In semiconductor devices and biological samplknowledge of the local electric potential distribution issignificant interest because it helps in linking the specimeobserved function with its local structure and compositioWith the advent of scanning tunneling microscopy higresolution mapping of local potential distributions becafeasible.1,2 Due to the close proximity of the probe to thsample as required for electron tunneling, potential mwith a lateral resolution on the nanometer scale couldobtained, yet, inevitably, the technique was limited to coductive surfaces. Adaptation of the atomic force microsco~AFM! to electric potential measurements3,4 immediatelybroadened the application range to nonconducting sambecause now the probe could be kept close to their surwithout the necessity of a tunneling current. Although diffeences in electric potential between sample and probe cbe detected by simply monitoring the electrostatic contrition to the cantilever deflection, the employed modulattechniques resulted in a higher sensitivity. In particulvariations of the Kelvin probe force microscope5–7 ~KFM!have evolved into reliable tools to characterize specimranging from semiconductor devices8,9 to biologicalsamples.10,11
In our KFM setup12 ~modified Nanoscope III, DigitalInstruments, USA! we measure topography and electric ptential using the lift-mode technique to minimize crosstaTo this end, we first acquire the surface topography osingle line scan and then immediately retrace this topograover the same line at a set lift-height from the sample surfto measure the electric potential. Images are obtained bypeating this procedure for each line along the slow-scan aWe have already shown that this combination of KFM a
a!Also with Laboratory of Field Theory and Microwave Electronics, ETCenter/ETZ, CH-8092 Zurich, Switzerland.
b!Author to whom correspondence should be addressed; [email protected]
1160021-8979/98/84(3)/1168/6/$15.00
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lift-mode technique leads to potential maps where featuresmall as a few ten nanometers in size can be qualitativdistinguished based on variations in chemical compositio12
However, since the magnitude of the measured electrictential critically depends on the size of the feature, its sroundings, and the probe geometry, a clear understandinthe contrast transfer mechanism in KFM is required to enaa quantitative analysis and interpretation of potential imagThe knowledge of the contrast transfer mechanism will pmit the combination of high-resolution surface topograpand electric potential data which is likely to significantfacilitate the development of new and improved semicondtor devices.
To analyze the contrast transfer mechanism in KFMintroduce a model based on a set of ideal conductors wmutual capacitances between them. Using a numerical silation method we will derive the contrast transfer charactistics of KFM for ~i! small spots depending on their size, a~ii ! steps in the electric surface potential distribution. Tcontrast transfer characteristics are evaluated for diffeprobe geometries to establish guidelines for optimal prodesign. Finally, we will provide experimental evidence fthe postulated contrast transfer characteristics.
II. A MODEL FOR QUANTITATIVE KELVIN PROBEFORCE MICROSCOPY
A. Field energy, force and KFM potential
To establish the correlation between the actual surfpotential distribution and the measured quantities, we moour KFM setup as a sample surface consisting ofn ideallyconducting electrodes of constant potentialF i and a tip ofpotentialF t ~Fig. 1!. The electrostatic field energy is thegiven by
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whereQi is the total charge on theith electrode. Accordingto the generalized theory of capacitance13 there is a linearrelationship between the charges$Qi ,Qt% and the potentials$F i ,F t%, andQi can be expressed as
Qi5S (j 51
n
Ci j ~F i2F j !D 1Cit~F i2F t!. ~2!
Similarly, the total chargeQt on the tip is
Qt5(i 51
n
Cit~F t2F i !, ~3!
where the mutual capacitancesCi j andCit describe the electrostatic coupling between different electrodes on the samitself and with the tip, respectively.
Introducing Eqs.~2! and ~3! into Eq. ~1! and using thereciprocity relationCi j 5Cji we obtain
We51
2F (i 51
n21 S (j 5 i 11
n
Ci j ~F i2F j !2D G
11
2 (i 51
n
Cit~F i2F t!2. ~4!
For a system with two electrodes on the sample surfacea tip of potentialF t , Eq. ~4! reduces to
We5 12 C12~F12F2!21 1
2 @C1t~F12F t!2
1C2t~F22F t!2#. ~5!
