Contact potential measurement: Spacingdependence errorsFrank Rossi Citation: Review of Scientific Instruments 63, 4174 (1992); doi: 10.1063/1.1143230 View online: http://dx.doi.org/10.1063/1.1143230 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/63/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A numerical study of geometry dependent errors in velocity, temperature, and density measurements fromsingle grid planar retarding potential analyzers Phys. Plasmas 17, 082901 (2010); 10.1063/1.3457931 Contact potential measurement: The preamplifier Rev. Sci. Instrum. 63, 3744 (1992); 10.1063/1.1143607 Contact Potential Measurements on Graphite J. Appl. Phys. 29, 1132 (1958); 10.1063/1.1723383 Measurement of Surface Potential or Contact Potential Differences Rev. Sci. Instrum. 17, 266 (1946); 10.1063/1.1770482 An Experiment to Demonstrate that “Frictional” Electricity Depends on Contact Potential Am. J. Phys. 4, 144 (1936); 10.1119/1.1999104

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Contact potential measurement: Spacing-dependence errors Frank Rossi School of Physics, The University of Melbourne, Parkville, Victoria 3052, Australia

(Received 28 October 1991; accepted for publication 25 May 1992)

We examine the causes of spacing dependence of the nulling bias voltage in the vibrating capacitor contact potential measurement technique. In addition to effects already recognized in the literature, namely, nonuniform work functions, nonparallel surfaces, fringe fields, and capacitive coupling to distant surfaces, we investigate the effects of finite gain and spurious signals in feedback loop systems. We argue that much of the spacing dependence reported in the literature may be due to microphonic signals, which are very difficult to eliminate. We also discuss the means by which existing spacing dependence can be minimized.

1. INTRODUCTlON

The work functions of metal surfaces are often inves- tigated by the measurement of contact potential differ- ences.’ In equilibrium, the electrostatic potential difference just outside any two metal surfaces is equal to the differ- ence of their work functions.’ Equilibrium is commonly attained by an electrical connection, which allows electrons to shift from one surface to the other. Thus, if the work function of one of the surfaces is known (the reference surface), that of the other sample surface can be deter- mined. Even in the absence of knowledge of the reference work function, provided it is stable, changes in the sample work function may be determinerd, e.g., by the admission of adsorbates, or by strain.3’4 The purpose of this article is to examine one source of error in (contact potential measure- ment, referred to as spacing dependence.

Since contact potentials are the result of equilibrium, they are not emfs and cannot be measured with a conven- tional voltmeter. The usual practice, which can be traced back to Kelvin’ and Zisman6 is to form a closely spaced time-varying parallel-plate capacitor between the two metal surfaces (Fig. 1). Assuming fringe fields are negli- gible, the capacitance is given by

C(t) =d/cw, (1) where E is the permittivity of the medium in the gap (usu- ally free space), A is the area of the capacitor plates, and d(t) is the spacing of the surfaces, made to oscillate sinu- soidally, viz.,

d(t) =do---a sin(wt+f#), (2)

where do is the average spacing, a is the amplitude of the oscillation, o is its frequency, and $ is its phase with re- spect to some fixed oscillator used to drive the vibrating plate. Often, the arrangement consists of a small vibrating capacitive probe (the reference surface) brought up to a larger planar sample surface. This modulated capacitance contains an infinite series of harmonics of w. For small values of the modulation index,

m=ff/&<l, (3) it may be shown that the fundamental component domi- nates,3*7 and is given by

C(ot) =mCo sin(ot+4), (4)

where C,,, the capacitance in the absence of modulation, is given by

Co= EA/& . (5)

The time-varying capacitance forces an ac current to flow through the circuit in Fig. 1, which can then be con- verted to an ac voltage with a suitable preamplifier. An increasingly common practice is to use a current amplifier (current-to-voltage converter, shown in Figs, 1 and 2), which, by virtue of its low input impedance, reduces the effects of parasitic input capacitance.317’8 While the contact potential Y could be inferred from this ac signal, a more elegant and commonly used technique is to insert a vari- able dc source of emf, I’,, into the circuit and adjust it until a null in the ac signal is detected. This emf is referred to as the bias or bucking voltage in the literature+ When the preamplifier input impedance is much smaller than that of C, at o, e.g., for a current amplifier, the fundamental com- ponent of the current is given by

