values of the r.h.s. are obtained from the data in table 1. Inspection of table4 shows that in all cases, in the framework of the formalism in use, the hypothesis that n > 1 is wildly improbable. Even in the n = 1 case the difference between the 1.h. and r.h. sides of the equation is at least 7 times the standard derivation of the experimental estimate; so, statistically, we have negligible probability that the two sides of the equation are equal within experimental error. That is, the experimentally-derived data are not compatible with the image-hump formalism under discussion. A further test of the simple image-hump formalism (including F2 terms) can be based on eq. (27). Values of the evaporation field F” predicted from the field-sensitivity data, for n = 1 to 3, are shown in table5 These can be compared with observed values of evaporation field as noted in ref. [ 191. Even allowing for 30% error in the “observed’ field values, there is only one
SO8
R.G. Forbes et ul. / Field evaporation theov
Table 4 Self-consistency test of the simple image-hump formalism (including F2 terms). based on eq. (29); values of the 1.h.s. of this equation have been derived as explained in the text, and are shown for values of n = 1 to 3; values of the r.h.s. of the equation are derived from the data in table 1; the last column expresses the difference between the 1.h.s. (for n = 1) and the r.h.s.. in terms of the standard deviation afor the r.h.s., in each case Species
Mo(ll0) Ru(ll?l) Hf( lOi0) W(ll0) Ir( 100) Pt( 100)
+E (ev)
5.12 4.86 3.65 5.14 5.27 5.84
K, -Q( F”) (eW n=l
n=2
n=3
8.59 8.93 9.50 11.30 10.56 8.81
19.6 20.8 20.8 24.2 22.3 21.5
41.7 44.4 40.4 43.0 44.0 43.1
$a, +P* W)
Deviation
11.7kO.3 3.520.4 5.6*0.4 13.2kO.3 3.520.4 3.520.4
120 130 90 70 70 13a
case (Hf) in which the observed value is compatible with any value predicted from the field-s~siti~ty data using an image-hump formalism. 5.3. Conclusion From these comparisons we may conclude with considerable confidence that, if Tsong’s data [I91 concerning evaporation&x field sensitivity may be taken as good estimates of the corresponding data concerning evaporation rate-constant field sensitivity, then these data are not compatible with the conventional simple image-hump formalisms used to analyse the Mtillerkhottky mechanism of field evaporation. Table 5 Values of the evaporation field F”, as derived from eq. (27), using the data in table 1. for values of n= 1 to 3; the last column gives the direct experimental estimate of F”, at liquid-nitrogen temperature, made by Tsong [ 191 Species
Mo(ll0) Ru(ll?l) Hf( loio) W(ll0) Ir( 100) Pt(rO0)
Derived field (V nm-
’)
n=l
n=2
n=3
Observed field (V nm-‘f
148 *2 11.6rtO.6 32.6 * 1 188 23 12.2~0.6 11.610.6
18.6 20.3 1.45kO.07 4.1 kO.1 23.5 kO.3 1.53~0.07 1.45-co.07
5.49r0.08 0.43 eO.02 1.21-co.04 6.97kO.08 0.45 rt 0.02 0.43 f 0.02
46 42 40 55 52 48
R.G. Forbes et al. / Field evaporation
theory
509
Obviously, this result has been proved only in the six cases for which data have been taken. However, there is nothing particularly “special” about these six cases, and the presumption must be that these image-hump formalisms are not generally appropriate for metal field evaporation. The inability of simple image-hump formalisms to explain field-sensitivity data has a further implication. It suggests that the use of such formalisms to predict evaporation fields (by setting Q in eq. (22) equal to zero) is without proper scientific legitimacy - even though this procedure leads to reasonable agreement with experiment. Closer investigation [29] in fact shows that equally good agreement can be achieved on the basis of a simple general argument concerning the energetics of field evaporation; it seems that ability of a formalism to approximately predict evaporation field is not a useful test of the corresponding mechanism of field evaporation. 6. Comparison with theory: III. Parabolic jug formalism 6.1. Derivation
of formulae
By working with a formalism in which the neutral atom is assumed to be bound in a parabolic potential well, of force-constant K, Forbes [18] has been able to derive expressions for partial energies that can be put in the following form: CL,=Kr”(a+ro)=2Q(1 p2
=
+a/r’),
-p,(2 + a/r”),
(32) (33)
where a is the distance of the neutral-atom bonding point from the metal’s electrical surface, and r” is the distance of the “point of escape” (at the crossing-point of neutral and ionic potential-energy curves) from the neutralatom bonding point, at the evaporation field F” in question. These expressions are derived from eqs. (43) and (52) in ref. [18]. Certain inessential mathematical approximations have been made in the derivation of eqs. (32) and (33), and hence these equations constitute a basic version of the formalism. Ref. [ 181in fact works in a context where the evaporation field is defined by a unity-rate-constant criterion, but the same expressions are relevant to the present context (where the evaporation field is defined by an evaporation-flux criterion), although obviously p,, c(~ and Q have slightly different meanings. By algebraic manipulation of eqs. (32) and (33) we may derive expressions for Q and for r”/a as follows:
Q= -id/h r”/a=
-P~/(~P~
+d
(34) +cL~).
Note that all the expressions derived in this section are independent charge state of the escaping ion.
