AFM tip characterization by Kelvin probe force microscopy - Supporting information -

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1 AFM tip characterization by Kelvin probe force microscopy - Supporting information - C. Barth 1, T. Hynninen,4, M. Bieletzki, C. R. Henry 1, A. S. Foster,4, F. Esch and U. Heiz 1 Centre Interdisciplinaire de Nanoscience de Marseille (associated with the Universities of Aix-Marseille II and III), CNRS, Campus de Luminy, Case 91, 188 Marseille Cedex 09, France Lehrstuhl für Physikalische Chemie, Technische Universität München, Department Chemie, Lichtenbergstr. 4, D Garching, Germany Department of Physics, Tampere University of Technology, P.O. Box 69, FIN-101 Tampere, Finland and 4 Department of Applied Physics, Aalto University School of Science and Technology, POBox FI Aalto, Finland In this supporting information, three dierent models for the calculation of the mean Kelvin voltage are considered. All models assume that the macroscopic part of the AFM tip and the surface can be represented by a metallic sphere (tip) and a metal plane (surface), respectively. Analytical calculations describe the rst two models that treat the electrostatic tip-surface interaction in an exact and in an approximate way, respectively. A charged or a polarized nanotip is introduced in these models by adding a xed point charge or a xed electric dipole below the tip apex (conducting sphere). In the third model, a complete atomistically described nanotip is included instead of just a charge or a dipole and the problem is solved numerically. It is shown that all models perfectly agree with each other and that they represent similar s as observed in experiments. SPHERE-PLATE CAPACITOR The AFM tip and the sample surface can be considered as a perfect conducting sphere and perfect conducting plate, respectively. If a bias voltage is applied between the tip and sample, as is done in Kelvin probe force microscopy (KPFM), the tip-sample system forms a simple sphere-plate capacitor. In the following, the quantities R and z are the radius of the sphere and the tip-sample (sphere-plate) distance, respectively. The model geometry is depicted in Figure 1. sphere center to plate surface and the sphere radius) and ε is the dielectric constant. As z/r 0 (large sphere, small tip-surface distance), the convergence of the series becomes quite slow so that an analysis of the capacitance using equation (1) is not straightforward. However, an excellent approximation for (1) was derived by Hudlet et al. [] ( ( C πεr + ln 1 + R )). () z The capacitance is plotted as a function of the relative distance z/r in Figure, obtained from both equations (1) and (). The agreement is outstanding. ANALYTICAL MODELS FOR KELVIN VOLTAGE Charge at the tip Figure 1: The model geometry, including denitions of the symbols used in the supporting information. The capacitance of a sphere-plate capacitor can be precisely calculated by using the method of image charges [1, ]. The capacitance can be written as sinh ( ln(λ + λ 1) ) C = 4πεR sinh ( k ln(λ + ), (1) λ 1) k=1 where λ = (R + z)/r (ratio between the distance from Next, a charged AFM tip is considered. It is assumed that a point charge q is localized at distance δ below the conducting sphere, mimicking a charged tip. The eective energy of the whole capacitor-charge system, the gradient of which gives the total force acting on the AFM tip, is given by [4, 5] E = 1 q φ q(q) + q φ C (U) 1 C U, () = u 0 + u 1 U + u U. (4) where U is the total voltage between the tip and surface, φ C (U) U is the potential due to the capacitor, and φ q (q) is the potential due to the image charges induced on the capacitor by the presence of the external charge q. That is, the rst term in () describes the interaction