I(M) = -(Y+ V,)dC(oJt)/dt

z -moC,(Y+ V&os(wt+$$). (6)

Ideally, the null requires V,= -Y, so that measure- ment of F’s with a voltmeter yields the contact potential directly. This procedure can be automated with a feedback loop system (Sec. II).

It should be noted that automated off-null techniques are also in use, as reviewed by Baikie et uE.~*‘~ In their system, the total ac signal is measured as V, is scanned around the null point, and optionally includes correction for noise and other spurious effects. Significant advantages have been claimed for such systems; however, it is not clear that their response time would allow them to be used to track fast changes in the sample work function, as was necessary in the author’s work.3,4 In this article, we will not examine off-null techniques in great detail as these have been covered in depth by Baikie et ~1.~~”

Since the contact potential W is given simply by the difference of the work functions of the two capacitor sur- faces, the nulling bias voltage V,, should be independent of the geometry of the capacitor. In practice, one finds that it

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I(t)

- Contact Potentiol Voltoge

FIG. 1. Standard contact potential measurement technique, using a vi- brating capacitor C(t) with nulling bias voltage Y,

varies with the capacitor spacing. This can be a serious problem, since it is then unclear which value of I’, gives the “correct” contact potential. Several effects contribute to this spacing dependence. Those that have been previ- ously studied in the literature’-I7 include (i) nonuniform work functions of the capacitor surfaces, (ii) nonparallel capacitor surfaces, (iii) the side faces of the capacitor plates, other nearby surfaces and fringe fields, and (iv) capacitive coupling to distant surfaces.

During the development of a modified contact poten- tial apparatus,3*4”8 it was necessary to reconsider these problems. Apart from the limitations imposed by noise, one often finds that the ac signal extracted from the capac- itor cannot be perfectly nulled with the bias voltage, i.e., there is a small residual signal. This is due to spurious ac signals coupled into the measurement circuit, which gen- erally have a different phase from the genuine signal. Such spurious signals have been recognized’-I7 as a source of systematic error, but it has not been appreciated that they can far outweigh (i)-(iv) above as sources of spacing de- pendence. One example is the high level signal used to drive the vibrating capacitor plate, although with careful shielding and grounding technique, coupling of this signal may be rendered negligible. Particularly problematic are the microphonic signals generated by vibration in the sys- tem. Microphonics are a general class of poorly understood phenomena associated with vibrating signal conductors and insulators.” Triboelectric and piezoelectric effects in insulators that are in contact with signal leads can generate spurious currents. Vibrating wires that are at different volt- ages (because of circuit biasing, or other contact poten- tials) produce signals akin to the contact potential under study. The main contention of this paper is that such spu-

LOCK-IN AMPUFLER r---------------------------~

R I I I sin(wf + $9”) I I , I I

I(t) t - ns ; AZ 4 F;

c(wr) f _lIi

C”,,cnt AC GO,” ; AC GO,” wrer oc Goin ‘o;,t~rss *“p’l,,O

; L_-_-----_-----------------~~

Y

i

FFcdbOCk

v. Log/hod etc mtcr Network

FIG. 2. An example of a feedback loop which automates the contact potential technique shown in Fig. 1.

4175 Rev. Sci. Instrum., Vol. 63, No. 9, September 1992

rious ac signals are generally the dominant source of spac- ing dependence.

In Sec. II we briefly discuss feedback loop systems for automating the null method discussed above, pointing out that spacing dependence can arise even in the absence of the effects we have discussed so far. In Sets. III and IV, we critically review earlier analyses of spacing dependence, discussing the problems (i)-( iv) above. Then in Sec. V, we show that the same phenomena can be produced by spuri- ous ac signals (usually microphonic) coupled into the measurement circuit. We will also show that this under- standing offers a practicable means of reducing spacing dependence.