(35)
of the
R.G. Forbes et al. / Field evaporation theory
510
6.2. Detailed cornpqrisons The experimental estimates of CL,and p2, listed in tables 2 and 3, can be used in various ways. In the original treatment in ref. 1181, the Arrhenius equation was used to make an a-priori estimate of Q as 0.2 eV, and this was then combined with the experimental estimate of EL,to make an estimate of a/r0 and hence a semi-theoretical estimate of p2/p,. This semi-theoretical estimate compared favourably with the experimental estimate derived from the data in table 3. An alternative approach is used here. Table6 shows values of Q and r”/a derived from the data in table2, using eqs. (34) and (39, and the question is whether the derived values are “physically reasonable”. An a-priori estimate of activation energy may be made by combining eq. (1) with the Arrhenius equation: k,
=Aexp(-Q/kT),
(36)
and taking: J= 0.01 layer/s, nhr =O.Ol layers (which implies k,, = 1 SC’); T = 77.3 K. This leads to the result: Q = 0.18 eV. If the field-evaporation pre-exponential A were “anomalously” low, which has been suggested for some materials in the past [30,31], then slightly lower values of Q would be expected. For example, taking A = lo8 s-i would lead to the result Q = 0.12 eV. The experimental estimates of Q are clearly compatible with these theoretical estimates, to within the limits of experimental error, and would still be compatible with the theoretical estimates if the limits of error were substantially reduced. Thus the activation-en~gy estimates are certainly “reasonable”. It remains to consider the experimental estimates of rO/a, which range from 0.2 to 1.0. The values for MO, W, and I-If are less th&n 0.5 and we feel these to be entirely reasonable. The remaining values, particularly the Pt and Ru values (both l.O), are higher than one would perhaps expect; however, the error limits on these particular estimates are very wide, and in consequence the stated
Table 6 Values of the activation energy Q(F”)
and the ratio r”/a,
using the data in table 2 Species
Q WI
P/a
M&l t0) Ru(1 lzi) Hf( loio) W(1 IO) 103) WC@)
0.13~0.06 0.2OkO.2 0.16*0.15 0.15~0.06 0.14*0.15 0.20*0.2
0.22-to.02 1.0 +I 0.460.1 0.2lAO.02 0.64kO.6 1.0 *1
as derived from
eq. (34) and eq. (35),
R.G. Forbes et a&. / Field emporntion
theory
511
values of r*/a could well be somewhat higher than the true values; so we do not regard these high estimated values as casting doubt on the formalism. In general, we conclude that a charge-exchange mechanism, in the form of a basic version of the parabolic-jug formalism, is apparently well able to provide a physically satisfactory explanation of partial energies derived from flux field-~siti~ty data. At the same time, it should be realized that - because this paper employs only the basic version of the formalism - the tabulated values of Q and r”/a are not necessarily the best estimates that could be derived from the data. 7. General discussion The conclusions of this paper need to be approached with care. Our immediate objective has been, not to decide the mechanism of metal field evaporation, nor to derive reliable atomic-level information from published experimental data, but the more modest one of testing the consistency of experimental data with the,existing published theoretical formalisms. What we have found is that the evaporation-flux field-sensitivity data are not compatible with existing image-hump formalisms, but are compatible with a charge-exchange formalism. Because the existing image-hump foamy are manifestly incomplete for example, past numerical analyses do not in&de any term relating to the repulsive ion-metal interaction that must physically be present - it is logically inappropriate to conclude from the present discussion that the field-sensitivity data are necessarily incompatible with the Miiller-Schottky mechanism. For example, the proposition might be advanced that the demonstrated inconsistencies could be removed by including a repulsive term, or by taking a more sophisticated approach to the calculation of the correlation interaction. Or it might be argued that the inconsistencies result from the neglect of field dependence in the pre-exponential But, intuitively, these propositions look unpromising, particularly when set against the demonstrated compatibility of the field-sensitivity data with a simple charge-exchange formalism. Charge-exchange thus looks much more likely than the Mtiller-Schottky mechanism. In view of often-expressed past doubts about the self-consistency of the simple image-hump formalisms, this conclusion is unsurprising. The new feature of the present discussion is that the ~~mpatib~ty of experimental data with these image-hump formalisms has been established by means of statistical tests. Appendix. Differentiation of the tern pseudo-liar
in F
There appears in the expression for field-evaporation activation energy Q, an “electrostatic” term (denoted here by u) that has the form: o = neFxP,
(37)
512
R.G. Forbes et ai. / Field ~up~rario~
theory
where xi’ is the distance of the “point of escape” from the metal’s electrical surface, and the other symbols have their usual meanings. In the context of a charge-exchange formalism, xi’ corresponds to the crossing-point of neutral and ionic potential energy curves. The expression for the total differential of Q with respect to F has (-do/dF) as one of its terms. Because xp is itself a function of F, the total differential do/d F is given by: (38) That is dx* FF = ptex* + neFdF * As stated in seetion4, the second term in this expression has often been neglected. In the context of a model in which the neutral atom moves in a parabolic potential-energy well, an expression for this second term, and hence for du/dF, can be derived as follows. (The treatment is based on that in ref. WI.) The field-evaporation activation energy can be written in the alternative forms: Q=fKr2
=
(A”+H,-~~E)-nel;xP+P(F)+C(~P),
w
where r is the distance of the point-of-escape (the crossing point) from the bottom of the potential well, K is the force constant for the well, P(F) is a (primarily) field-dependent correction term relating to the polarisation of the neutral and the ion, or equivalent effects, and C(x) is a (primarily) distancedependent function describing an interaction between the ion and the surface due to correlation and (in principle) repulsive forces. C(x) is usually approximated by the image-potenti~. The other symbols have their usual meanings, as listed in section 5.1. Taking differentials we obtain: ~rdr=nex*dF+
(dP/dF)
dF-neFdxP
+ (dC/dxP)
dxr.
(41)
The relationship of r and xi’ is: xp=a+r.
(42)
It is a good appro~ma~on to assume that a, the distance of the well bottom from the metal’s electrical surface, is independent of field. Hence dr = dxr and we obtain: dxP/dF=
- (nexp -dP/dF)/(neF+~-dC/dxP).
As a first approximation repulsion terms. Thus dxP/dF-
we may neglect the polarisation
-nexP/(neF+rcr).
(43) and correlation + WI
R.G. Forbes et al. /
Field evaporation theory
513
Substituting into eq. (39) then gives du/dF-nexP[l-neF/(neF+kr)]
=nexbcr/(neF+~r).
(45)
At this stage we can neglect or in comparison with neF, the basic result stated in section4 follows: du/d F = KrxP/F.
(W
Obviously, higher-order approximations can be obtained by including the omitted terms, but these do not dramatically affect the result, numerically.
Acknowledgement
This work forms part of a research project funded by the UK Science and Engineering Research Council. In addition, one of us (K.C.) wishes to thank the Ministry of Higher Education and Scientific Research of the Republic of Algeria for personal financial support.