2 h 1 = 0. (9) Capacitance Since the time dependence of the energy is entirely in U, it is easily found from equations (4), (5) and (7) that ( u1 h 1 = U ac + u ) (U dc + U CPD ). (10) Inserting the latter equation into (9) yields the Kelvin voltage / Udc Kelvin = u 1 u U CP D. (11) Figure : Relative capacitance C/(ε R) of a sphere-plate capacitor in dependency on the relative tip-surface distance z/r on a log-log scale. The blue curve describes the approximated model from Hudlet et al. [] whereas the red squares belong to the exact model [1, ]. of the xed charge with its images, the second term is the energy of the charge in the electric eld of the capacitor, and the third term is the energy of the capacitor itself (including the power source maintaining the bias voltage). In frequency modulated KPFM, the total voltage between tip and surface is composed of U = U CPD + U dc + U ac sin (ω t), (5) where U CPD is the tip-sample contact potential dierence and U dc, U ac are the externally applied dc and ac voltages. The electrostatic force is obtained by dierentiation of the electrostatic energy () and includes both, a constant part and parts oscillating at frequencies ω and ω: F = E (6) = h 0 + h 1 sin (ω t) + h cos (ω t). (7) Since frequency modulated KPFM is sensitive to the force gradient [6], the force gradient has to be calculated, which includes again a constant term and terms oscillating at ω and ω F = h 0 + h 1 sin (ω t) + h cos (ω t). (8) The oscillating terms are called the rst harmonic ( h 1 /) and second harmonic ( h /) in the following. In KPFM, the dc bias is regulated such that the rst harmonic vanishes, i.e., so that Comparing equations () and (4) yields solutions of the coecients u 1 and u : u 1 = q φ C (U)/U z δ (1) q (1 z δ ) = q δ z z (1) u = 1 C. (14) The energy u 1 is the potential energy of the charge in the capacitor eld, divided by U, and u is half the negative capacitance. The form (1) is obtained by assuming that the capacitive potential changes linearly from U to 0 from the surface of the plate (potential = U) to the surface of the sphere (potential = 0), and evaluating the potential at the xed charge (at z δ). Comparison with the exact potential calculated by the image charge method veries this to be an excellent approximation on the symmetry axis of the system (not shown). By inserting (1) and (14) into (11), a simple expression is obtained for the : U Kelvin dc = q δ π ε R (1 + z/r) (1 + z/r) (z/r) U CPD. (15) Figure shows the dependence of the (15) on the relative tip-sample separation z/r for both the exact image charge model and the approximate one. For simplicity, the contact potential U CPD is set to zero. Fixed dipole Instead of having net charge on the AFM tip, the tip may be merely polarized. Such a case can be analyzed in a similar fashion by replacing the point charge with a dipole p. The energy of the system may be written analogously with () as E = 1 p E p(p) p E C (U) 1 C U, (16)

3 charge is at the tip apex and vice-versa. The same applies for a negative dipole. Figure : Relative (U dc ε R /(q δ)) as a function of the relative tip-sample separation z/r when a point charge is xed at the tip. The curves are plotted on a log-log scale. with E C (U) U denoting the electric eld due to the capacitor and E p (p) the eld due to the charges induced by the dipole. With the coecients u 1 and u u 1 = p E C (U)/U z δ (17) p cos (θ), z (18) u = 1 C, (19) the following expression for the is obtained: U Kelvin dc = p cos (θ) π ε R (1 + z/r) (1 + z/r) (z/r) U CPD.(0) The parameter θ is the angle between the dipole vector and the sphere normal vector (θ = 0: dipole pointing from the sphere to the plate, positive end towards the surface). The dierence to the (15) obtained for a point charge is a substitution q δ p cos (θ). Interpretation of the A couple of conclusions can be drawn from (15) and (0): A charge or dipole modulates the contact potential dierence U CPD between tip and surface. For experiments it means that the contact potential dierence between tip and surface, which is normally measured in KPFM, deviates as soon as a charge/dipole is attached at the tip. The at the sample is negative (assuming U CPD is zero for simplicity), if a negative The dipole contribution depends on the orientation of the dipole with respect to the sample surface. The voltage has its maximum, when the dipole is oriented perpendicular to the surface. It is zero, when the dipole is parallel to the surface. Values between U CP D (θ = ±90 ) and ± p (1+z/R) π ε R (1+z/R) (z/r) U CPD (θ = 0, 180 ) can be obtained. The net charge contribution depends on the size of the nanotip and the distribution of charge within. The farther the charge is from the conducting part of the tip, the greater the. MODEL WITH ATOMISTIC TIP To compare the simple analytical model with a more realistic system, s within a model where the nanotip is included in atomic detail are calculated with empirical interatomic interactions, using the SciFi AFM simulation tool [5, 7]. Various MgO tips from single ions to clusters of 64 atoms are considered. Parameters for the MgO interactions are the same as in previous studies [8]. The conducting macrotip and sample are treated as continuous media with their contribution analyzed using the image charge scheme. The total force acting on the tip in a dense grid of dierent tip heights, z tip = Å, is calculated for three bias voltages U = U 0, 0, U 0. Applying simple numeric dierentiation with respect to z tip, one obtains the gradient of the force in the (z tip, U) grid. Following (4), the expression F = u 0 + u 1 U + u U (1) is obtained allowing identication of the coecients as u 1 = F U, () U=0 u = 1 F U. () U=0 These coecients are obtained by numeric dierentiation of the force gradient with respect to U, and the Kelvin voltage is found via equation (11) for the whole z tip grid. CALCULATED KELVIN VOLTAGES s obtained by using (15) and (0) with realistic parameters are plotted in Figure 4. The absolute values of U dc depend on the size of the tip (sphere radius) and the tip-surface distance. Assuming single ions