II. FEEDBACK LOOP GAIN

The contact potential technique of Fig. 1 is often au- tomated by using a feedback loop to provide V,. This al- lows continuous measurement of Y, and is essential, for example, in time-dependent adsorption studies, or surface- voltage profile measurements. Several variations have ap- peared in the literature (see Ref. 1, for a review); Fig. 2 shows one implementation, used by the author,3 which will serve for subsequent discussion.

For now we assume the ideal case, free of the sources of error outlined in the introduction. In Fig. 2, feedback of V, is accomplished by applying this dc voltage to the pos- itive input terminal of the current amplifier, which in turn maintains the negative input terminal, and hence the vi- brating capacitor plate, at very nearly the same voltage.3’7 The ac current, given by Eq. (6), is converted to an ac voltage, which is then synchronously demodulated by the lock-in amplifier (referenced at the vibration frequency w ). The output is then filtered and the resulting phase-sensitive dc voltage provides V,. Assuming conditions are appropri- ate for stable operation of this 10op,~ V, will be given by

V,= - GT, (7) where G< 1 is the dc closed loop gain. In turn, G is given by the dc open loop gain H, viz.,

(8)

(Note that H is defined as the ratio - V,/T when the feedback connection is broken.) For the loop in Fig. 2, it may be readily shown3 that

H= -mmwRCd1A2A3F1F2 sin($-#M)/2

=KmCo

a l/d:, (9) where the nature of the parameters in this equation should be clear by reference to Fig. 2, and K is a lumped constant. Ideally H> 1, GZ 1, and VBz - ?‘“. Table I summarizes typical values of the parameters used by the author.

It is important to note that H, and hence G, varies with the capacitor spacing do through m and Co [Eqs. (8) and (9)]. This implies that V, will show spacing dependence even in the absence of all the nonideal effects discussed in

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TABLE I. Typical operating parameters for the apparatus in Fig. 2, as discussed in the text.

Parameter Approximate value

Probe diameter

so CO w/2n R 444f2 FIFZ sin(+h) H

5mm 20 pm peak

100 pm 2 PF

500 Hz 70 MR 1.5x 104

-1 (at dc) -1

1300

the introduction, a point often loverlooked in the literature. As H-+m, G becomes insensitive to spacing [Eq. (S)]. Figure 3 shows the variation of G with d,, for the system in Fig. 2. The top curve corresponds to normal operation (Table I, H= 1300 at da= 100 pm), for which G shows only moderate spacing dependence even up to de= 10 pm. For do < 150 pm, which is the range normally used, spacing dependence is quite negligible compared with other sources, as shall become evident. The two lower curves in Fig. 3 show the increased spacing dependence when H is 10 and 100 times smaller.

It is impossible to make H’ indefinitely large, as insta- bility will eventually result. Achieving high open loop gain requires careful compensation for poles present in the com- plex frequency plane for the ac open loop gain. This is one of the functions of the filter block, I;‘, (Fig. 2), which contains a lag/lead compensation network. Other factors can lim it H. To increase the spatial resolution in surface profile studies, a smaller diameter probe may be used. However, this decreases Ce and hence H in Eq. (9), which requires the gain to be made ulp later, often in stages after the m ixer (As in Fig. 2), as otherwise input noise would overload the m ixer. However, synchronous hash generated by the m ixer is then amplified to the point where instability

1.0 0.9 0.8

0.7

0.6 0.5 0.4 0.3 0.2

0.1 0.01 * ' . s ' ' . s ' '

0 100 200 300 400 500 600 700 800 900 1000 do (mkrons)

FIG. 3. Spacing dependence of the closed loop gain G in the feedback system used by the author (Fig. 2). The top curve is the normal case where the open loop gain II(&) is given by Eq. (9) and the parameter values in Table I, i.e., for do= 100 pm, H= 1300. The two lower curves show the effect on G when His 10 and 100 times smaller.

