Analytical Approach to the Local Contact Potential Difference on (001) Ionic Surfaces: Implications for Kelvin Probe Force Microscopy
Abstract
An analytical model of the electrostatic force between the tip of a noncontact Atomic Force Microscope (ncAFM) and the (001) surface of an ionic crystal is reported. The model is able to account for the atomic contrast of the local contact potential difference (CPD) observed while ncAFMbased Kelvin Probe Force Microscopy (KPFM) experiments. With the goal in mind to put in evidence this shortrange electrostatic force, the Madelung potential arising at the surface of the ionic crystal is primarily derived. The expression of the force which is deduced can be split into two major contributions: the first stands for the coupling between the microscopic structure of the tip apex and the capacitor formed between the tip, the ionic crystal and the counterelectrode; the second term depicts the influence of the Madelung surface potential on the mesoscopic part of the tip, independently from its microscopic structure. The former has the lateral periodicity of the Madelung surface potential whereas the latter only acts as a static component, which shifts the total force. These shortrange electrostatic forces are in the range of ten picoNewtons. Beyond the dielectric properties of the crystal, a major effect which is responsible for the atomic contrast of the KPFM signal is the ionic polarization of the sample due to the influence of the tip/counterelectrode capacitor. When explicitly considering the crystal polarization, an analytical expression of the bias voltage to be applied on the tip to compensate for the local CPD, i.e. to cancel the shortrange electrostatic force, is derived. The compensated CPD has the lateral periodicity of the Madelung surface potential. However, the strong dependence on the tip geometry, the applied modulation voltage as well as the tipsample distance, which can even lead to an overestimation of the real surface potential, makes quantitative KPFM measurements of the local CPD extremely difficult.
pacs:
07.79.Lh, 41.20.Cv, 73.40.CgPublished in Phys. Rev. B 78, 035410 (2008)
I Introduction
Electrostatic forces play a key role in noncontact Atomic Force Microscopy (ncAFM), not only in the imaging process (1) but also for the investigation of the surface electronic properties. Electronic properties such as the work function and surface charges can be acquired by Kelvin Probe Force Microscopy (KPFM) (2); (3) simultaneously to imaging topography by ncAFM. In KPFM, a feedback is used to apply a voltage between the tip and the sample in order to minimize the electrostatic tipsample interaction. For metals, this voltage is equal to the contact potential difference (CPD), i.e. the bias voltage to be applied between the tip and the surface to align their fermi levels. It is connected to the difference between the work functions of the two surfaces, and thereby to their local electronic properties, according to:
(1) 
being the elementary electrical charge: C.
Nowadays, two KPFMbased techniques provide facilities to map the spatial variations of the CPD on the nanometer scale, namely Frequency (4) or AmplitudeModulationKPFM (5); (6); (7) (FM or AMKPFM, respectively). These methods were primarily applied to metallic and semiconducting surfaces to study the distribution of dopants in semiconductors (8), or the adsorption of organic molecules (for an overview see ref.[(9)]). In a few experiments, even molecular (10) or atomic (11); (12); (13) contrast has been reported. The extension of the technique to insulating surfaces, was performed more recently, as demonstrated by the results reported on thin ionic films on metals (14); (15), or on the contribution of bulk defects to the surface charge state of ionic crystals (16); (17).