To calculate the force acting on the tip we keep the potentof all the electrodes and the tip fixed by external voltasources, and move the tip along thez axis ~see Fig. 1!. Usingthe relation14 Fz5]We(z)/]z and Eq.~4! we obtain
Fz51
2F (i 51
n21 S (j 5 i 11
n
Ci j8 ~F i2F j !2D G
11
2 (i 51
n
Cit8 ~F i2F t!2, ~6!
whereCi j8 5]Ci j /]z are the derivatives of the capacitancat the actual tip location. Thus, the electrostatic force inaction between tip and sample surface depends both onderivatives of the capacitancesCi j between different regionson the surface and the derivatives of the capacitancesCit
between tip and sample. Note that when the tip is moalong thez axis towards the surface~Fig. 1! the electrostatic
FIG. 1. Model of the KFM setup: System of ideal conductors with electstatic interactions represented by mutual capacitancesCi j .
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coupling between different sample regions under the tipdisturbed and the corresponding coefficientsCi j decrease.Hence, the derivativesCi j8 5]Ci j /]z are not zero and contribute to the electrostatic tip force at the actual tip locatio
In KFM an external ac voltage with an adjustableoffset is applied to the conducting tip:F t5FDC
1UAC cos(vt). Hence, the electrostatic forceFz acting onthe tip has spectral components both at dc and at the freqcies v and 2v. It is worth noting that the first harmoniccomponent of the tip force,Fv , depends only on the mutuacapacitances between tip and surfaceCit and not on the mu-tual capacitances between different surface elementsCi j :
Fv52(i 51
n
Cit8 •~F i2FDC!•UAC . ~7!
In KFM the magnitude of this force component is measuand the feedback electronics adjust the dc potential offFDC, until Fv vanishes. SettingFv 50 we obtain
FDC5( i 51
n ~Cit8 •F i !
( i 51n Cit8
. ~8!
Equation ~8! demonstrates that resolution and accuracyKFM are defined by the electrostatic coupling betweentip and the different surface regions. The measured KpotentialFDC does not exactly match the surface potentbelow the tip, rather it is a weighted average over all pottials F i on the surface, the derivatives of the capacitancCit8 , being the weighting factors.12
At the tip location shown in Fig. 1 the potential spot~3!on the right has a smaller weighting factor due to its smasize and its larger distance from the tip compared to etrodes to the left of the tip~1! and below~4!. Therefore, itonly adds a small contribution to the measured KFM pottial. Furthermore, the potential of an isolated area will aproach the value of the surrounding surface potential asarea decreases in size. Moving the tip to the left acrosspotential step, the weighting factorC1t8 will increase whereasall other weighting factors will decrease. Thus, an ideal ptential step on the surface will appear as a smoothedgradually approaching the value ofF1 in the KFM potentialimage.
Subdividing an infinitely large, ideally conducting anperfectly flat surface into equally sized elements of aDx* Dy at locations$xi ,yj%, the KFM potential@Eq. ~8!# canbe expressed as
FDC~xt ,yt!5( j 52`
` ( i 52`` @C8~xi2xt ,yj2yt!•F~xi ,yj !#
( j 52`` ( i 52`
` @C8~xi2xt ,yj2yt!#,
~9!
where$xt ,yt% is the tip location. As the denominator of Eq~9! is independent of the lateral tip location and equal toderivative of the total tip–surface capacitance,Ctot8 , Eq. ~9!simplifies to
FDC~xt ,yt!5 (j 52`
`
(i 52`
` S C8~xi2xt ,yj2yt!
Ctot8•F~xi ,yj ! D
~10!
and
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1170 J. Appl. Phys., Vol. 84, No. 3, 1 August 1998 Jacobs et al.
FDC~xt ,yt!5E2`
` E2`
`
h~x2xt ,y2yt!F~x,y!dx dy,
~11!
h~x2xt ,y2yt!5 limDx,Dy→0
S C8~xi2xt ,yi2yt!
Ctot8 DxDy Dfor infinitely small surface elementsDx* Dy.
Equation~11! shows that KFM potential maps of flaideally conducting surfaces are two-dimensional convotions of the actual surface potential distributionF~x,y! withthe corresponding transfer functionh(x,y).
