Mathematical analysis of the operation of a scanning Kelvin probe
MMathematical analysis of the operation of a scanning Kelvin probe
Edgar A. Pina-Galan a) Manchester Institute of Biotechnology, University of Manchester, Manchester M13 9PL,United Kingdom b) (Dated: July 18, 2019) The scanning Kelvin probe is a tool that allows for the contactless evaluation of contact potential differencesin a range of materials, permitting the indirect determination of surface properties such as work function orFermi levels. In this paper, we derive the equations governing the operation of a Kelvin probe and describethe implementation of the off-null method for contact potential difference determination, we conclude with ashort discussion on design considerations.PACS numbers: 07.79.–v, 81.70.–qKeywords: scanning Kelvin probe, contact electrification, contact potential difference, work function
I. INTRODUCTION
The Kelvin probe is a non-contact, non-destructivetechnique based on the so-called contact electrificationthat occurs in a parallel-plate capacitor when electricalcontact between its plates is established. This tool isused to evaluate information on a surface topography; typically used to measure work function in metals, itcan also quantify surface potential difference for non-metals or Fermi levels in organic thin films. The principle of operation was first proposed by Kelvinin 1898 as a means to measure the contact potential dif-ference V CPD of two electrically connected metals withoutphysical contact between each other inner surfaces. Themethod was based on the measurement of the surfacecharge variation ∆ Q s between the sample surface and anelectrically conductive reference electrode in a parallel-plate capacitor configuration in order to determine the V CPD between both materials. A backing potential V b was applied to one of the plates, then, displacing the ref-erence electrode a certain distance ∆ d from the samplesurface will induce a change in capacity ∆ C , Kelvin thenused a galvanometer to measure the surface charge Q s .This process should be repeated until ∆ Q s = 0 whichindicated that the potential difference between the sam-ple material and reference electrode was equal to the biasvoltage ( i.e. , − V CPD = V b ).Zisman improved this experiment in 1932 by usinga piano wire to set the reference electrode into vibra-tion, generating an oscillating capacitive charge C K ( t )between the two surfaces. The result is an electric cur-rent i ( t ) = δQ s ( t ) /δt induced on the reference electrodewhich was further converted into an audio-frequency sig-nal. A backing potential V b was applied and manuallyadjusted until no signal was heard, indicating that V b equalled − V CPD . Copyright (2019) Edgar Alexis Pina Galan. This article is dis-tributed under a Creative Commons Attribution (CC BY) Li-cense.
Later, in 1999, Baikie presented a Kelvin probe sys-tem fully controlled by a computer. The inclusion of acomputer allowed for the introduction of the so-calledoff-null method for determining V CPD . Additionally, bymaintaining a constant probe-to-sample distance, it be-came possible to automatically scan a relatively irregularsurface.In this paper we provide the background and derivethe equations governing the operation of a Kelvin probeand the off-null method for determining V CPD . The equa-tions provided can be readily applied to the simulationor design of a scanning Kelvin probe system capable ofobtaining reliable measurements.
