- Email: [email protected]
Calculation of the frequency shift in dynamic force microscopy
Applied Surface Science 140 Ž1999. 344–351
Calculation of the frequency shift in dynamic force microscopy ) H. Holscher , U.D. Schwarz, R. Wiesendanger ¨ Institute of Applied Physics and Microstructure Research Center, UniÕersity of Hamburg, Jungiusstrasse 11, D-20355, Hamburg, Germany Received 22 August 1998; accepted 1 September 1998
Abstract A theoretical study of the quality and the range of validity of different numerical and analytical methods to calculate the frequency shift in dynamic force microscopy is presented. By comparison with exact results obtained by the numerical solution of the equation of motion, it is demonstrated that the commonly used interpretation of the frequency shift as a measure for the force gradient of the tip–sample interaction force is only valid for very small oscillation amplitudes and leads to misinterpretations in most practical cases. Perturbation theory, however, allows the derivation of useful analytic approximations. q 1999 Elsevier Science B.V. All rights reserved. PACS: 61.16.Ch; 07.79.Lh; 02.60.Cb Keywords: Atomic force microscopy; Frequency modulation force microscopy; Dynamic force microscopy; Tip–sample interaction; Frequency shift; Oscillating cantilever
1. Introduction In 1981, the invention of the scanning tunneling microscope ŽSTM. by Binnig et al. w1x enabled the direct observation and manipulation of individual surface atoms w2x. In spite of this experimental success, the question of the interpretation of the observed atomic-scale contrast remained, which was answered 1983 by the finding of Tersoff and Hamann w3,4x that the tunneling current between tip and sample is in first approximation proportional to the local density of states of the surface at the position of the tip. Contrary to the STM, the scanning force microscope ŽSFM. w5x, invented in 1986, is believed to
)
Corresponding author. Tel.: q49-40-4123-6201; Fax: q4940-4123-5311; E-mail: [email protected]
image in the so-called contact mode the ‘topography’ of the surface Ži.e., the total density of states of the surface atoms. w6x. It therefore produces results complementary to STM investigations and has the advantage that it is not restricted to conducting samples. Theoretical considerations suggest, however, that a single atom at the tip end is not stable at loading forces which are practicable in contact mode under ambient conditions or in ultrahigh vacuum ŽG 10y9 N. w7x. On the other hand, blunt tips prevent the observation of point-like defects, although the periodicity of the atomic lattice can still be resolved w8x. This limitation was overcome in 1995 by Giessibl w9x who obtained atomic resolution in the so-called non-contact or dynamic mode of the scanning force microscope. In this mode, the observation of atomic defects on semiconductors w10x as well as on insula-
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 5 5 2 - 2
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
tors w11x could be realized. The measurement principle is thereby based on the fact that the interaction force between tip and sample causes a shift of the resonance frequency of the cantilever which can be detected and used for image formation. This frequency shift is usually interpreted in terms of simple models w12–14x and associated with the gradient of the tip–sample interaction force. Thus, it is said that the dynamic force microscopy measures the force gradient, whereas contact mode force microscopy is directly sensitive to the force. In this paper, we will discuss and compare different methods which have been used in the past for the calculation of the frequency shift. As a general result, it can be concluded that an interpretation of the frequency shift in terms of force gradients is unjustified in most cases, and physical arguments based on this interpretation are consequently misleading.
2. The principle of dynamic force microscopy The principle of dynamic force microscopy and the notations used in this article are shown in Fig. 1. A cantilever is vibrated with a fixed resonance amplitude A at its resonance frequency f. This resonance frequency is different from the eigenfrequency f 0 of the free cantilever due to the influence of the interaction force between the tip and sample and
Fig. 1. Scheme of the notations used in this article. The cantilever oscillates with a resonance amplitude A. During one cycle, the tip approaches the sample to the nearest tip–sample distance D. The distance between the tip and the sample at the point where the cantilever in undeflected is called the support–sample distance d.
