JOURNAL OF APPLIED PHYSICS 103, 114311 共2008兲
Characteristic capacitance in an electric force microscope determined by using sample surface bias effect G. C. Qi,1 H. Yan,1 L. Guan,1 Y. L. Yang,1 X. H. Qiu,1,a兲 C. Wang,1,b兲 Y. B. Li,2 and Y. P. Jiang3 1
National Center for Nanoscience and Technology, Beijing 100190, People’s Republic of China College of Chemistry and Life Science, Gannan Normal University, Ganzhou 341000, Jiangxi Province, People’s Republic of China 3 Department of Physics, Tsinghua University, Beijing 100871, People’s Republic of China 2
共Received 18 December 2007; accepted 9 April 2008; published online 11 June 2008兲 A method to determine the dependence of characteristic capacitance of an electric force microscopy tip on tip-sample separation is presented. It is demonstrated that by introducing sufficient voltage to the sample surface, the first derivative of the characteristic capacitance for tip-sample complex could be obtained and, subsequently, the characteristic capacitance versus tip-sample separation could be determined. In addition, the effective charge position on the tip relative to sample surface could also be identified. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2938846兴 I. INTRODUCTION
Electric force microscopy 共EFM兲 has been widely pursued as a promising technique for probing the microscopic distribution of dielectric properties and electric charges.1–4 In addition to retrieving qualitative information on the electrical properties of surfaces at nanometer scale, extensive efforts have been dedicated to explore the potentials of getting quantitative electrical characteristics which is crucial for studying properties of nanostructures.5,6 Among many experimental and theoretical EFM studies, the probe structure is one of the central issues. It has been recognized that the capacitance between the tip and sample is an effective avenue to characterize the geometry of probe and cantilever. The EFM tip is an integrated structure including the tip apex, cone, and cantilever. With different contributing interactions due to the nature of forces between the tip and sample surface, it is nontrivial to obtain the characteristic capacitance between the tip and sample. The analytical models for the electrostatic interaction between the tip and sample surface have been developed both theoretically and experimentally for different kinds of tip geometries7,8 and at various tip-sample distance. It was found that the apex and conical portions of a typical EFM tip are responsible to the majority of the interactions when the tip is not far away from the sample surface, and the contribution from the cantilever plate becomes more appreciable at larger tip-sample distances.7 Nevertheless, the experimental determination of the characteristic tip-sample capacitance is a complex process although highly desirable in quantitative measurements of properties such as the dielectric constant of nanomaterials9 and the charge density on sample surfaces. Previous studies have shown that the total electrostatic forces acting on an EFM probe over a sample can be separated into two components: Capacitive forces associated with a兲
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the tip-sample capacitance and Coulombic forces due to the static charges and/or multipoles on sample surface. A generalized expression was given by4 F=
1 dCs-t 2 V + E sQ t . 2 dz t
共1兲
Here, dCs-t / dz is the derivative of the empty tip-sample capacitance with respect to tip-to-sample separation. Vt is the voltage applied to the tip. Es represents the electric field at the tip location that is only created by the charges and/or multipoles in the sample surface. In the surface normal direction, Es is expected to be relatively constant in the vicinity of a flat sample surface with a uniform charge distribution. Qt denotes the effective charge on the tip Qt = Cs-tVt + Qim, where Cs-tVt is the charge stored on the tip at an applied voltage Vt with respect to a grounded sample, and Qim is the image charge on the tip induced by the static charge distribution on the sample surface. For planar sample, Qim could be regarded as a constant at a small tip-to-sample separation. Therefore, the force gradient F⬘ acting on the tip can be approximated as F⬘ =
F 1 ⬘ 共Vt兲2 + EsCs-t ⬙ Vt . ⬇ Cs-t 2 z
共2兲
The electric field Es is normally caused by the static charges and/or multipole distribution on sample surface. In the case when a voltage of Vs is applied to a conducting sample, as illustrated schematically in Fig. 1, additional charges built up on the sample surface could account for the main contribution to the total electric field Es. We expect that the average electric field experienced by the tip could be written as Es = g共s兲Vs, where g共s兲 is a factor related to the tip geometry. It is noticeable in Eq. 共2兲 that as Vs increases, the Coulombic term EsCs-t ⬘ Vt increases linearly and contributes more to the force gradient F⬘, while the contribution from the capacitive interaction remains unchanged. This expectation is validated by our experimental results, as shown in
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© 2008 American Institute of Physics
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FIG. 1. 共Color online兲 Schematic illustration of the setup of EFM measurements in the lift mode: The tip is oscillated at a constant distance h over the sample with a fixed voltage applied to sample and the voltages applied to the tip vary in a limited range from −4 to 4 V. The phase shift ⌬ represents the lag between the drive signal voltage applied to the cantilever and its oscillation.
Sec. III, where the force gradient F⬘ shows a linear dependence on tip voltage Vt at the conditions of Vs Ⰷ Vt. In this case, Eq. 共2兲 could be approximated as
⬘ Vt . F⬘ ⬇ g共s兲VsCs-t
共3兲
In EFM phase measurements, the forces on the tip give rise to a phase shift ⌬, which represents the phase lag between the drive signal voltage to the cantilever and its oscillation, providing a direct measure of the force gradient F⬘ exerted on the tip.10–12 For small oscillation amplitude, the phase shift ⌬ could be approximated as ⌬ ⬇ −QF⬘ / k, where Q is the quality factor of the cantilever 共typically a few hundreds in air兲, and the effective spring constant k of the cantilever is typically of the order of a few N/m. By combining it with Eq. 共3兲, we can obtain
⬘ V t = K 0V t , ⌬ ⬀ − g共s兲VsCs-t
共4兲
where K0 = −g共s兲VsCs-t ⬘ . The formula suggests an interesting approach to characterizing the capacitance between the tip and sample, i.e., the derivative of the tip-sample characteristic capacitance could be obtained by measuring the slope K0 of the phase shift 共⌬兲 versus tip voltage 共Vt兲 characteristic. The characteristics of tip-sample capacitance would be further determined by fitting the curve of K0 versus lift height h as illustrated in this study. II. EFM MEASUREMENTS
The EFM measurements are performed, as schematically illustrated in Fig. 1. The Au coated doped Si 共100兲 substrate is used as the sample. Images are obtained by a Dimension 3100 atomic force microscope 共Veeco Metrology Group, USA兲. Co/Cr-coated tips 共MESP兲 from Veeco are with resonance frequency of 75 kHz, spring constant k of 2.8 N/m, and the tip height h in the range from 10 to 15 m. For typical EFM operation, each line is scanned twice. The first pass consisted of a standard tapping mode scan without voltages applied to the tip to get topography images with the feedback loop active. In the second pass, the topographic data are used to retrace the first line. The tip is lifted to a fixed height above the sample surface and external voltages are applied to the tip as it scans with feedback loop disabled.
FIG. 2. The characteristics of phase shift ⌬ vs tip voltage Vt acquired when voltages of 共a兲 0 V and 共b兲 20 V are applied on the sample. The tip is lifted from the sample surface by 700 nm.
The recorded EFM signal is the phase difference between the driving force and the actual oscillation of the cantilever, which is the phase shift ⌬. The recorded EFM phase shift images are proportional to the gradient F⬘ of the tip-sample force in the surface normal direction. This method allows the maximum removal of the topographic contribution to the EFM phase signal. In order to precisely and instantly change the voltages applied to a tip in our experiments, the software was programed to successively and automatically change the tip voltages during scanning. The EFM images are original without being flatten or other processing operation. III. RESULTS AND DISCUSSIONS
The current study is based on the phase shift ⌬ which resulted from the interaction between charges on a tip and electric field created by sample voltages. In order to enhance the contribution from the Coulombic term, a voltage of 20 V was applied to the sample surface, while the bias voltages applied to a tip are limited in the range from −4 to 4 V. Figures 2共a兲 and 2共b兲 show the phase shift 共⌬兲 versus tip voltage 共Vt兲 characteristics when the sample voltage is at Vs = 0 and 20 V, respectively. When the sample voltage Vs = 20 V, the curve approaches to a linear characteristic, while
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sis on the tip-sample interaction to include an effective height heff to correct the tip-sample distance h to accommodate the effective charge position. We thus suggest a general expression for the first derivative of tip-sample capacitance as
⬘ = − A0共h + heff兲a . Cs-t
FIG. 3. 共Color online兲 Phase shift ⌬ vs tip voltage Vt characteristics acquired at different tips lift heights. The sample voltage is 20 V.
the quadratic part associated with the capacitive interaction becomes negligible. This observation is consistent with the facts that the linear term in Eq. 共2兲 becomes dominant with increasing voltages applied to the sample. It was worth noting that the slight asymmetry of the characteristic curves in Fig. 2共a兲 could come from the work function differences between the tip and sample, which do not affect the current results. The experiments were performed under different lift heights to obtain a series of phase shift 共⌬兲 versus tip voltage 共Vt兲 spectra, as shown in Fig. 3.12 The traces G through Q in Fig. 3 correspond to different tip-sample separations increasing from 500 to 1600 nm. The phase shift 共⌬兲 versus tip voltage 共Vt兲 characteristics were fitted to obtain the corresponding slopes K0, as defined in Eq. 共4兲. The relationship of K0 versus lift heights h was plotted in Fig. 4. A number of previous studies3,7 have simulated the function of the tip-sample characteristic capacitance as lift height h to be Cs-t ⬙ ⬀ 1 / ha, with the index ␣ in the range from 1.0 to 3.0, irrespective of the complex shape of the tips. However, it is unreasonable to consider the lift height as the effective distance between the two plates of tip-sample capacitor because the tip apex has an irregular shape, in addition to the effect of cone and cantilever.7,13 It is essential for the analy-
共5兲
Here, ␣ is an index related to the geometrical shape of a given tip, and A0 is a proportional coefficient. For a concrete tip, the value of g共s兲Vs is fixed when the voltage applied to a sample is constant. The function of the characteristic capacitance of the tip as the effective tip-sample distance 共h + heff兲 could be obtained by integrating Eq. 共5兲, and we get Cs-t = −
A0 共h + heff兲a+1 + A1 . ␣+1
共6兲
Here, A1 resulted from the integration of Eq. 共5兲. The curve in Fig. 4 is fitted over Eq. 共5兲 and then obtained the values of ␣ = −1.3 and heff = 2322 nm. The value of ␣ is in the range between the reported sphere-plane geometry 共2Cs-t / z2 ⬀ z−2兲 共Ref. 14兲 and parallel-plate geometry 共2Cs-t / z2 ⬀ z−3兲.15 The effective charge position locates at about the point of 2322 nm from the tip apex, which is consistent with the results that the most forces derive from the tip apex.16 The merits of the above presented method could be summarized as follows: 共1兲 the functions of tip characteristic capacitance as the effective charge distance 共h + heff兲 could be obtained via integrating only once, avoiding introducing additionally an ambiguous linear term which resulted from a second integration, as reported in the existing studies. The model can widely reproduce previously reported results and 共2兲 the correcting factor of effective height heff is reasonably introduced to correct the effective charge position. IV. SUMMARY
This work reports a simple method to characterize the tip-sample characteristic capacitance as a function of the tipsample distance h. We introduce an effective height heff to represent the effective charge position, compensating the effect which resulted from the irregular shape of the AFM tips. The method based on sample surface bias effect could be beneficial for calibrating EFM tips. ACKNOWLEDGMENTS
This work was supported by the National Basic Research Program of China 共No. 2007CB936800兲 and the Chinese Academy of Sciences 共No. KJCX2-YW-M04兲. Financial support from the National Science Foundation of China 共No. 90406019兲 is also gratefully acknowledged. S. W. Howell and D. B. Janes, J. Appl. Phys. 97, 043703 共2005兲. B. D. Terris, J. E. Stern, D. Rugar, and H. J. Mamin, Phys. Rev. Lett. 63, 2669 共1989兲. 3 T. D. Krauss and L. E. Brus, Phys. Rev. Lett. 83, 4840 共1999兲. 4 O. Cherniavskaya, L. W. Chen, V. Weng, L. Yuditsky, and L. E. Brus, J. Phys. Chem. B 107, 1525 共2003兲. 5 T. Mélin, H. Diesinger, D. Deresmes, and D. Stiévenard, Phys. Rev. B 69, 035321 共2004兲. 6 R. Krishnan, M. A. Hahn, Z. Yu, J. Silcox, P. M. Fauchet, and T. Krauss, 1 2
FIG. 4. 共Color online兲 Slope K0 vs lift height h obtained from Fig. 3. The slope K0 is proportional to the derivative of the tip-sample capacitance in the surface normal direction of the sample. The fitting line is plotted in red.
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Phys. Rev. Lett. 92, 216803 共2004兲. S. Belaidi, P. Girard, and G. Leveque, J. Appl. Phys. 81, 1023 共1997兲. 8 G. M. Sacha, A. Verdaguer, J. Martinez, J. J. Sáenz, D. F. Ogletree, and M. Salmeron, Appl. Phys. Lett. 86, 123101 共2005兲. 9 A. V. Krayev, G. A. Shandryuk, L. N. Grigorov, and R. V. Talroze, Macromol. Chem. Phys. 207, 966 共2006兲. 10 Y. Martin, D. W. Abraham, and H. K. Wickramasinghe, Appl. Phys. Lett. 52, 1103 共1988兲. 11 T. Albrecht, P. Grutter, D. Horne, and D. Rugar, J. Appl. Phys. 69, 668 共1991兲. 12 We take into account the fact that the EFM spatial resolution will dramatically degrade at large tip-sample distance and consequently compromise 7
J. Appl. Phys. 103, 114311 共2008兲 its application capability. We hereby limited the lift height in the measurement to a moderate range from 500 to 1600 nm. On the other hand, strong nonlinear effects caused by the large electric field in tip-sample gap were observed when the tip is set close to the sample surface, usually occurring at a lift height of a few hundreds of nanometers. 13 A. Alessandrini, G. Valdrè, B. Morten, and M. Prudenziati, J. Appl. Phys. 92, 4705 共2002兲. 14 J. Jiang, T. D. Krauss, and L. E. Brus, J. Phys. Chem. B 104, 11936 共2000兲. 15 M. J. Gordon and T. Baron, Phys. Rev. B 72, 165420 共2005兲. 16 E. Tevaarwerk, D. G. Keppel, P. Rugheimer, M. G. Lagally, and M. A. Eriksson, Rev. Sci. Instrum. 76, 053707 共2005兲.
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