B. Calculating tip–sample capacitances Cit
We considered two fundamental electrode configuratiin the sample planez 5 0 ~Fig. 1!, the ‘‘spot-potential’’ andthe ‘‘step-potential.’’ The spot-potential was modeled asdisk of variable diameterd and potentialF1 embedded in aplane of potentialF2 with the tip kept above the center of thdisk. The step-potential was modeled as the half planx,0 with potentialF1 and the half planex.0 with potentialF2 . For both electrode configurations, Eq.~8! translates into
FDC5C1t8 •F11C2t8 •F2
C1t8 1C2t8, ~12!
where the valuesC1t8 and C2t8 depend on geometry parameters, i.e., the location of the tip and the diameterd of thedisk for the spot-potential.
According to Eq.~3! the tip’s charge is given by
Qt5C1t~F t2F1!1C2t~F t2F2!. ~13!
ProvidedQt(z) is known,C1t can be found by selecting thboundary conditionsF t5F250, F1Þ0; similarly, C2t isobtained via the boundary conditionsF t5F150 and F2
Þ0:
C1t52Qt~z!
F1, C2t52
Qt~z!
F2. ~14!
The derivatives of the capacitances$C1t8 , C2t8 % required topredict the potentialFDC @Eq. ~12!# were obtained numerically via the difference in capacitance due to a small heichangeDz of the tip:
C1t8 5C1t~z1Dz!2C1t~z!
Dz, C2t8 5
C2t~z1Dz!2C2t~z!
Dz.
~15!
Thus, the task of determining the derivatives of the mutcapacitances,Cit8 (z), reduces to the problem of calculatinthe tip chargeQt for each electrode configuration and eaboundary condition. However, this requires a highly accurknowledge of the three-dimensional electrostatic field disbution.
The multiple multipole program15 ~MMP! is a powerfultool for solving Maxwell’s equations in piecewise linear ahomogeneous materials and allowed us to calculatechargeQt for a given set of electrodes with potentialsF i .For our purposes only the electrostatic module was neede16
MMP uses a linear combination of vector-valued expansfunctions which are selected by the user to easily solveelectrostatic field problem. Each expansion function h
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seven components: three for the electric field, three fordielectric displacement, and one for the scalar potential.most simple expansion functions are point chargesspherical geometries or line charges for cylindrical ones. Dpending on the geometry and the complexity of the corsponding field distribution, more sophisticated expansfunctions are necessary. Before MMP can solve the fiproblem the user defines the boundary conditions usinmesh of surface elements. In the next step, location andof each expansion function are chosen. The expansion futions to describe the influence of the tip were multipoplaced on the axis of the tip. A total of 16 multipoles of ord10 ~66 unknowns for each multipole! was chosen and resulted in N51056 unknowns. Special expansion functiowere developed for the exact description of the field opolygon-shaped potential spot.17 Boundary conditions for thecontinuity of the electric potential, the tangential componeof the electric field, and the normal component of the dieltric displacement were evaluated on 3480 surface elemon the tip. Using least-squares fitting techniques MMP cculated the 66 parameters of each multipole by matchingelectrostatic field to the boundary conditions. Due to MMPsymmetry feature the mesh only covered one quadrant oftip. Nonetheless, each evaluation of a single valueCit for aparticular tip position took 346 CPU seconds on a Sun SpUltra1/Creator~167 MHz! computer.
Figure 2 displays particular tip–sample geometries atheir associated electrostatic field distributions~planey50)obtained by MMP for the spot-potential@Figs. 2~a! and 2~c!#and the step-potential@Figs. 2~b! and 2~d!#. The field vectorsand the contours of constant potential clearly demonstthat the boundary conditions are met for the entire electrconfiguration.
FIG. 2. Modeled tip and electrostatic field distribution for a spot-potenlocated below the tip~a!,~c!, and a step-potential shifted inx direction~b!,~d!. Tip length l 521 mm, opening anglea534°, apex radiusr a
5100 nm, and cantilever widthdtop518 mm match the dimensions of thetip used in our experiments~Figs. 4–6!. In ~a! and ~c! the tip is located 60nm above the spot of diameterd59 mm. In ~b! and~d! the tip is positioned150 nm beside and 150 nm above the potential step. Boundary cond(F t5F250, F1 Þ0).
