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Shape dependence of the capacitance of scanning capacitance microscope probes
ARTICLE IN PRESS
Ultramicroscopy 108 (2008) 712–717 www.elsevier.com/locate/ultramic
Shape dependence of the capacitance of scanning capacitance microscope probes Sˇ. La´nyi Institute of Physics, Slovakian Academy of Sciences, Du´bravska´ cesta 9, SK-845 11 Bratislava, Slovakia Received 5 March 2007; accepted 2 November 2007
Abstract The capacitance of approximately conical scanning capacitance microscope probes placed perpendicularly over a conducting plane has been modelled using the finite element method. The dependence on tip/surface distance, radius of curvature of the tip apex, cone angle and height has been analysed. Both shielded and unshielded probes have been considered. The fits of obtained dependences have been combined into an analytic approximation of the capacitance as a function of tip/surface distance, radius of curvature, cone angle and height. The results can be used to estimation of stray capacitance, achievable lateral resolution and contrast. r 2008 Elsevier B.V. All rights reserved. PACS: 07.79.v; 41.20.Cv; 02.70.Dh; 07.05.Tp Keywords: Computer simulation; Scanning capacitance microscope probe; Finite element method
1. Introduction One of the first scanning probe microscopes, the scanning capacitance microscope (SCM) [1–6] is able to image and analyse structures on the surface of conductors, both free and coated with insulating films, and buried structures in semiconductors and dielectrics. It has been identified as one of the tools expected to satisfy the needs of semiconductor industry for high-resolution and high-accuracy determination of dopant concentration in semiconductor structures on the nanometre scale, needed for future generations of ULSI integrated circuits [7,8]. Though lateral resolution of 5 nm on conducting surfaces [9] and sub-10 nm on MOS capacitors [10] has been already achieved, high resolution and high accuracy of carrier concentration determination at a time seems rather demanding. The probe of the SCM is a sharp conducting tip forming with the conducting surface or insulator coated substrate a capacitor. It is raster scanned over the surface or in contact with the insulating film. The sensed capacitance depends on Tel.: +421 2 59410525; fax: +421 2 54776085.
E-mail address: [email protected] 0304-3991/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2007.11.002
the tip-to-counter electrode distance, e.g. the film thickness or semiconductor depletion layer width, and from the permittivity of the material between them. Their local variations are utilised to create a picture of topography of conducting surfaces or material property (semiconductors and dielectrics). The most representative applications comprise the analysis of free carrier concentration [11], delineation of p–n junctions [12,13], visualisation of crosssection of semiconductor structures [14–16] or controlling wafer processing [17]. The lateral resolution and the formed contrast depend on the distribution of the electrostatic field between the probe and the sample. The useful information is provided by the tip apex, in the vicinity of which the electrostatic field is most concentrated. The rest contributes only a resolution gradually deteriorating with distance, ending up in a featureless background. Unfortunately, this contribution is orders of magnitude larger than the desired one. It reduces the achieved contrast of small features and may effectively preclude quantitative analysis [18]. In most transducer solutions some noise components are proportional to the overall capacitance, hence a large stray capacitance may seriously reduce the signal/noise ratio [9].
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
The mapping of the electrostatic field is also used for estimation of electrostatic force acting on the scanning force microscope (SFM) probe [19], e.g. using the generalised image charge method [20], and for calculation of capacitance vs. distance in Ref. [21]. The knowledge of the capacitance is useful in interpretation of Scanning Kelvin Probe Microscopy images [22,23]. However, there is an important difference between the two cases—force is a vector, whereas capacitance is a scalar quantity. So, the forces acting to opposite sides of the probe or of a topographic feature subtract, whereas fluxes, i.e. capacitance contributions add. The exact shape of a probe is usually unknown. The height of the tip and the cone angle can be evaluated using an optical microscope. The manufacturers’ specifications of the radius of curvature of the tip apex are frequently wide, first of all of metal-coated tips. The measurement of radii on the order of tens of nanometres is possible by means of scanning electron microscopes. This is expensive, not accessible in each laboratory and not without risk of damaging the cantilever. In the present paper, the capacitance of probes of varying shape and size has been analysed using the finite element method (FEM) [24]. The results are presented in the form of graphs and as an analytic approximation. They may help to estimate the expectable performance and to select the optimal shape for a given task. If the height and cone angle are provided, the tip radius can be deduced from a simple capacitance vs. distance measurement. 2. Analysed probes The probes used in SCMs comprise sharpened wires [2,25], conducting SFM cantilevers [26], platinum cantilevers [27] and shielded sharpened wires [5,28]. Electrochemically etched tips are approximately conical, with small cone angles (10–201) [21,27]. Those of micromachined conducting SFM cantilevers have larger cone angles (301) and are significantly shorter (15 mm) than the tips etched from wires (tenths of millimetre and more). The mutual capacitance of two conductors is the ratio of accumulated charge and the potential difference between them. Its calculation requires solving the Laplace equation DV=0 in three dimensions. The surface charge density sr, induced in the electrodes by the applied voltage, is coupled with the electrostatic field Er through the Poisson equation (r is the radial variable). In general, no closed-form analytic solution of this problem is known, except of some relatively simple cases. Configurations like sphere/plane, cone/plane, paraboloid/plane or hyperboloid/plane combinations can be solved in prolate spheroidal coordinates. For instance, the effect of an electron on the potential and tunnelling [29] and the electrostatic potential of a hyperbolic probe [30] were calculated using this method. The FEM is a flexible tool for the calculation of complex geometries. It can be used to arbitrary tip shapes as well as surface morphology. However, until now axially symmetric
