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Ion current generation in the field ion microscope
Surface Science 52 (1975) 597-614 0 North-Holland Publishing Company
ION CURRENT GENERATION
IN THE FIELD ION MICROSCOPE
II. Quasi-static approach Hiroshi IWASAKI and Shogo NAKAMURA Institute of Scientific and Industrial Research, Osaka University, Suita, Osaka, Japan
Received 18 March 1975; manuscript received in final form 11 August 1975
The field ion current is calculated for helium on tungsten, for various fields, and for various values of tip temperature and gas temperature on the basis of the balance equation developed by Van Eekelen. The expression of ion current by rate constants for ionization and for escape, the total supply and the capture probability is derived. The behaviour of ion current as a function of other parameters is discussed in the light of these equilibrium properties of the system. Field adsorption effects are also considered. The increase of the mass ratio of the gas atom to the metal atom causes the increase of ion current even if ionization probability is decreased by the field adsorption. Anomalous features of the field ion image at 4.2 K are discussed.
1. Introduction Experimental measurements of field ion current versus other parameters of interest have been reported by a number of authors [l-S] . The increase of ion current by the field adsorption [S] of imaging-gas has also been reported by McLane et al. [7]. These experimental results provide data for the improvement of our understanding of the whole process of ion current generation, which becomes of greater significance for the interpretation of the image. There are two different ways for calculating the field ion current, which are called by Miiller and Tsong [8 3, the dynamic and the quasi-static approach. The former one has been discussed in the authors’ previous paper [9], henceforth referred to as I, in treating the purely dynamical calculation of ion current. The quasistatic approach developed in the paper of Van Eekelen [lo], henceforth referred to as VE, enabled us to calculate the velocity distribution function of gas particles and to explain many experimental features in finer details. We follow Van Eekelen [ 10 ] to compute the field ion current with some modefications and extensions as follows. (1) The expression of the velocity distribution function of the supplied particles derived in I is used. (2) The tip temperature and the gas temperature are taken independently. (3) Field adsorption effects are considered.
H. Iwasaki, S. Nakamura/Ion
598
current generation in FIM. II
In the present study, equilibrium properties such as rate constants for ionization and for escape are formulated as functionals of the distribution. The expression of the total ion current by these terms [11,12], which is familiar but not well founded, is reformulated on the basis of the balance equation derived in VE. The computed results are discussed in the light of these rate constants and the capture probability
P21. 2. The balance equation It will be assumed for simplicity that the emitter is spherical. The local field variations at the tip are not taken into account. Also we disregard supply of gas particles from the shank of the tip. It is shown in I that the velocities of the arriving particles at the tip surface from a field free region should not be expected to be purely radial as VE did. It is assumed that all newly incident particles on the tip surface have tangential velocity to the surface plane, ut, equal to the most probable tangential velocity, utm, given by eq. (8) in I. The velocity distribution of particles that have hit the surface with velocity (u ,,‘, ut’), where u,’ is radial velocity, is described by the collision matrix b (un, ut, u,‘, ut’) [9,10]. We base the collision matrix on the hard cube model [ 131 (see Appendix in I). As it conserves the tangential velocity of a particle, it may be justified to assume that the tangential velocities of all particles are equal to utm . Hereafter, u means u, in I and the collision matrix is written as b(u, u’). Particles which left the tip with radial velocity u smaller than u,, which is given by eq. (11) in I, go on trajectories returning to the surface. We obtain, for ut = utm, UC= up - (kTp/2m)“2
)
(1)
where up is a dipole attraction velocity (cu/m)“*Ft, k is the Boltzmann constant, Tg is the temperature of the ambient gas, Ft is the electric field at the tip surface, (Yis the polarizability of a gas atom and m its mass. Now, following VE, we call the numbers of gas particles that in unit time hit or leave a unit tip surface with radial velocity between u and u + du, in equilibrium, N’(u) du and N(u) du, respectively. Particles which hit the tip with u > ue come only from a field free region. Particles which hit the tip with u < ue, on the other hand, are composed of two components: particles attiving from a field free region and particles which have previously hit the tip and return from a round trip passing through the ionization zone twice. The number of particles which arrive at a unit tip surface in unit time from a field free region with radial velocity u between u and u + du, N,,(u), is given by eq. (7a) in I. Thus, in equilibrium we have N’(u) = N,(u) [l N’(u)=N,(u)[l-Q(u)]
@(41 +N(u)[l-Q(u)]~
for u > uc, foru
(2)
H. Iwasaki, S. Nakamurallon current generation in FLU. II
599
where Q(U) is the probability for a particle, which leaves the tip with velocity (LJ,utm), to be ionized in passing through the ionization zone once. Q(u) is given by eqs. (1 l), (12) and (13) in I. We have, by definition of a collision matrix,
N(u) = y N’(u’) b(u, u’) du’.
(3)
0
By substituting
(2) into (3), we obtain the balance equation
for N(u):
N(u) = jc N(d) [l - Q(u’)] 2 b(u,u’) du’ W,(u),
(4)
0
where
N,(u) = i
N,,(u’) [l - Q(u’)] b(u,u’) du’.
(5)
0
If we put Q = 0 in N,(u) and integrate over u using normalization
condition
of b, we
get f
N,(u) du = 7 N,(u) du = S,
0
(6)
0
where S is the supply function and the second equation of (6) is given by eq. (9) of I. Following VE, for free particles with u > u,, the ionization probability Q(u) is replaced by Q, = Q(u,). The derivation of the total ion current I as an example of the equilibrium quantities of the system has been shown in VE. There, the part of I due to the ionization of bound particles with u < u,, was shown to be
I’ = r
N(u) [2Q(u) - Q(u)~] du.
(7)
0
We have, from eq. (4) N(u) = j+ &‘(u’) b(u, d) du’ + N,(u),
(8)
0
where &l(u)
= N(u) [ 1 - Q(u)] 2.
(9)
600
H. Iwasaki, S. Naka~~raj~on current generation in FIM. II
Nt ‘(u) gives the contribution to N’(u) from “bound” particles, after they have passed the ionization zone twice. We call the probability of escape and capture of a particle, which hit the tip with radial velocity u’, Pe(u’) and P&u’), respectively. They are given by the following relation
b(u, u')du = 1 -Pt(u’).
P,(u’) = j
(10)
UC
If we integrate N(u) from 0 to u, using eqs. (9) and (lo), we get i” N(u) du = jc N,‘(u) f 1 -P,(u)] 0 0
du + i
N,(u) [l - Q(u)] &(u)du.
(11)
0
One finds, by transposition VC [N(u)
-
&‘(u)f
du + 2&A+ = ( 1 - CQ>,)PCS,
s
(12)
0 where UC A$’ = I- N,'(u) du,
034
0
(13c)
Pc = 7 4,(u)[I-Qt~)l~&)du/f(~-@2),,~1~ 0 Thebrackets indicate an average, k, gives the rate constant the capture probability for the supply. From (9) we have
(13d) for escape and P, gives
UC
s
[N(u) -N,‘(u)]
du =N, 4,’
0
"c = where
s 0
N(U)
[~Q(u)-Q(u)~] du=2kiNt,
(14)
vc ivt = s
0
N(u) du,
(lsa)
H. Iwasaki, S. N~kamura~Ion current generation
2ki= ~N(v)iZp(v)-Q(v)‘}
in FIM. II
601
05b)
dUlNt.
0
Ki gives the rate constant for ionization. We obtain, from [2ki + (1 - 2ki)2kJ
Nt=(l
-(Q),)PcSs
eqs.
(12) and (14) 06)
From eq. (7) ki(1 -CQ>,)P,S Z’ = 2kINt = ki+(I_2ki)k,
(17)
For ki -4 1, the number of bound particles that in unit time hit and leave unit tip surface, N, is given by N=Nt i-N,‘-2Nt.