II. ANALYSIS OF THE OPERATION OF A KELVINPROBEA. Operation principle
The scanning Kelvin probe operates by creating aparallel-plate capacitor between an electrically conduc-tive probe vibrating perpendicularly and the nearest sur-face it approaches (Fig. 1). When external electricalcontact between probe and sample takes place by meansof the application of a backing potential V b , the Fermilevels of the probe and sample surfaces start to equalize,resulting in a charge flow which in turn generates a con-tact potential difference V CPD . Determination of V CPD permits the indirect calculation of surface properties suchas work function, surface potential, or Fermi levels. B. Contact potential difference
The contact potential difference V CP D between thesurfaces of two materials arises from a phenomenon re-ferred to as contact electrification. Whenever two dissim-ilar materials are brought into contact, thermodynamicequilibrium mechanisms cause the electrochemical poten-tials of their charged particles to equalize. Therefore, the a r X i v : . [ phy s i c s . i n s - d e t ] J u l Figure 1. Schematic of the principle of operation of a Kelvinprobe. Where V CPD is the probe-to-sample contact potentialdifference, d is the mean probe-to-sample distance, d is theprobe oscillation amplitude, i ( t ) is the current induced on theprobe tip, V out is the voltage output of the probe circuit, and V b is the backing potential. Since the capacitance is a func-tion of the separation between the plates, the vibration of theprobe results in an oscillating capacitive charge, generatinga current i ( t ) that can be converted into a voltage V out forfurther processing of the signal. contact potential difference V CP D is related to the Fermilevel and work function of the material.We can define the work function of a surface as thedifference in potential energy of an electron between thevacuum level and the Fermi level; with a magnitudegiven by Φ = − eφ − (cid:15) F , (1)where Φ is the work function of the surface, − e is theelementary charge, φ is the electrostatic potential in thevacuum, and (cid:15) F is the Fermi level of the material. There-fore, the term − eφ represents the energy of an electronat rest in the vacuum.With this in mind, a contact electrification scenariocan then be depicted by the system in Fig. 2. Whentwo dissimilar materials, A and B, are approached witha distance d separating their inner surfaces, and consid-ering that Φ A (cid:54) = Φ B : (a) if there is no physical contactbetween materials A and B, electrons remain at theirrespective Fermi levels, (cid:15) AF and (cid:15) BF , with work functionsΦ A and Φ B , respectively; (b) when electrical contact isestablished, highest energy electrons on the outermostlevel bands will flow from the lowest- to the highest-workfunction material until all particles at the Fermi levels (cid:15) AF and (cid:15) BF are distributed uniformly and equilibrium isreached. Work function remains unchanged as it beingan intrinsic property of materials. For the case whereΦ A > Φ B , electrons flow from surface B to A, this fluxpolarizes the inner faces of the materials with the nega- tive pole at the surface receiving the electrons, resultingin a contact potential difference V CPD equivalent to thedifference in work functions of the materials A and B;this is expressed as V CPD = (Φ B − Φ A ) e = ( (cid:15) AF − (cid:15) BF ) e . (2) Figure 2. Electron energy level diagrams of two dissimilarmaterials, A and B, separated by a distance d : (a) withoutcontact and (b) with external electrical contact, allowing asurface charge flow Q s that originates a contact potential dif-ference V CPD . (cid:15) AF and (cid:15) BF refer to the Fermi levels, and Φ A and Φ B are the work functions of the materials A and B,respectively. The contact potential difference V CP D can then be de-fined as the energy required to transfer an electron fromone surface to another between two materials with dis-similar work functions. Since V CP D is related to Φ and (cid:15) F , it is possible to indirectly determine Φ or (cid:15) F whenmeasuring V CP D between a sample and a reference ma-terial with known Φ or (cid:15) F , respectively.In order to determine the contact potential difference V CPD , a known voltage can be applied between the probeand sample and varied until there is no measurable cur-rent induced on the probe, indicating that its magnitudehas equalled V CPD , this traditional concept is referredto as the null method. However, in modern devices, aso-called off-null approach is rather employed due to thehigher signal-to-noise ratio provided as well as allowingthe automatic scanning of an irregular surface.
C. Off-null method