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [IO] [II] [I21 [I31 [I41
E.W. Mtiller, J.A. Pa&z and S.B. McLane, Rev. Sci. Instr. 39 (1968) 83. J.A. Panitz, CRC Critical Rev. Solid State Sci. 5 (1975) 153. R. Clampitt, Nucl. Instr. Methods 189 (1981) I I I. L.W. Swanson, in: Microcircuit Engineering ‘80. Amsterdam, 1980. N. Ernst, Surface Sci. 87 (1979) 469. N. Ernst and G. Bozdech, in: Proc. 27th Intern. Field Emission Symp., Tokyo, 1980. R. Haydock and D.R. Kingham, Phys. Rev. Letters 44 (1980) 1520. D.R. Kingham, Surface Sci. 108 (1981) L460. T.T. Tsong, Surface Sci. 70 (1978) 211. E.W. Mttller and T.T. Tsong, Progr. Surface !I& 4 (1974)1. E.W. Mtlller, Phys. Rev. 102 (1956) 618. E.W. Muller, Advan. Electron. Electron Phys. 13 (1960) 83. R. Gomer and L.W. Swanson, J. Chem. Phys. 38 (1%3) 1613. D. McKinstry, Surface Sci. 39 (1972) 37. [IS] D. Brandon, Brit. J. Appl. Phys. 14 (1%3) 474. [Ia] E.W. Mttller and T.T. Tsong, Field Ion Microscopy: Principles and Applications (Elsevier, Amsterdam, 1969). [I71 T.T. Tsong, J. Chem. Phys. 54 (1971) 813. [ 181 R.G. Forbes, Surface Sci. 70 (1978) 239. [ 191 T.T. Tsong, J. Phys. F (Metal Phys.) 8 (1978) 1349. [ZO] R.G. Forbes, Surface Sci. 46 (1974) 577. [Zl] R.G. Forbes, in: Proc. 7th Intern Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, p. 387. [22] This assumption is reasonable at temperatures near 80 K, but would not be reasonable at temperatures at which metal atom diffusion could occur. [23] C. Patel, MSc Thesis, University of Aston in Birmingham (1974). [24] R.G. Forbes and C. Patel, 22nd Intern. Field Emission Symp., Atlanta, 1975 (unpublished).
514
R.G. Forbes et ul. / Field euaporution theory
[25] T.T. Tsong and G. Kellogg, Phys. Rev. 12B (1975) 1343.
[26] The field that appears in the conventional definition of polarisability is the self-consistent local field acting on.the atom. At a metal surface this field may be significantly different from the external field. [27] J.C. J&i&e, in: Solid State Science, Vol. 1, Ed. M. Green (Dekker, New York, 1969) p. 179. [28] Z. Knor, in: Surface and Defect Properties of Solids, Vol. 6 (1977)p. 139. [29] R.G. Forbes, Appl. Phys. Letters, to be published. 1301 D.G. Brandon, Phil. Msg. I4 (1966) 803. [31] In our view, Brandon’s deductions concerning the value of the field evaporation preexponential should now be treated with reservations.
Surface Science 114 (1982) 498-514 North-Holland Publishing Company
FIELD EVAPORATION THEORY: A REANALYSIS OF PUBLISHED FIELD SENSITIVITY DATA Richard G. FORBES, R.K. BISWAS and K. CHIBANE Department
of Physics, University of Aston, Costa Green, Birmingham 84 7ET, UK
Received 2 July 1981; accepted for publication
29 September
1981
Data concerning the field sensitivity of field evaporation flux were published for six refractory metals by Tsong in 1978. These data are converted into the form of partial energies, and used in a re-examination of the consistency of published field evaporation formalisms describing the escape step. The general conclusion is that the data are not compatible with the conventional simple image-hump formalisms, even when experimental error is taken into account, but are compatible with an analysis of charge exchange based on a simple parabolic-jug formalism.
1. Introduction Field evaporation has an established significance as the emission process in the atom-probe field-ion microscope [1,2] and related techniques, and a new importance as a possible emission mechanism in connection with liquid-metal field-ion sources [3,4]. Following recent work by Ernst [5,6] and by Kingham [7,8], it now seems virtually certain that for most metals field evaporation is a two-stage process. First a metal atom escapes as a singly-charged or possibly doubly-charged ion; subsequently, it may be post-ionized into higher charge states. Kingham’s demonstration, that theoretically post-ionisation is a likely process, provides a most plausible exljlanation of the high charge states sometimes observed amongst field-evaporation products [9], and serves to redirect theoretical attention towards the nature of the escape process. As is well known [lo], there are two commonly discussed alternatives for the escape process; the Mtiller-Schottky (or “image-hump”) mechanism [ 11,121 in which ionization precedes escape; or some form of charge-exchange process [ 131 in which ionization and escape occur together. These proposed mechanisms are incompatible alternatives, that lead to quantitatively different physical interpretations of basic field evaporation data [14], and it would be useful to decide firmly between them. Detailed comparison of the mechanisms has, however, been handicapped by two things. First, the Mtiller-Schottky mechanism is often discussed in terms of simple image-hump formalisms that are of dubious validity [ 15- 171(but do 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
R.G. Forbes et al. / Field ~uporafion
theory
499
apparently lead to reasonable predictions of evaporation field-see, for example, ref. [9]). Second, most mathematical analyses of the charge-exchange mechanism prior to 1978 contain a disabling mathematical flaw [ 14,181. Recently, Tsong [ 191 has reported some measurements on the field dependence of evaporation flux (“evaporation current”). The objectives of this note are to retabulate these in a form analogous to that used by Forbes [18,20], and to explore whether the resulting data are compatible with the existing fieldevaporation formalisms. The structure of the paper is as follows. In section 2 we set down some basic theory relating to the concept of “partial energies”. Section3 converts the experimental data into the form of partial energies_ Sections 4, 5 and 6 then discuss this data in the contexts of the published formalisms. Section 7 draws our conclusions together.
2. Background thmy It is first necessary to deal with some matters of notation and terminology. The amount of material evaporated per unit time from a field-ion emitter may be expressed in the form [21]: J = n&h,,
(1)
where nhr is the amount of material (or “count of atoms”) at high risk of evaporation, and k, is the fi&euaporation rate-constant for the atoms at ‘high risk” sites (which in practice are the kink sites). The rate-constant k,, is measured in s - ‘, The quantity nhr is regarded as having the dimension [amount-of-substance], but is measured in the units “atoms” or “layers” (these being regarded as non-S1 units of amount of substance). The quantity J is regarded as having the dimensions ~amount-of-subs~ce/time], is measured in layers/s (or atoms/s), and is here called the ~~~d-~uF~rutiun flux. J has previously been called “evaporation current” [20], this notionally being an abbreviation for “amount-of-substance current”. However, it now seems to us that the term “current” is better used to refer to the electric current associated with field evaporation, particularly in the context of liquid-metal field-ion sources. Thus we have adopted the term “flux” as a name for rate-like quantities having dimensions [amount-of-substance/time], and the name “field -evaporation flux” for J. In the literature, J is also called “evaporation rate”, “measured evaporation rate”, and “relative evaporation rate”, but we prefer not to use these names. Our system has been adopted in order to avoid ambiguity, to be consistent with the revised SI system introduced in 1971, and to keep k as having the traditional chemical meaning of “rate-constant”. Note that Tsong’s k is our J, and Tsong’s K is our k,,. The three parameters in eq. (1) are all functions of the field strength F at the
R.G. Forbes et al. / Field evuporutlon theov
500
high-risk sites. Consider some reference situation, in which F has the value F”, and let the corresponding values of these parameters be Jo, nL andkk. It follows that: ln( J/Jo)
= In( n,/n&)
+ ln( k,/kk).