4176 Rev. Sci. Instrum., Vol. 63, No. 9, September 1992 Contact potential 4176

occurs, unless it is selectively attenuated by custom de- signed notch and low-pass filters ( lumped into F2). Even so, in the author’s case, increasing H in Table I by a factor of 3 brings the system to the point of instability. Operation with H= 1300 provided a safe margin before instability, with m inimal spacing dependence over the typical range of spacing do < 150 pm. The important point is that without care, the magnitude of H can be severely lim ited, resulting in significant spacing dependence for G.

III. SURROUNDING CONDUCTORS

It is very difficult to eliminate stray capacitance be- tween the vibrating capacitor plate, or probe, and other conductors in the environment. This additional capaci- tance is also modulated, and generally involves different contact potentials. The total ac current is then a weighted measure of ah these contact potentials. At best, one can make the weighting favor the two surfaces under study by making their capacitance the largest of all.

D’Arcy and Surplice’2.13 and Ritty et aLI advocated connecting the preamplifier to the stationary plate of the capacitor to reduce stray time-dependent capacitance. However, this arrangement is rarely used by experiment- ers, due to various experimental requirements. In any case, it should be noted that the vibrating plate will still produce time-dependent screening of background conductors, and so generate some unwanted signal, as verified by Baikie et ai.9

Others have investigated guard conductors to reduce stray capacitance. Danyluk2’ used a guard actively mnin- tained at the same potential as the vibrating probe and rigidly connected to it, with the preamplifier connected to the vibrating probe. The data presented still show spacing dependence. Ritty et al. ” investigated a closely spaced, stationary, and grounded guard surrounding the vibrating probe, with the preamplifier connected to the stationary sample. They compare the spacing dependence with and without the guard. Although the effects are changed, there is no improvement. de Boer et al. l5 connected the preamp- lifier to the stationary electrode and surrounded the whole vibrating capacitor with a widely spaced, stationary screen of variable potential. This potential was adjusted to null out contact potentials of stray time-varying capacitances. With some surfaces, they obtained significant improve- ments. Engelhardt et al. l6 used a similar variable potential screen, but connected the preamplifier to the vibrating probe, claiming the effects were made negligible. Baikie et ai. 9 also advocated the use of shields for some configura- tions.

Early experiments conducted by the author used the configuration advocated by D’Arcy and SurpIice,‘2”3 with the sample connected to the current amplifier and without a guard. Once residual m icrophonics and other spurious external couplings had been eliminated, very little spacing dependence was found, as shown by curve 1 in Fig. 4. However, experimental necessities required the sample to be physically grounded, so that the current amplifier had to be connected to the vibrating probe. This resulted in the increased spacing dependence in curve 2 in Fig. 4, as one

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. s -70 - !E

s” -8o- /*

P FIG. 5. Multiple capacitor model for the patch effect, and nonparallel probe with fringe fields and side faces. Physical model at left, equivalent circuit at right.

-90 ‘....-• .A’

4 ,(;,l,im’l=]~/---:

-------• -.- -----X5

3 -100 + 0 ' ' * ' 1 ' ' c 1 a 1

0 40 80 120 160 200 240 280 do (microns)

FIG. 4. Spacing dependence of the dc bias voltage, measured on an early version of the apparatus. In curve 1, the current amplitier is connected to the sample, as advocated by D’Arcy and Surplice (see Refs. 12 and 13), resulting in minimal spacing dependence. In curve 2, the current amplifier is connected to the vibrating reference plate, resulting in increased spacing dependence due to a spurious microphonic signal in the now vibrating input cable. By injecting a similar signal of opposite sign into the capac- itor, the microphonic signal could be approximately cancelled, as dem- onstrated by curves 3, 4, and 5. Sample: brass, probe: brass.

might expect. However, the change was traced to micro- phonics in the (now vibrating) input cable, diagnosed by opening the feedback loop and examining the ac output of the current amplifier. As the spacing was increased (i.e., the probe to sample capacitance decreased), the signal quickly decreased, but instead of approaching zero, asymp- toted to a constant value by about 1 mm spacing (normal spacings are about 0.1 mm). By injecting a similar signal of opposite sign into the capacitor, this microphonic signal could be approximately cancelled, resulting in curves 3, 4, and 5. Curve 5 is as flat as curve 1 with the original (sup- posedly optimum) connection scheme.