In this work, atomic corrugation of the CPD signal is reported for the first time on the (001) surface of a bulk ionic crystal of KBr. For that purpose, KPFM experiments were carried out in ultrahigh vacuum with a base pressure below mbar using a home built noncontact atomic force microscope operated at room temperature (18). A highly doped silicon cantilever with a resonance frequency kHz, and a spring constant N.m was used. The typical oscillation amplitude of the fundamental bending resonance was nm. The cantilever was annealed (30 min @ 120) and gently sputtered with Ar ions (12 min @ 680 eV). The KBr crystal was cleaved in ultrahigh vacuum along the (001)plane and subsequently annealed at 120 during an hour. The KPFM signal was detected using the AM mode, as described in detail in Ref. (5). While these experiments, atomicscale contrast was as well visible in the topography channel (data not shown). The CPD measurements are reported in figs.1a and b. In fig.1a, the image exhibits atomic features, the measured period of which is 0.63 nm, which is visible in the joint cross section. This value matches to a good agreement the lattice constant of KBr, 0.66 nm. The vertical contrast yields about 100 mV around an average value of 3.9 V, the origin of which will be discussed in section IV.2. A striking aspect of those results is the robustness of the imaging process in terms of stability and reproducibility upon the tips used. These results suggest an intrinsic imaging process relying on the microscopic origin of the contact potential arising at the sample surface. In this case, the CPD rather turns into local CPD, consistently with the concept of local work function which has been introduced by Wandelt on metals(19). By “intrinsic imaging process”, it is meant that the contrast can be accounted for with a tip consisting of a single material, namely a metal. Thus, the atomic contrast neither relies on adsorbed nor on unstable species at the tip apex, as often reported for topographic(20) or dissipation(21) imaging by ncAFM.
In this work, in order to understand the local CPD contrast formation, an electrostatic model is proposed that allows us to derive an expression of the shortrange electrostatic force occurring between a biased metallic tip of a ncAFM microscope and the surface of a bulk dielectric. On the contrary to more refined numerical methods proposed in the literature for almost similar systems(22); (23), the analytical development is restricted to a simple tip geometry and a classical, continuous electrostatic approach. Notwithstanding, the model allows us to define a general frame, able to put in relation the surface electrostatic properties with the imaging process yielding atomic contrast of the CPD on ionic surfaces. Obviously, the main results presented here remain qualitatively correct for more complex tip geometries, although numerical methods are then required to get quantitative numbers.
The motivations for that work are twofold. On the one hand, to our knowledge, a compact modelization of the shortrange electrostatic forces responsible for the atomic contrast in KPFM on a bulk ionic crystal is still lacking. On the other hand, when evaluating KPFM experiments, the relative complexity of FM or AMKPFM experimental setups, both including four electronic controllers, makes the interpretation of the experimental images, and primarily CPD images, tedious, especially when dealing with atomic resolution. Several groups analyzed the KPFM imaging mechanism in order to evaluate their data in terms of quantitative values and lateral resolution(24); (25); (26); (27); (28); (29); (30); (31); (32); (33), a few of them also compared AM and FMKPFM in terms of evaluating the force and its gradient, respectively(7); (31). The fact that many of the above mentioned calculations point out that KPFM results, especially for nanoobjects, show a strong distance dependence, clearly points out that for a careful analysis, it is not sufficient to only calculate the electrostatic tipsample interaction. It is rather important to perform simulations including all imaging mechanisms and in particular also the distance control in order to exclude artifacts due to the feedback circuits. In order to explore the origin of the CPD atomic contrast, we are aiming to closely mimic a real KPFM setup by means of an earlier developed ncAFM simulator(34). In the present case of an ionic surface, it is required to consider a large slab of ions, virtually infinite, to describe properly the electrostatic interaction. This is hardly feasible by means of ab initio calculations which fail to describe tipsurface systems involving a too big number of atoms. Therefore, prior to simulating the CPD contrast on ionic surfaces by means of our simulator, which is the scope of a future work, we have to find an analytical expression for the electrostatic tipsample interaction.
The following section details the boundaryvalue electrostatic problem leading to the expression of the force (section III). In section IV, the analytical expression of the local CPD is derived and discussed to be put in relation with the experimental observations. The implications for KPFM experiments are discussed as well.