(2)
Each of the terms in eq. (2) has value zero at F = F”, and it is convenient to Taylor-expand them about F”. If we introduce a variable g by: g= (F-
FO)/FO,
(3)
then the expansion of the rate-constant form: ln(&/W
= (C.c&Wg+t(I-(JkT)g*
term (for example) can be put in the +....
(4)
The (p,/kT), (p2/kT), etc., are the Taylor coefficients, which are written in this way because the quantities p,, cl2 are expected to be temperature independent if field evaporation is a thermally-activated process obeying the Arrhenius equation. By analogy with eq. (4), we may expand the other terms in a similar way, thus: ln(J/J’)
=(A,/kT)g++(X,/kT)g’+...,
ln(n,,/&)
= (v,/kT)g+i(v2/kT)g2
(5) + . .. .
(6)
Clearly we have: x, =jJ, + Y,, A, =jJ, + v*.
(7)
Existing theories of field evaporation are, of course, formalisms that deal with rate-constant field sensitivity rather than evaporation-flux field sensitivity. Each formalism provides an expression for the field-evaporation activation energy Q(F); expressions for the “partial energies” /.L, and cl2 can then be obtained as follows [l&20]. The Arrhenius equation may be written in the form: k,, = exp(M/kT),
(8)
where M= kTln{A}
- Q,
(9)
where {A} is the numerical value of the field evaporation pre-exponential expressed in s-‘. It follows that: ln(k,/kk)
=[M-M(F’)]/kT.
Taylor-expanding
A, (10)
M in terms of F, about F = F”, we obtain:
A4=M(F”)+(F-F’)+$l
= F
++(F-F’)‘$ F”
_ F-F’
+....
(11)
R. G. Forbes et ul. /
501
Field euuporation theory
Substituting from eq. (3) and then substituting into eq. (10) gives: ln(k,/k&)
= (F”g/kT)+$lpO
++[(F”)2g2\kT]$IF0
+....
(12)
Hence, by comparison with eq. (4): OdM
C1,=F
dFFO
N
-FOde
034
dF p’
p, = (F”)‘s
N - (F”12s FO
(13b)
FO’
The second step in each of these relationships assumes that the field dependence of the pre-exponential A may be neglected. Previous analyses, for example refs. [ 14,17,18], have taken this to be an adequate approximation, particularly at temperatures near 80 K and above, and we continue with this assumption. The arguments principally involved are as follows. First, the pre-exponential A involves a factor relating to the vibrational frequency of the bound neutral. This frequency .depends on the shape of the neutral potential well, which (in comparison with the activation energy) is unlikely to be sensitively affected by a small field variation. Second, the pre-exponential involves a factor that relates to the possibility of “ion tunneling” [ 161 and is in principle field-dependent for this reason. It has always been supposed that, at temperatures near 80 K and above, any field dependence due to this cause is insignificant. Neglect of posssible field dependence in the pre-exponential A is, in practice, an integral part of the formalism under discussion in later sections. The question .of whether this neglect is really justified is far from straightforward and would require detailed mathematical analysis that regretfully is beyond the scope of the present paper.
3. Experimental data 3.I. Conversion of data Tsong’s experimental results [ 191 are in the form of a plot of lg(J/JO) versus F/F’, where F” is the field at which the evaporation flux is 10 -2 layers/s. He seems to have treated this data by malting the (perfectly reasonable [22]) assumption that lg(nk/&) is effectively zero, and hence that: lg( J/Jo) = lg( k,/kk)
= a0 + a,( F/F’)
+ a2(F/Fo)‘.
(14)
He does not explicitly mention any assumption equivalent to taking lg(nk/&) zero, but he does state that a, and a2 are best-fit coefficients to the experimental data, so we shall assume that the correct logical identification between his
R.G. Forbes et ul. / Aeld evaporution theory
502
parameters and ours is: A, = ln(10) X kT- (u, + 2a,), A, =2ln(lO)XkT-a,.
For convenience in dealing with errors, it is useful to define working parameters (Y,,01~by: cy, =ln(lO)
X kT+a,,
@a)
a2 = In{ 10) X kT- a2.
(16b)
Tsong’s values of a, and a2, together with corresponding values of (Y, and (Ye are shown in table 1; T has been put equal to 77.3 K. Existing theories of field evaporation deal with rate-constant field sensitivity rather than evaporation-flux field sensitivity, so we require p1 and pcIzrather than A, and A,. We shah thus follow Tsong and other workers (for example, Brandon [15]) in assuming that there is no significant field dependence in the amount of material at high risk of evaporation. This implies that ZJ,and Yecan be neglected, and hence that experimental estimates of partial energies are given by: p,(exp) +(exp)
= ai + 2or,
(17a)
pz(exp) = X,(exp) = 2im,.