(18)
Then, for ki g 1 I’ = kiN = kiP,Sf(ki + ke)t
(19)
ki = jc N(U) Q(U) dU/Nt,
(2Oa)
where
0
k, = $ i” N(u)P,.u)
du/Nt,
(20b)
0
PC = i
N,(u)P,(u)
du/S.
(2Oc)
0
Eq. (19) is the formula by Southon [ 121. It may be noticed that ki and k, are functions of N(u). The total ion current I is given by Z = I’ +
(21)
The second term and the third term give the contribution to f from incoming free particles and outgoing free particles, respectively.
3. The collision matrix
Let us define c(u, u‘) as the probability that the normal velocity of a gas particle is changed by the collision with the surface, from u’ to u. On the basis of the classical hard cube model, c(u, u’) is found from a one-dimensional, head-on collision of
602
H. Iwasaki,S. Nakamura/Ion current generation in FM II
n
H.
r
WTIP 5,o V/A
Fig. 1. The collision matrix elements, b(u, u’) for helium with a tungsten tip, at a field strength of S.OV/A and Tg = 80 K. The incident normal velocity v’ for each curve is shown by the arrow. The miue of vc is 0.380 in eV’” (see I for the unit of velocity).
HI!
wTIP ---- 2,s V/A 4.5 V/A
38140 39140
1
41/40 42/40 43140 44140
v / v, Fig. 2. Probabilities of escape, P,(u) for helium with a tungsten tip at Tg = 80 K, and at F = 2.5 V/A (dashed curves) and at F = 4.3 V/A (solid curved). The values of vc are 0.169 and 0.338eV1’2 for F = 2.5 and 4.5 V/A, respectively.
H. Iwasaki, S. Nakamurajlon current generation in FIM. II
603
a particle with the surface atoms that have a Maxwellian velocity distribution [ 131. So c(u, u’) is identical with that derived by VE in a one-dimensional model. We use the collision matrix b (u, u’) which is constructed by VE from c(u, u’) for higher values of u and from a Maxwell distribution at a partially accommodated temperature for lower values of u. The derived matrix b (u, u’) satisfies the condition of detailed balance. Some collision matrices are depicted in fig. 1. The dependence of the matrix on the tip temperature T, may be noticed. The probabilities of escape, P,(u), are shown in fig. 2. The escape probability, P,, for the bound particle becomes small as the tip temperature decreases or as the field strength increases. The situation is reversed for the particle which hit the surface with larger normal velocity. This may be understood as follows. Particles have some probability of colloding with the metal atoms moving in the same direction, losing the necessary amount of velocity to be trapped, when the surface is at a high temperature. As for P, , it changes only slightly when Ts is changed.
4. Results and discussion The balance equation (4) is the second kind Fredholm type integral equation. Following VE, it is replaced by matrix equation and solved by iteration. Some particle distributions for helium on tungsten are depicted in figs. 3,4 and 5. Fig. 3 shows the dependence of the particle distributions on the tip temperature at very low field of 2.5 V/A. The curves and those found from Maxwell distributions tit together except for 300 K. At 300 K, the population of high-energy particles is lower than that found from Maxwell distribution, because high-energy particles are easy to escape by thermal activation. Shown in fig. 6 are the ratios ofN, to the thermal equilibrium value of the supply function, So exp (Ep/kTs). So and Ep are given by (7~) and (2~) in I respectively. For low F, e.g. 2.0 V/A, and for T, > 80 K, k, is much greater than ki. In this case, as is seen from fig. 6, Nr is often nearly equal to So exp (Ep/kTs). Then, I = 2 kiN, N 2 kiS0 exp (Ep/kTs),
(22)
k, -P,Sexp(-Ep/kTs)/2So.
(23)
At 20 K and F = 2.0 V/A, k, is comparable to ki and the concentration of gas particles at the tip surface is much smaller than that in thermal equilibrium. The structure of particle distribution for low temperature discussed by VE is seen from fig. 4. It can be seen from fig. 5 that the low-energy peak virtually disappears at 80 K. This has been also indicated by VE. It can be also seen from fig. 5 that the particles are ionized before they are well accommodated to the tip temperature for a high field. It can be seen from figs. 5 and 6 that the gas concentration in the working-range of the field is far from the concentration in thermal equilibrium as Forbes [ 141 has assumed.
604
H. Iwasaki, S. NakamumjIon
current generation in FIM. II
//-----
He W TIP 2.5 V/A I'
0.25
0.5 V
/
1.0 V,
Fig. 3. Particle distribution for helium with a tungsten tip under isothermal conditions, at a field strength of 2.5 V/A, from the present calculation, solid curves and the Maxwell distribution, dashed curves. The latter curves are normalized so that their lower velocity parts fit together with those of the former curves. The peak values of the solid curves (the values of u,) are 1.3 X lo* (0.196), 5.3 X IO4 (0.169) and 2.4 X lo2 (0.131) inPI/l~(eV1’2) units for Tequal to 20 K, 80 K and 300 K. Where P is the gas pressure and V its volume.
0525
0.5 V
i
cl.75
I,0
V,
Fig. 4. Particle distribution,N(u) together with Ns(u) and Q(v), for helium with a tungsten tip at Ts= Tg= 20KandF=4.0V/A.ThepeakvalueofQ(v) is0.3;vc= 0.317eV’n.
H. Iwasaki, S. NakQmura~~o~current generation in FIM. Ii
He
W TIP
0.25
80 K
0,5 V
605
0.75
1.0
/ V,( FOR F=5,5V/A)
Fig. 5. Particle distribution for helium with a tungsten tip at Ts = Tg = 80 K. The thermal velocity, Uth (a) is shown by an arrow. The value of uc for F = 5.5 V/A is 0.423 eV”*. The peak valuesare 2.3 X 105,4.4X 104, 2.6 X lo4 and 1.1 X 104(PV/G) forFequa1 to 3.0,4.0, 4.4 and 5.5 V/A.
2.0
I
I
2.5
3.0
3.5
4.0
FIELD STRENGTH F ( V/A )
Fig. 6. The ratio Nt/S, exp (Ep/kTs), for helium with a tungsten tip, under isothermal conditions.
When T, rises, ke increases both by the increase of P, for the bound particles and by the shift of the peak of the particle distribution to the larger velocity shown in fig. 3. When the field is increased, in spite of the decrease of P, for the bound parti-
606
H. Iwasaki,S. Nakamurallon
current generation in FIM. II
cles (see fig. 2), k, increases by the shift of the peak of the particle distribution to the larger velocity (see fig. 5). The values of k,, ki and N, for various temperatures and fields are collected in table 2. 4.1. Current-voltage
characteristics
Logarithmic plots of the total ion current I versus the field strength F, for helium on tungsten, are given in fig. 7. The curves exhibit most of the features observed experimentally [l-5,15] in the way similar to VE. The values of the slope of an almost straight high-field region and of the cut-off field strength [ 121 are in good agreement with those of VE. The slopes of low-field region are 46,34 and 3 1 at T equal to 20,80 and 300 K respectively in isothermal conditions. This increase of the slope with decreasing tem-
2.0
2.5
3,o
3.5
4.0
5.0
6.0
FIELD STRENGTH F (V/A)
Fig. 7. Logarithmic plot of the total ion current per unit tip surface helium with a tungsten tip, under the same gas pressure.
versus the field strength
for
H. iwasaki, S. Nukarnura/Ioncurrentgeneration
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in FIM. II
tie W TIP -
I/S
---" PC
,//.MM
0.6
0.4
cl,2
0 !.O
3.0
4.0
5.0
6.0
FIELD STRENGTH F (V/A)
Fig, 8. The ratios Z/5’ (solid curves) and the capture probability Pc (dashed strength for helium with a tungsten tip.
curves)versusfield
perature is in good agreement with the experiment by Chen and Seidman 1151. They explained this temperature dependence of the slope of very low field region by the assumed expression of the ion current similar to eq. (22). Anyway, for low field where k, 3 ki, the ion current is expressed by I = kiP,S/kes
(24)
On the other hand, in the straight high-field region, the ion current must be expressed by eqs. (19) or (17) and (21). In fig. 8, the ratios of the ion current to the supply function, S, together with Pe, have been plotted. As has been shown in I, P, increases when field strength increases. As discussed by Tsong and Miiller [5], the number of atoms escaping from the tip region without ionization is indeed comparable to or larger than the ionized fraction under the usual experimental conditions. As VE has stressed, the straight highfield region is an ~~rrne~ate region where the current does not equal the supply. At 20 K, k, is much smaller than ki for F > 2.25 V/A. Then I=P,S.