713
cases, sections of three-dimensional problems in Cartesian coordinates [31,32] and combination of both have been analysed [33–36]. The FEM analysis has been performed using the MEP 6.0 code [37]. To enable reduction of the three-dimensional problem to two dimensions, axial symmetry, i.e. the probe axis perpendicular to the plane, has been assumed. This condition can be usually easily fulfilled and a few degrees deviation has negligible effect. Both shielded and unshielded probes have been analysed. Under shielding active shielding is assumed, i.e. the voltage sensed by the input of the transducer electronics imposed to the shield [5]. The radius of curvature of approximately hemispherical apex r was varied from 25 nm to 1 mm, the cone angle a from 101 to 301, the height L from 150 to 300 mm. They would correspond to tips formed from wires with diameter from 26 to 160 mm, or thicker ones but shielded, e.g. by a pierced foil placed above the sample or placing the rest of the wire into a tube. These probes were denoted unshielded. The shielded ones protruded 5 mm from a shield with orifice radius of 25 mm. The attribute ‘‘approximately’’ means that due to the smooth transition to the cone the apex is less than a hemisphere. The shape of electrochemically etched wires sometimes deviates from exact conical shape. Therefore, slightly concave or convex shape with principal radius of 2 mm has also been considered. The assumed tip-to-plane distance d was 5–25 nm. Fig. 1 shows the assumed tip shapes and the varied dimensions. Since shielding limits the spreading of the electrostatic field of the tip, thus rendering a good definition of boundary conditions possible, the stray field of unshielded probes is not limited. Even worse, the capacitance between an infinite plane and a cone perpendicular to it grows approximately linearly with the height of the cone. However, simulation has shown that the flux density near the tip apex did not differ by more than 2.5% when the size of the cone has been doubled. Therefore, the tasks were limited by a force line (Neumann boundary condition) in the form of a circular arc, ending perpendicularly on both
L
d
α
R
R
r
Fig. 1. The shape of tips: (a) conical, (b) concave and (c) convex. The tip height L, cone angle a, apex radius r and distance from plane d have been changed. The radius of forming arcs of concave and convex shape R was 2 mm.
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
714
9 Cone angle 10° 20° 30° d = 5 nm r = 100 nm
8
CAPACITANCE (fF)
7 N
D
D
N
6 5 4 3 2 1
N 0 0
50
100
150 200 TIP HEIGTH (μm)
250
300
D
D
Fig. 3. Dependence of capacitance on tip height L and cone angle a. Tip radius r was 100 nm.
D N
D
D
N
8.0
D 7.5
the cone and the plane, in three dimensions approximately a hemisphere. The employed boundary conditions are shown in Fig. 2.
cone angle:
CAPACITANCE (fF)
Fig. 2. The employed boundary conditions: (a) unshielded tip and (b) shielded tip. D denotes Dirichlet boundary conditions (equipotentials) and N Neumann boundary conditions (force lines).
7.0
d:
10° 20° 30° 5 15 25 nm
6.5 6.0 5.5
3. Results 5.0
The capacitance of a cone with vertex in the planar counter-electrode should be a linear function of the cone height. Fig. 3 shows that, for relatively high cones, the deviation caused by rounding of the apex, and by its small separation from the plane, is negligibly small. In Fig. 4, the dependence of the capacitance of unshielded probes on radius of curvature r and tip/plane separation d is seen. The results with shielded probes, except the capacitance magnitude, are similar but the C vs. r dependences are less steep (Fig. 5). Active shielding, even with a large orifice compared to the diameter of the protruding part of the tip, significantly reduces the stray capacitance. In most of the interior of the shield there is practically no field. The capacitance corresponds to that of a 16.8-mm high unshielded tip. Important is the nearly perfect linear dependence of the capacitance on the tip radius. Minor deviations from linearity appeared at the smallest radii. However, they were more a scatter than systematic and correspond to small capacitance differences, most affected by computational errors. Some of the C vs. r dependences could be fitted with higher correlation to a second-order polynomial than to a line. However, for our purpose the difference is of no
4.5 0
200
400 600 TIP RADIUS (nm)
800
1000
Fig. 4. Dependence of capacitance on tip radius r and cone angle a of 250 mm high unshielded tip.
importance. For a flat apex an approximately parabolic C vs. r would result. The slopes of linear fits of C vs. r dependences are shown in Fig. 6. All curves are remarkably similar. The capacitance– distance dependence of a parallel plate capacitor is hyperbolic. Such dependence cannot hold for the approximately conical tip over a plane. However, the curves have been fitted with satisfactory correlation to a shifted hyperbola: CðdÞ ¼ a þ
b . dþc
(1)
The difference [C(d)a] is displayed in Fig. 7. The employed fit convincingly represents both the shielded and unshielded probes.