Ii Electrostatic model
The geometry of the problem of FM or AMKPFM experiments applied to bulk insulating materials is reported in fig.2b. The dielectric sample is an alkali halide crystal like NaCl, KBr, KCl with a fcc structure and a lattice constant (cf. fig.2a). In the area where the tip is, the model assumes that the crystal carries neither net charge, nor local dipole. Its thickness is much larger than all other distances of the problem, typically a few millimeters. Below the surface, the crystal is treated as a continuous dielectric medium with a dielectric permittivity . At the surface, the atomic corrugation of the crystal is described by a single layer of alternate point charges arranged with a fcc structure perpendicular to the [001] direction. The layer extends infinitely in the plane direction. The motivation for such a rationalization of the problem will be justified in section (II.2).
The crystal lies on a metallic sample holder (hereafter referred to as the counterelectrode), with respect to which the tip is biased. The counterelectrode is a planar and perfect conductor. The tip is also assumed to be a perfect conductor which is biased at . In order to preserve the analyticity of the model, the electrostatic boundaryvalue problem is restricted to a tip with a very simple apex geometry, namely: a hemispherical mesoscopic part (radius nm) on top of which is halfembedded a small spherical asperity (radius ). The contributions of the cantilever and of the macroscopic part of the tip to which the apex is connected to are assumed to be negligible. This issue will be justified in section III.1. The tipsurface distance is denoted (typically a few Å).
The electric field produced locally between the tip, the dielectric, and the counterelectrode polarizes the ionic crystal, which acquires a macroscopic polarization oriented along . In the former equation, is the number of polarizable species per unit volume and , the local dipolar moment per unit cell. In KPFM, the bias voltage is modulated at frequencies which do not influence the electronic polarization of the ions. The major part of the crystal polarization has rather an ionic character, i.e. a net displacement in opposite directions of the ions due to their charge with respect to their equilibrium positions, , with , being the elementary electrical charge. The polarization effect occurs as well at the crystal surface, where the positions of the ions become modulated perpendicularly to the surface plane, i.e. , as sketched in figs.2b and d. is proportional to the local electric field at any ionic site and to the total polarizability of the dielectric restricted, in our approach, to the ionic polarizability (ref.[(35)]). Usually, differs from the external electric field due to the biased tip because explicitly depends on the polarization of the dielectric. The Lorentz’s model links and (ref.[(36)]), thus, is written:
(2) 
with , the dielectric susceptibility of the sample. In the former equation, it is important to notice that depends on and hence on the bias voltage . Consequently, this is also true for . For the sake of discussions, the bias dependence will henceforth be explicitly outlined . Despite cannot be estimated at this point, it is crucial to keep in mind that the sample surface is polarized by the influence of the bias since this is a key aspect of the origin of the CPD atomic contrast.
Our approach of the electrostatic boundaryvalue problem relies on an ad hoc assumption. The tip, being a perfect conductor, develops a surface charge density , the origin of which is split into two main contributions , namely:

a charge density due to the capacitor formed between the biased tip and the counterelectrode with the dielectric in between (cf. fig.2c). Owing to the distance between the electrodes, i.e. roughly the dielectric thickness , has a mesoscopic character. It is not influenced by the local structure of the tip apex, but rather by its overall shape.

a charge density originating from the Madelung potential that expands at the crystal surface. When the tip is at a distance where the potential is effective, typically a few Ångströms, then it develops, in addition to , a surface charge density (cf. fig.2d). has a microscopic character and must strongly depend on the local structure of the tip apex and on .
Despite the simple tip geometry that has been assumed, the calculation of the electrostatic force acting on it due to the combined influence of the capacitive coupling and of the Madelung surface potential has no exact analytical solution. Nevertheless, one can build up an approximate solution to the boundaryvalue problem relying on the superposition principle. For that purpose, the problem is split up into two boundaryvalue subproblems schemed in figs.2c and d:

problem : the tip biased at in front of a dielectric continuous medium (height , dielectric permittivity ) held on an infinite planar conductor, the counterelectrode (fig.2c). The local structure of the dielectric is not supposed to influence the tip. This is the description of the “capacitive” problem, the solution of which provides the surface charge density .