(17b)
Partial energies derived via eq. (17) are shown in table2, together with corresponding values of their ratio. For simplicity in later argument, we label the columns here and later in terms of c,, p2 and derived expressions. However, it should be remembered that strictly (except in the case of table 3) the columns show experimental estimates of X,, A, and derived expressions. For comparison with the tungsten estimates in table 2, we show in table 3 values of &, p; and & /r_1T,derived by Forbes and. Pate1 [23,24] from Tsong’s measurements [17] on the field sensitivity of evaporation rate-constant, in the
Table I Values of F”, a, and az, as given in ref. [19], together with working parameters q and a2 derived via eq. (16); the error in a, is not explicitly stated in ref. [19], but is assumed here to be * 5 in each case; for working purposes, the values and errors for a, and az are given to more places than are physically significant Species
F” (V/nm)
(I*
Mo(ll0)
46
Ru(ll~1) Hf(lOi0) W(l10) Ir(100) Pt(100)
42 40 55 52 48
715~5 20015 335*5 805*5 205r5 200*5
-310*10 -75220 - 135*20 -350* 10 - 80-‘20 -75’20
*I (evt
a2 (ev)
10.%7r+O.O77 3.068*0.077 5.138*0.077 12.347*0.077 3.i44*0.077 3.068-cO.077
-4.754*0.153 - 1.150~0.307 -2.071 -co.307 -5.368*0.153 - 1.227*0.307 - 1.150*0.307
503
R.G. Forbes et (11./ Field euuporution theory
Table 2 Experimental estimates of the partial energies c, and as, and their ratio, derived from the data in table I, using eq. (17); it has been assumed that the quantities Y, and Y*appearing in eq. (7) are negligibly small Species
CI (eV)
Mo(1 IO) Ru(l111) Hf( loio) W(110) Ir( 100) Pt(100)
1.46kO.4 0.77 -co.7 1.00*0.7 1.61kO.4 0.69kO.7 0.77kO.7
~2
WI
C2/CI
-6.5~2 -3.0*3 -4.2k3.5 -6.7*2 -3.6k3.5 -3.o-c3
-9.5120.3 -2.3OkO.6 -4.14-tO.6 - 10.74*0.3 -2.45kO.6 -2.3020.6
case of tungsten. (The suffix c on the partial energies indicates that these relate to an evaporation field defined by a unity-rate-constant criterion [20] rather than the evaporation-flux criterion used in table 2.) Bearing in mind that different criteria may lead to different field values being identified as the “evaporation field” [20], and that the value of p, will depend on the choice of “evaporation field”, we feel that the comparison of table 3 (based on rate-constant field sensitivity) with the tungsten estimates in table2 (based on flux field sensitivity) is entirely satisfactory, and goes some way towards confirming that Y, and Ye can be neglected in the context of low-temperature field evaporation. 3.2. Treatment of errors Some comment on our treatment of errors is necessary. In deriving the errors on the p’s from those on the u’s we have used simple linear formulae. For example, the standard deviation o(p,) is obtained from the formula, based on eq. (17a). 4%)
= 4%)
+ 2 4%).
(18)
Table 3 Experimental estimates of the partial energies pt and ~‘2. and their ratio, derived by Pate1 [23] from original data concerning rate-constant field sensitivity taken by Tsong [ 171; in the analysis of the fig. 9 data, certain deviant points have been excluded from the analysis on statistical grounds; the original data were obtained from pulse-type experiments on the field evaporation of tungsten adatoms from a tungsten substrate Data source
PC (ev)
PEG (eV
W, ref. [ 171,fig. 9 W, ref. [17], fig. 10
1.43-to.07 1.31*0.08
- 1222 -923
-8.5% 1 -7 23
504
R.G. Forbes et al. / Field evaporatron theory
This simple procedure will usually give an error estimate larger than any other reasonable procedure, and - as will become clear later - is adequate for the purposes of this paper. But in reality this procedure is not statistically correct, because a, and (Ye are not statistically independent variables. A better estimate of error would be obtained from: ~*(cL,) = e*(ai)
+ 4c2(02)
+ 4p12 e(ot)
4a2),
(19)
where p,* is the correlation coefficient between (Y,and 02. Information about the covariance of a, and a, is not available in ref. [19], and to obtain such information it would be necessary to carry out new regression calculations on the original data sets. New regression calculations should result in estimates of error lower than those given in table 2. However, because our objective is to test formalisms, it has not been found necessary to perform these calculations. A formalism that fails when errors are overestimated would fail when the errors were correctly estimated; on the other hand, if derived parameter values are “reasonable”, then the error estimate is of lesser significance. In deriving the error estimates in tables 4 to 6 simple methods have been used, and in consequence the errors are probably overestimated. For this reason, in some places parameter values are given to more significant figures than the stated error limits would seem to justify. Finally, we would point out that if the temperature of the emitter tip is in fact higher than the boiling point of liquid nitrogen (77.3 K), then there will be a small systematic error in some of the parameter values stated.
4. Compivlaon with theory: I. Tsong’s fomda In his paper, Tsong [19] has interpreted his coefficients a, and a2 in terms of an analysis of the charge-exchange mechanism that he gave some years earlier [ 171. There is, however, a mathematical flaw in this earlier analysis. This arises in the differentiation of the “pseudolinear” term in F, in the expression for activation energy. The correct expression [ 14,181 is:
&
( neFxP) = nexP + neF%
,
where e is the elementary (proton) charge, ne is the charge on the ion, and xp is the distance of the “crossing point” (of neutral and ionic standard potential curves) from the emitter’s electrical surface. Tsong has neglected the second term in eq. (20), apparently thinking it to be relatively small [17]. But in reality the second term is almost equal to the first, but opposite in sign. A corrected analysis [ 181, assuming the bound neutral to be moving in a parabolic potential well, with force-constant K, leads to the
R. G. Forbes et al. /
Field evaporation theory
505
result (see appendix):
(21)
-& (neFx*) = urx*/F,
where r is the distance of the crossing-point from the bottom of the well. This term is smaller than nex* by about an order of magnitude. The physical implication of this flaw is that Tsong’s derivation of values of “polarisability” and of “xc + A” from his tabulated regression data is meaningless. His values may be disregarded.
5. Comparison with theory: II. Simple image-hump formalisms 5.1. Data manipulation With the simple image-hump formalism, it may be shown by elementary analysis that, to second order in F, the activation energy Q(F) for the field evaporation of an atom bound as a neutral is given by: Q(F)=(A’+H,
-n~E)-(n3e3F/41rro)“2+~(co-~~)F2,
(22)
where H,, is the energy needed to form a n-fold ion, from the neutral, in remote field-free space, being given by the sum of the first n ionization energies; A0 is the binding energy of the neutral to the emitter surface, in the absence of the field; +E is the relevant local work-function of the emitter; and co is the electric constant. co is a parameter associated with polarisation of the neutral in an external field F (or, possibly, with partial charge transfer from evaporating atom to emitter [25]), and c, is the corresponding parameter for an n-fold ion. Note that, although these coefficients are sometimes called “polarisability”, neither coefficient is a polarisability in the sense of the ordinary text-book definition [26]. Using eq. (20) in eqs. (13a) and( 13b), we obtain: p, =~(n3e3F?/41n0)1’2-(~o p2 = -~(n3e3F0/4wco)1’2Pz/P,
=
-40
-c,,)Fo2,
(23)
(co -c,)Fo2,
(24)
+@,
(29
where 8 is a “polarisation correction factor” given by: B=6(c,
-c.)F02/[(n3e3F0/4mo)1’2-
By further manipulation
2(c, - c.)Fo2].
of eqs. (23) and (24) we obtain:
( n3e3F0/4wco)“* = $(p, - p2) =$(Y,, $(c, -c,,)Fo2
= +(p,
(26)
+2/~,)-
-(a2
(27) +4,/6).