The off-null method can be employed in order to de-termine the probe-to-sample contact potential difference V CPD as well as to maintain a constant mean probe-to-sample distance d during automatic scanning witha Kelvin probe. Maintaining a constant d is essentialto prevent erroneous V CPD readings arising from spacingchanges. Its implementation involves lower costs whencompared with the traditional null method, which re-quires the use of a lock-in amplifier (widely employed inatomic force microscopes). Additionally, it offers a highersignal-to-noise ratio by avoiding common sources of mea-surement errors related to the null point such as straycapacitance or talk-over noise from the actuator. The method consists on applying at least two valuesof a backing potential V b , the value for V CPD can subse-quently be retrieved from the linear relation between V b and the voltage V out delivered by the device, which is alinear function of the form V out = m K V b + c, (3)where the value for V b corresponding to V out = 0 isequal to V CPD , i.e. , V b = − V CPD , and m K is termed theKelvin gradient (Fig. 3). Figure 3. Schematic diagram representing the output voltage V out vs backing potential V b function: curve (a) representsthe plotted function of the form V out = m K V b + c , (b) showsa change in the gradient m K induced by a mean probe-to-sample distance d variation, and (c) shows the effect of avariation in work function Φ on the sample material. Therefore, given the two backing potential values V and V , that elicit the responses V and V , respec-tively, and assuming that the value of V CPD lies anywherebetween V and V , V CPD is calculated by simple linearinterpolation as V CPD = V + ( V − V ) − V V − V . (4)Additionally the Kelvin gradient m K of this function isinversely proportional to the mean probe-to-sample dis-tance squared ( i.e. , m K ∝ d − ); this will be deduced in section III. Hence, a constant value for m K can bemaintained during scanning by calculating its value as m K = V − V V − V (5)and instructing a translation system to compensateany variations with a corresponding adjust of d via afeedback loop. Maintaining a constant mean probe-to-sample distance d is an essential requirement to obtainreliable measurements over an irregular surface. III. KELVIN PROBE ELECTRIC CIRCUIT ANALYSIS
Let us consider the electric circuit in Fig. 4, whichrepresents a parallel-plate capacitor formed between asample surface and an electrically conductive referenceelectrode, where V CPD is the contact potential differencegenerated between the plates, d is the mean probe-to-sample distance, C K ( t ) is the time-varying capacitance,and i ( t ) is the current induced due to the oscillation ofthe reference electrode. A backing potential V b is intro-duced to close the circuit, while V out is measured in orderto determine V CPD . The Kelvin probe usually consistsof an operational amplifier as a current-to-voltage
I/V converter component whose output is further amplifiedand filtered before being input to a data acquisition sys-tem. For this analysis, the Kelvin probe circuit has beenreduced by replacing the
I/V converter for its effectiveresistance R in . Additionally, current lost due to straycapacitance C s is neglected under the assumption thatthe contribution of C s in the magnitude of the effectiveimpedance of the circuit is minimal. An analysis of straycapacitance on the Kelvin probe circuit can be found inRefs. 11 and 14.
Figure 4. Reduced circuit of a Kelvin probe, where the
I/V converter is represented by its effective resistance R in andcurrent lost due to stray capacitance C s is neglected. According to Gauss’s law, the electric field between theplates of a parallel-plate capacitor is given by −→ E = Q s ε r ε A p , (6)which for our purpose can be rewritten as −→ E d = V CPD = Q s d ε r ε A p , (7)where −→ E is the electric field magnitude, Q s is the sur-face charge, A p is the area of the probe interacting withthe sample surface, and ε r and ε are the relative per-mittivity of the medium and vacuum, respectively.Since charge is defined as Q s = V CPD C K , (8)it can be noted from Eqs. 7 and 8 that C K = ε r ε A p d . (9)For a parallel-plate capacitor with an average gap d between its plates, when a sinusoidal displacement withamplitude d and frequency f is introduced on one of theplates, the distance d ( t ) separating the plates over timeis described by d ( t ) = d + d cos 2 πf t = d (1 + γ cos ϕ ) , (10)where γ ≡ d /d is termed the modulation index, and ϕ = 2 πf t . Equation 9 then becomes C K ( t ) = ε r ε A p d (1 + γ cos ϕ ) , (11)where C K is now referred to as the time-varying C K ( t ).When a backing potential V b is introduced, Eq. 8 takesthe form ∆ Q s = ( V CPD + V b )∆ C K . (12)If the probe is set into vibration, the oscillation of C K generates a current i ( t ) on the probe due to the periodicchange in charge δQ s ( t ) /δt . Equation (12) can then berewritten as δQ s ( t ) δt = i ( t ) = ( V CPD + V b ) δC K ( t ) δt . (13)Equation 11 is now substituted into Eq. 13 to obtain i ( t ) = ( V CPD + V b ) ε r ε A p d δδt (cid:20) γ cos ϕ ) (cid:21) . (14)The derivative of i ( t ) yields a function that describesthe current induced at the vibrating probe tip as i = ( V CPD + V b ) (cid:20) ε r ε A p d (cid:21)(cid:20) πf γ sin ϕ (1 + γ cos ϕ ) (cid:21) . (15)The output voltage is then given according to Ohm’slaw by V out = GR f C ωγ ( V CPD + V b ) (cid:20) sin( ϕ + α )(1 + γ cos( ϕ + α )) (cid:21) , (16)where G = − R /R is the pre-amplifier gain of theKelvin probe circuit, R f is the feedback resistance of the I/V converter, ω = 2 πf is the angular vibration fre-quency, α is a phase angle, and C ≡ ε r ε A p d (17)is termed the mean capacitance.The sensitivity S directly affects V out intensity on aKelvin probe circuit (Fig. 5) and can be defined fromEq. 