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
0.9
1.6e-4 cone angle: 10° 20° 30°
0.8
d: 5 10 15 20 25 nm
0.7
unshielded shielded angle: 10° 20° 30°
1.4e-4 REDUCED SLOPE
CAPACITANCE (fF)
715
0.6 0.5
1.2e-4
1.0e-4
0.4 8.0e-5 0.3 6.0e-5
0.2 0
200
400 600 TIP RADIUS (nm)
800
Fig. 5. Dependence of capacitance on tip radius r and cone angle a of 250 mm high shielded tip.
5
10 15 20 DISTANCE (nm)
ΔC/Δd (fF/ 5 nm)
4.e-4
30
cone angle: tip heigth: 10° 30° 200 250 300 μm
4.e-3
5.e-4
25
Fig. 7. Reduced slopes of C vs. r dependences of both unshielded and shielded 250 mm high tips.
unshielded shielded angle: 10° 20° 30°
6.e-4
SLOPE (fF/nm)
0
1000
3.e-3
2.e-3
3.e-4 1.e-3 5 0
5
10 15 20 DISTANCE (nm)
25
30
Fig. 6. Slopes of C vs. r dependences of unshielded tips from Fig. 4 and of shielded tips from Fig. 5.
At small radii, the relative contribution of the tip apex may become very small. However, it is accentuated in a difference of capacitances measured in two tip/sample distances. Fig. 8 shows that the effect of cone angle and height is reduced but still present. The comparison of conical tip with cone angle 201 with the convex (limiting angle 241) and concave tips (limiting angle 161) in Fig. 9 demonstrates that the capacitance difference is nearly independent from the actual shape of the tip. 4. Approximation of results Within the analysed ranges, the dependences of capacitance on both cone height (Fig. 2) and angle (Figs. 4 and 5)
10
15 20 MEAN DISTANCE (nm)
25
Fig. 8. Dependence of capacitance difference, resulting from displacement of unshielded and shielded tips by 5 nm, on cone angle and tip height. The influence of tip height at small cone angle is negligibly small.
are reasonably linear. By reducing the cone height toward zero, the capacitance would drop to that of the spherical apex. An extrapolation of cone angle to zero would turn the tip to a hemispherically terminated cylinder, the capacitance of which, with respect to the plane, would also remain non-zero. Therefore, the lines representing the C vs. L and C vs. a dependences are expected to intersect the ordinate axis above zero. The C vs. r dependences in Figs. 4 and 5 for various cone angles and same d are not parallel. The slopes also depend on the cone height. The linear fits of dependences of C on L, a and r, the hyperbolic fit of dependence on d, as well as the linear fits of the previous fits’ parameters on the remaining variables, can be
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
716
ΔC/Δd (fF/ 5 nm)
on tip/surface spacing and makes it better suitable as an input quantity to control the probe position in constant-capacitance mode, than the capacitance itself. The expression
tip radius: shape: 100 nm 25 nm concave conical convex
4.e-3
3.e-3
dC rðc2 þ c3 a þ c4 L þ c5 aLÞ ¼ dd ðc6 þ c7 a þ c8 L þ c9 aL þ dÞ2
(3)
2.e-3
shows that by this the first two terms in Eq. (2) are eliminated.
1.e-3
5. Discussion
0e+0 5
10
15 20 MEAN DISTANCE (nm)
25
Fig. 9. Effect of tip shape, concave, conical and convex, on the capacitance difference resulting from tip displacement by 5 nm. The cone angle was 201 and the height and base diameter of all three tips was the same. The limiting angle at the apex of concave tip is 161 and of the convex tip 241.
Table 1 Valuesa of coefficients in expressions (2) and (3) Slopes of
a
L
aL
c0 0.01298
c1 5.195e4
r
ra
rL
raL
c2 2.080e3
c3 5.261e5
c4 4.644e7
c5 7.178e8
a
L
aL
c7 1.476e4
c8 1.951e6
c9 1.168e7
c6 9.722e3
a The tip height L, radius of curvature r and distance d are in mm, the angle a in degrees and the result in fF.