(25)
608
H. Iwasaki, S. Nakamurallon current generation in FM. I1
20
100
200
300
TIP TEMPERATURE( K)
Fig. 9. Relative ion current versus tip temperature for helium with a tungsten tip with 20 K and 80 K gas temperature. Solid lines are calculated curves for various field strength. Dashed lines are experimental curves at F = 4.5 V/A by Tsong and Miiller (51.
The curves I/S and PC for T = 20 K in fig. 8 fit together for F from 2.5 V/A to 4.0 V/A. For higher field, the contribution from free particles becomes large. 4.2. Temperature
effects
Tsong and Miiller [5] have investigated the effects of the tip temperature on the ion current at a given field strength and gas temperature. Plots of the ion current versus the tip temperature, for helium on tungsten, are given in fig. 9. The calculated curves for F = 3.25 V/A seem to fit with the experimental curves. The tip temperature dependence of the ion current is explained by the behaviour of k, as a function of T, discussed in this section. For low field, when Ts rises, the ion current decreases rapidly by the rapid increase of k,. For a high field, on the other hand, when T, rises, the ion current decreases only slowly, in agreement with the experiment, by three reasons as follows. The increase of k, with Ts becomes small as the shift of the peak of the distribution with T, is less remarkable for the high field. Next, ki becomes larger than k, for the high field and the ion current is pro-
H. Iwasaki, S. Nakamura/h
current generation in FIM. II
609
H.
I
WTIP
I
40 \
---- EXP5,
\ 80”/300” \ \ \ \ \
0 30 5
20 -
lo-:
2,O
3.0
4.0
5.0
6.0
FIELD STRENGTH F (V/A)
Fig. 10. Ratio of ion current at two different temperatures under isothermal conditions versus field strength for helium with a tungsten tip. Solid lines are calculated curves and dashed lines are experimental curves by Tsong and Miiiler [ 51.
portional
to (ki + k,)-’ not to k;‘. Finally, the contribution to the ion current from free particle, which is insensitive to T,, becomes large for the high field. It is observed experimentally that the ion current with 78 K gas temperature decreases only slightly at Ts = 78 K. This effect is more remarkable for neon and hydrogen on tungsten at 3.8 and 4.5 V/A, respectively [S]. The calculations suggest that this may be explained as follows. For the temperature and field strength in discussion, k, is much smaller than ki and so (ki f k&-l, and hence I, decreases only slightly as ke increases. It can be seen from fig. 7 that the shifts of the cut-off fields towards higher fields are caused mainly by the increase of T,. This effect may be explained as follows. From the discussion of section 4.1, the cut-off may be considered as the field at which ki becomes comparable to k,. When Ts rises, k, increases and then the cutoff field, where ki - k,, shifts towards higher fields. The gas temperature has effects on I mainly through its effects on 5’ and Pc. The total suppIy to the tip, S, is proportional to Tg under the same gas pressure. Our calculation showed that PC increases the number by 2-4 times when Tg decreases from 300 to 20 K. This dependence of PC on Tg has been discussed in I.
H. Iwasaki, S. Nakamurallon
610
current generation in FIM. II
Ratios of ion current at two different temperatures under isothermal conditions are plotted as a function of field strength in fig. 10. The curves exhibit most of the features observed experimentally [5]. However, the critical field strength where the value of the ratio increases abruptly is smaller by 0.5 V/A for 1zo/180 and 1 .O V/A for I~o/I~oo. 4.3. The effects of field adsorption on ion current Tsong and Miiller [16] have shown that the probability Pa that at any instant time an inert-gas atom is field adsorbed [6] on the apex of the surface atom is
of
given by P, = [1 + (VCT,/Pp,,F)exp(-H/kT,)]
(26)
-l,
where H is the short-range binding energy and C is a constant which can be estimated from experimental conditions. For field adsorption of helium on tungsten, H is chosen as (f, - l)E, = 1.399 E,, where f, is an enhancement factor, and C as C(Ts/Pg,,F) = lop5 set at 20 K, 2 mTorr and 4.5 V/A [16]. For Pgas = 2 mTorr, the values of P, for a variety of temperatures and fields have been collected in table 1. The field adsorption may have effects on I by changing the ionization probability [ 18,191 and by changing the gas-surface interaction. So, we calculated the ion current for the two cases where the ionization life time, r [see eq. (14) in I] is assumed to be r1 = (1 - 0.9Pa) r and r2 = (1 + 9.OPa) r. The field ionization is enhanced and suppressed by a factor of ten by field adsorption in cases 1 and 2, respectively. It is assumed that the change of the gas-surface interaction by field adsorption is taken into account by taking the mass ratio of a gas particle to a metal atom, ~1, as a function of P,. Here p is assumed to be (1 + P,)p, for both cases. Then, the accommodation coefficient is increased by two times by field adsorption. The calculated results are shown in fig. 11.
Table 1 Probability Pa for helium with a tungsten tip for 2 mTorr F
Ts W
W/A)
20
80
2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.98 1.0 1.0 1.0 1.0 1.0 1.0
3.6 x 1.2 x 7.3 x 8.7 x 0.96 1.0 1.0
300 lo-’ 10-S 104 1O-2
1.4 4.1 1.4 5.8 2.5 1.6 1.3
x x x x x x x
10-g 10-g 10-8 lo-* 10-7 lo+’ 10-s
H. Iwasaki, S. Nakamura/Ion
He
2,o
W TIP
2,5
current generation in FM
I1
611
(T,,Tg)
3,o
4,O
5.0
6.0
FlELD STRENGTH F N/A)
Fig. 11. Logarithmic plot of the tota ion current versus the field strength for helium with a tungsten tip. Solid lines 1 and 2 are calculated curves for case (1) and (21, respectively. Dashed lines are the curves when no adsorption effect is considered. The total supplies are plotted also (dot-dash lines).
The curves exhibit many interesting features. (I) In the straight hip-field region, ion currents are increased equally for both cases. The values of k, (ki) are 2.2 X 10e5 (4.0 X 10P3), 4.7 X 10m6 (4.0 X 1.0em4), 2.2 X 10W7 (4.0 X 10P5), and 2.2 X 10v8 (4.0 X IOW6) for 7 equal to 7 X 10-l, ~,~X10and~X10~atF=3.0V/A,T~=20KandT~=80K.Theseshowthatk, strongly depends on ki and k, < ki for the field range in discussion. Then Z is expessed by P,Sand the shifts of Z towards higher values are solely caused by the increase of PC by field adsorption. It may be noticed that, in general, some part of the straight high-field region is independent on the magnitudes of kit though narrow for higher tip temperature (see the curve for T, = Tg = 80 K in fig. 1 I). The ion current, which is proportional to ki/(ki + ke) for the field region in discussion, is kept constant for the change of the ionization probability on account of the following change of k,, as discussed above. (2) The enhancement and the suppression of the ion current due to those of the field ionization adsorption are seen in both the extremely low field region where Z = kiPcS/ke and in the relatively high field region where the contribution to Z from free particles, ((z&S’, becomes large.