problem : the tip now biased at 0 V with an infinite plane of alternate point charges located at the same distance than the surface of the dielectric in problem A, i.e. . The layer of point charges is polarized under the electric field that occurs in problem A (fig.2d). This describes the “microscopic” problem. The solution provides the surface charge density .
Besides, in order to carry out the calculations more easily, it is convenient to distinguish the two following geometrical areas on the tip (cf. fig.2d), namely: the asperity, area (1), i.e. a halfsphere with radius and the mesoscopic tip apex around it, area (2), a sphere with radius (typically ). These two areas do not overlap, but the continuity between them is ensured. The vertical force acting on the tip(37) is written in terms of and as described above:
(3) 
and are the normal to the tip surface and the unitary vector along the vertical axis of the problem, respectively. Doing so, we only focus on the vertical resultant of the force acting onto the tip. The above expression can be expanded into three components: a purely capacitive part, originating from the tip/dielectric/counterelectrode capacitor; a coupling part, , which can be interpreted in terms of the resulting force of all the elementary forces due to the electric field onto each elementary charge produced on the tip by the influence of the Madelung potential of the crystal, ; and a purely microscopic part, , standing for the shortrange electrostatic force due to .
ii.1 Problem A: estimation of
Although the boundaryvalue problem of a planar conductor biased with respect to another one with an incomplete dielectric layer in between yields an expression of the surface charge density, the problem with the sphere does not. However, one can argue that the expression of must be a combination between a configuration in which there is no dielectric medium in the sphere/counterelectrode interface and an opposite one, where the interface is completely filled with it. One can therefore postulate an effective dielectric permittivity , where is a constant to be set. Owing to the fact that the mesoscopic part of the tip apex, referred to as area (2), is located at a distance from the counterelectrode, the analytical expression of , explicitly calculated in refs.[(37); (38)], asymptotically trends towards the surface charge density of an isolated, biased sphere (37):
(4) 
To get the surface charge density on area (1), , we seek the potential which equals over the asperity. For that purpose, a spherical coordinate system centered on the asperity is chosen. The problem having an azimuthal symmetry, the sought potential can be expanded in Legendre polynomials (37):
(5) 
At large distance from the asperity, , the potential must be similar to the one of a sphere with radius biased at , namely (37):
(6) 
Hence, the coefficients of the expansion are known. The coefficients are deduced from the property of orthogonality of the Legendre polynomials at the boundary condition . The potential is finally written:
(7) 
The former equation rigorously describes the potential of a system apex/asperity where the junction point between the two spheres is smoothed and not singular, as sketched in fig.2d. Nevertheless, for , the influence of the singular area is negligible. Therefore, equ.7 is a good approximation to the boundaryvalue problem. The normal derivation along the surface of the asperity yields the expression of :
(8) 
Owing to the condition , the sum can be restricted to the first term. Therefore:
(9) 
Thus, despite develops on the asperity, its strength is governed by the radius of the mesoscopic part of the tip apex, and not by the local radius of curvature of the asperity, .