(28)
506
R.G. Forbes et ui, / Field evupo~utron theory
Finally, by substituting eqs. (27) and (28) back into eq. (22), we obtain the consistency relationship, at field F”: K, -Q(F’)
=$r,
where, for notational
-142 +,
+q,
W-4
convenience, we have defined a new symbol K,, by:
K”=A”+H,-n+E.
(29b)
5.2. Detailed comparisons The simplest version of the image-hump formalism ignores the F2 (“polarisation”) terms in eqs. (22) to (25). If this were a reasonable approximation, then we should expect p2 /,u, ~5:- 3. Inspection of table 2, however, shows that experimental estimates (ignoring error limits) lie between -3 and -7. Because of the relatively wide error limits stated for pa/p,, it is only in the cases of MO and W that it is possible to say decisively that the in~~dual experimental results are incompatible with the simple no-polarisation formalism. This incompatibility can also be illustrated graphically. From earlier equations it follows that: lg(J/JO)
+(F”)
- Q(F)]/kTln(lO).
(30)
Hence, on using eq. (22), but ignoring the F2 term, we obtain: lg(J/‘J”)
=B[(F/F’)“*-
11,
(3la)
where B is a constant defined by B = ( ~3=3Fo/4~~o)“2/~~~(
10).
(3lb)
Fig. 1 shows Tsong’s results [ 191 for tungsten, replotted with a superimposed theoretical curve obtained by adjusting B to fit the observed slope at F/F0 = 1. The discrepancy is obvious and it also deserves note that the F” values derived from eq. (3lb), using the observed B, are much less than the observed field. (The derived values are: for n = 1, F” = ‘7V/nm; for n = 2, F” = 0.9 V/run: the observed value is 55 V/m.) When the observed field is used to plot a theoretical curve (assuming n = 2}, the discrepancy is even more obvious. If all the six ratio estimates are taken together as a group a clear result is also achieved. Because the simple formalism, without F2 terms, makes a prediction that is independent of species and of charge-state at escape, the six ratio values in table 2 can be taken as six independent estimates of a quantity whose value is known. From these estimates (ignoring the a-priori errors) we may deduce that p2/p,(exp) = -4.48 + 0.69. Using a Student t test, it is found that even at the 0.5% level there is a significant difference between this “experimental” mean and the %ue” mean of -0.5. Thus there is at least a 99.5% probability that these experimental results are not compatible with the simple formalism that neglects the F2 terms.
R.G. Forbes et al. / Field evaporation theory
501
The next simplest image-hump formalism, in which the F2 terms are included, as in eqs. (22) to (25), has been widely used in the literature. A test of its self-consistency may be based on eq. (29). Values of the 1.h.s. of this
g
(= F/F"-11
Fig. I. A comparison of Tsong’s (1978) experimental field sensitivity data [ 191for tungsten with the predictions of the simple image-hump (no F* terms) formalism. Two theoretical comparisons are shown: one in which the parameter F” in eq. (31b) is taken as equal to the observed field, with n = 2 assumed; the other in which the parameter F” is adjusted so that the theoretical curve has the same slope at the origin as does the experimental curve. (In making these comparisons it has been assumed that the quantity Y is effectively constant.)
equation, shown in table 4 for n = 1 to 3, are obtained by: (a) setting Q( F’) = 0.2 eV; (b) using the p values shown, which have been taken from refs. [27] and [28]; (c) using values of A0 and ionization energy I, taken from ref. [9]. Values of the r.h.s. are obtained from the data in table 1. Inspection of table4 shows that in all cases, in the framework of the formalism in use, the hypothesis that n > 1 is wildly improbable. Even in the n = 1 case the difference between the 1.h. and r.h. sides of the equation is at least 7 times the standard derivation of the experimental estimate; so, statistically, we have negligible probability that the two sides of the equation are equal within experimental error. That is, the experimentally-derived data are not compatible with the image-hump formalism under discussion. A further test of the simple image-hump formalism (including F2 terms) can be based on eq. (27). Values of the evaporation field F” predicted from the field-sensitivity data, for n = 1 to 3, are shown in table5 These can be compared with observed values of evaporation field as noted in ref. [ 191. Even allowing for 30% error in the “observed’ field values, there is only one
SO8
R.G. Forbes et ul. / Field evaporation theov
Table 4 Self-consistency test of the simple image-hump formalism (including F2 terms). based on eq. (29); values of the 1.h.s. of this equation have been derived as explained in the text, and are shown for values of n = 1 to 3; values of the r.h.s. of the equation are derived from the data in table 1; the last column expresses the difference between the 1.h.s. (for n = 1) and the r.h.s.. in terms of the standard deviation afor the r.h.s., in each case Species
Mo(ll0) Ru(ll?l) Hf( lOi0) W(ll0) Ir( 100) Pt( 100)
+E (ev)
5.12 4.86 3.65 5.14 5.27 5.84
K, -Q( F”) (eW n=l
n=2
n=3
8.59 8.93 9.50 11.30 10.56 8.81
19.6 20.8 20.8 24.2 22.3 21.5
41.7 44.4 40.4 43.0 44.0 43.1
$a, +P* W)
Deviation
11.7kO.3 3.520.4 5.6*0.4 13.2kO.3 3.520.4 3.520.4
120 130 90 70 70 13a
case (Hf) in which the observed value is compatible with any value predicted from the field-s~siti~ty data using an image-hump formalism. 5.3. Conclusion From these comparisons we may conclude with considerable confidence that, if Tsong’s data [I91 concerning evaporation&x field sensitivity may be taken as good estimates of the corresponding data concerning evaporation rate-constant field sensitivity, then these data are not compatible with the conventional simple image-hump formalisms used to analyse the Mtillerkhottky mechanism of field evaporation. Table 5 Values of the evaporation field F”, as derived from eq. (27), using the data in table 1. for values of n= 1 to 3; the last column gives the direct experimental estimate of F”, at liquid-nitrogen temperature, made by Tsong [ 191 Species
Mo(ll0) Ru(ll?l) Hf( loio) W(ll0) Ir( 100) Pt(rO0)
Derived field (V nm-
’)
n=l
n=2
n=3
Observed field (V nm-‘f
148 *2 11.6rtO.6 32.6 * 1 188 23 12.2~0.6 11.610.6
18.6 20.3 1.45kO.07 4.1 kO.1 23.5 kO.3 1.53~0.07 1.45-co.07
5.49r0.08 0.43 eO.02 1.21-co.04 6.97kO.08 0.45 rt 0.02 0.43 f 0.02
46 42 40 55 52 48
R.G. Forbes et al. / Field evaporation
theory
509
Obviously, this result has been proved only in the six cases for which data have been taken. However, there is nothing particularly “special” about these six cases, and the presumption must be that these image-hump formalisms are not generally appropriate for metal field evaporation. The inability of simple image-hump formalisms to explain field-sensitivity data has a further implication. It suggests that the use of such formalisms to predict evaporation fields (by setting Q in eq. (22) equal to zero) is without proper scientific legitimacy - even though this procedure leads to reasonable agreement with experiment. Closer investigation [29] in fact shows that equally good agreement can be achieved on the basis of a simple general argument concerning the energetics of field evaporation; it seems that ability of a formalism to approximately predict evaporation field is not a useful test of the corresponding mechanism of field evaporation. 6. Comparison with theory: III. Parabolic jug formalism 6.1. Derivation
of formulae
By working with a formalism in which the neutral atom is assumed to be bound in a parabolic potential well, of force-constant K, Forbes [18] has been able to derive expressions for partial energies that can be put in the following form: CL,=Kr”(a+ro)=2Q(1 p2
=
+a/r’),
-p,(2 + a/r”),
(32) (33)
where a is the distance of the neutral-atom bonding point from the metal’s electrical surface, and r” is the distance of the “point of escape” (at the crossing-point of neutral and ionic potential-energy curves) from the neutralatom bonding point, at the evaporation field F” in question. These expressions are derived from eqs. (43) and (52) in ref. [18]. Certain inessential mathematical approximations have been made in the derivation of eqs. (32) and (33), and hence these equations constitute a basic version of the formalism. Ref. [ 181in fact works in a context where the evaporation field is defined by a unity-rate-constant criterion, but the same expressions are relevant to the present context (where the evaporation field is defined by an evaporation-flux criterion), although obviously p,, c(~ and Q have slightly different meanings. By algebraic manipulation of eqs. (32) and (33) we may derive expressions for Q and for r”/a as follows:
Q= -id/h r”/a=
-P~/(~P~
+d
(34) +cL~).
Note that all the expressions derived in this section are independent charge state of the escaping ion.
(35)
of the
R.G. Forbes et al. / Field evaporation theory
510
6.2. Detailed cornpqrisons The experimental estimates of CL,and p2, listed in tables 2 and 3, can be used in various ways. In the original treatment in ref. 1181, the Arrhenius equation was used to make an a-priori estimate of Q as 0.2 eV, and this was then combined with the experimental estimate of EL,to make an estimate of a/r0 and hence a semi-theoretical estimate of p2/p,. This semi-theoretical estimate compared favourably with the experimental estimate derived from the data in table 3. An alternative approach is used here. Table6 shows values of Q and r”/a derived from the data in table2, using eqs. (34) and (39, and the question is whether the derived values are “physically reasonable”. An a-priori estimate of activation energy may be made by combining eq. (1) with the Arrhenius equation: k,
=Aexp(-Q/kT),
(36)
and taking: J= 0.01 layer/s, nhr =O.Ol layers (which implies k,, = 1 SC’); T = 77.3 K. This leads to the result: Q = 0.18 eV. If the field-evaporation pre-exponential A were “anomalously” low, which has been suggested for some materials in the past [30,31], then slightly lower values of Q would be expected. For example, taking A = lo8 s-i would lead to the result Q = 0.12 eV. The experimental estimates of Q are clearly compatible with these theoretical estimates, to within the limits of experimental error, and would still be compatible with the theoretical estimates if the limits of error were substantially reduced. Thus the activation-en~gy estimates are certainly “reasonable”. It remains to consider the experimental estimates of rO/a, which range from 0.2 to 1.0. The values for MO, W, and I-If are less th&n 0.5 and we feel these to be entirely reasonable. The remaining values, particularly the Pt and Ru values (both l.O), are higher than one would perhaps expect; however, the error limits on these particular estimates are very wide, and in consequence the stated
Table 6 Values of the activation energy Q(F”)
and the ratio r”/a,
using the data in table 2 Species
Q WI
P/a
M&l t0) Ru(1 lzi) Hf( loio) W(1 IO) 103) WC@)
0.13~0.06 0.2OkO.2 0.16*0.15 0.15~0.06 0.14*0.15 0.20*0.2
0.22-to.02 1.0 +I 0.460.1 0.2lAO.02 0.64kO.6 1.0 *1
as derived from
eq. (34) and eq. (35),
R.G. Forbes et a&. / Field emporntion
theory
511
values of r*/a could well be somewhat higher than the true values; so we do not regard these high estimated values as casting doubt on the formalism. In general, we conclude that a charge-exchange mechanism, in the form of a basic version of the parabolic-jug formalism, is apparently well able to provide a physically satisfactory explanation of partial energies derived from flux field-~siti~ty data. At the same time, it should be realized that - because this paper employs only the basic version of the formalism - the tabulated values of Q and r”/a are not necessarily the best estimates that could be derived from the data. 7. General discussion The conclusions of this paper need to be approached with care. Our immediate objective has been, not to decide the mechanism of metal field evaporation, nor to derive reliable atomic-level information from published experimental data, but the more modest one of testing the consistency of experimental data with the,existing published theoretical formalisms. What we have found is that the evaporation-flux field-sensitivity data are not compatible with existing image-hump formalisms, but are compatible with a charge-exchange formalism. Because the existing image-hump foamy are manifestly incomplete for example, past numerical analyses do not in&de any term relating to the repulsive ion-metal interaction that must physically be present - it is logically inappropriate to conclude from the