4 4 Figure 4: s Udc distance z for sphere radii R = nm and q δ or p cos (θ) = e Å obtained from (15) and (0). or dipoles at the tip (q δ or p cos (θ) = e Å) and considering realistic tip-surface distances in KPFM, which are in between 1 and nm, the is of the order of some hundreds of millivolts. This agrees perfectly well with values from experiment. Note that the plotted separation z is the distance between the sample and the macrotip (plate and sphere). If a nanotip is present, the true tip-surface distance is smaller than z by the height δ of the nanotip. In Figures 5 and 6, the distance dependent Kelvin voltages obtained from the numerical calculations are shown for four characteristic tips: purely ionic (Mg + with q δ = 1.8 eå, red squares), strongly polar dipole (MgO molecule with p = 14. Debye, blue diamonds), weakly polar (stoichiometric Mg O cluster, black triangles), and mixed type (OMg 4+, grey crosses). The macrotip radius was xed at R = 5 nm in Fig. 5 and 0 nm in Fig. 6. Note that the dipole of p = 14. Debye for the MgO molecule is a result of the parameters used in the atomistic simulations, specically the formal charges of ± used for Mg and O, which overexaggerates the ionicity of the molecule (larger clusters are more ionic and, hence, much better represented within the atomistic model). A more realistic charge distribution, such as that provided in rst principles calculations [9], results in very good agreement with the experimental value of p = 6. Debye [10]. In order to represent a real MgO molecule at the tip, the complete curve has to be devided by a factor of because the is directly proportional to strength of the dipole. The curves from atomistic calculations represent a t to (15) or (0), showing a match between the dierent levels of theory. For both a point charge and a strong dipole, s of a few hundred mv are observed, in agreement with Figure 4. A stoichiometric Mg O Figure 5: s Udc distance z obtained for atomistic nanotips attached to a macrotip of radius R = 5 nm. For each tip, the calculated values are for the tip height range z tip = Å from which the plotted z values are obtained via a shift by the height of the nanotip. The curves show a t to equations (15) and (0). The curves are shown on a linear (a) and log-log scale (b) for a better reading. tip has only a weak dipole moment and so the Kelvin voltage drops to some tens of millivolts. If the charges are inversed (Mg's and O's are swapped), only the sign of the voltage changes (not shown). If the tip houses both a net charge and a dipole moment, their contributions are added together. For instance, an OMg 4+ tip has a net positive charge and a dipole moment pointing away from the sample. The former creates a positive while the latter results in a negative one. The combined eect is a positive with the relative strength U Kelvin U Kelvin Kelvin OMg 4+ OMg +UMg + U Kelvin MgO Kelvin + UMg. + Electronic address: barth@cinam.univ-mrs.fr

5 5 Distance [1] E. Durand, Électrostatique II Problêmes Gênêraux Conducteurs (Masson, Paris, 1966). [] J. C. Maxwell, A treatise on electricity and magnetism (Clarendon Press, Dover, 1876). [] S. Hudlet, M. Saint Jean, C. Guthmann, and J. Berger, Eur. Phys. J. B, 5 (1998). [4] A. S. Foster, Ph.D. thesis, University College London (UK), University College, London (000). [5] L. N. Kantorovich, A. I. Livshits, and M. Stoneham, J. Phys. Cond. Mater. 1, 795 (000). [6] P. Girard, Nanotechnology 1, 485 (001). [7] Kantorovich, L. N., Foster, A. S., Shluger, A. L., Stoneham, and A. M., Surf. Sci. 445, 8 (000). [8] Livshits, A. I., Shluger, A. L., Rohl, A. L., Foster, and A. S., Phys. Rev. B 59, 46 (1999). [9] M. Bieletzki, T. Hynninen, T. M. Soini, M. Pivetta, C. R. Henry, A. S. Foster, F. Esch, C. Barth, and U. Heiz, Phys. Chem. Chem. Phys. 1, 0 (010). [10] R. Weast and C. R. Company, CRC Handbook of Chemistry and Physics (CRC press Boca Raton, FL, 1988). Distance Figure 6: s Udc distance z obtained for atomistic nanotips attached to a macrotip of radius R = 0 nm. For each tip, the calculated values are for the tip height range z tip = Å from which the plotted z values are obtained via a shift by the height of the nanotip. The curves show a t to equations (15) and (0). The curves are shown on a linear (a) and log-log scale (b) for a better reading.