ii.2 Problem B: estimation of
The calculation of the surface charge density on the mesoscopic sphere+asperity is more tedious, primarily because it relies on the estimation of the Madelung potential of the ionic crystal, . The boundaryvalue problem is now restricted to the determination of the surface charge density arising on a metallic tip at zero potential under the influence of an infinite planar slab of point charges. Again, the solution of such a problem has no straightforward analytical solution. However, we can again use the condition , as depicted, to some extend, in fig.2d. Consequently, the mesoscopic part of the apex can be assumed as equivalent to an infinite planar conductor, at least in a small area along the sides of the asperity (light grey area in fig.2d). The new boundaryvalue problem defined by an infinite planar conductor influencing another infinite planar conductor at zero potential carrying a hemispherical bump with a radius , now yields a solution (38). The method of the image charges is used to solve it. The first set of image charges ensuring a zeropotential value on the counterelectrode(39) produces an electric field which influences the tip. But as a matter of fact, this contribution can be neglected because the distance between the tip and the counterelectrode is on the millimeter range and the image charges originate from the Madelung potential, which is known to decay exponentially fast(40); (41) (cf. also hereafter). The second set of image charges(42) is quasipunctual and located at the center of the asperity. Hence, owing to the simplified geometry of the electrode, the problem is reduced to a sphere with radius at zero potential in the influence of two infinite planes, i.e. the slab and its image, which are antisymmetrically spaced with respect to the sphere. This procedure ensures a zeropotential on the approximated plane within which the asperity is embedded. Again, it is more convenient to use a spherical coordinate system centered on the asperity. Then, the surface charge density is derived by normal derivation along areas (1) and (2), namely:
(10) 
The potential is derived from the Madelung potential of the ionic crystal, previously referred to as , by the method of the image charges and the Kelvin transform (influence on a sphere biased at zero potential)(38), namely:
(11) 
The former equation fulfills the boundary condition everywhere along the surface asperity or along the surface of the local planar area around it.
can be estimated on the base of the work by Watson et al. (40). When considering an infinite planar slab of point charges, the authors state that the potential, socalled Madelung surface potential, reaches its asymptotic value in a very short distance normal to the slab. Consequently, the ions within the crystal at a distance only one lattice constant from the surface have Madelung potentials which are indistinguishable from those of the bulk. In other words, the tip will mainly be influenced by the surface potential and not by the one arising from the bulk part of the ionic crystal. This is why a single, infinite, layer of point charges is enough to describe the influence of the Madelung surface potential on the tip, which motivates our initial assumption. The potential is written:
(12) 
where is the polar vector of any ion of the surface slab in an orthogonal basis , being the projection of the center of the asperity on the surface, and the unitary vectors of the fcc unit cell. The summation is performed over the vectors of the reciprocal lattice of an arbitrarily defined unit cell and is proportional to the lattice constant of the fcc unit cell. is a structure factor:
(13) 
It is summed over the ions within the defined unit cell. The k ion carries an electrical charge . Its planar and perpendicular coordinates from the origin of the cell are given by the two vectors and . The latter reflects the polarization effect felt by the ion within the unit cell, previously referred to as . In equ.13, it is assumed that the polarization of the ions extends all over the (001) surface plane. In any case, it must extend over an much larger area than the tip asperity. This assumption is consistent with the electric field produced by area (2), which is constant within an area roughly scaling as the mesoscopic tip radius . is calculated from the unit cell defined in fig.2a (light grey). It consists of 4 anions and a cation weighting for a fourth and one, respectively. The vectors of the direct lattice are and , where . Owing to the exponential decay of the potential with , visible in equ.12, the calculation of the structure factor can be restricted to the first four reciprocal vectors, namely: ). The calculation yields:
(14) 
with: and , a spatial modulation term. and are the and coordinates of the center of the asperity projected onto the unit cell. Setting locates the asperity and therefore the tip on top of an anion, the reference ion within the defined unit cell. The above expression exhibits the expected exponential decaying behavior as a function of . The potential is reported in fig.3 for nm, pm and Å. The value of will be justified in section IV.
The expression of can now be derived from equ.10. The calculation of yields:
(15) 
with , the expression of in the spherical coordinate system centered on the asperity and , the spherical expression of another spatial modulation term given by: . The functions and are written:
(16) 
and:
(17) 
with: (see also the appendix). Thus, it was necessary to assume area (2) as an infinite plane in order to derive the expression of by means of the method of the images. In such a case, the expression of is rigorously derived from equ.10 and is explicitly given in ref.[(43)]. For and positions large compared to however, this description does not fit with the geometry of the tip apex defined in problem B. The “infinite plane” must actually be shrunk down to a spatially limited area around the asperity, as sketched in fig.2d. This is made possible when assuming that any planar area is the asymptotic limit of a sphere with large radius compared to its extension. The former statement is fulfilled by the condition . The surface charge density of a sphere with radius under the influence of the surface potential is also derivable from the Kelvin transform. Here, the spherical coordinate system is centered on the sphere with radius . is written:
(18) 
The functions and are written:
(19) 
and:
(20) 
Since and decrease exponentially fast as one moves away from the foremost position of the tip apex, it can readily be verified that equ.18 and the expression given in ref.[(43)] do fit for or . Thus, even in the vicinity of the asperity, expression 18 can be used instead of ref.[(43)]. Therefore, the most part of the contribution of the sphere to is restricted to a small area that can be assumed as locally planar.