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For our experiments we used commercially availablen-doped silicon tapping-mode tips from a single wafer~Nano-probes, Digital Instruments, USA!. All investigated tips hadthe same opening anglea534° and tip lengthl 521 mm asdetermined by a calibrated scanning electron microsc~Hitachi S900!. At the base of the tip the cantilever measur18 mm in width. The apex radiusr a of the tips varied be-tween 10 and 200 nm. Hence we approximated this geomby a conical tip with apex radius, opening angle, andlength matching the measured parameters. The cantilwas modeled as a disk of diameterdtop518 mm and allparts were considered to have zero resistivity.
C. Contrast transfer in KFM
To study the total system response a series of simtions was run with different tip locations and different spdiameters. To this end, we calculated the net tip chargethe expected KFM potential from the three-dimensional~3D!electrostatic field using Eqs.~12!–~15!. The field distribu-tions shown in Fig. 2 are particular examples correspondto one point of the response characteristic of spotlikesteplike potentials, respectively. Figure 3 shows the modespot-size~b! and step~c! response in KFM for several different tip geometries~a!. An immediate conclusion to bedrawn from comparing geometries~k! versus ~m! and ~j!versus~l!, respectively, is that the cantilever surface affethe measured potential despite being 15mm above thesample surface. Thus, a minimal cantilever width and surfarea is desirable because it results in a steeper responseacteristic. A larger apex size as shown with geometries~o!and ~p! further steepens this response.
To summarize, optimum performance of KFMachieved when the sum of local electrostatic interactionsdominates the sum of nonlocal ones. This ratio is favoredlong and slender tips provided the tip apex is not too sm~geometry p!.
III. EXPERIMENTAL DATA
Test structures to characterize contrast transfer~Fig. 4!were fabricated using optical and electron beam lithogratechniques. GeNiAu ohmic contacts were deposited ontoInP-based depletion-type high electron mobility transisheterostructure and annealed at 340 °C in a nitrogen enviment. The contacts were electrically isolated by a mesausing a nonselective H3PO4– H2O2– H2O solution. The mesadefines a 100mm3100 mm area of heterostructure matericonnected to one ohmic contact, which is separated fromsecond ohmic contact by a 10mm wide gap. Metallic struc-tures with linewidths of 0.2–10mm were fabricated using atwo layer PMMA/P~MMA–MAA ! electron beam resist ana lift-off metallization technique. Ti and subsequently Awith a total thickness of 150 nm was evaporated and forma Schottky contact to the underlying semiconduc~In0.53Ga0.47As).
Figure 4~a! shows a composite light microscope imaof our microfabricated test sample with Au lines varyingwidth from 200 nm to 10mm. Typical examples of our experimental data are shown for topography in Figs. 4~b! and
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4~d! and for the corresponding KFM potential measureddifferent tip–sample separations in Figs. 4~c! and 4~e!, re-spectively. For clean surfaces without any surface contanations, oxides, isolated charges, or condensed water fithe potential difference~contact potential! between two dif-ferent materials is equal to the difference in work functio5
The potential difference measured in air at ambient pressbetween the large Au pad~P1! and the surrounding InGaAsubstrate~P2! is 330 mV which is smaller than the 450 m(Wvac,Au55.1 eV,Wvac,In0.53Ga0.47As'4.65 eV) reported forclean surfaces.18–20 To verify our modeled spot-size response, we measured the potential difference between thmetallization and the substrate for different tip–sample serations and normalized the data with the contact potenmeasured between P1 and P2@Fig. 4~a!#. The collected re-sults of all measurements are shown in Fig. 5. The error-bindicate the standard deviation of the four measurements
FIG. 3. ~a! Tip geometries,~b! modeled spot-size, and~c! step response for15 nm tip–sample separation. The insets show a close up view of thecontrast transfer characteristics. The solid lines are splines fitted to the meled normalized potential valuesFm /Fspot andFm /Fstep ~discrete points!,respectively.
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eraged for each data point. The modeled spot-size resp~solid lines! and the observed response are in good agment although the measurements were taken on lines anon spots.
FIG. 4. Microfabricated test structures to measure the contrast transfdifferent sized Au lines.~a! Composite light microscope image of differenchip regions. Topography~b!,~d! and KFM potential for different tip–sample separations~c!,~e! taken on the regions marked in~a!.