16 as S ≡ GR f C ωγ. (18) t V ou t S S S S Figure 5. Representation of the effect of the sensitivity S ona typical output signal V out of a Kelvin probe circuit. Fourarbitrary S values are employed, where S < S < S < S . Additionally, V out is modulated by the last term of Eq.16, the effect of the modulation index γ on V out can beappreciated in Fig. 6. t V ou t = 0 = 0.25 = 0.5 = 0.75 = 1 Figure 6. Effect of the modulation index γ on the outputsignal V out of a Kelvin probe circuit. For lower γ values ( ∼ γ < . V out approaches a sine function, higher γ values( ∼ γ >
1) result in unreadable V out outputs, where V out ischaracterized by large peaks over short time periods and nomeasurable V out for the majority of the signal. Therefore, it isdesirable to operate under low modulation index conditions, i.e. , ∼ γ < . If required, with the assumption of a low modulationindex ( ∼ γ < . d < d , anal-yses can be simplified by reducing Eq. 16 to V out = GR f C ωγ ( V CPD + V b ) sin( ϕ + α ) . (19)Finally, the implementation of the off-null method for V CPD determination (section II C) is based on the linearfunction V out = m K V b + c originated from the relationbetween V b and V out , where the Kelvin gradient m K isgiven by m K = 2 GR f C ωγ. (20)Since C ≡ ε r ε A p /d and γ ≡ d /d , m K can berewritten as m K = 2 GR f ε r ε A p ωd d , (21)where it is observed that all the parameters in the nu-merator are constants, revealing the relation m K ∝ d − , (22)which can be exploited to implement the automaticscanning of irregular surfaces on a Kelvin probe by com-pensating d variations with a corresponding m K adjust via a feedback loop. IV. DESIGN CONSIDERATIONSA. Lateral resolution
The lateral resolution can be defined as the ability of animaging system to differentiate a parameter between twoseparate points in space. In a Kelvin probe, this propertyis limited mainly by the dimensions and shape ofthe probe tip, while also being strongly related to themean probe-to-sample distance d . High lateral res-olutions are characterized by flat probe tips with smallarea A p and small d . In this paper we analyzed onlythe scenario of a flat probe tip.
B. Sensitivity
The output signal V out , and therefore the feasibility ofthe determination of V CPD , is influenced by the sensitiv-ity S of the Kelvin probe. Equation 18 shows that reduc-ing the tip area A p , while increasing the lateral resolu-tion, also causes a decrease in S . Nonetheless, is inferredthat an S value can be maintained as long as any reduc-tion in A p is compensated by a proportional increase invibration frequency f . Equation 18 also shows that asmaller mean probe-to-sample distance d will increase S . However, a smaller d will also increase the modula-tion index γ , which is preferably kept low ( ∼ γ < . d for increasing the sensitivity is thuslimited by the relation d < d , necessary to guarantee alow γ value. C. Signal-to-noise ratio
The signal-to-noise ratio plays an important role in theaccuracy of V CPD determination on a Kelvin probe, forthis aspect, the implementation of the off-null methodmight suffice. Selecting backing potential V b values farenough from the null point increases the accuracy byavoiding random noise generated when V b values ap-proach the null point. Additionally, while it is possibleto retrieve V CPD by linear extrapolation or interpolation,the later is preferred as the former is subject to greateruncertainty. Furthermore, while any number of backingpotentials can be applied, two has been found to be theoptimal value that allows for the lowest uncertainty in V CPD determination. The actuator selection and wiring of the system is alsoa factor to account for, e.g. , the high voltages requiredto drive piezoelectric actuators can constitute a sourceof noise. A discussion on sources of noise and trou-bleshooting strategies can be found in Refs. 12 and 13.
ACKNOWLEDGMENTS
The author would like to acknowledge the financialsupport from the Tsinghua-Berkeley Shenzhen Institute(TBSI) and the National Council of Science and Technol-ogy of Mexico (CONACyT) with the TBSI PhD Schol-arship and the CONACyT Scholarship No. 424671, re-spectively.
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