The graph of the projection of on the tip apex (areas (1) and (2)) is reported in fig.4 on top of an anion (positive charge density on the asperity) at a distance Å from the surface and for nm and Å. The oscillations of at the surface of the tip due to the image charges of the crystal are readily visible, although their amplitude decreases exponentially fast along the sides of the asperity. The relevant oscillations in area (2) spread out in a circular area with only two unit cells radius. Owing to the exponential decay of the Madelung surface potential farther away from the asperity, the tip surface is too far from the crystal surface to produce relevant image charges. On top of the asperity, the surface charge density reaches about C.m. For a halfsphere with radius Å and with pm, this results in a local charge carried by the tip of about .
Iii Estimation of the force
iii.1 Estimation of
Owing to the geometry of the problem and since is equivalent to the surface charge density of an isolated, conducting sphere, we can now fully evaluate the purely capacitive component to the total force. The force effectively acting on the mesoscopic part of the tip apex can be derived from the image charge of the sphere placed at a symmetric position with respect to the counterelectrode, i.e. at a distance from the mesoscopic apex:
(21) 
With (ref.[(44)]) and (arbitrarily), then . For a typical ac modulation of the bias of about 0.5 V and considering nm and mm, the former equation gives an estimation for the mesoscopic component of the electrostatic force: pN. As suspected, this contribution is negligible compared to the other two components(45). Therefore the expression of the total force simplifies to:
(22) 
The reason why the tip has been restricted to its apex is now clear. Regarding the former capacitive force, owing to the distance between the tip and the counterelectrode, the contribution of a macroscopic body in addition to the mesoscopic apex would not change notably the total force. Regarding the other force components, and , owing to the exponential decay of the Madelung surface potential, the influence of the macroscopic body of the tip is expected to be negligible as well.
iii.2 Estimation of
Let us first focus on . The integral must be performed over the asperity and the mesoscopic sphere around it, areas (1) and (2), respectively. It is recalled that the two areas do not overlap each together. For each of them, the spherical coordinate system must be centered on the corresponding sphere. Thus:
(23) 
The radial coordinates are fixed to and in areas (1) and (2), respectively. The polar integration in area (1) is performed with , whereas in area (2), it is performed with , where , which ensures the continuity from (1) to (2). Regarding area (2), the choice of the beginning angle of the interval () does not influence notably the result of the integration. In other words, owing to the exponential decay of , the exact shape of the mesoscopic part of the tip apex far from the asperity is not relevant. This justifies a posteriori the choice, although simple, of a spherical geometry. Both integrations over the azimuthal angle must be performed over . Note that in each of the former integrals, the term stands for the vertical projection of the force, as stated initially. Although the integration over the azimuthal angle yield an analytical result, the integration over does not, which requires to evaluate some integrals numerically. The expression of is written:
(24) 
with: and , the reduced coordinates of the tip above the crystal surface. The integral forms of coefficients , and are reported in the appendix as functions of and . Taking a typical lattice constant for alkali halides nm and assuming Å, we get:
(25) 
Thus, explicitly depends on the spatial modulation of the surface potential. Note also the exponential decay with the distance, actually faster than the distance dependence of the Madelung surface potential and also the doubling spatial period term.