On physical grounds we should expect the coupling to relatively distant (compared to the probe-sample spacing) surrounding conductors to be negligible. Examination of formulae appearing in most standard texts on electrostatics for the capacitance between a conductor and a surrounding conductor, e.g., two concentric spheres or cylindrical co- axial conductors, reveals that, to first order, it is insensitive to small relative displacements. Unless the contact poten- tial between the probe and its distant environment is very nonuniform, we would expect similar insensitivity.

In some cases, e.g., Ritty et a1.,17 the probes were of order 1 mm diameter with sample spacings not much smaller. The smallest probe used by the author was 5 mm in diameter with spacing of order 0.1 mm or less, so that the ac part of the capacitance was typically a factor of 100 larger. This might explain what appears to be a greater sensitivity to distant couplings in their apparatus.

In a variety of apparatus, with different surfaces, the most significant source of spacing dependence was found to be the presence of microphonics in the measurement cir- cuit, as exemplified by Fig. 4. We shall examine this in detail in Sec. V. In contrast to the elaborate guard arrange- ments used by others, a simple guard-free probe design was

found to produce less spacing dependence. The most im- portant issue was to minimize contact of the vibrating probe with insulators, which seemed to be the major cause of microphonics. The only shield, for the capacitor as a whole, was widely spaced, stationary and grounded.

IV. MULTIPLE CAPACITOR MODEL

Metal surfaces have been known to exhibit nonuniform work functions since the early days of thermionic emission studies.2’ This is attributed to the variation of the work function with the crystal facets and nonuniform contami- nation (the patch effect). Consequently, the contact poten- tial difference between the electrodes of a vibrating capac- itor will generally vary over the surfaces. Parker and Warren” measured the contact potential variation over a gold sample with a small gold coated probe, at 10v9 Torr. Spatial variations of order 100 mV were reported, even after bakeout at 350 “C. Experience shows that surfaces are rarely more uniform than about 10 mV. Thus, it becomes meaningless to speak of nulling “the contact potential” with the bias voltage.

Several authors1’-17 have modeled the vibrating capac- itor as several smaller vibrating capacitors C’i( t) each with a different contact potential pi as shown in Fig. 5. This may be used to account for the patch effect, nonparallel plates, side faces, and fringe fields. Any coupling to other surfaces in the environment can be treated in the same way. The usual form taken for the capacitance elements is the parallel-plate type, as described by Eq. ( 1). Although this is not quantitatively accurate, it is an expedient mea- sure taken to avoid analytical difficulties with more realis- tic models. As shown by Ritty et al.,17 the bias voltage in such a model is given by

VB= - ,f+l F”~(A/d&l/ 5 (A/&), i=l

(10)

which explicitly shows the weighting of the Y/s according to area Ai, and spacing doi, of each capacitor element. Clearly, it is important to keep the capacitor plates as par- allel as possible. Externally, the pi give the appearance of a single spacing-dependent contact potential, F( do) :

- V,(do) = - Vdo). (11)

An interesting phenomenon was exploited to investi- gate the effects of an extremely nonuniform surface on spacing dependence. If the vibrating capacitive probe is left at close spacing over the same spot on the sample for ex- tended periods, i.e., a few days, then a circular contact

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(4

-100 E

-200

2 5 ' -300

s= e s””

302

-394

-308

-248

-250

-252

N-F;,;‘, /--, d: f: 1 ' c’b : ‘e

\c>dd-,’ \s-,’ --

Contact Potential

Patch Profile

-15 -10 -5 0 5 10 15

Y (mm>

40 80 a0 loo 120 do (micm)