combined to expression C ¼ ðc0 þ c1 aÞL þ
rðc2 þ c3 a þ c4 L þ c5 aLÞ . c6 þ c7 a þ c8 L þ c9 aL þ d
The experimentally found capacitance of unshielded probes is frequently much larger than the results in Fig. 4. The reason is that the present analysis was restricted to a cone of limited height. Probes with a stray capacitance of 0.275 pF [38] or 0.5 pF [21] would hardly yield an acceptable signal/noise ratio. The noise in capacitance images presented in Ref. [38] was on the order of 10 fF, comparable with the signal from 40 nm deep carbon grating. The derived approximation of capacitance (Eq. (2)) enables a simple evaluation of the effect of tip geometry on the stray capacitance of a conical probe. The tip height and cone angle can be evaluated using an optical microscope. Then, the only relevant unknown parameter is the radius of the tip apex. Using expression (2) and the capacitance measured in two known tip/conducting surface distances, the radius r can be calculated. An alternative solution is vibrating the tip and application of expression (3). Unfortunately, this procedure would not work in the case of unshielded conducting cantilevers. The contribution of the cantilever beam and fixture would have to be considered separately. In this way, the integral capacitance could be calculated, however, the capacitance difference estimated in different positions, or the modulation produced by vibration, would be dominated by the beam and the fixture rather than by the radius of curvature. 6. Summary
(2)
It represents a second-order approximation of the C(L, a, r, d) hypersurface. As it is evident from Fig. 4, the change of the capacitance of unshielded probes with distance d is very small. To reduce the scatter of data, for the fit of C(r) vs. d the average of the reduced slopes (cf. Fig. 7) has been used. The coefficients c0–c9, valid for lengths (r, L and d) given in mm, angle in degrees and the capacitance in fF, are summarised in Table 1. Vibrating the tip perpendicularly to the surface (cf. Fig. 8) and using the rectified capacitance modulation suppresses the effect of stray capacitance and improves the resolution [2]. It also increases the dependence of signal
Finite element analysis of the capacitance of a conical SCM probe with approximately hemispherical apex, with respect to a conducting plane, has been used to compute its dependence on tip height, cone angle, radius of curvature and distance from the plain. The obtained dependences have been fitted to lines, except the dependence on distance, which has been fitted to a hyperbola. The obtained fits, and the fits of their parameters, could be combined to a formula enabling to calculate the capacitance for a wide range of the respective dimensions. Knowing the cone height and angle, which can be estimated using an optical microscope, the radius of curvature can be derived from the capacitance measured in different tip/plane distances.
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
Acknowledgements Partial support of the VEGA Grant Agency, and the Research and Development Support Agency (Project no. APVT-51-013904) is kindly acknowledged. References [1] J.R. Matey, US Patent 4 481 616. [2] C.D. Bugg, P.J. King, J. Phys. E 21 (1988) 147. [3] H.P. Kleinknecht, J.R. Sandercock, H. Meier, Scanning Microsc. 2 (1988) 1839. [4] C.C. Williams, W.P. Hough, S.A. Rishton, Appl. Phys. Lett. 55 (1989) 203. [5] Sˇ. La´nyi, J. To¨ro¨k, P. Rˇehu˚rˇ ek, Rev. Sci. Instrum. 65 (1994) 2258. [6] D.T. Lee, J.P. Pelz, B. Bhushan, Rev. Sci. Instrum. 73 (2002) 3525. [7] International Technology Roadmap for Semiconductors, 1999 ed., Semiconductor Industry Association, San Jose, CA, 1999. [8] P. De Wolf, R. Stephenson, T. Trenkler, T. Clarysse, T. Hantschel, W. Vandervorst, J. Vac. Sci. Technol. B 18 (2000) 361. [9] Sˇ. La´nyi, Acta Phys. Slovaca 52 (2002) 55. [10] E. Bussmann, C.C. Williams, Rev. Sci. Instrum. 75 (2004) 422. [11] Y. Huang, C.C. Williams, J. Vac. Sci. Technol. B 12 (1994) 369. [12] V.V. Zavyalov, J.S. McMurray, C.C. Williams, J. Appl. Phys. 85 (1999) 7774. [13] H. Edwards, R. McGlothlin, R. San Martin, E.U.M. Gribeluk, R. Mahaffy, C.K. Shih, R.S. List, V. Ukraintsev, Appl. Phys. Lett. 72 (1998) 698. [14] C.Y. Nakakura, D.L. Hetherington, M.R. Shaneyfelt, P.J. Shea, A.N. Ericson, Appl. Phys. Lett. 75 (1999) 2319. [15] S. Shin, J.-I. Kye, U.H. Pi, Z.G. Khim, J.W. Hong, S. Park, S. Yoon, J. Vac. Sci. Technol. B 18 (2000) 2664. [16] V. Rainieri, S. Lombardo, J. Vac. Sci. Technol. B 18 (2000) 545. [17] O. Jeandupeux, V. Marsico, A. Acovic, P. Fazan, H. Brune, K. Kern, Microelectron. Reliab. 42 (2002) 325.
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[18] Sˇ. La´nyi, Tip-Shape Effect on the Accuracy of Capacitance Determination by Scanning Capacitance Microscopes, Nanoscale Calibration Standards and Methods, in: G. Wilkening, L. Koenders (Eds.), Wiley-VCH, Weinheim, Germany, 2005 (Chapter 34). [19] S. Watanabe, K. Hane, T. Ohye, M. Ito, T. Goto, J. Vac. Sci. Technol. B 11 (1993) 1774. [20] G. Mesa, E. Dobado-Fuentes, J.J. Sa´enz, J. Appl. Phys. 79 (1996) 39. [21] S. Kurokawa, A. Sakai, J. Appl. Phys. 83 (1998) 7416. [22] M. Nonnenmacher, M.P. O’Boyle, H.K. Wickramasinghe, Appl. Phys. Lett. 58 (1991) 2921. [23] G. Koley, M.G. Spencer, Appl. Phys. Lett. 79 (2001) 545. [24] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, New York, 1977. [25] H. Tomiye, H. Kawami, T. Yao, Appl. Surf. Sci. 117/118 (1997) 166. [26] R.C. Barrett, C.F. Quate, J. Appl. Phys. 70 (1991) 2725. [27] E. Bussmann, C.C. Williams, in: Four Corners Meeting, 4–5 October 2002, Salt Lake City, UT. [28] Sˇ. La´nyi, M. Hrusˇ kovic, Rev. Sci. Instrum. 73 (2002) 2923. [29] R.M. Feenstra, J. Vac. Sci. Technol. B 21 (2003) 2080. [30] V.J. Peridier, L.-H. Pan, T.E. Sullivan, J. Appl. Phys. 78 (1995) 4888. [31] J.F. Marchiando, J.J. Kopanski, J.R. Lowney, J. Vac. Sci. Technol. B 16 (1998) 463. [32] J.F. Marchiando, J.J. Kopanski, J. Vac. Sci. Technol. B 11 (2004) 411. [33] Sˇ. La´nyi, Surf. Interface Anal. 27 (1999) 348. [34] N.C. Bruce, A. Garcia-Valenzuela, D. Kouznetsov, J. Phys. D: Appl. Phys. 33 (2000) 2890. [35] L. Ciampolini, M. Ciappa, P. Malberti, W. Fichtner, Solid State Electron. 46 (2001) 445. [36] J.F. Marchiando, J.J. Kopanski, J. Appl. Phys. 92 (2002) 5798. [37] L. Deˇdek, L. Deˇdkova´, MEP 6.0, Department of Electrical Engineering and Computer Science, Technical University of Brno, Brno, Czech Republic. [38] H. Arakawa, R. Nishitani, J. Vac. Sci. Technol. B 19 (2001) 1150.
Ultramicroscopy 108 (2008) 712–717 www.elsevier.com/locate/ultramic
Shape dependence of the capacitance of scanning capacitance microscope probes Sˇ. La´nyi Institute of Physics, Slovakian Academy of Sciences, Du´bravska´ cesta 9, SK-845 11 Bratislava, Slovakia Received 5 March 2007; accepted 2 November 2007
Abstract The capacitance of approximately conical scanning capacitance microscope probes placed perpendicularly over a conducting plane has been modelled using the finite element method. The dependence on tip/surface distance, radius of curvature of the tip apex, cone angle and height has been analysed. Both shielded and unshielded probes have been considered. The fits of obtained dependences have been combined into an analytic approximation of the capacitance as a function of tip/surface distance, radius of curvature, cone angle and height. The results can be used to estimation of stray capacitance, achievable lateral resolution and contrast. r 2008 Elsevier B.V. All rights reserved. PACS: 07.79.v; 41.20.Cv; 02.70.Dh; 07.05.Tp Keywords: Computer simulation; Scanning capacitance microscope probe; Finite element method
1. Introduction One of the first scanning probe microscopes, the scanning capacitance microscope (SCM) [1–6] is able to image and analyse structures on the surface of conductors, both free and coated with insulating films, and buried structures in semiconductors and dielectrics. It has been identified as one of the tools expected to satisfy the needs of semiconductor industry for high-resolution and high-accuracy determination of dopant concentration in semiconductor structures on the nanometre scale, needed for future generations of ULSI integrated circuits [7,8]. Though lateral resolution of 5 nm on conducting surfaces [9] and sub-10 nm on MOS capacitors [10] has been already achieved, high resolution and high accuracy of carrier concentration determination at a time seems rather demanding. The probe of the SCM is a sharp conducting tip forming with the conducting surface or insulator coated substrate a capacitor. It is raster scanned over the surface or in contact with the insulating film. The sensed capacitance depends on Tel.: +421 2 59410525; fax: +421 2 54776085.
E-mail address: [email protected] 0304-3991/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2007.11.002
the tip-to-counter electrode distance, e.g. the film thickness or semiconductor depletion layer width, and from the permittivity of the material between them. Their local variations are utilised to create a picture of topography of conducting surfaces or material property (semiconductors and dielectrics). The most representative applications comprise the analysis of free carrier concentration [11], delineation of p–n junctions [12,13], visualisation of crosssection of semiconductor structures [14–16] or controlling wafer processing [17]. The lateral resolution and the formed contrast depend on the distribution of the electrostatic field between the probe and the sample. The useful information is provided by the tip apex, in the vicinity of which the electrostatic field is most concentrated. The rest contributes only a resolution gradually deteriorating with distance, ending up in a featureless background. Unfortunately, this contribution is orders of magnitude larger than the desired one. It reduces the achieved contrast of small features and may effectively preclude quantitative analysis [18]. In most transducer solutions some noise components are proportional to the overall capacitance, hence a large stray capacitance may seriously reduce the signal/noise ratio [9].
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
The mapping of the electrostatic field is also used for estimation of electrostatic force acting on the scanning force microscope (SFM) probe [19], e.g. using the generalised image charge method [20], and for calculation of capacitance vs. distance in Ref. [21]. The knowledge of the capacitance is useful in interpretation of Scanning Kelvin Probe Microscopy images [22,23]. However, there is an important difference between the two cases—force is a vector, whereas capacitance is a scalar quantity. So, the forces acting to opposite sides of the probe or of a topographic feature subtract, whereas fluxes, i.e. capacitance contributions add. The exact shape of a probe is usually unknown. The height of the tip and the cone angle can be evaluated using an optical microscope. The manufacturers’ specifications of the radius of curvature of the tip apex are frequently wide, first of all of metal-coated tips. The measurement of radii on the order of tens of nanometres is possible by means of scanning electron microscopes. This is expensive, not accessible in each laboratory and not without risk of damaging the cantilever. In the present paper, the capacitance of probes of varying shape and size has been analysed using the finite element method (FEM) [24]. The results are presented in the form of graphs and as an analytic approximation. They may help to estimate the expectable performance and to select the optimal shape for a given task. If the height and cone angle are provided, the tip radius can be deduced from a simple capacitance vs. distance measurement. 2. Analysed probes The probes used in SCMs comprise sharpened wires [2,25], conducting SFM cantilevers [26], platinum cantilevers [27] and shielded sharpened wires [5,28]. Electrochemically etched tips are approximately conical, with small cone angles (10–201) [21,27]. Those of micromachined conducting SFM cantilevers have larger cone angles (301) and are significantly shorter (15 mm) than the tips etched from wires (tenths of millimetre and more). The mutual capacitance of two conductors is the ratio of accumulated charge and the potential difference between them. Its calculation requires solving the Laplace equation DV=0 in three dimensions. The surface charge density sr, induced in the electrodes by the applied voltage, is coupled with the electrostatic field Er through the Poisson equation (r is the radial variable). In general, no closed-form analytic solution of this problem is known, except of some relatively simple cases. Configurations like sphere/plane, cone/plane, paraboloid/plane or hyperboloid/plane combinations can be solved in prolate spheroidal coordinates. For instance, the effect of an electron on the potential and tunnelling [29] and the electrostatic potential of a hyperbolic probe [30] were calculated using this method. The FEM is a flexible tool for the calculation of complex geometries. It can be used to arbitrary tip shapes as well as surface morphology. However, until now axially symmetric
713
cases, sections of three-dimensional problems in Cartesian coordinates [31,32] and combination of both have been analysed [33–36]. The FEM analysis has been performed using the MEP 6.0 code [37]. To enable reduction of the three-dimensional problem to two dimensions, axial symmetry, i.e. the probe axis perpendicular to the plane, has been assumed. This condition can be usually easily fulfilled and a few degrees deviation has negligible effect. Both shielded and unshielded probes have been analysed. Under shielding active shielding is assumed, i.e. the voltage sensed by the input of the transducer electronics imposed to the shield [5]. The radius of curvature of approximately hemispherical apex r was varied from 25 nm to 1 mm, the cone angle a from 101 to 301, the height L from 150 to 300 mm. They would correspond to tips formed from wires with diameter from 26 to 160 mm, or thicker ones but shielded, e.g. by a pierced foil placed above the sample or placing the rest of the wire into a tube. These probes were denoted unshielded. The shielded ones protruded 5 mm from a shield with orifice radius of 25 mm. The attribute ‘‘approximately’’ means that due to the smooth transition to the cone the apex is less than a hemisphere. The shape of electrochemically etched wires sometimes deviates from exact conical shape. Therefore, slightly concave or convex shape with principal radius of 2 mm has also been considered. The assumed tip-to-plane distance d was 5–25 nm. Fig. 1 shows the assumed tip shapes and the varied dimensions. Since shielding limits the spreading of the electrostatic field of the tip, thus rendering a good definition of boundary conditions possible, the stray field of unshielded probes is not limited. Even worse, the capacitance between an infinite plane and a cone perpendicular to it grows approximately linearly with the height of the cone. However, simulation has shown that the flux density near the tip apex did not differ by more than 2.5% when the size of the cone has been doubled. Therefore, the tasks were limited by a force line (Neumann boundary condition) in the form of a circular arc, ending perpendicularly on both
L
d
α
R
R
r
Fig. 1. The shape of tips: (a) conical, (b) concave and (c) convex. The tip height L, cone angle a, apex radius r and distance from plane d have been changed. The radius of forming arcs of concave and convex shape R was 2 mm.
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714
9 Cone angle 10° 20° 30° d = 5 nm r = 100 nm
8
CAPACITANCE (fF)
7 N
D
D
N
6 5 4 3 2 1
N 0 0
50
100
150 200 TIP HEIGTH (μm)
250
300
D
D
Fig. 3. Dependence of capacitance on tip height L and cone angle a. Tip radius r was 100 nm.
D N
D
D
N
8.0
D 7.5
the cone and the plane, in three dimensions approximately a hemisphere. The employed boundary conditions are shown in Fig. 2.
cone angle:
CAPACITANCE (fF)
Fig. 2. The employed boundary conditions: (a) unshielded tip and (b) shielded tip. D denotes Dirichlet boundary conditions (equipotentials) and N Neumann boundary conditions (force lines).
7.0
d:
10° 20° 30° 5 15 25 nm
6.5 6.0 5.5
3. Results 5.0
The capacitance of a cone with vertex in the planar counter-electrode should be a linear function of the cone height. Fig. 3 shows that, for relatively high cones, the deviation caused by rounding of the apex, and by its small separation from the plane, is negligibly small. In Fig. 4, the dependence of the capacitance of unshielded probes on radius of curvature r and tip/plane separation d is seen. The results with shielded probes, except the capacitance magnitude, are similar but the C vs. r dependences are less steep (Fig. 5). Active shielding, even with a large orifice compared to the diameter of the protruding part of the tip, significantly reduces the stray capacitance. In most of the interior of the shield there is practically no field. The capacitance corresponds to that of a 16.8-mm high unshielded tip. Important is the nearly perfect linear dependence of the capacitance on the tip radius. Minor deviations from linearity appeared at the smallest radii. However, they were more a scatter than systematic and correspond to small capacitance differences, most affected by computational errors. Some of the C vs. r dependences could be fitted with higher correlation to a second-order polynomial than to a line. However, for our purpose the difference is of no
4.5 0
200
400 600 TIP RADIUS (nm)
800
1000
Fig. 4. Dependence of capacitance on tip radius r and cone angle a of 250 mm high unshielded tip.
importance. For a flat apex an approximately parabolic C vs. r would result. The slopes of linear fits of C vs. r dependences are shown in Fig. 6. All curves are remarkably similar. The capacitance– distance dependence of a parallel plate capacitor is hyperbolic. Such dependence cannot hold for the approximately conical tip over a plane. However, the curves have been fitted with satisfactory correlation to a shifted hyperbola: CðdÞ ¼ a þ
b . dþc
(1)
The difference [C(d)a] is displayed in Fig. 7. The employed fit convincingly represents both the shielded and unshielded probes.
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0.9
1.6e-4 cone angle: 10° 20° 30°
0.8
d: 5 10 15 20 25 nm
0.7
unshielded shielded angle: 10° 20° 30°
1.4e-4 REDUCED SLOPE
CAPACITANCE (fF)
715
0.6 0.5
1.2e-4
1.0e-4
0.4 8.0e-5 0.3 6.0e-5
0.2 0
200
400 600 TIP RADIUS (nm)
800
Fig. 5. Dependence of capacitance on tip radius r and cone angle a of 250 mm high shielded tip.
5
10 15 20 DISTANCE (nm)
ΔC/Δd (fF/ 5 nm)
4.e-4
30
cone angle: tip heigth: 10° 30° 200 250 300 μm
4.e-3
5.e-4
25
Fig. 7. Reduced slopes of C vs. r dependences of both unshielded and shielded 250 mm high tips.
unshielded shielded angle: 10° 20° 30°
6.e-4
SLOPE (fF/nm)
0
1000
3.e-3
2.e-3
3.e-4 1.e-3 5 0
5
10 15 20 DISTANCE (nm)
25
30
Fig. 6. Slopes of C vs. r dependences of unshielded tips from Fig. 4 and of shielded tips from Fig. 5.
At small radii, the relative contribution of the tip apex may become very small. However, it is accentuated in a difference of capacitances measured in two tip/sample distances. Fig. 8 shows that the effect of cone angle and height is reduced but still present. The comparison of conical tip with cone angle 201 with the convex (limiting angle 241) and concave tips (limiting angle 161) in Fig. 9 demonstrates that the capacitance difference is nearly independent from the actual shape of the tip. 4. Approximation of results Within the analysed ranges, the dependences of capacitance on both cone height (Fig. 2) and angle (Figs. 4 and 5)
10
15 20 MEAN DISTANCE (nm)
25
Fig. 8. Dependence of capacitance difference, resulting from displacement of unshielded and shielded tips by 5 nm, on cone angle and tip height. The influence of tip height at small cone angle is negligibly small.
are reasonably linear. By reducing the cone height toward zero, the capacitance would drop to that of the spherical apex. An extrapolation of cone angle to zero would turn the tip to a hemispherically terminated cylinder, the capacitance of which, with respect to the plane, would also remain non-zero. Therefore, the lines representing the C vs. L and C vs. a dependences are expected to intersect the ordinate axis above zero. The C vs. r dependences in Figs. 4 and 5 for various cone angles and same d are not parallel. The slopes also depend on the cone height. The linear fits of dependences of C on L, a and r, the hyperbolic fit of dependence on d, as well as the linear fits of the previous fits’ parameters on the remaining variables, can be
ARTICLE IN PRESS Sˇ. La´nyi / Ultramicroscopy 108 (2008) 712–717
716
ΔC/Δd (fF/ 5 nm)
on tip/surface spacing and makes it better suitable as an input quantity to control the probe position in constant-capacitance mode, than the capacitance itself. The expression
tip radius: shape: 100 nm 25 nm concave conical convex
4.e-3
3.e-3
dC rðc2 þ c3 a þ c4 L þ c5 aLÞ ¼ dd ðc6 þ c7 a þ c8 L þ c9 aL þ dÞ2
(3)
2.e-3
shows that by this the first two terms in Eq. (2) are eliminated.
1.e-3
5. Discussion
0e+0 5
10
15 20 MEAN DISTANCE (nm)
25
Fig. 9. Effect of tip shape, concave, conical and convex, on the capacitance difference resulting from tip displacement by 5 nm. The cone angle was 201 and the height and base diameter of all three tips was the same. The limiting angle at the apex of concave tip is 161 and of the convex tip 241.
Table 1 Valuesa of coefficients in expressions (2) and (3) Slopes of
a
L
aL
c0 0.01298
c1 5.195e4
r
ra
rL
raL
c2 2.080e3
c3 5.261e5
c4 4.644e7
c5 7.178e8
a
L
aL
c7 1.476e4
c8 1.951e6
c9 1.168e7
c6 9.722e3
a The tip height L, radius of curvature r and distance d are in mm, the angle a in degrees and the result in fF.
combined to expression C ¼ ðc0 þ c1 aÞL þ
rðc2 þ c3 a þ c4 L þ c5 aLÞ . c6 þ c7 a þ c8 L þ c9 aL þ d
The experimentally found capacitance of unshielded probes is frequently much larger than the results in Fig. 4. The reason is that the present analysis was restricted to a cone of limited height. Probes with a stray capacitance of 0.275 pF [38] or 0.5 pF [21] would hardly yield an acceptable signal/noise ratio. The noise in capacitance images presented in Ref. [38] was on the order of 10 fF, comparable with the signal from 40 nm deep carbon grating. The derived approximation of capacitance (Eq. (2)) enables a simple evaluation of the effect of tip geometry on the stray capacitance of a conical probe. The tip height and cone angle can be evaluated using an optical microscope. Then, the only relevant unknown parameter is the radius of the tip apex. Using expression (2) and the capacitance measured in two known tip/conducting surface distances, the radius r can be calculated. An alternative solution is vibrating the tip and application of expression (3). Unfortunately, this procedure would not work in the case of unshielded conducting cantilevers. The contribution of the cantilever beam and fixture would have to be considered separately. In this way, the integral capacitance could be calculated, however, the capacitance difference estimated in different positions, or the modulation produced by vibration, would be dominated by the beam and the fixture rather than by the radius of curvature. 6. Summary
(2)
It represents a second-order approximation of the C(L, a, r, d) hypersurface. As it is evident from Fig. 4, the change of the capacitance of unshielded probes with distance d is very small. To reduce the scatter of data, for the fit of C(r) vs. d the average of the reduced slopes (cf. Fig. 7) has been used. The coefficients c0–c9, valid for lengths (r, L and d) given in mm, angle in degrees and the capacitance in fF, are summarised in Table 1. Vibrating the tip perpendicularly to the surface (cf. Fig. 8) and using the rectified capacitance modulation suppresses the effect of stray capacitance and improves the resolution [2]. It also increases the dependence of signal
Finite element analysis of the capacitance of a conical SCM probe with approximately hemispherical apex, with respect to a conducting plane, has been used to compute its dependence on tip height, cone angle, radius of curvature and distance from the plain. The obtained dependences have been fitted to lines, except the dependence on distance, which has been fitted to a hyperbola. The obtained fits, and the fits of their parameters, could be combined to a formula enabling to calculate the capacitance for a wide range of the respective dimensions. Knowing the cone height and angle, which can be estimated using an optical microscope, the radius of curvature can be derived from the capacitance measured in different tip/plane distances.
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