FIG. 5. KFM potential~open circles! measured on microfabricated Au lineof width d @Fig. 4~a!# and modeled spot-size response~solid line! for elec-trodes of diameterd plotted at different tip–sample separationsh. For com-parison, measured KFM potentials are normalized to the potential differebetween P1 and P2 in Fig. 4~a!. Inset: tip model used for simulation withcharacteristics of actual tip used in experiment.
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To obtain a test structure for the step response we facated a perforated metal film on a GaAs substrate followpublished protocols.12,21 To achieve an even distribution othe 30% aqueous NaCl solution on the hydrophobic Gasurface, the latter was covered with a piece of rice pabefore a drop of the salt solution was added. The rice pahelped spread the drop and was removed once the salttion had completely dried. After evaporation of 8 nm Pt–the whole structure was rinsed with ultrapure water to dsolve the salt crystals and create the desired structure.
Figure 6 displays topography and KFM potential of sua perforated Pt–C film taken in air at ambient pressure. Tmeasured potential difference between the 8 nm thick Pfilm and the GaAs substrate is 540 mV which is smaller ththe 800 mVWvac,Pt–C'5.6 eV, Wvac,GaAs'4.8 V) reportedfor clean surfaces19,20 and thick films.22 We measured theKFM potential along the three lines indicated in Fig. 6~b!,normalized the data using the total measured potential difence of 540 mV, and calculated the mean value of the msured responses. Figure 6~c! compares these mean respons~solid lines! with the modeled step responses~discretepoints! for different tip–sample separationsh. Again, we ob-tain good agreement between the predicted step responsthe measured data. Near the transition (usu,500 nm) thepredicted response is slightly steeper than the experimenobserved response. This is not surprising because we useideal step potential for our simulations. In an actual potendistribution the transition is defined by the length of tspace charge region, which always has a finite length.
of
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FIG. 6. Step-response measured on perforated 8 nm thick Pt–C filmGaAs substrate:~a! topography,~b! KFM potential at 15 nm tip–sampleseparation,~c! step-response~solid line! averaged from measurements alonthe three lines indicated in~b! and normalized to the maximum potentiadifference between the Pt–C film and the GaAs substrate. Open symmark the modeled step-response calculated with a tip~inset! matching thecharacteristics of the actual tip used in the experiment.
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IV. CONCLUSION
Kelvin probe force microscopy offers an attractivmethod to obtain high-resolution maps of the surface potial distribution on conducting and nonconducting samplThe technique is nondestructive and minimally invasive atherefore can be applied as a viable tool in integrated cir~IC! analysis.8,9,12 Once the technique becomes quantitatand the actual surface potential distribution can be extrafrom the measured values, KFM could substantially facilitthe design and experimental validation of new semicondtor devices. We have taken a first step in this directionintroducing a model which links the actual surface potendistribution to the measured KFM potential on ideally coducting and perfectly clean surfaces. It was shown thatobserved KFM potential is a weighted average over alltentialsF i on the surface, the derivatives of the capacitanbetween specimen and tip,Cit8 , being the weighting factors@Eq. ~8!#. Although, measured KFM potential maps candeconvoluted in the case of perfectly flat and ideally coducting surfaces, such a linear and space-invariant relatship no longer exists for structured surfaces as typicallycountered on integrated circuits. Nevertheless formoderately structured test specimens reported here,model could accurately predict the experimental data withany additional fitting parameters.
To improve resolution and accuracy of KFM evenhighly structured surfaces special attention must be paithe geometry of the tip and the cantilever. Our numerisimulations have shown that the cantilever surface predonates the local electrostatic interaction when the apex siztoo small. In this case the cantilever9arm9 will contribute tothe observed electrostatic interaction in an orientation depdent manner. In KFM resolution, accuracy of the measupotential can be improved using a long and slenderslightly blunt tip supported by a cantilever of minimal widand surface area.
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ACKNOWLEDGMENTS
The authors thank Dr. H. F. Knapp for critically readinthe manuscript. This work was supported by the ETH Zurand the Swiss National Science Foundation, NRP Nasciences, Grant No. 4036-044062.
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