Similar integration on area (2) yields:
(26) 
The integral forms of coefficients , and are derived from those of coefficients , and , by replacing , and with , and , respectively. The integration is now performed with . With similar parameters than before and setting nm, we now get:
(27) 
The striking discrepancy between , and is due to the combined contribution of the Bessel functions and the exponential decay of the functions and occurring in the integral forms of the coefficients and . Hence, the force simplifies to:
(28) 
The spatial modulation of the potential does not influence the mesoscopic part of tip while scanning the surface. This contribution acts as a static shift to the total electrostatic force, similarly as the Van der Waals longrange interaction for the shortrange chemical interactions which are responsible for the topographic atomic contrast in ncAFM.
iii.3 Estimation of
The geometrical splitting in terms of areas (1) and (2) used for the estimation of , can equivalently be applied to . Thus:
(29) 
and are the surface charge densities on areas (1) and (2) within the frame of problem A. The calculation of yields:
(30) 
The integral form of is reported in the appendix. The numerical integration results in . Similar integration over area (2) gives a similar expression of by replacing indexes (1) with indexes (2) and with . The integral form of the coefficient is similar to the one of , except that the term is replaced by . The numerical integration of yields almost zero. This was expected since the integration is performed over the oscillations of the charge density due to the image charges of the surface potential (cf. fig.4). On the opposite, this was not observed in because the integration was performed on the square of . Therefore is finally written:
(31) 
Hence, the former equation states that the coupling of the force to the bias actually occurs only by means of the microscopic surface charge density at the foremost part of the tip, i.e. on the asperity. Owing to the integration over the mesoscopic part of the tip apex and the subsequent cancellation due to the oscillations of , no relevant coupling between the bias voltage and the mesoscopic part of the tip apex can occur. This aspect strongly suggests that the effect we are reporting is mainly controlled by the foremost structure of the tip. The effect is expected to be much enhanced in case of tips with sharper geometries, particularly those with apexes including atomically sharp edges.
The expression for the vertical contribution of the total force acting on the tip due to the combined influence of the capacitive coupling and of the Madelung surface potential is finally written: , with given by equ.30, by equ.24 and by equ.28. The graph of the total force and of its components is reported in fig.5 for similar parameters than previously, namely: nm, pm and Å. A typical value of bias has been set, V. Thus, it is visible that the term is negligible compared to others. This is due to the prefactor vs. for . The total electrostatic force finally simplifies to:
(32) 
In fig.5, the force reaches an average value of about 9 pN (absolute value) and a corrugation of about 2 pN (peak to peak).
Iv Implications for Kpfm
iv.1 Estimation of
The net displacement of the topmost ionic layer induced by the polarization can be estimated out of the electric field between the tip and the surface (cf. equ.2). Section II.1 has shown that the field induced by the bias voltage was essentially controlled by the mesoscopic radius of the tip, , and not by the foremost asperity. Therefore, owing to equs.4 or 8: . The expression of can now be deduced using equ.2:
(33) 
An order of magnitude for can now be calculated as follows: with V, nm, m (ref.[(36)]) and m (number of polarizable ionic species per volume unit in a fcc crystal of KBr with a lattice constant nm), we get V.m, F.m and therefore pm.
iv.2 Detected signal in KPFM: connection with the local CPD
When performing KPFM experiments, the bias voltage is modulated with a frequency and may as well include a static component to compensate for the longrange electrostatic forces. This is the reason why the average value of the experimental CPD image shown in fig.1a reaches 3.9 V. It is not rare that, on ionic surfaces, many volts are required to compensate for the longrange electrostatic forces due to trapped charges while the cleavage of the cristal (16). Thus:
(34) 
The electrostatic force is thus triggered at and then detected as an additional low or high frequency component when doing FM or AMKPFM, respectively. In both techniques, a proper dc bias voltage produced by an external controller, hereafter referred to as , is applied between the tip and the counterelectrode to cancel the modulated component at , i.e. the oscillation amplitude of the second bending eigenmode of the cantilever in AMKPFM, or the one of the frequency shift in FMKPFM. When applied to the tip (i.e. with the counterelectrode grounded), this dc bias is the opposite of the local CPD defined in equ.1: .