612
H. Iwasaki,S. Nakamurallon current generation in FM II
(3) In case (2), where the field ionization is suppressed by field adsorption, the curve for Ts = 20 K and Ts = 80 K intersects with the curve for no adsorption. This reveral of the values of ion current from field adsorbed surface and from noadsorbed surface, when the field is increased, suggests the mechanism to explain the unusual features of FIM image at 4.2 K [19]. At 4.2 K, the brightness of the image spots in the same plane changes as the field strength is varied. The metastable site atom A in fig. 12 is imaged brighter than the atoms B and C in the very high field region (F & 5.7 V/A). When the applied voltage is lowered, the image brightness of the atom A is dimini~ed and the images shown by the broken lines in fig. 12 become brighter [ 191. At 4.2 K, the sites B and C may be more adsorbed than the site A, because atoms are supplied to the former sites by the migration from the inner part of the (011) plane. The field-adsorbed or physisorbed gas atoms both increase the time spent by a gas atom in the ionization zone by improving the accommodation and suppress the ionization probability of the gas atom [17,18]. An analog to (3), the latter effect is expected to be more effective than the former effect at very high field and vice versa in the working-range of the field. Namely, at the very high field the ionization life time of a gas particle, T, is so short that even fast particles are almost certainly ionized in the first pass through the ionization zone and so, it does not matter for ion current generation whether the lost momentum of the particle is large or not. In this field region, the site A, where ionization is lesser suppressed, is brighter than the sites B and C. On the other hand, in the workingrange of the field where the ionization life time is long, the change of the staying time of particles in the ionization zone by field adsorption is more effective than the change of the ionization probability itself. So, in this field region, the sites B and C, where the time duration of the gas atom in the ionization zone is longer, are brighter than the site A. I 100 ZONE
(011) PLANE
Fig. 12. Schematic diagram of the ledge of the (Oil) plane of tungsten.
H. Iwasaki,S. Nakamurallon current generation in FIM. II
613
Table 2 Values of k,, ki, and Nt for helium with a tungsten tip a
F
Ts (K)
V/A -
20
80
300
2.0
20 80 300
16-6 17-4 16 -2
24-7 60-7 40-7
.58+1 40+6 48 +3
55 -2 10-l 37 -1
29-l 49-7 41-7
82+4 68+3 21 +2
52 -1 67 -1 14
26-7 40-7 39 -7
82+3 11 +3 60 +4
2.5
20 80 300
48-6 26-6 11 -4
19-4 10-4 19 -4
64+1 19+7 63 +S
88-3 21-2 87-2
18 -4 12-4 17-4
70+5 50 +4 1s +3
29 -1 37-l 62-l
12-4 10-4 15-4
20+4 30+3 23+2
3,o
20 80 300
72-S 47 -5
84-s
5s -3 40 -3 51 -3
29 +6 68 +5 78 i-4
18-3 34 -3 18-2
72-3 60 -3 65 -3
15 +6 22+5 98 +3
15 -1 20-l 33--l
51 -3 48-3 55 -3
47 +4 72+3 65 +2
3.5
20 80 300
29-4 42-4 49-4
43-2 68-2 65-l
49 +5 52+4 86+3
68 -3 19 -3 10 -2
80 -2 96 -2 93 -2
22 +S 33 +4 51 +3
94 -2 12 -1 19-1
66 -2 7s --2 78-2
72 +4 12+4 13+3
4.0
20 80 300
1s -3 11-3 20-3
44-l 35-l SO-l
59 +4 13 +4 15+3
25 -2 22 -2 28-2
45 -1 43 -1 53-l
49 +4 94 +3 13 +3
14-1 14 -1 18-l
43-l 41 -1 so-1
31+4 65 +3 90+2
4.5
20 80 300
20-3 21 -3 31 -3
12 13 15
25 +4 40 +3 58 +2
43 -2 44-2 48-2
12 13 15
22 +4 38 +3 5s +2
22-l 23-l 25 -1
12 13 16
15 +4 30+3 4s +2
5.0
20 80 300
23 -3 26 -3 38 -3
29 31 35
11+4 18 +3 26 +2
68 -2 69 -2 13 -2
29 31 35
98 +4 17 +3 25 +2
32 -1 29 -1 32 -1
30 32 36
76 +3 15 +3 24 +2
s The values of k,, ki and Nt are arranged in order. For example, 76 -6 means 0.76 X lo-‘. is in the units of PV/G
Nt
5. Conclusions The rate constants for ionization ki and for escape k, are formulated as functionals of the distribution function, N(v). The formula of the total ion current, I, which is expressed by the rate constants, the total supply, S, and the capture probability, PC, is derived. The formula coincides with that given by Southon [ 121 for ki -4 1. The behaviour of the ion currents is analyzed by using the equilibrium properties of the system. In the very low field region, I is equal to kiP,S/ke and in straight high-iield region, I is equal to kiP,S~(ki + k,). In the latter region, there exists a part where I is independent of the values of ki.
614
H. Iwasaki, S. Nakamurallon
current generation
in FM.
II
The effects of the tip temperature, T, are discriminated from those of the gas temperature, Tg. The dependence of I on T, is qualitatively explained by the T, dependence of k,. The gas temperature has also been shown to have considerable effects on the ion current, in agreement with the results of I. It is shown that the ion current is indeed increased by the field adsorption of inert gas atoms even if the field ionization probability were suppressed [ 181. Moreover, the proposed mechanisms that the ionization probability is decreased and the staying time of a particle in the ionization zone is increased by the field adsorption of an inert gas atom, enabled us to explain the experimentally observed anomalous features of field ion images at 4.2 K satisfactorily.
Acknowledgements We wish to thank Mr. T. Adachi for discussions about the experimental 4.2 K.
results at
References [l] [2] [3] [4] [5] [6] [7] [8]
E.W. Miiller and K. Bahadur, Phys. Rev. 102 (1956) 624. K. Bahadur, J. Sci. Ind. Res. (India) 19b (1960) 177. M.J. Southon and D.G. Brandon, Phil. Mag. 8 (1963) 579. U. Feldman and R. Comer, J. Appl. Phys. 37 (1966) 2380. T.T. Tsong and E.W. Miiller, J. Appl. Phys. 37 (1966) 3065. E.W. Miiller, Quart. Rev. (London) 23 (1967) 177; Naturwissenschaften 57 (1970) 222. S.B. McLane, E.W. Miiller and S.V. Krishnaswamy, Surface Sci. 27 (1971) 367. E.W. Mtiller and T.T. Tsong, Field Ion Microscopy, Field Ionization and Field Evaporation, in: Progress in Surface Science, Ed. S. Davison (1973) Vol. 4, Part 1. [9] H. Iwasaki and S. Nakamura, Surface Sci. 52 (1975) 588. [lo] H.A.M. Van Eekelen, Surface Sci. 21 (1970) 21. [ 1 l] R. Gomer, Field Emission and Field Ionization (Harvard Univ. Press, 1961). [ 121 M.J. Southon, Ph.D. Thesis, University of Cambridge (1963) (unpublished). (131 R.M. Logan and R.E. Stickney, J. Chem. Phys. 44 (1966) 195. [14] R.G. Forbes, Vacuum 22 (1972) 517. [15] Y.C. Chen and D.N. Seidman, Surface Sci. 27 (1971) 231. [ 161 T.T. Tsong and E.W. Miller, J. Chem. Phys. 55 (1971) 2284. [17] D.A. Nolan and R.M. Herman, Phys. Rev. B8 (1973) 4099; BlO (1974) 50. 1181 H. Iwasaki and S. Nakamura, Japan. J. Appl. Phys. Suppl. 2, Part 2 (1974) 43; Surface Sci. 49 (1975) 664. [19] T. Adachi and K. Ohnishi and S. Nakamura, Japan. J. Appl. Phys. 13 (1974) 549.
ION CURRENT GENERATION
IN THE FIELD ION MICROSCOPE
II. Quasi-static approach Hiroshi IWASAKI and Shogo NAKAMURA Institute of Scientific and Industrial Research, Osaka University, Suita, Osaka, Japan
Received 18 March 1975; manuscript received in final form 11 August 1975
The field ion current is calculated for helium on tungsten, for various fields, and for various values of tip temperature and gas temperature on the basis of the balance equation developed by Van Eekelen. The expression of ion current by rate constants for ionization and for escape, the total supply and the capture probability is derived. The behaviour of ion current as a function of other parameters is discussed in the light of these equilibrium properties of the system. Field adsorption effects are also considered. The increase of the mass ratio of the gas atom to the metal atom causes the increase of ion current even if ionization probability is decreased by the field adsorption. Anomalous features of the field ion image at 4.2 K are discussed.
1. Introduction Experimental measurements of field ion current versus other parameters of interest have been reported by a number of authors [l-S] . The increase of ion current by the field adsorption [S] of imaging-gas has also been reported by McLane et al. [7]. These experimental results provide data for the improvement of our understanding of the whole process of ion current generation, which becomes of greater significance for the interpretation of the image. There are two different ways for calculating the field ion current, which are called by Miiller and Tsong [8 3, the dynamic and the quasi-static approach. The former one has been discussed in the authors’ previous paper [9], henceforth referred to as I, in treating the purely dynamical calculation of ion current. The quasistatic approach developed in the paper of Van Eekelen [lo], henceforth referred to as VE, enabled us to calculate the velocity distribution function of gas particles and to explain many experimental features in finer details. We follow Van Eekelen [ 10 ] to compute the field ion current with some modefications and extensions as follows. (1) The expression of the velocity distribution function of the supplied particles derived in I is used. (2) The tip temperature and the gas temperature are taken independently. (3) Field adsorption effects are considered.
H. Iwasaki, S. Nakamura/Ion
598
current generation in FIM. II
In the present study, equilibrium properties such as rate constants for ionization and for escape are formulated as functionals of the distribution. The expression of the total ion current by these terms [11,12], which is familiar but not well founded, is reformulated on the basis of the balance equation derived in VE. The computed results are discussed in the light of these rate constants and the capture probability
P21. 2. The balance equation It will be assumed for simplicity that the emitter is spherical. The local field variations at the tip are not taken into account. Also we disregard supply of gas particles from the shank of the tip. It is shown in I that the velocities of the arriving particles at the tip surface from a field free region should not be expected to be purely radial as VE did. It is assumed that all newly incident particles on the tip surface have tangential velocity to the surface plane, ut, equal to the most probable tangential velocity, utm, given by eq. (8) in I. The velocity distribution of particles that have hit the surface with velocity (u ,,‘, ut’), where u,’ is radial velocity, is described by the collision matrix b (un, ut, u,‘, ut’) [9,10]. We base the collision matrix on the hard cube model [ 131 (see Appendix in I). As it conserves the tangential velocity of a particle, it may be justified to assume that the tangential velocities of all particles are equal to utm . Hereafter, u means u, in I and the collision matrix is written as b(u, u’). Particles which left the tip with radial velocity u smaller than u,, which is given by eq. (11) in I, go on trajectories returning to the surface. We obtain, for ut = utm, UC= up - (kTp/2m)“2
)
(1)
where up is a dipole attraction velocity (cu/m)“*Ft, k is the Boltzmann constant, Tg is the temperature of the ambient gas, Ft is the electric field at the tip surface, (Yis the polarizability of a gas atom and m its mass. Now, following VE, we call the numbers of gas particles that in unit time hit or leave a unit tip surface with radial velocity between u and u + du, in equilibrium, N’(u) du and N(u) du, respectively. Particles which hit the tip with u > ue come only from a field free region. Particles which hit the tip with u < ue, on the other hand, are composed of two components: particles attiving from a field free region and particles which have previously hit the tip and return from a round trip passing through the ionization zone twice. The number of particles which arrive at a unit tip surface in unit time from a field free region with radial velocity u between u and u + du, N,,(u), is given by eq. (7a) in I. Thus, in equilibrium we have N’(u) = N,(u) [l N’(u)=N,(u)[l-Q(u)]
@(41 +N(u)[l-Q(u)]~
for u > uc, foru
(2)
H. Iwasaki, S. Nakamurallon current generation in FLU. II
599
where Q(U) is the probability for a particle, which leaves the tip with velocity (LJ,utm), to be ionized in passing through the ionization zone once. Q(u) is given by eqs. (1 l), (12) and (13) in I. We have, by definition of a collision matrix,
N(u) = y N’(u’) b(u, u’) du’.
(3)
0
By substituting
(2) into (3), we obtain the balance equation
for N(u):
N(u) = jc N(d) [l - Q(u’)] 2 b(u,u’) du’ W,(u),
(4)
0
where
N,(u) = i
N,,(u’) [l - Q(u’)] b(u,u’) du’.
(5)
0
If we put Q = 0 in N,(u) and integrate over u using normalization
condition
of b, we
get f
N,(u) du = 7 N,(u) du = S,
0
(6)
0
where S is the supply function and the second equation of (6) is given by eq. (9) of I. Following VE, for free particles with u > u,, the ionization probability Q(u) is replaced by Q, = Q(u,). The derivation of the total ion current I as an example of the equilibrium quantities of the system has been shown in VE. There, the part of I due to the ionization of bound particles with u < u,, was shown to be
I’ = r
N(u) [2Q(u) - Q(u)~] du.
(7)
0
We have, from eq. (4) N(u) = j+ &‘(u’) b(u, d) du’ + N,(u),
(8)
0
where &l(u)
= N(u) [ 1 - Q(u)] 2.
(9)
600
H. Iwasaki, S. Naka~~raj~on current generation in FIM. II
Nt ‘(u) gives the contribution to N’(u) from “bound” particles, after they have passed the ionization zone twice. We call the probability of escape and capture of a particle, which hit the tip with radial velocity u’, Pe(u’) and P&u’), respectively. They are given by the following relation
b(u, u')du = 1 -Pt(u’).
P,(u’) = j
(10)
UC
If we integrate N(u) from 0 to u, using eqs. (9) and (lo), we get i” N(u) du = jc N,‘(u) f 1 -P,(u)] 0 0
du + i
N,(u) [l - Q(u)] &(u)du.
(11)
0
One finds, by transposition VC [N(u)
-
&‘(u)f
du + 2&A+ = ( 1 - CQ>,)PCS,
s
(12)
0 where UC A$’ = I- N,'(u) du,
034
0
(13c)
Pc = 7 4,(u)[I-Qt~)l~&)du/f(~-@2),,~1~ 0 Thebrackets indicate an average, k, gives the rate constant the capture probability for the supply. From (9) we have
(13d) for escape and P, gives
UC
s
[N(u) -N,‘(u)]
du =N, 4,’
0
"c = where
s 0
N(U)
[~Q(u)-Q(u)~] du=2kiNt,
(14)
vc ivt = s
0
N(u) du,
(lsa)
H. Iwasaki, S. N~kamura~Ion current generation
2ki= ~N(v)iZp(v)-Q(v)‘}
in FIM. II
601
05b)
dUlNt.
0
Ki gives the rate constant for ionization. We obtain, from [2ki + (1 - 2ki)2kJ
Nt=(l
-(Q),)PcSs
eqs.
(12) and (14) 06)
From eq. (7) ki(1 -CQ>,)P,S Z’ = 2kINt = ki+(I_2ki)k,
(17)
For ki -4 1, the number of bound particles that in unit time hit and leave unit tip surface, N, is given by N=Nt i-N,‘-2Nt.
(18)
Then, for ki g 1 I’ = kiN = kiP,Sf(ki + ke)t
(19)
ki = jc N(U) Q(U) dU/Nt,
(2Oa)
where
0
k, = $ i” N(u)P,.u)
du/Nt,
(20b)
0
PC = i
N,(u)P,(u)
du/S.
(2Oc)
0
Eq. (19) is the formula by Southon [ 121. It may be noticed that ki and k, are functions of N(u). The total ion current I is given by Z = I’ +
(21)
The second term and the third term give the contribution to f from incoming free particles and outgoing free particles, respectively.
3. The collision matrix
Let us define c(u, u‘) as the probability that the normal velocity of a gas particle is changed by the collision with the surface, from u’ to u. On the basis of the classical hard cube model, c(u, u’) is found from a one-dimensional, head-on collision of
602
H. Iwasaki,S. Nakamura/Ion current generation in FM II
n
H.
r
WTIP 5,o V/A
Fig. 1. The collision matrix elements, b(u, u’) for helium with a tungsten tip, at a field strength of S.OV/A and Tg = 80 K. The incident normal velocity v’ for each curve is shown by the arrow. The miue of vc is 0.380 in eV’” (see I for the unit of velocity).
HI!
wTIP ---- 2,s V/A 4.5 V/A
38140 39140
1
41/40 42/40 43140 44140
v / v, Fig. 2. Probabilities of escape, P,(u) for helium with a tungsten tip at Tg = 80 K, and at F = 2.5 V/A (dashed curves) and at F = 4.3 V/A (solid curved). The values of vc are 0.169 and 0.338eV1’2 for F = 2.5 and 4.5 V/A, respectively.
H. Iwasaki, S. Nakamurajlon current generation in FIM. II
603
a particle with the surface atoms that have a Maxwellian velocity distribution [ 131. So c(u, u’) is identical with that derived by VE in a one-dimensional model. We use the collision matrix b (u, u’) which is constructed by VE from c(u, u’) for higher values of u and from a Maxwell distribution at a partially accommodated temperature for lower values of u. The derived matrix b (u, u’) satisfies the condition of detailed balance. Some collision matrices are depicted in fig. 1. The dependence of the matrix on the tip temperature T, may be noticed. The probabilities of escape, P,(u), are shown in fig. 2. The escape probability, P,, for the bound particle becomes small as the tip temperature decreases or as the field strength increases. The situation is reversed for the particle which hit the surface with larger normal velocity. This may be understood as follows. Particles have some probability of colloding with the metal atoms moving in the same direction, losing the necessary amount of velocity to be trapped, when the surface is at a high temperature. As for P, , it changes only slightly when Ts is changed.
4. Results and discussion The balance equation (4) is the second kind Fredholm type integral equation. Following VE, it is replaced by matrix equation and solved by iteration. Some particle distributions for helium on tungsten are depicted in figs. 3,4 and 5. Fig. 3 shows the dependence of the particle distributions on the tip temperature at very low field of 2.5 V/A. The curves and those found from Maxwell distributions tit together except for 300 K. At 300 K, the population of high-energy particles is lower than that found from Maxwell distribution, because high-energy particles are easy to escape by thermal activation. Shown in fig. 6 are the ratios ofN, to the thermal equilibrium value of the supply function, So exp (Ep/kTs). So and Ep are given by (7~) and (2~) in I respectively. For low F, e.g. 2.0 V/A, and for T, > 80 K, k, is much greater than ki. In this case, as is seen from fig. 6, Nr is often nearly equal to So exp (Ep/kTs). Then, I = 2 kiN, N 2 kiS0 exp (Ep/kTs),
(22)
k, -P,Sexp(-Ep/kTs)/2So.
(23)
At 20 K and F = 2.0 V/A, k, is comparable to ki and the concentration of gas particles at the tip surface is much smaller than that in thermal equilibrium. The structure of particle distribution for low temperature discussed by VE is seen from fig. 4. It can be seen from fig. 5 that the low-energy peak virtually disappears at 80 K. This has been also indicated by VE. It can be also seen from fig. 5 that the particles are ionized before they are well accommodated to the tip temperature for a high field. It can be seen from figs. 5 and 6 that the gas concentration in the working-range of the field is far from the concentration in thermal equilibrium as Forbes [ 141 has assumed.
604
H. Iwasaki, S. NakamumjIon
current generation in FIM. II
//-----
He W TIP 2.5 V/A I'
0.25
0.5 V
/
1.0 V,
Fig. 3. Particle distribution for helium with a tungsten tip under isothermal conditions, at a field strength of 2.5 V/A, from the present calculation, solid curves and the Maxwell distribution, dashed curves. The latter curves are normalized so that their lower velocity parts fit together with those of the former curves. The peak values of the solid curves (the values of u,) are 1.3 X lo* (0.196), 5.3 X IO4 (0.169) and 2.4 X lo2 (0.131) inPI/l~(eV1’2) units for Tequal to 20 K, 80 K and 300 K. Where P is the gas pressure and V its volume.
0525
0.5 V
i
cl.75
I,0
V,
Fig. 4. Particle distribution,N(u) together with Ns(u) and Q(v), for helium with a tungsten tip at Ts= Tg= 20KandF=4.0V/A.ThepeakvalueofQ(v) is0.3;vc= 0.317eV’n.
H. Iwasaki, S. NakQmura~~o~current generation in FIM. Ii
He
W TIP
0.25
80 K
0,5 V
605
0.75
1.0
/ V,( FOR F=5,5V/A)
Fig. 5. Particle distribution for helium with a tungsten tip at Ts = Tg = 80 K. The thermal velocity, Uth (a) is shown by an arrow. The value of uc for F = 5.5 V/A is 0.423 eV”*. The peak valuesare 2.3 X 105,4.4X 104, 2.6 X lo4 and 1.1 X 104(PV/G) forFequa1 to 3.0,4.0, 4.4 and 5.5 V/A.
2.0
I
I
2.5
3.0
3.5
4.0
FIELD STRENGTH F ( V/A )
Fig. 6. The ratio Nt/S, exp (Ep/kTs), for helium with a tungsten tip, under isothermal conditions.
When T, rises, ke increases both by the increase of P, for the bound particles and by the shift of the peak of the particle distribution to the larger velocity shown in fig. 3. When the field is increased, in spite of the decrease of P, for the bound parti-
606
H. Iwasaki,S. Nakamurallon
current generation in FIM. II
cles (see fig. 2), k, increases by the shift of the peak of the particle distribution to the larger velocity (see fig. 5). The values of k,, ki and N, for various temperatures and fields are collected in table 2. 4.1. Current-voltage
characteristics
Logarithmic plots of the total ion current I versus the field strength F, for helium on tungsten, are given in fig. 7. The curves exhibit most of the features observed experimentally [l-5,15] in the way similar to VE. The values of the slope of an almost straight high-field region and of the cut-off field strength [ 121 are in good agreement with those of VE. The slopes of low-field region are 46,34 and 3 1 at T equal to 20,80 and 300 K respectively in isothermal conditions. This increase of the slope with decreasing tem-
2.0
2.5
3,o
3.5
4.0
5.0
6.0
FIELD STRENGTH F (V/A)
Fig. 7. Logarithmic plot of the total ion current per unit tip surface helium with a tungsten tip, under the same gas pressure.
versus the field strength
for
H. iwasaki, S. Nukarnura/Ioncurrentgeneration
607
in FIM. II
tie W TIP -
I/S
---" PC
,//.MM
0.6
0.4
cl,2
0 !.O
3.0
4.0
5.0
6.0
FIELD STRENGTH F (V/A)
Fig, 8. The ratios Z/5’ (solid curves) and the capture probability Pc (dashed strength for helium with a tungsten tip.
curves)versusfield
perature is in good agreement with the experiment by Chen and Seidman 1151. They explained this temperature dependence of the slope of very low field region by the assumed expression of the ion current similar to eq. (22). Anyway, for low field where k, 3 ki, the ion current is expressed by I = kiP,S/kes
(24)
On the other hand, in the straight high-field region, the ion current must be expressed by eqs. (19) or (17) and (21). In fig. 8, the ratios of the ion current to the supply function, S, together with Pe, have been plotted. As has been shown in I, P, increases when field strength increases. As discussed by Tsong and Miiller [5], the number of atoms escaping from the tip region without ionization is indeed comparable to or larger than the ionized fraction under the usual experimental conditions. As VE has stressed, the straight highfield region is an ~~rrne~ate region where the current does not equal the supply. At 20 K, k, is much smaller than ki for F > 2.25 V/A. Then I=P,S.
(25)
608
H. Iwasaki, S. Nakamurallon current generation in FM. I1
20
100
200
300
TIP TEMPERATURE( K)
Fig. 9. Relative ion current versus tip temperature for helium with a tungsten tip with 20 K and 80 K gas temperature. Solid lines are calculated curves for various field strength. Dashed lines are experimental curves at F = 4.5 V/A by Tsong and Miiller (51.
The curves I/S and PC for T = 20 K in fig. 8 fit together for F from 2.5 V/A to 4.0 V/A. For higher field, the contribution from free particles becomes large. 4.2. Temperature
effects
Tsong and Miiller [5] have investigated the effects of the tip temperature on the ion current at a given field strength and gas temperature. Plots of the ion current versus the tip temperature, for helium on tungsten, are given in fig. 9. The calculated curves for F = 3.25 V/A seem to fit with the experimental curves. The tip temperature dependence of the ion current is explained by the behaviour of k, as a function of T, discussed in this section. For low field, when Ts rises, the ion current decreases rapidly by the rapid increase of k,. For a high field, on the other hand, when T, rises, the ion current decreases only slowly, in agreement with the experiment, by three reasons as follows. The increase of k, with Ts becomes small as the shift of the peak of the distribution with T, is less remarkable for the high field. Next, ki becomes larger than k, for the high field and the ion current is pro-
H. Iwasaki, S. Nakamura/h
current generation in FIM. II
609
H.
I
WTIP
I
40 \
---- EXP5,
\ 80”/300” \ \ \ \ \
0 30 5
20 -
lo-:
2,O
3.0
4.0
5.0
6.0
FIELD STRENGTH F (V/A)
Fig. 10. Ratio of ion current at two different temperatures under isothermal conditions versus field strength for helium with a tungsten tip. Solid lines are calculated curves and dashed lines are experimental curves by Tsong and Miiiler [ 51.
portional
to (ki + k,)-’ not to k;‘. Finally, the contribution to the ion current from free particle, which is insensitive to T,, becomes large for the high field. It is observed experimentally that the ion current with 78 K gas temperature decreases only slightly at Ts = 78 K. This effect is more remarkable for neon and hydrogen on tungsten at 3.8 and 4.5 V/A, respectively [S]. The calculations suggest that this may be explained as follows. For the temperature and field strength in discussion, k, is much smaller than ki and so (ki f k&-l, and hence I, decreases only slightly as ke increases. It can be seen from fig. 7 that the shifts of the cut-off fields towards higher fields are caused mainly by the increase of T,. This effect may be explained as follows. From the discussion of section 4.1, the cut-off may be considered as the field at which ki becomes comparable to k,. When Ts rises, k, increases and then the cutoff field, where ki - k,, shifts towards higher fields. The gas temperature has effects on I mainly through its effects on 5’ and Pc. The total suppIy to the tip, S, is proportional to Tg under the same gas pressure. Our calculation showed that PC increases the number by 2-4 times when Tg decreases from 300 to 20 K. This dependence of PC on Tg has been discussed in I.
H. Iwasaki, S. Nakamurallon
610
current generation in FIM. II
Ratios of ion current at two different temperatures under isothermal conditions are plotted as a function of field strength in fig. 10. The curves exhibit most of the features observed experimentally [5]. However, the critical field strength where the value of the ratio increases abruptly is smaller by 0.5 V/A for 1zo/180 and 1 .O V/A for I~o/I~oo. 4.3. The effects of field adsorption on ion current Tsong and Miiller [16] have shown that the probability Pa that at any instant time an inert-gas atom is field adsorbed [6] on the apex of the surface atom is
of
given by P, = [1 + (VCT,/Pp,,F)exp(-H/kT,)]
(26)
-l,
where H is the short-range binding energy and C is a constant which can be estimated from experimental conditions. For field adsorption of helium on tungsten, H is chosen as (f, - l)E, = 1.399 E,, where f, is an enhancement factor, and C as C(Ts/Pg,,F) = lop5 set at 20 K, 2 mTorr and 4.5 V/A [16]. For Pgas = 2 mTorr, the values of P, for a variety of temperatures and fields have been collected in table 1. The field adsorption may have effects on I by changing the ionization probability [ 18,191 and by changing the gas-surface interaction. So, we calculated the ion current for the two cases where the ionization life time, r [see eq. (14) in I] is assumed to be r1 = (1 - 0.9Pa) r and r2 = (1 + 9.OPa) r. The field ionization is enhanced and suppressed by a factor of ten by field adsorption in cases 1 and 2, respectively. It is assumed that the change of the gas-surface interaction by field adsorption is taken into account by taking the mass ratio of a gas particle to a metal atom, ~1, as a function of P,. Here p is assumed to be (1 + P,)p, for both cases. Then, the accommodation coefficient is increased by two times by field adsorption. The calculated results are shown in fig. 11.
Table 1 Probability Pa for helium with a tungsten tip for 2 mTorr F
Ts W
W/A)
20
80
2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.98 1.0 1.0 1.0 1.0 1.0 1.0
3.6 x 1.2 x 7.3 x 8.7 x 0.96 1.0 1.0
300 lo-’ 10-S 104 1O-2
1.4 4.1 1.4 5.8 2.5 1.6 1.3
x x x x x x x
10-g 10-g 10-8 lo-* 10-7 lo+’ 10-s
H. Iwasaki, S. Nakamura/Ion
He
2,o
W TIP
2,5
current generation in FM
I1
611
(T,,Tg)
3,o
4,O
5.0
6.0
FlELD STRENGTH F N/A)
Fig. 11. Logarithmic plot of the tota ion current versus the field strength for helium with a tungsten tip. Solid lines 1 and 2 are calculated curves for case (1) and (21, respectively. Dashed lines are the curves when no adsorption effect is considered. The total supplies are plotted also (dot-dash lines).
The curves exhibit many interesting features. (I) In the straight hip-field region, ion currents are increased equally for both cases. The values of k, (ki) are 2.2 X 10e5 (4.0 X 10P3), 4.7 X 10m6 (4.0 X 1.0em4), 2.2 X 10W7 (4.0 X 10P5), and 2.2 X 10v8 (4.0 X IOW6) for 7 equal to 7 X 10-l, ~,~X10and~X10~atF=3.0V/A,T~=20KandT~=80K.Theseshowthatk, strongly depends on ki and k, < ki for the field range in discussion. Then Z is expessed by P,Sand the shifts of Z towards higher values are solely caused by the increase of PC by field adsorption. It may be noticed that, in general, some part of the straight high-field region is independent on the magnitudes of kit though narrow for higher tip temperature (see the curve for T, = Tg = 80 K in fig. 1 I). The ion current, which is proportional to ki/(ki + ke) for the field region in discussion, is kept constant for the change of the ionization probability on account of the following change of k,, as discussed above. (2) The enhancement and the suppression of the ion current due to those of the field ionization adsorption are seen in both the extremely low field region where Z = kiPcS/ke and in the relatively high field region where the contribution to Z from free particles, ((z&S’, becomes large.
612
H. Iwasaki,S. Nakamurallon current generation in FM II
(3) In case (2), where the field ionization is suppressed by field adsorption, the curve for Ts = 20 K and Ts = 80 K intersects with the curve for no adsorption. This reveral of the values of ion current from field adsorbed surface and from noadsorbed surface, when the field is increased, suggests the mechanism to explain the unusual features of FIM image at 4.2 K [19]. At 4.2 K, the brightness of the image spots in the same plane changes as the field strength is varied. The metastable site atom A in fig. 12 is imaged brighter than the atoms B and C in the very high field region (F & 5.7 V/A). When the applied voltage is lowered, the image brightness of the atom A is dimini~ed and the images shown by the broken lines in fig. 12 become brighter [ 191. At 4.2 K, the sites B and C may be more adsorbed than the site A, because atoms are supplied to the former sites by the migration from the inner part of the (011) plane. The field-adsorbed or physisorbed gas atoms both increase the time spent by a gas atom in the ionization zone by improving the accommodation and suppress the ionization probability of the gas atom [17,18]. An analog to (3), the latter effect is expected to be more effective than the former effect at very high field and vice versa in the working-range of the field. Namely, at the very high field the ionization life time of a gas particle, T, is so short that even fast particles are almost certainly ionized in the first pass through the ionization zone and so, it does not matter for ion current generation whether the lost momentum of the particle is large or not. In this field region, the site A, where ionization is lesser suppressed, is brighter than the sites B and C. On the other hand, in the workingrange of the field where the ionization life time is long, the change of the staying time of particles in the ionization zone by field adsorption is more effective than the change of the ionization probability itself. So, in this field region, the sites B and C, where the time duration of the gas atom in the ionization zone is longer, are brighter than the site A. I 100 ZONE
(011) PLANE
Fig. 12. Schematic diagram of the ledge of the (Oil) plane of tungsten.
H. Iwasaki,S. Nakamurallon current generation in FIM. II
613
Table 2 Values of k,, ki, and Nt for helium with a tungsten tip a
F
Ts (K)
V/A -
20
80
300
2.0
20 80 300
16-6 17-4 16 -2
24-7 60-7 40-7
.58+1 40+6 48 +3
55 -2 10-l 37 -1
29-l 49-7 41-7
82+4 68+3 21 +2
52 -1 67 -1 14
26-7 40-7 39 -7
82+3 11 +3 60 +4
2.5
20 80 300
48-6 26-6 11 -4
19-4 10-4 19 -4
64+1 19+7 63 +S
88-3 21-2 87-2
18 -4 12-4 17-4
70+5 50 +4 1s +3
29 -1 37-l 62-l
12-4 10-4 15-4
20+4 30+3 23+2
3,o
20 80 300
72-S 47 -5
84-s
5s -3 40 -3 51 -3
29 +6 68 +5 78 i-4
18-3 34 -3 18-2
72-3 60 -3 65 -3
15 +6 22+5 98 +3
15 -1 20-l 33--l
51 -3 48-3 55 -3
47 +4 72+3 65 +2
3.5
20 80 300
29-4 42-4 49-4
43-2 68-2 65-l
49 +5 52+4 86+3
68 -3 19 -3 10 -2
80 -2 96 -2 93 -2
22 +S 33 +4 51 +3
94 -2 12 -1 19-1
66 -2 7s --2 78-2
72 +4 12+4 13+3
4.0
20 80 300
1s -3 11-3 20-3
44-l 35-l SO-l
59 +4 13 +4 15+3
25 -2 22 -2 28-2
45 -1 43 -1 53-l
49 +4 94 +3 13 +3
14-1 14 -1 18-l
43-l 41 -1 so-1
31+4 65 +3 90+2
4.5
20 80 300
20-3 21 -3 31 -3
12 13 15
25 +4 40 +3 58 +2
43 -2 44-2 48-2
12 13 15
22 +4 38 +3 5s +2
22-l 23-l 25 -1
12 13 16
15 +4 30+3 4s +2
5.0
20 80 300
23 -3 26 -3 38 -3
29 31 35
11+4 18 +3 26 +2
68 -2 69 -2 13 -2
29 31 35
98 +4 17 +3 25 +2
32 -1 29 -1 32 -1
30 32 36
76 +3 15 +3 24 +2
s The values of k,, ki and Nt are arranged in order. For example, 76 -6 means 0.76 X lo-‘. is in the units of PV/G
Nt
5. Conclusions The rate constants for ionization ki and for escape k, are formulated as functionals of the distribution function, N(v). The formula of the total ion current, I, which is expressed by the rate constants, the total supply, S, and the capture probability, PC, is derived. The formula coincides with that given by Southon [ 121 for ki -4 1. The behaviour of the ion currents is analyzed by using the equilibrium properties of the system. In the very low field region, I is equal to kiP,S/ke and in straight high-iield region, I is equal to kiP,S~(ki + k,). In the latter region, there exists a part where I is independent of the values of ki.
614
H. Iwasaki, S. Nakamurallon
current generation
in FM.
II
The effects of the tip temperature, T, are discriminated from those of the gas temperature, Tg. The dependence of I on T, is qualitatively explained by the T, dependence of k,. The gas temperature has also been shown to have considerable effects on the ion current, in agreement with the results of I. It is shown that the ion current is indeed increased by the field adsorption of inert gas atoms even if the field ionization probability were suppressed [ 181. Moreover, the proposed mechanisms that the ionization probability is decreased and the staying time of a particle in the ionization zone is increased by the field adsorption of an inert gas atom, enabled us to explain the experimentally observed anomalous features of field ion images at 4.2 K satisfactorily.
Acknowledgements We wish to thank Mr. T. Adachi for discussions about the experimental 4.2 K.
results at
References [l] [2] [3] [4] [5] [6] [7] [8]
E.W. Miiller and K. Bahadur, Phys. Rev. 102 (1956) 624. K. Bahadur, J. Sci. Ind. Res. (India) 19b (1960) 177. M.J. Southon and D.G. Brandon, Phil. Mag. 8 (1963) 579. U. Feldman and R. Comer, J. Appl. Phys. 37 (1966) 2380. T.T. Tsong and E.W. Miiller, J. Appl. Phys. 37 (1966) 3065. E.W. Miiller, Quart. Rev. (London) 23 (1967) 177; Naturwissenschaften 57 (1970) 222. S.B. McLane, E.W. Miiller and S.V. Krishnaswamy, Surface Sci. 27 (1971) 367. E.W. Mtiller and T.T. Tsong, Field Ion Microscopy, Field Ionization and Field Evaporation, in: Progress in Surface Science, Ed. S. Davison (1973) Vol. 4, Part 1. [9] H. Iwasaki and S. Nakamura, Surface Sci. 52 (1975) 588. [lo] H.A.M. Van Eekelen, Surface Sci. 21 (1970) 21. [ 1 l] R. Gomer, Field Emission and Field Ionization (Harvard Univ. Press, 1961). [ 121 M.J. Southon, Ph.D. Thesis, University of Cambridge (1963) (unpublished). (131 R.M. Logan and R.E. Stickney, J. Chem. Phys. 44 (1966) 195. [14] R.G. Forbes, Vacuum 22 (1972) 517. [15] Y.C. Chen and D.N. Seidman, Surface Sci. 27 (1971) 231. [ 161 T.T. Tsong and E.W. Miller, J. Chem. Phys. 55 (1971) 2284. [17] D.A. Nolan and R.M. Herman, Phys. Rev. B8 (1973) 4099; BlO (1974) 50. 1181 H. Iwasaki and S. Nakamura, Japan. J. Appl. Phys. Suppl. 2, Part 2 (1974) 43; Surface Sci. 49 (1975) 664. [19] T. Adachi and K. Ohnishi and S. Nakamura, Japan. J. Appl. Phys. 13 (1974) 549.