C. Barth1, T. Hynninen3,4, M. Bieletzki2, C. R. Henry1, A. S. Foster3,4, F. Esch2 and U. Heiz2
1Centre Interdisciplinaire de Nanoscience de Marseille (associated with the Universities of Aix-Marseille II and III),
CNRS, Campus de Luminy, Case 913, 13288 Marseille Cedex 09, France
2Lehrstuhl für Physikalische Chemie, Technische Universität München,
Department Chemie, Lichtenbergstr. 4, D-85748 Garching, Germany
3Department of Physics, Tampere University of Technology,
P.O. Box 692, FIN-33101 Tampere, Finland and
4Department of Applied Physics, Aalto University School of
Science and Technology, POBox 11100 FI-00076 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 Kelvin voltages 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 mi-croscopy (KPFM), the tip-sample system forms a simple sphere-plate capacitor. In the following, the quantities R andz are the radius of the sphere and the tip-sample (sphere-plate) distance, respectively. The model geome-try is depicted in Figure 1.
Figure 1: The model geometry, including denitions of the symbols used in the supporting information.
The capacitance of a sphere-plate capacitor can be pre-cisely calculated by using the method of image charges [1, 2]. The capacitance can be written as
C= 4πεR ∞ X k=1 sinh ln(λ+√λ2−1) sinh k ln(λ+√λ2−1), (1) where λ= (R+z)/R (ratio between the distance from
sphere center to plate surface and the sphere radius) and εis 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. [3] C≈2πεR 2 + ln 1 +R z . (2)
The capacitance is plotted as a function of the relative distancez/R in Figure 2, obtained from both equations (1) and (2). The agreement is outstanding.
ANALYTICAL MODELS FOR KELVIN VOLTAGE Charge at the tip
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 2q φq(q) +q φC(U)− 1 2C U 2, (3) = u0+u1U+u2U2. (4) whereU is the total voltage between the tip and surface, φC(U) ∝ U is the potential due to the capacitor, and φq(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 (3) describes the interaction
Separation C a pa ci ta n ce
Figure 2: Relative capacitanceC/(ε R) of a sphere-plate
ca-pacitor in dependency on the relative tip-surface distancez/R
on a log-log scale. The blue curve describes the approximated model from Hudlet et al. [3] whereas the red squares belong to the exact model [1, 2].
of the xed charge with its images, the second term is the energy of the charge in the electric eld of the ca-pacitor, 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 be-tween tip and surface is composed of
U =UCPD+Udc+Uac sin (ω t), (5) where UCPD is the tip-sample contact potential dier-ence and Udc, Uac are the externally applied dc and ac voltages. The electrostatic force is obtained by dieren-tiation of the electrostatic energy (3) and includes both, a constant part and parts oscillating at frequenciesωand 2ω:
F = −∂E
∂z (6)
= h0+h1 sin (ω t) +h2 cos (2ω 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 oscillat-ing atω and2ω ∂F ∂z = ∂h0 ∂z + ∂h1 ∂z sin (ω t) + ∂h2 ∂z cos (2ω t). (8) The oscillating terms are called the rst harmonic (∂h1/∂z) and second harmonic (∂h2/∂z) in the follow-ing. In KPFM, the dc bias is regulated such that the rst harmonic vanishes, i.e., so that
∂h1
∂z = 0. (9)
Since the time dependence of the energy is entirely inU, it is easily found from equations (4), (5) and (7) that
h1=−Uac ∂u 1 ∂z + 2 ∂u2 ∂z (Udc+UCPD) . (10)
Inserting the latter equation into (9) yields the Kelvin voltage UdcKelvin=− ∂2u 1 ∂z2 2∂ 2u 2 ∂z2 −UCP D. (11) Comparing equations (3) and (4) yields solutions of the coecientsu1andu2: u1 = q φC(U)/U|z−δ (12) ≈ q(1−z−δ z ) =q δ z (13) u2 = − 1 2C. (14)
The energyu1is the potential energy of the charge in the capacitor eld, divided byU, andu2is half the negative capacitance. The form (13) is obtained by assuming that the capacitive potential changes linearly fromUto 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 (atz−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 (13) and (14) into (11), a simple expression is obtained for the Kelvin voltage:
UdcKelvin = q δ π ε R2
(1 +z/R)2
(1 + 2z/R) (z/R)−UCPD. (15) Figure 3 shows the dependence of the Kelvin voltage (15) on the relative tip-sample separation z/R for both the exact image charge model and the approximate one. For simplicity, the contact potentialUCPD 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 (3) as E=−1 2p·Ep(p)−p·EC(U)− 1 2C U 2, (16)
Separation K e lvin vol ta ge
Figure 3: Relative Kelvin voltage (Udcε R2/(q δ)) as a
func-tion of the relative tip-sample separafunc-tion z/R when a point
charge is xed at the tip. The curves are plotted on a log-log scale.
with EC(U) ∝ U denoting the electric eld due to the capacitor andEp(p)the eld due to the charges induced
by the dipole. With the coecientsu1 andu2
u1 = −p·EC(U)/U|z−δ (17) ≈ pcos (θ) z , (18) u2 = − 1 2C, (19)
the following expression for the Kelvin voltage is ob-tained:
UdcKelvin = pcos (θ) π ε R2
(1 +z/R)2
(1 + 2z/R) (z/R)−UCPD.(20) The parameter θ is the angle between the dipole vec-tor and the sphere normal vecvec-tor (θ= 0: dipole pointing from the sphere to the plate, positive end towards the sur-face). The dierence to the Kelvin voltage (15) obtained for a point charge is a substitutionq δ→pcos (θ).
Interpretation of the Kelvin voltage
A couple of conclusions can be drawn from (15) and (20):
A charge or dipole modulates the contact poten-tial dierence UCPD between tip and surface. For experiments it means that the contact potential dierence between tip and surface, which is nor-mally measured in KPFM, deviates as soon as a charge/dipole is attached at the tip.
The Kelvin voltage at the sample is negative (as-suming UCPD is zero for simplicity), if a negative
charge is at the tip apex and vice-versa. The same applies for a negative dipole.
The dipole contribution depends on the orienta-tion of the dipole with respect to the sample sur-face. The voltage has its maximum, when the dipole is oriented perpendicular to the surface. It is zero, when the dipole is parallel to the sur-face. Values between UCP D (θ = ±90°) and ± p
π ε R2
(1+z/R)2
(1+2z/R)·(z/R) −UCPD (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 Kelvin voltage.
MODEL WITH ATOMISTIC TIP
To compare the simple analytical model with a more realistic system, Kelvin voltages 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 an-alyzed using the image charge scheme. The total force acting on the tip in a dense grid of dierent tip heights, ztip = 5. . .10 Å, is calculated for three bias voltages U =−U0, 0, U0. Applying simple numeric dierentia-tion with respect toztip, one obtains the gradient of the force in the(ztip, U)grid. Following (4), the expression
−∂F ∂z = ∂2u 0 ∂z2 + ∂2u 1 ∂z2 U+ ∂2u 2 ∂z2 U 2 (21) is obtained allowing identication of the coecients as
∂2u 1 ∂z2 = − ∂ ∂U ∂F ∂z U=0 , (22) ∂2u 2 ∂z2 = − 1 2 ∂2 ∂U2 ∂F ∂z U=0 . (23)
These coecients are obtained by numeric dierentiation of the force gradient with respect toU, and the Kelvin voltage is found via equation (11) for the wholeztipgrid.
CALCULATED KELVIN VOLTAGES
Kelvin voltages obtained by using (15) and (20) with realistic parameters are plotted in Figure 4. The absolute values of Udc depend on the size of the tip (sphere ra-dius) and the tip-surface distance. Assuming single ions
Separation K e lvin vol ta ge
Figure 4: Kelvin voltagesUKelvin
dc as a function of tip-surface
distance z for sphere radii R = 5. . .20nm and q δ or pcos (θ) = 2e Å obtained from (15) and (20).
or dipoles at the tip (q δ orpcos (θ) = 2e Å) and consid-ering realistic tip-surface distances in KPFM, which are in between1 and 2 nm, the Kelvin voltage is of the or-der of some hundreds of millivolts. This agrees perfectly well with values from experiment. Note that the plotted separationz is the distance between the sample and the macrotip (plate and sphere). If a nanotip is present, the true tip-surface distance is smaller thanz by the height δof 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 (Mg2+ with q δ = 1.8eÅ, red squares), strongly polar dipole (MgO molecule with p = 14.3 Debye, blue diamonds), weakly polar (stoichiometric Mg32O32 cluster, black tri-angles), and mixed type (OMg4+
3 , grey crosses). The macrotip radius was xed at R = 5 nm in Fig. 5 and 20 nm in Fig. 6. Note that the dipole of p= 14.3 De-bye for the MgO molecule is a result of the parameters used in the atomistic simulations, specically the formal charges of±2used 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 ofp= 6.2 Debye [10]. In order to represent a real MgO molecule at the tip , the complete curve has to be devided by a factor of∼2because the Kelvin voltage is directly proportional to strength of the dipole.
The curves from atomistic calculations represent a t to (15) or (20), showing a match between the dierent levels of theory. For both a point charge and a strong dipole, Kelvin voltages of a few hundred mV are observed, in agreement with Figure 4. A stoichiometric Mg32O32
Figure 5: Kelvin voltagesUKelvin
dc as a function of tip-surface
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 rangeztip= 5. . .10Å from which
the plottedz values are obtained via a shift by the height of
the nanotip. The curves show a t to equations (15) and (20). 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 contribu-tions are added together. For instance, an OMg4+
3 tip has a net positive charge and a dipole moment point-ing away from the sample. The former creates a posi-tive Kelvin voltage while the latter results in a negaposi-tive one. The combined eect is a positive Kelvin voltage with the relative strengthUKelvin
OMg4+3 ≈U Kelvin
OMg +2UMgKelvin2+ ≈ −UKelvin
MgO + 2UMgKelvin2+ .
Distance K e lvin vol ta ge Distance K e lvin vol ta ge
Figure 6: Kelvin voltagesUdcKelvin as a function of tip-surface
distance z obtained for atomistic nanotips attached to a
macrotip of radius R= 20nm. For each tip, the calculated
values are for the tip height rangeztip= 5. . .10Å from which
the plottedz values are obtained via a shift by the height of
the nanotip. The curves show a t to equations (15) and (20). The curves are shown on a linear (a) and log-log scale (b) for a better reading.
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