-18Q - do W=-) (d)

-231 40 80 100 120

do (n%ons)

FIG. 6. Spacing dependence measured in the case of an extremely nonuniform surface. (a) A step change in the contact potential profile across the Cu surface, due to a circular patch produced by a prolonged shadowing effect of the probe on the surface. The center of the probe is given by they coordinate, the boundary of the probe in (b)-(f) are marked as dashed circles. (b) y=O mm, directly over patch. (c) y= -0.1 mm. (d) y= -5.0 mm, probe just outside left edge of patch. (e) y=7.8 mm, probe well to right of patch. (f) y= -2.7 mm, probe center directly over left edge of patch.

potential patch often forms on the sample due to a kind of shadowing effect. Because the mean-free path for back- ground gases in the vacuum system is very much larger than the probe-sample spacing, the region covered by the probe is relatively isolated. Thus, background adsorbates differ significantly between the covered and uncovered re- gions. If left uncovered for a sufficient period, this shadow patch is eventually smoothed out by adsorbates.14 Figure 6(a) shows the one-dimensional profile of one such patch, measured by scanning the probe across it. Note that this shadow patch has the same lateral extent as the 5-mm- diam probe, and is a rather sudden step jump in the contact potential of some 100 mV. Figure 6(b) shows VB(do) when the probe is positioned directly over the patch, and Fig. 6(c) when it is fractionally to the left of it, both show- ing similar moderate spacing-dependence. In Figs. 6(d) and 6(e), the probe has been moved completely off the patch, where the contact potential profile is relatively uni- form. As expected, the spacing-dependence is significantly

4176 Rev. Sci. Instrum., Vol. 6:3, No. 9, September 1992

reduced. In contrast, we would expect the most extreme spacing-dependence to arise when the probe is positioned directly over the edge of the patch, as is confirmed by Fig. 6(f) . Experimentally, it is quite clear that nonuniformity of the contact potential produces spacing dependence, how- ever, we note that even in the very extreme case of Fig. 6(f), this spacing dependence is quite small compared to that produced by microphonics (Figs. 4, 8, and 9).

Ritty et al. I7 used a three-capacitor model (N= 3) to qualitatively account for the patch effect, nonparallel sur- faces, the side faces of electrodes and coupling to distant surfaces. The first three can be lumped together and rep- resented by two capacitor elements with commensurate area and spacing, while the last was represented by a third capacitor element with much larger spacing and area. This combination reproduces the general behavior in VB(do> that is seen by many experiments. Two general observa- tions could be drawn from these numerica studies:

(i) The patch effect, nonparallel surfaces, and the side

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- (/

1,(f) = I,sm(wf +-jl,

I -- I(0 f

-J-$fJlp.~~~~-

Cbl) t- V,(t) = V,sin(ot+J,) V

Tb

OS

?,

FIG. 7. The feedback loop system shown in Fig. 2 with the addition of several spurious voltage and current sources.

faces of electrodes tend to produce asymptotic behavior in V,( de), much like curve 1 in Fig. 4 (cf. Fig. 4 in Ref. 17).

(ii) The third capacitor, representing coupling to dis- tant surfaces, tends to produce monotonic behavior in VB(dO), much like curve 2 in Fig. 4 (cf. Fig. 3 in Ref. 17).

Despite this qualitative agreement with observations, a parallel-plate form cannot be a realistic model for the ca- pacitance to relatively distant surfaces, as argued in the last section. Such monotonic spacing dependence could even result from insufficient gain in a feedback loop system (Sec. II), although one would presume that other investigators were careful in this respect. In the next section, we will show that spurious ac signals in the measurement circuit also produce similar monotonic spacing dependence.