In order to stick to the AMKPFM experiments (fig.1), it is necessary to estimate the amplitude of the second eigenmode of the cantilever and then derive a condition on the dc value of the bias ultimately able to nullify it. On the one hand, it is therefore mandatory to check carefully all the occurrences of the modulated component of the bias voltage in the expression of the force. This includes explicit dependencies, such as those due to the polarization, but also implicit ones, as discussed hereafter. On the other hand, an expression of the oscillation amplitude of the mode modulated at must be derived.
We first address the problem of explicit and implicit bias dependencies in the expression of the electrostatic force. Owing to the explicit dependence in , which has been kept throughout the description of the model, it can be seen that the polarization effect is mainly included in the function of and through a linear and a quadratic dependence, respectively (cf. equs.30 and 28). Furthermore, since is small compared to , can be expanded in series. To first order:
(35) 
with . Replacing this expansion in the expressions of the components of the force and keeping the linear and quadratic terms in yields, with compact notations:
(36) 
and:
(37) 
where and are two dimensionless coefficients standing for geometrical factors of areas (1) and (2), respectively:
(38) 
, are also two dimensionless coefficients carrying the spatial dependence of each force component:
(39) 
and:
(40) 
Implicit dependencies are now discussed. In the two former equations, particular attention must be paid to . So far, this parameter was defined as the tipsurface distance and set to an arbitrary, constant, value. However, when dealing with AMKPFM, is not static but actually coupled to the bias and to the oscillation amplitude of the fundamental eigenmode of the cantilever. In the following, for the sake of clarity, the variables related to the fundamental eigenmode of the cantilever will be denoted with index “0” and those of the second eigenmode with index “1”. Thus, let , and be the instantaneous position of the fundamental eigenmode of the cantilever, the instantaneous position of the second eigenmode of the cantilever and the distance between the surface and the equilibrium position of the cantilever at rest, respectively. Therefore, . Hence, if has the form given in equ.34, one can postulate, to first order: . and stand for the oscillation amplitude of this mode and its phase lag with respect to the electrostatic actuation, respectively. Their exact expressions are not easily derivable, but it must be noticed that must be connected to the amplitude of the modulation, namely . When accurately matches the actual resonance frequency of the second eigenmode(46), then . is experimentally driven by the control electronics of the microscope at the actual resonance frequency of the fundamental mode of the cantilever, . It is known that it has an almost harmonic behavior of the form: . Thus, and have a component at which is further modulated by the dynamics of the second eigenmode, electrostatically actuated at .
We can now propose a selfconsistent approximated solution to the equation of motion for and thus derive the expression of the oscillation amplitude . This equation has the standard form:
(41) 
where is an external force oscillating at which controls the dynamics of . Let us assume: i , i.e. the amplitude of the mode is much smaller than the one of the fundamental mode, ii , where , being the lattice constant of the crystal and iii that the dynamics of is mainly influenced by components at . Assumptions i and ii are not too strong, since the experimental estimations of yield a few tens of picometers. Assumption i implies that the dynamics of the fundamental mode is not much influenced by the one of the second eigenmode. Hence, the solution of the equation of motion of has indeed the form postulated above. Assumption ii allows us to linearize and with respect to . Finally, assumption iii, which is consistent with the postulated solution for , , simplifies further the above equation of motion. Now, owing to the former assumptions, equ.41 can be solved by injecting the postulated expressions of both eigenmodes and keeping only the terms oscillating at . For that purpose, the exponential term wherein occurs must be expanded in Fourier series. Then, the only possibility to preserve terms at is to keep the lone static component of the Fourier expansion, hereafter referred to as (expansion of ) and (exansion of ). After linearization, and can finally be written as:
(42) 
and: