Tip−Sample Interactions in Kelvin Probe Force Microscopy - American

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J. Phys. Chem. C 2008, 112, 17368–17377

Tip-Sample Interactions in Kelvin Probe Force Microscopy: Quantitative Measurement of the Local Surface Potential Andrea Liscio,† Vincenzo Palermo,*,† Klaus Mu¨llen,‡ and Paolo Samorı`*,†,§ Istituto per la Sintesi Organica e la FotoreattiVita` (ISOF), Consiglio Nazionale delle Ricerche (CNR), Via Gobetti 101, 40129 Bologna, Italy, Max-Planck Institute for Polymer Research, Ackermannweg 10, 55124 Mainz, Germany, and Nanochemistry Laboratory, ISIS-CNRS 7006, UniVersite´ Louis Pasteur, 8 alle´e Gaspard Monge, 67083 Strasbourg, France ReceiVed: July 27, 2008; ReVised Manuscript ReceiVed: September 04, 2008

We study the influence of different experimental parameters on the interaction between the probe and the sample in Kelvin probe force microscopy (KPFM) measurements. We provide a precise and reproducible determination of the local surface potential (SP) of clean macroscopic highly oriented pyrolytic graphite (HOPG) samples and of organic semiconducting nanostructures of an alkyl-substituted perylene-bis(dicarboximide) (PDI) self-assembled at surfaces. We distinguish two different terms in the measured SP, intrinsic and extrinsic, containing the electrical properties of the studied object and the experimental artifacts, respectively. We investigate the effect of the most relevant experimental parameters including tip-sample distance, relative humidity (RH), and potential applied to the tip, which govern the extrinsic term of the measured SP. Moreover, we devise a theoretical description of the tip-sample interaction taking into account the extra modulation of the probe due to the applied ac potential during the KPFM scan. A deep understanding of all the terms which contribute to the measured SP in air environment made it possible to devise a new protocol to quantify the electrical properties of nano-objects leading to an improvement of the achieved lateral resolution, as demonstrated by the good agreement between the proposed model and the experimental results. 1. Introduction The quantitative determination of the electrical potential of very small objects, such as micro- and nanoelectrodes, thin layers, nanowires as well as nanocrystals, with a spatial resolution on the nanoscale represents a crucial issue for technological applications in micro- and nanoelectronics.1,2 Performing a reliable nanoscale-resolved electrical potential measurements using poorly or noninvasive approaches is still a great challenge. Such measurement is particularly complicated when attempted both on organic nanostructures, due to their complex electronic properties, and on working devices, such as transistors and solar cells, because charges are continuously created and/or transported across the device. A very important technique used to measure potential on the nanoscale is Kelvin probe force microscopy (KPFM), also known as Kelvin probe microscopy (KPM) or scanning Kelvin probe microscopy (SKPM). This technique makes it possible to obtain a quantitative measurement of the surface potential (SP)3,4 of nanostructures both in air and in vacuum with a lateral resolution below 50 nm and a potential resolution lower than 10 mV. Furthermore it provides quantitative insight into other electronic properties of the investigated architecture including the work function of metals,4 band bending of semiconductors,5 and electrical polarization due to adsorbed molecules at the surface.6,7 Being contactless, it does not perturb significantly the system under study; thus, it enables in situ exploration of operating electronic devices under both vacuum and air environment. * Corresponding authors. E-mail: [email protected] (V.P.), samori@ isis-ulp.org (P.S.). † Consiglio Nazionale delle Ricerche. ‡ Max-Planck Institute for Polymer Research. § Universite ´ Louis Pasteur.

KPFM has been successfully employed to investigate various systems,8 including inorganic9-11 and organic thin films12-14 as well as proteins,15-17 across multiple length scales, i.e., from meso- to nanoscopic scale. In particular, KPFM was proved to be a viable tool to explore dynamic properties of the working devices. The technique has been used to explore the photocharge density variation in donor-acceptor blends at surfaces.18-21 The flat source-drain architecture of organic transistors can be suitably studied by the KPFM while the transistor is operating, thus allowing the measurement of the current flow and potential drops for different source and gate voltages.22-27 However, so far the mechanisms and details of the tip-sample interaction as well as the effects of the various experimental parameters on the performed measurements are not completely understood. A more detailed comprehension is especially needed in order to gain insight into working systems and devices such as transistors and solar cells, which are characterized by a dynamic generation and transport of charges. The physical principle of the Kelvin probe technique is relatively simple and was introduced in the 19th century by Lord Kelvin.28 It relies on the measurement of the SP difference between two metal plates: the sample and the probe. KPFM was first reported in 1991,29 and it is based on the atomic force microscopy (AFM) setup. In addition to the conventional feedback circuit used to map the topography of the surface with an AFM,30-34 a second feedback is exploited to record the local variations in SP during scanning. This measurement mode is typically called the amplitude-sensitive method (AM-KPFM), but in this work we omit the suffix for the sake of simplicity. The presence of a scanning probe tip with a nanometric apex leads to the potential measurement of artifacts, especially when the dimension of the measured nano-objects is smaller than the tip size. Albeit both theoretical and experimental approaches

10.1021/jp806657k CCC: $40.75  2008 American Chemical Society Published on Web 10/11/2008

Tip-Sample Interactions in KPFM have been devised and reported aiming at modeling KPFM measurements,35-41 strong approximations have been considered including the fixed tip-sample distance and the description of the tip either as a flat plate, hemisphere, or cone. The overall electrical potential of a surface measured by KPFM depends on two contributions: the intrinsic and the extrinsic ones. The first term represents the potential of the object under study which depends on its work function, on the presence of charges, and on band-bending effects, and thus it does not depend on the experimental method and setup employed. The second term includes contributions and artifacts determined by the measurement technique. We have shown that in KPFM measurements the detected surface potential signal is averaged over a finite area of the surface of the sample interacting with the tip (i.e., effective area).37 The highest lateral resolution42 value obtained for measurements performed in air amounts to about 30 nm, although KPFM makes it possible to measure isolated systems having a smaller lateral size. In these scales, the averaging of the potential of the sampled area becomes crucial.43 We have proposed a semiquantitative procedure to separate these two contributions and to extract the intrinsic SP of the sample from the measured one, for the case of nano-objects. The extrinsic contribution contains all the information on the performed measurement, such as the details of the tip-sample interaction as well as the acquisition electronic circuits. In contrast to AFM, which uses primarily van der Waals interactions to map the surface, in KPFM quasi-electrostatic forces between tip and sample are recorded. These interactions, due their long-range nature, involve a large volume of scanning tip, i.e., not just its nanometric apex. Thus, a much more accurate description of the tip-sample interaction is needed to understand the KPFM results. In this work, we focus our attention on the effects of the most relevant experimental parameters on the measured SP of both macroscopic and nanoscopic structures. We present a thorough study that by combining the experimental evidence and a more generalized model to describe the tip-sample interaction in KPFM casts light onto the origin of the experimental artifacts in order to remove them, thus to ultimately obtain lateral resolution only limited by the size of the used probe. To accomplish this goal, we have systematically explored the dependence of the SP signal on different experimental parameters in KPFM measurements of organic (nano)structures at surfaces including (i) the tip-sample distance, (ii) the relative humidity, and (iii) the potential applied to the tip. Finally, we compare our model with experimental data and estimate the quasi-Fermi energy level of an organic nanosystem. It is worth noting that the KPFM technique when operated in air environment is typically used to map the change of the SP of the substrate while a water layer from the humidity present in the air is adsorbed on its top,44,45 or corrosion phenomena occur on the microscopic scale.46,47 In contrast, in this work the humidity and other experimental parameters are used to modulate and study in detail the tip-sample interactions. 2. Model The electrostatic force (FTS) between the conducting tip and a metallic sample is proportional to the square of the bias voltage (U), and it can be described as

FTS )

1 ∂C 2 U 2 ∂z

with

U ) ∆VTS - (φtip - φsample) (1)

where the z-axis is perpendicular to the surface of the sample,

J. Phys. Chem. C, Vol. 112, No. 44, 2008 17369 and the capacitance C depends on the tip-sample distance (d). The bias voltage between the two electrodes is given by the sum of the applied bias potential on both tip and sample (∆VTS) and the work function difference (φtip - φsample). In eq 1, the force is the product of two different contributions: the first one (∂ C/∂ z) contains the geometrical description of the tip-sample system, whereas the second one (U2) describes its electrical properties. In the case of parallel plates, the geometrical term reduces to a parallel plate capacitor, and it can be described using the well-known relation εS/d2, with ε being the dielectric constant of the sample, S the area, and d the distance between plates. In the KPFM measurements, both an alternating current (ac) and a direct current (dc) bias voltage signal are applied between the tip and the sample. For the sake of simplicity, we considered only the case in which the sample is grounded. Hence, the voltage ∆VTS is only applied to the tip, and the tip-sample potential difference can be expressed by

∆VTS ) Vdc + Vac sin(ωt)

(2)

with Vdc being the dc offset potential applied to the tip and Vac and ω being the amplitude and frequency of the applied ac voltage signal, respectively. Generally, the frequency of the ac bias is set to the resonance frequency of the cantilever. Further information is reported in ref 37. KPFM measurements can be usually performed using the lift-mode technique. In such a mode the topography and the potential signals are subsequently recorded. First the surface topography is acquired in tappingmode along a single line profile.37 Then, turning off the mechanical excitation of the cantilever, a second scan is executed along the same line following the topographic profile at a set lift-height (LH) from the sample surface, recording local variations in work function (and SP, as detailed below). During the second scan, the tip-sample distance is constant and is given by dAFM + LH, where dAFM represents the tip-sample distance during the topographic scan. Hence, in the case of a flat sample or when the surface roughness results much smaller than the LH value, the geometrical term ∂C/∂z is constant and, by using simple trigonometric formulas, the electrostatic force (FTS) expressed in eq 1 can be easily divided into three components: FTS ) F0 + Fω + F2ω, depending on the dc potential and the ω and the 2ω frequencies, respectively.48 All the time-dependent characteristics of the electrostatic interaction are contained in the U2 term. Combining eq 2 with eq 1, Fω assumes the common form:

Fω )

∂C [V - (φtip - φsample)]Vac sin ωt ∂z dc

(3)

The ω-component results as the product of three independent terms, and when it is set to zero, the only one nontrivial solution corresponds to the case in which Vdc is equal to the difference in work function between the tip and the sample surface (i.e., Vdc ) φtip - φsample). Therefore, by calibrating the work function of the tip, the work function value of the sample can be obtained by monitoring the voltage Vdc applied by the feedback signal. KPFM exploits the first-harmonic component (Fω) of the electrostatic force interaction to determine the local work function on the sample surface. It is important to note that the work function difference estimated from eq 3 is independent both on the tip-sample distance and on the applied Vac bias. It is remarkable that the expression shown in eq 3 has been calculated considering the tip-sample distance as constant during the KPFM scan. However, the applied ac bias induces a vibration of the tip, and this effect has to be taken into account

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in the tip-sample description. The tip-sample distance d is not constant but varies with the frequency ω, and its time dependence can be written as

d(t) ) dAFM + LH + dosc sin(ωt)

(4)

with dosc being the amplitude of the cantilever oscillation due to the Vac bias. This amplitude shows a further dependence on the tip-sample distance, dosc ) dosc(d, Vac), which will be treated in detail in the following section. It is clear that the dosc increases with the amplitude of ac bias, whereas it decreases when the tip moves far from the surface. Equation 4 shows that the distance d(t) oscillates around its mean value which is constant during the KPFM scan. When the amplitude dosc is not neglected, eq 3 becomes more complex and cannot be calculated using only trigonometric formulas also because the geometrical term ∂C/∂z exhibits a time dependence. By expressing the tip-sample interaction FTS as a Fourier series, the ω-component results described by the first Fourier coefficient:

Fω )

1 4π

U2e-iωt dt ∫-ππ ∂C ∂z

(5)

In this case, the Vdc value which nullifies the ω-component of the force depends on several parameters that have to be taken into account such as the tip-sample distance, the applied Vac bias, as well as the behavior of the tip-sample interaction. It is evident that the Fourier expansion is a general representation which also describes the case of nonoscillating cantilever (i.e., Vac , Vdc and dosc , d). In such a case, the geometrical term may be assumed as a scaling factor, the integral contains only the U2 term, and eq 5 assumes the expression reported in eq 3. When this approximation is valid, the measured work function represents the intrinsic potential value of the sample (referred to the tip). By increasing the Vac bias potential or by decreasing the tip-sample distance (i.e., by increasing the amplitude of the cantilever oscillation dosc) the signal-to-noise ratio of the measurements improves, but the fixed distance approximation is not longer valid. The model presented for conductive samples may be also extended to semiconductors as well as thin oxide films.4 For these kinds of materials the work function (WF) as well as the charge density at the surface contribute to the measured potential, which is called SP. Charge density takes into account the presence of permanent dipoles and those induced by the presence of a charged tip. Moreover, local SP provides a map of the local energy distribution of surface and surface states.5,49 A more detailed description on the interplay between these terms has been reported in a previous work.37 In general, the contribution to the SP of induced surface dipoles is considered as an extrinsic term because of the presence of the tip. 3. Experimental Procedures The experiments have been carried out using a homemade sealed chamber to control the relative humidity (RH). The RH was modulated by blowing a gentle flow of N2 in the chamber either in presence or in absence of a water reservoir and measured with a commercial RH meter (Honeywell, HIH-4000001) having an operating range from 0% to 100% RH with an accuracy of (3.5%. RH values between 0% and 80% could be in this way obtained and kept stable within (2% during the data acquisition period. Thus, we define nominal 0%, the measured RH ) 0% value.

The KPFM measurements have been performed by employing a commercial microscope Multimode (Veeco) with the extender electronics module. In order to obtain a sufficiently large and detectable mechanical deflection, we used (k ) 2.8 N/m) Pt/ Ir-coated Si ultralevers (SCM, Veeco) with oscillating frequencies in the range of 60 < ω < 90 KHz. Both sides of cantilever are coated with 20 nm of Pt/Ir, with a buffer layer (3 nm) of Cr to improve the adhesion. The work function of the used tip has been obtained by calibration with respect to freshly cleaved highly oriented pyrolytic graphite (HOPG). 4. Results and Discussion The effect of the experimental parameters on the measured SP is determined on a test sample, i.e., a macroscopic surface of freshly cleaved HOPG (0001) which has been chosen because it possesses a good electrical conductivity which is constant over a scale of several hundreds of micrometers due to the crystalline nature of the sample and its flatness. Moreover, we have chosen as representative samples different architectures of perylene-bis(dicarboximide) (PDI), including thin layers and strongly anisotropic crystals. This molecule is a well-known n-type organic semiconductor50-52 widely employed for optoelectronic applications.53-56 The aliphatic substitutions covalently linked in the peripheral positions grant a good solubility in organic solvents, enabling thin film processing from solutions.57,58 Previous reports revealed that PDI can selfassemble into highly ordered supramolecular architectures and morphologies depending on the interplay of molecules-molecule, molecule-solvent, molecule-substrate, and solvent-substrate interactions.59-67 4.1. Effects of Experimental Parameters on the Measured Potential. The reproducible KPFM measurement requires a full control over the environment in which the experiments are performed, the geometry of the interaction between the sample and the probe, as well as the feedback circuit that modulates the potential variations on the surface of the sampled object. To this end we have varied systematically (i) the RH, (ii) the tip-sample distance (by tuning LH), and (iii) the applied Vac. Thus, the measured SP can be described as a function in a multidimensional space, i.e., one dimension for each parameter. For the sake of simplicity, we start exploring the effect of RH variation for fixed tip-sample distance and Vac bias. Then, we describe the SP dependence by fixing one parameter and varying the other two. It is remarkable to point out that, by varying the experimental parameters, the electronic properties (i.e., the intrinsic term) of the studied samples do not change. In particular, the induced band bending at the surface due to the charged tip is negligible.68 4.1.1. RH Effect. The changes in surface potential of the hydrophobic HOPG substrate due to the water film adsorbed on the sample surface at the room temperature are shown in Figure 1, by plotting the measured SP as a function of the RH. In order to reduce the polarization of the adsorbed water due to the charged tip, measurements have been carried out keeping a constant low ac bias (Vac ) 500 mV) and a large tip-sample distance (LH ) 300 nm). The choice of the Vac value will be considered and discussed in detail in the following paragraphs. The measured SP of HOPG decreases with the increasing RH and a -120 mV potential drop is measured upon varying the RH from 0% to 80%. Changes in the SP of the tip due to the humidity cannot be neglected. However, by comparing all the collected data points, the mechanical response of the tip does not vary. The free amplitude and fundamental oscillating frequency of the tip remain unchanged, suggesting that the SP

Tip-Sample Interactions in KPFM

Figure 1. Relative humidity (RH) dependence of the surface potential of an HOPG surface measured by KPFM at room temperature. The tip-sample distance during measurements was kept constant at 40 nm.

Figure 2. Tip-sample distance dependence of the measured SP measured (data points) and fitted using the proposed model (continuous lines) by varying the applied Vac bias: 500 (0), 1000 (O), 2000 (4), 3000 (3), and 4000 ()) mV.

contribution of the water adsorbed on the tip is surprisingly smaller than that adsorbed on the surface of the sample. Thus, the negative trend suggests that the dipole moments of the water molecules point on toward HOPG, the positive charges are close to the surface, whereas the electron-rich oxygen atom is exposed on the surface. This result is in good agreement with the RH trend observed on macroscopic Kelvin probe measurements performed on hydrophilic mica surfaces.69 Thus, it suggests that an additional contribution to the measured SP due to the experimental artifacts given by changing in the tip-sample interaction cannot be neglected. As introduced in the theoretical section, the measured SP is not an absolute value. We have chosen to set to zero the SP measured at Vac ) 500 mV and LH ) 400 nm, when RH ) 0% (i.e., SP0). All the measured SP values will be referred to SP0. 4.1.2. Effect of LH and Vac. We have fixed the RH to a nominal value of 0% in order to minimize the effect of the adsorbed water on the hydrophilic Pt/Ir tip and studied the Vac dependence of the measured SP. The measured SP is plotted in Figure 2 (squares) as a function of the tip-sample distance and by varying the applied Vac from 500 to 4000 mV. The tip-sample distance is calculated employing the eq 4, by neglecting the amplitude of the cantilever oscillation due to the Vac bias (dosc), whereas the tip-sample distance during the topographic scan dAFM ) 17 ( 3 nm has been obtained performing force-distance curve measurements70 using the same tip and scanning parameters employed for the KPFM measurements. All the potential distributions show a monotonic decreasing trend that converges to the SP0 reported in Figure 1. Moreover,

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Figure 3. Tip-sample distance of the measured SP measured (data points) and fitted using the proposed model (continuous lines) by varying the RHs: 60% (0), 30% (O), 10% (4), and 0% (3).

we observe that the SP variation decreases with the applied ac voltage. While it amounts to 90 mV for Vac ) 4000 mV, this value is reduced to about 15 mV when Vac ) 500 mV is applied. 4.1.3. Effect of LH and RH. Figure 3 displays the tip-sample distance dependence of the measured SP in the range included between 20 and 220 nm for an HOPG sample. By fixing the applied Vac potential at 4000 mV, the SP data are reported for four different RH values: nominal 0% (3), 10% (4), 30% (O), and 60% (0). All the measured data sets show a monotonic decrease of the potential with an asymptotic regime for distances larger than 200 nm. The asymptotic SP values measured at fixed RH correspond to those reported in Figure 1. In the asymptotic regime, the geometrical factor which can be described by LH and the artifacts due to the technique (i.e., Vac) may be neglected; thus, the corresponding asymptotic SP value only reflects the electrical properties of the sample (and the tip). In this view, the RH dependence of the asymptotic SP may be totally ascribed to the variation in the electrical properties of the water film adsorbed on the surface of the sample. These results suggest that the suitable conditions to extract the electronic information of the sample can be achieved when the tip is far enough from the surface of the sample (i.e., about 200 nm). However, several studies showed that the lateral resolutionincreaseswiththedecreasingtip-sampledistance.38,40,48,71 In order to study nanosystems adsorbed on the surface, the asymptotic regime cannot be used and a more accurate description of the geometrical factors is required. 4.2. Description of Tip-Sample Interaction. During the KPFM mapping of the surface the tip-sample electrostatic force is not uniform because of the three-dimensional (3D) nature of both the tip and sample. The contribution of the sample may be neglected in the case of flat surfaces or thin nanosystems and, more generally, when the sample roughness is smaller than the tip-sample distance. Although an extensive analysis of the electrostatic force due to the tip shape has been performed by Colchero et al. describing the tip as a truncated cone with a hemisphere and taking into account also the contribution of the cantilever,72 the real tip features a much more complex geometry. In general, all the geometrical details may be parametrized using a modified parallel plate description. Whereas for a parallel plate capacitor the geometrical term of the electrostatic tip-sample force shows a quadratic inverse d dependence (i.e., ∂C/∂z ∝ d-β, with β ) 2), more detailed modeling of the tip geometry72,73 exhibits a more complex analytical d dependence in which the tip-sample force decreases much slowly than in the parallel

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Figure 4. Electrostatic tip-sample force (FTS) calculated with FE analysis (symbols) and the corresponding fitting curves (continuous lines) obtained using the modified parallel plate model to describe the tip-sample geometry: (a) Vdc dependence of the FTS and (b) d dependence of the FTS.

plate case (i.e., β < 2). According to Takahashi and Ono,74 we thus assume that the tip-sample interaction in the modified parallel plate description (∼FTS) may be written as

1 ∂C R 1 ε0S F˜TS ) U ) β [Vdc - (φtip - φsample) + Vac sin ωt]R 2 ∂z 2d eff

(6) with deff being the effective tip-sample distance which takes into account the finite size of the tip. R and β represent the electrostatic force dependence on the sample bias and on the tip-sample distance, respectively. In the case of parallel plates with infinite size, deff is equivalent to d, and both R and β are equal to 2. We modeled the static electric field of the tip on the surface of the sample using finite element (FE) analysis. The 3D simulation has been performed in the tip-sample distance range between 20 and 250 nm, and assuming the used Pt/Ir tip as a truncated cone with 25° of aperture angle with a semisphere apex with 15 nm radius. All the geometrical information have been obtained from electron microscopy measurement performed by the manufacturing company (SCM-PIT tip, Veeco). The potential signal change due to the cantilever contribution has been the subject of many quantitative studies.75-78 In the range of distances between 20 and 250 nm, the d dependence of the electrostatic force between the cantilever and the sample varies less than 9%.72 Thus, the main variations can be ascribed to the apex and the cone contribution, and the d dependence component due to the cantilever may be considered as a constant value. From eq 5, it is evident that when the geometrical term ∂C/∂z does not depend on the tip-sample distance, the resulting ω-component of the electrostatic force is independent of d. Hence, the contribution of the cantilever may be neglected. This result is confirmed by preliminary simulations executed by neglecting the cantilever contribution, and by varying the height of the tip, indicating that the greatest contribution to the detected force signal is due to the last portion of the tip, i.e., about 3 µm in height, interacting with the sample. In fact the remaining part of the tip (i.e., the upper portion of cone and the cantilever) was found to contribute less than 10% to the entire tip-sample force in the performed experimental conditions. Figure 4 shows the simulated electrostatic force (symbols) depicted (a) as a function of the applied Vdc at a fixed tip-sample geometry and (b) as a function of the tip-sample distance at a fixed potential applied to the tip. Both simulated distributions are fitted with eq 6 (lines) using R (Figure 4a), as well as β and deff (Figure 4b) as free parameters. We underline that R and β are uncorrelated parameters, related to geometrical and electric properties of the system.

Concerning the R parameter, the parabolic dependence of the electrostatic force on the sample bias at the constant tip-tosample distance is confirmed (R ) 2.00 ( 0.01) for different tip-sample distance, indicating that the parallel plate approximation is valid even for this complex geometry to describe the potential variations. In contrast, the tip-sample distance dependence results weaker (β ) 1.10 ( 0.03) than the case of parallel plates, whereas the effective tip-sample distance is larger than the distance between the bottom of the tip and the surface: (deff-d ) 43 ( 2 nm). The effective tip-sample distance is strongly correlated with the geometry of the tip. In the case of spherical tip, deff-d results equal to radius of the probe. In general, the larger value obtained by the simulations can be ascribed to the contribution of the cone which cannot be neglected. All the obtained parameters are calculated considering the electrical interaction between tip and sample as static (i.e., electrostatic limit). This assumption remains valid because the frequency of the applied Vac is in the 104-105 Hz range, and it results much smaller than the resonance frequencies of vibrational and stretching modes of C-H or C-O which lie in the gigahertz range. This suggests that the dielectric properties of the organic material are not altered by the presence of a charged tip at few tens of nanometers distance. Thus, we describe the ω-component of tip-sample interaction in the potential scan (∼Fω) as a quasi-electrostatic one with a slow time dependence. When we apply a Vac bias, the tip oscillates with dosc amplitude as shown in eq 4. Its periodical time dependence may be obtained by solving the Newton second law:

z¨ )

F˜ω ; m0

z(0) ) 0,

z˙(0) )

∫ F˜ω|z)0,t)0

(7)

where the damping term is neglected and m0 describes the mass of the portion of the tip which interacts with the sample as explained before. Equation 7 is a second-order nonlinear differential equation not having exact solutions. The numerical solutions are based on implemented Runge-Kutta algorithms for oscillatory problems.79 For the sake of simplicity, the numerical solutions are fitted by analytical curves. In the section 2 (Model) above, we have introduced that the tip oscillation amplitude dosc increases with the amplitude of ac bias, whereas it decreases with the tip-sample distance deff. Thus, we have fitted the numerical results by combining two power functions: a positive one for the ac bias dependence and a negative one for the distance dependence. Both power scaling factors are treated as free parameters, and the resulting analytical curve can be described as follows:

Tip-Sample Interactions in KPFM

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Figure 5. Tip oscillation amplitude dosc value estimated by the measurements by using the proposed model (symbols) by varying (a) the applied Vac biases and (b) the RH values. The line represents the calculated Vac dependence by solving the second law of Newton.

dosc ) γ

Vac deff

(8)

0.8

where γ is a scale factor. The amplitude of the tip oscillation results proportional to the applied Vac bias, whereas the tip-sample distance dependence is described by a negative power law with a scaling factor of 0.8. As introduced in the theoretical section for a given Vac bias, the SP measured in the KPFM scan is equal to Vdc which nullifies the ω-component of the tip-sample force that has been parametrized in eq 6. By substituting eq 6 in eq 5, we obtain the measured SP function which depends on both the Vac bias and the deff expressed by

Fω )

1 2π

∫-ππ F˜TS(ω)e-iωt dt ) 0 w SPmeas ) Vdc(Vac, 〈deff〉) (9)

where 〈deff〉 represents the time-independent effective tip-sample obtained by combining eqs 4, 7, and 8 and averaging over the period T ) 2π/ω:

( 〈 〉 ) 〈[ 1 deff

1.1

)

1 dAFM + LH + ∆ + dosc sin ωt

]〉 1.1

(10) T

where ∆ represents the difference between the effective and the geometric tip-sample distance amounting to 43 nm, as demonstrated previously. It is evident that the average of term dosc sin ωt performed over the period T ) 2π/ω amounts to zero. However, the time averaging is calculated by solving the Newton equation where the tip-sample interaction (∼Fω) does not linearly depend on dosc sin ωt term, but it shows a negative power law dependence with a scaling factor of 1.1 as evinced before by using the modified parallel plate description. Thus, the dosc component differs from zero representing a further analytical component due to the ac bias which tends to reduce the tip-sample distance. All the terms shown on the right side of eq 10 are known parameters or may be directly estimated except for the scale factor γ (eq 8) which is treated as a free parameter. The tip-sample distance dependences shown in Figure 2, measured on HOPG substrates by varying the Vac bias, are fitted with the function SPmeas obtained by solving eq 9 and taking into account eq 10. A dedicated MatLab-based procedure was employed to perform all the numerical calculations. LH is an experimental parameter guided by the software which controls the microscope, whereas dAFM amounts to 17 ( 3 nm. This value is measured by force-distance curve measurements performed with the same tip and scanning parameters employed for the KPFM measurements.

The good agreement between the theoretical and the measured trends suggests that the measured differences in SP observed by applying different ac biases may be entirely determined by experimental artifacts due to geometrical effects of dosc amplitudes. On the other hand, the systematic differences achieved observed for d > 30 nm may be principally ascribed to a poor description of the tip-sample geometry as well as the film of water adsorbed on the surface of the sample which may be polarized by the charged tip.37 Both contributions tend to increase with the tip-sample distance decreasing. Figure 5a shows the tip oscillation amplitude dosc value for a fixed Vac bias, obtained by the measurements (symbols) which agree with those calculated using the second law of Newton (line). The average is performed on all the tip-sample distance for each Vac bias. The model is applied also to reproduce the d dependence for different RHs. In this case, together with the scale factor γ, also the asymptotic SP value is set as a free parameter. The curves obtained by the fitting procedures are reported in Figure 3 (continuous lines), showing a good agreement with the measured ones. Thus, by varying the RH both the intrinsic SP of the surface and the extrinsic potential terms due to the tip-sample interactions are modified. The proposed model makes it possible to obtain the intrinsic SP term by decoupling the two terms and by calculating the extrinsic one. Furthermore, the correlation between the dosc amplitude and the RH value revealed an inverse proportional behavior as shown in Figure 5b. 4.3. Studies of Functional Nanostructures. The proposed model, which has been exploited above to describe the tip-sample interaction on the macroscopic scale, can be applied also to meso- and nanoscale systems featuring a roughness of the studied surface, or more generally a height of the adsorbed objects, which is smaller than the tip-sample distance. Moreover, for studies of nanoscale objects, it is desirable to achieve the highest lateral resolution (LR). To bestow information onto the role played by the different experimental parameters we have performed measurements on PDI nanostructures adsorbed on the HOPG substrate. In order to neglect the tip-broadening effects, we have studied organic structures featuring a lateral size much larger than the effective area, thus having a diameter of about 150 nm. Thus, we can rule out the contribution of the SP of the substrate to the SP measured for the PDI nanostructures. It is well-known that both lateral and potential resolutions achieved by KPFM measurements increase with the decreasing tip-sample distance. Moreover, it can be expected that the applied Vac bias plays a crucial role. To cast light onto this aspect, Figure 6 shows both the topographical and the corresponding KPFM images of a PDI nanoribbon acquired at

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Figure 6. Organic PDI strip self-assembled on HOPG substrate: (a) gradient of topographic image and (b-h) corresponding KPFM images acquired varying Vac, while the tip-sample distance amounts to about 40 nm. Different ac potentials: (b) 50, (c) 100, (d) 200, (e) 500, (f) 1000, (g) 2000, and (h) 3000 mV. Sizes: 1 × 1 µm2. Z-scales: (a) 1 nm/nm, (b-h) 60 mV.

different ac amplitudes and RH ) 20%. Figure 6a shows the gradient of the topographic image of a PDI ribbon having micrometric length which is self-assembled on HOPG substrate. The width and height of the anisotropic architecture are 180 ( 10 nm and 10 ( 1 nm, respectively. Parts b-h of Figure 6 show the corresponding KPFM images acquired varying Vac from 50 to 3000 mV by keeping a fixed tip-sample distance of about 40 nm. In general, the SP noise level decreases with the increasing Vac. For ac potentials smaller than 100 mV (Figure 6, parts b and c), the SP of the ribbon results similar to the substrate one and the main SP variations are measured at the edges of the stripes. When the Vac bias is increased (Figure 6, parts d and e), both the SP difference between organic anisotropic architecture and substrate (i.e., ∆SP ) SPPDI - SPHOPG) and the LR increase. When the ac bias is larger than 500 mV (Figure 6e-h) no significant ∆SP and LR variations are observed. Thus, this ac value is fixed for the study of the effects of the other two experimental parameters (i.e., RH and tip-sample distance). As previously shown, using an ac polarization of at least 500 mV, the measured SP of clean HOPG depends on the tip-sample distance. Thus, the approximation of nonoscillating cantilever results insufficient to describe the tip-sample interaction. Figure 7 reports both the (parts a and c) topographical and the (parts b and d) corresponding KPFM images of PDI crystals. PDI molecules self-aggregate forming flat and stepped crystalline lamellae with heights ranging from 18 up to ca. 100 nm. The observed steps in height are multiples of 1.4 ( 0.2 nm, in accordance to the value measured on similar PDI architectures adsorbed on mica substrate.61 Widths and lengths of the PDI crystalline lamellae vary from ca. 50 to 500 nm and from ca. 400 nm to some micrometers, respectively. The four images have been acquired at two different RH values, nominal 0% and 60%, respectively, and with tip-sample distance amounts of 40 nm. In both cases, the measured SP of PDI structures results negative with respect to the substrate one. For RH ) 0%, the achieved LR results much larger than those measured for RH ) 60% and a strong SP variation is localized at the edge of the PDI structures. The measured ∆SP differences

between the two RHs may be ascribed to the adsorbed water film and to geometrical factors. Although the first one may be neglected for RH ) 0%, the second component may be estimated by the model presented in the previous sections. Figure 8 shows the d dependence of measured SP of both substrate (data points are marked with dots) and PDI structures (circles) corresponding to two different RH values: (a) 0% and (b) 60%. In all the cases, the measured SP distributions show a decreasing monotonic trend. The HOPG curve agrees with that observed for the macroscopic sample (Figure 3), whereas the SP of the adsorbed PDI structures results smaller than the substrate potential. For both RH values, the asymptotic regime may be localized for a tip-sample distance exceeding about 200 nm. It is remarkable that the measured SP on the uncovered HOPG in Figure 7 agrees with the potential measured on macroscopic HOPG (see Figures 2 and 3). The SP of HOPG does not depend on the presence of PDI structures beside the areas exposing the neat substrate surface, and the corresponding d dependence of the potential is purely described by extrinsic terms, i.e., it depends only on the tip-sample interactions and does not depend on the electronic properties of the sample. Thus, all the measured variations in this nanosystem are due to the intrinsic property of the organic nanostructure. The tip-induced polarization may be neglected in the asymptotic regime (i.e., d > 200 nm). Thus, it is possible to focus only on the intrinsic SP term, and the ∆SP difference between the PDI lamellae and the uncovered HOPG substrate may be written as

∆SPRH ) SPPDI + ∆PDI,RH - SPHOPG

(11)

where ∆PDI,RH represents the potential, depending on RH, induced by the water film adsorbed on PDI. Therefore, the d dependence of the intrinsic term of the measured ∆SP depends on the water film deposited on the PDI structures. In the limit case of no adsorbed water (i.e., RH ) 0%), the measured ∆SP does not depend on d. By increasing the RH value, the asymptotic ∆SP varies and the potential spread increases.

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Figure 7. (a and c) Topographical and (b and d) corresponding KPFM image of PDI deposited on HOPG. The images are acquired at a welldefined RH which amounted to (a and b) 0% and (c and d) 60%. Z-range: (a) 40 nm, (b) 100 mV, (c) 40 nm, and (d) 50 mV.

Figure 8. d-Dependence of the measured SP of both PDI structures (squares) and HOPG substrate (circles) for (a) RH ) 0% and (b) RH ) 60%. (c and d) Corresponding SP difference: ∆SP ) SPPDI - SPHOPG (triangles). The asymptotic ∆SP is indicated by a dashed line, whereas the ∆SP data set acquired for RH ) 60% is fitted by a two-degree inverse polynomial (continuous line).

Parts c and d of Figure 8 display the SP difference (4) between the adsorbate and the substrate, i.e., ∆SP ) SPPDI SPHOPG, performed at RH ) nominal 0% and 60%, respectively. The ∆SP trends strongly depend on the environment. At RH ) 60% the ∆SP decreases with a monotonic slope of potential variation of ca. 40 mV. The adsorbed water dipoles induced by the tip represent the main contribution to the tip-sample

interaction. The measured ∆SP distribution shows a two-degree inverse polynomial trend (continuous line),37 where the fitting curve is calculated taking into account the effective tip-sample distance presented in the model. A more complex behavior is observed for nominal RH ) 0%. In this case, the total potential variation of about 20 mV is measured and it is possible to identify two different regions: a

17376 J. Phys. Chem. C, Vol. 112, No. 44, 2008 zone with a quick decrease of the SP difference for d < 40 nm and an asymptotic plateau achieved for larger tip-sample distances in which the mean ∆SP amounts to -77 ( 3 mV (dashed line). Therefore, the SP variation of the sample due to the film water adsorbed on the surface turns out to be reduced and it may be localized in the early 40 nm of tip-sample distance. By comparing the measured asymptotic SP value of PDI lamellae with the HOPG work function reference,80 we estimate the pseudo-Fermi level of the PDI nanostructure which amounts to about 4.40 ( 0.01 V. This value is consistent with the mean HOMO-LUMO gap values of PDI reported in literature.81 This is a purely qualitative agreement because conductive and valence bands of PDI lamellae are likely to differ from the levels of the isolated molecule. Thus, further theoretical calculations at the ensemble level will be important to simulate the energy levels of the well-packed PDI molecules forming the observed ribbons. 5. Conclusions A comprehensive investigation of the influence of the main experimental parameters during a KPFM measurement has been performed. The effects of (i) humidity, (ii) applied ac bias, and (iii) tip-sample distance on the measured SP have been studied on both macroscopic samples and organic semiconducting nanosystems. We have devised a general model which describes the tip-sample geometry effects on the measured SP suggesting that the humidity varies the electronic properties of the sample (and the tip) although changes in the mechanical properties of the probe cannot be neglected. In particular, the model separates these two aspects indicated as intrinsic and extrinsic, respectively. By using the proposed model it is possible to separate and to calculate the extrinsic term of the measured SP for the nanoscopic systems as well as the macroscopic ones. In particular, it is possible to estimate the SP variation due to the experimental artifact (tip-sample distance and Vac bias) and those due to the electronic properties. The capability to separate these two terms is crucial for the quantification of electronic properties of (nano)device at work with a high degree of precision. By removing all the artifacts due to the technique and the environment where the device is working, it is possible to extract with a high precision the intrinsic potential of the object under study (i.e., work function, potential drop, charge carrier concentrations) governing the characteristics of the device. Our findings make it possible to gain a reliable quantitative characterization of the electronic properties of semiconducting nanostructured materials; thus, they represent a step forward toward both the optimization of their electronic characteristics and the tuning of their interfaces with metallic electrodes, as key steps in the fabrication of (nano)electronic devices. Acknowledgment. This work was supported by the ESFSONS2-SUPRAMATES project, the Regione Emilia-Romagna PRIITT Nanofaber Net-Laboratory, the German Science Foundation (Mu 334/28-1), and the EU through the projects Marie Curie EST-SUPER (MEST-CT-2004-008128), the RTNs PRAIRIES (MRTN-CT-2006-035810). References and Notes (1) 605. (2) (3) (4)

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