present discussion that the field-sensitivity data are necessarily incompatible with the Miiller-Schottky mechanism. For example, the proposition might be advanced that the demonstrated inconsistencies could be removed by including a repulsive term, or by taking a more sophisticated approach to the calculation of the correlation interaction. Or it might be argued that the inconsistencies result from the neglect of field dependence in the pre-exponential But, intuitively, these propositions look unpromising, particularly when set against the demonstrated compatibility of the field-sensitivity data with a simple charge-exchange formalism. Charge-exchange thus looks much more likely than the Mtiller-Schottky mechanism. In view of often-expressed past doubts about the self-consistency of the simple image-hump formalisms, this conclusion is unsurprising. The new feature of the present discussion is that the ~~mpatib~ty of experimental data with these image-hump formalisms has been established by means of statistical tests. Appendix. Differentiation of the tern pseudo-liar
in F
There appears in the expression for field-evaporation activation energy Q, an “electrostatic” term (denoted here by u) that has the form: o = neFxP,
(37)
512
R.G. Forbes et ai. / Field ~up~rario~
theory
where xi’ is the distance of the “point of escape” from the metal’s electrical surface, and the other symbols have their usual meanings. In the context of a charge-exchange formalism, xi’ corresponds to the crossing-point of neutral and ionic potential energy curves. The expression for the total differential of Q with respect to F has (-do/dF) as one of its terms. Because xp is itself a function of F, the total differential do/d F is given by: (38) That is dx* FF = ptex* + neFdF * As stated in seetion4, the second term in this expression has often been neglected. In the context of a model in which the neutral atom moves in a parabolic potential-energy well, an expression for this second term, and hence for du/dF, can be derived as follows. (The treatment is based on that in ref. WI.) The field-evaporation activation energy can be written in the alternative forms: Q=fKr2
=
(A”+H,-~~E)-nel;xP+P(F)+C(~P),
w
where r is the distance of the point-of-escape (the crossing point) from the bottom of the potential well, K is the force constant for the well, P(F) is a (primarily) field-dependent correction term relating to the polarisation of the neutral and the ion, or equivalent effects, and C(x) is a (primarily) distancedependent function describing an interaction between the ion and the surface due to correlation and (in principle) repulsive forces. C(x) is usually approximated by the image-potenti~. The other symbols have their usual meanings, as listed in section 5.1. Taking differentials we obtain: ~rdr=nex*dF+
(dP/dF)
dF-neFdxP
+ (dC/dxP)
dxr.
(41)
The relationship of r and xi’ is: xp=a+r.
(42)
It is a good appro~ma~on to assume that a, the distance of the well bottom from the metal’s electrical surface, is independent of field. Hence dr = dxr and we obtain: dxP/dF=
- (nexp -dP/dF)/(neF+~-dC/dxP).
As a first approximation repulsion terms. Thus dxP/dF-
we may neglect the polarisation
-nexP/(neF+rcr).
(43) and correlation + WI
R.G. Forbes et al. /
Field evaporation theory
513
Substituting into eq. (39) then gives du/dF-nexP[l-neF/(neF+kr)]
=nexbcr/(neF+~r).
(45)
At this stage we can neglect or in comparison with neF, the basic result stated in section4 follows: du/d F = KrxP/F.
(W
Obviously, higher-order approximations can be obtained by including the omitted terms, but these do not dramatically affect the result, numerically.
Acknowledgement
This work forms part of a research project funded by the UK Science and Engineering Research Council. In addition, one of us (K.C.) wishes to thank the Ministry of Higher Education and Scientific Research of the Republic of Algeria for personal financial support.
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [IO] [II] [I21 [I31 [I41
E.W. Mtiller, J.A. Pa&z and S.B. McLane, Rev. Sci. Instr. 39 (1968) 83. J.A. Panitz, CRC Critical Rev. Solid State Sci. 5 (1975) 153. R. Clampitt, Nucl. Instr. Methods 189 (1981) I I I. L.W. Swanson, in: Microcircuit Engineering ‘80. Amsterdam, 1980. N. Ernst, Surface Sci. 87 (1979) 469. N. Ernst and G. Bozdech, in: Proc. 27th Intern. Field Emission Symp., Tokyo, 1980. R. Haydock and D.R. Kingham, Phys. Rev. Letters 44 (1980) 1520. D.R. Kingham, Surface Sci. 108 (1981) L460. T.T. Tsong, Surface Sci. 70 (1978) 211. E.W. Mttller and T.T. Tsong, Progr. Surface !I& 4 (1974)1. E.W. Mtlller, Phys. Rev. 102 (1956) 618. E.W. Muller, Advan. Electron. Electron Phys. 13 (1960) 83. R. Gomer and L.W. Swanson, J. Chem. Phys. 38 (1%3) 1613. D. McKinstry, Surface Sci. 39 (1972) 37. [IS] D. Brandon, Brit. J. Appl. Phys. 14 (1%3) 474. [Ia] E.W. Mttller and T.T. Tsong, Field Ion Microscopy: Principles and Applications (Elsevier, Amsterdam, 1969). [I71 T.T. Tsong, J. Chem. Phys. 54 (1971) 813. [ 181 R.G. Forbes, Surface Sci. 70 (1978) 239. [ 191 T.T. Tsong, J. Phys. F (Metal Phys.) 8 (1978) 1349. [ZO] R.G. Forbes, Surface Sci. 46 (1974) 577. [Zl] R.G. Forbes, in: Proc. 7th Intern Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, 1977, p. 387. [22] This assumption is reasonable at temperatures near 80 K, but would not be reasonable at temperatures at which metal atom diffusion could occur. [23] C. Patel, MSc Thesis, University of Aston in Birmingham (1974). [24] R.G. Forbes and C. Patel, 22nd Intern. Field Emission Symp., Atlanta, 1975 (unpublished).
514
R.G. Forbes et ul. / Field euaporution theory
[25] T.T. Tsong and G. Kellogg, Phys. Rev. 12B (1975) 1343.
[26] The field that appears in the conventional definition of polarisability is the self-consistent local field acting on.the atom. At a metal surface this field may be significantly different from the external field. [27] J.C. J&i&e, in: Solid State Science, Vol. 1, Ed. M. Green (Dekker, New York, 1969) p. 179. [28] Z. Knor, in: Surface and Defect Properties of Solids, Vol. 6 (1977)p. 139. [29] R.G. Forbes, Appl. Phys. Letters, to be published. 1301 D.G. Brandon, Phil. Msg. I4 (1966) 803. [31] In our view, Brandon’s deductions concerning the value of the field evaporation preexponential should now be treated with reservations.