V. SPURIOUS AC SOURCES

The vibrating capacitor technique relies on the mea- surement of an ac signal. If for some reason other ac sig- nals, unrelated to the contact potential, are mixed in with the genuine signal, the measurement will clearly be in er- ror. Note that there are then three unknown quantities: the contact potential itself, and the amplitude and the phase of the added spurious signal. What we measure is a total ac signal, which has an amplitude and a phase, i.e., only two pieces of information. Thus, unless there is some reliable additional information about the spurious signal, it will be impossible to determine the contact potential.

The same problem is present in the feedback loop in Fig. 2 since it responds to the total w signal going into the mixer that is in-phase with the genuine one, producing a net dc bias voltage at the output. When this net dc voltage is modulated by the capacitance, it cancels the total in- phase o signal entering the mixer. At best, any spurious signal that is in quadrature (900 phase) will be ignored [cf. Eq. (9)]. It is only in the absence of spurious signals that this ac null corresponds to a null in the contact potential.

Ideally, when there is no in-phase ac input to the mixer, there should be no dc output. In practice, the com- ponents that follow the mixer will have dc offset voltages. Depending on how much dc gain is present in this section, these may also produce significant errors in V,.

Figure 7 shows the feedback loop of Fig. 2 with the addition of a small number of lumped spurious sources, so chosen to demonstrate certain characteristic features. In reality, the spurious sources may be widely or differently

-900

2 - -910

9 -920

t

2 .~-*--*--+--*-* .

-930 l I

-940 ’ I 0 20 40 60 80 100 120 140 160

do (microns)

FIG. 8. Spacing dependence measured on another functionally similar version of the apparatus. Curve 1 shows the effect of microphonics in the input cable. Curve 2 shows the improvement obtained by using low- microphonic coaxial cable together with vibration damping and isolation. Sample: aluminum, probe: brass.

distributed to those shown. The ac voltage source V, and the ac current source 1, are primarily intended to represent microphonic sources, which could have either character. The ac voltage source V, could be the result of improper grounding technique. The dc offset voltage V,, lumps to- gether all dc offsets in components following the mixer.

Ignoring all other problems for now, it is readily shown3 that the above sources modify Eq. (7) for the bias voltage as follows:

V,= --G{y+ ~Jdo)), where the error term .YO is given by

(12)

Ye(&) = -{sin(B1-4M) V,/m+cos(8;-~M)ll/mwCo

+2V,,/moRCoA,A2)/sin(~-~~). (13) The important point is that Ye is a monotonic func-

tion of spacing due to the appearance of m and Co in Eq. ( 13). Apart from any spacing dependence due to G, the V, source will produce a linear variation with do, while the I,, V,, and V,,, sources will produce quadratic variations with do. Note that curve 2 in Fig. 4 appears to be roughly qua- dratic in shape. Given that diagnostic experiments revealed this to be due to microphonics, the above analysis would suggest that the microphonic source behaved more like the current source Ii than the voltage source VI. These sources need not be very large. If the spacing dependence in curve 2 in Fig. 4 was entirely due to Ii, then (assuming the worst case for its phase 6;) its magnitude was - 10 pA peak, which could pass casual inspection of the broad-band ran- dom noise present at the current amplifier output.

Curve 1 in Fig. 8 shows the initial spacing dependence observed in a physically different but functionally similar apparatus, in which the current amplifier was connected to the vibrating probe. This was determined to be due to mi- crophonics in the input signal cable. The use of low- microphonic coaxial cable together with vibration damping

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300

290

280

5‘ 270

.E. 260

s” 250

240

230

/’ 1

/

.I’. /*

.e-----• 2

.‘.-.-..---.--.---.-. .

l -.--..-.-.-..----.~-.-. 3

l -----.-.--* 4 l -. .

0 20 40 60 80 100 120 140 160 do (microns) VI. DISCUSSION

FIG. 9. Curve 1 shows the initial quadratic spacing dependence due to microphonics. Curves 2, 3, and 4 demonstrate the reduction of this spac- ing dependence by the introduction of a controlled dc offset voltage V, into the feedback loop. Sample: copper, probe: gold.

and isolation resulted in the greatly reduced spacing de- pendence shown in curve 2. In the author’s experience, attention to such matters is more likely to yield significant improvements than elaborate guarding arrangements for the vibrating probe. Typical microphonic signals produce much larger spacing dependetice than even severely non- uniform surfaces, as shown by comparison of the curves in Figs. 4, 8, and 9 with those in Fig. 6. While it is difficult to generalize to all apparatus, it is suggested that much of the spacing dependence attributed in the literature to capaci- tive coupling to distant surfaces is in fact due to such spu- rious ac signals.

While it is difficult to generalize to all apparatus, we have argued physically and on the basis of experiments, that much of the spacing dependence attributed to capac- itive coupling to distant surfaces is in fact due to spurious signals coupled into the measurement circuit. Particularly important offenders are microphonic signals generated by vibration in the system. O ther effects discussed in the lit- erature, such as nonuniform work functions of the capac- itor surfaces, nonparallel surfaces, the side faces of the capacitor plates and fringe fields, do produce significant spacing dependence. However, the effects of microphonics have been found to far outweigh all of these.

The potentially large effect of insufficient open loop gain in feedback loop systems has not been considered in the literature. Unless care was taken, much of the spacing dependence observed by other workers may in fact be due to this cause.

Another approach to minimizing spacing dependence is to deliberately inject controllable sources like VI, II, V,, or V,,, into the feedback loop. This allows the error term ??Jdc) to be modified [Pq. (13)]. The additional spacing dependence thereby introduced can then be used to com- pensate that due to existing sources. Since linear and qua- dratic dependences can be produced, there is considerable flexibility in flattening out the V, vs do curve. In the au- thor’s experience, it is convenient, and often sufficient, to simply introduce the V,,s dc offset. Many lock-in amplifiers contain an output rezeroing facility, which introduces an output dc offset voltage, eg., the Ithaca model 393 (ITHACO Inc., Ithaca, NY). This approach is illustrated in Fig. 9, where curve 1 shows the initial quadratic spacing dependence due to microphonics, i.e., due to an I, type of spurious signal. By applying a V, of a few volts dc, a compensating quadratic spacing dependence may be intro- duced into y,Jdo) and hence ,VB(do), as demonstrated in curves 2, 3, and 4 (see also Ref. 18).

Great care should be taken with sensitive areas, such as the input cable from the vibrating capacitor to the pre- amplifier. Much has been said about the use of elaborate guards around the vibrating probe to eliminate coupling to distant surfaces, but experience suggests that guard-free probes are preferable, so as to minimize microphonics.

Finally, we have shown that by judicious introduction of controllable ac and dc sources into a feedback loop sys- tem, existing spacing dependence may be minimized over a wide range of capacitor spacing. This provides a simple method for minimizing errors due to a change of spacing during the course of a measurement.

ACKNOWLEDGMENTS

I would like to thank G . I. Opat and A. Cimmino for many useful discussions during the course of this work. This work was supported by the Australian Research Grants Committee, and partly by a Commonwealth Post- graduate Research Award.

This ability to flatten out the VB(do) curve has impor- tant applications.3 For example, in surface-potential profile studies, if V,(d,) is not flat, any change in the spacing as the probe scans across the sample surface would lead to spurious changes in VB that are unrelated to genuine spa- tial variations in the contact potential. Palau and Bonnet23 employed an additional feedback loop to actively maintain constant spacing while scanning over a surface. Baikie

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4180 Rev. Sci. Instrum., Vol. 63, No. 9, September lQQ2 Contact potential

et ai. lo also described an active spacing control system. The technique outlined above offers a much simpler solution. Based on experience, with care, V,( do) can be made flat to a fraction of a mV over typically a 100 pm range. The spacing can usually be kept constant within about 10 pm without an elaborate feedback loop, and the resulting er- rors below 0.1 mV. The only problem with this technique is that an error ye constant for the whole profile, is intro- duced. However, such an error will be present unless all existing spurious sources have been eliminated. If the only interest is in measuring surface potential uff&ti~ilzs, this technique is eminently suitable.

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