345
Fig. 2. Graph illustrating the reason for the frequency shift occurring in dynamic force microscopy Žsee text..
changes while changing the resonance amplitude A or the distance d between the sample and the tip if the cantilever is undeflected. Since this distance is varied in the experiment by moving the position of the cantilever support relative to the sample surface, d will be called the support–sample distance in the following. Additionally, the tip–sample distance at the point of closest approach is denoted as D. The reason for the shift of the cantilever resonance frequency can easily be understood by looking at the potentials plotted in Fig. 2. If the cantilever is far away from the sample surface, the tip moves in a parabolic potential Ždotted line., and its oscillation is harmonic. In such a case, the tip motion is sinusoidal and the resonance frequency is given by the eigenfrequency f 0 of the cantilever. If, however, the support–sample distance d is reduced, the potential which determines the tip oscillation is modified and given by an effective potential Žsolid line. represented by the sum of the parabolic potential and the tip–sample interaction potential Ždashed line.. This effective potential differs from the original parabolic potential and shows an asymmetric shape, contrary to the symmetric shape of the parabolic potential. Consequently, the resulting tip oscillation becomes inharmonic, and the resonance frequency of the oscillation now depends on the oscillation amplitude A. Since the effective potential experienced by the tip changes with the support–sample distance d, the frequency shift D f depends on both parameters: D f Ž d, A..
346
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
3. Calculation of the frequency shifts
microscopy, the interaction force Fint is usually expanded into a Taylor series:
The aim of this chapter is to review three different methods which can be used for the calculation of the frequency shifts occurring in dynamic force microscopy, namely solving the equation of motion, the force gradient method and the calculation of the frequency shifts using perturbation theory.
at the equilibrium point d 0 which is given by the root of:
3.1. Equation of motion
c z Ž z y d . s Fint Ž z . .
An exact way to calculate the frequency shift is to solve the equation of motion of the tip in its potential and to determine the deviation of the actual resonance frequency from the eigenfrequency of the cantilever. Setting up the correct equation of motion is, however, not just straightforward since the damping mechanisms involved are difficult to consider exactly and small inharmonicities of the free cantilever oscillation might occur. Additionally, a correct simulation of the tip movement has to include the properties of the microscope feedback which drives the cantilever support w15x. Nevertheless, the basic principle of the frequency modulation technique w14x which is commonly used in dynamic force microscopy operated in vacuum includes the compensation of the cantilever damping by the driving mechanism in order to measure the resonance frequency f independently from the actual damping. Therefore, it is under most circumstances sufficient to neglect the above mentioned issues and to calculate the path of the tip from the conservative equation of motion: m z z¨ q c z Ž z y d . s Fint Ž z . ,
Ž 1.
where m z s c zrŽ2p f 0 . 2 is the effective mass and c z the spring constant of the cantilever; the term Fint Ž z . s yŽEVint Ž z ..rŽE z . describes the interaction force between tip and sample. The amplitude A is controlled through the initial conditions z 0 [ z Ž t s 0., Õ˙ 0 [ 0 s z˙Ž t s 0.. 3.2. Force gradients A widespread method to handle a non-linear equation of motion like Eq. Ž1. is to simplify the non-linear term Fint and to solve the resulting equation of motion analytically. In the case of dynamic force
Fint Ž z . s Fint Ž d 0 . q
E Fint Ž d 0 . Ez
Ž z y d0 . q . . . , Ž 2.
Ž 3.
Fint Ž z . can be approximated within the range of the oscillation Ž d 0 y A F z F d 0 q A. by the first two terms of the Taylor series if adequate oscillation amplitudes are applied. For general interaction forces Fint Ž z ., such oscillation amplitudes are normally very small. Eq. Ž1. can than be written as: m z z¨ q c z Ž z y d . s Fint Ž d 0 . q
E Fint Ž d 0 . Ez
Ž z y d0 . . Ž 4.
The solution of this differential equation is given by z Ž t . s d 0 q AcosŽ2p Ž f 0 q D f . t ., where: D f s f0
ž(
1y
1 E Fint Ž d 0 . cz
Ez
/
y1 .
Ž 5.
Using '1 y x f 1 y xr2, Eq. Ž5. reduces to: D ffy
f 0 E Fint Ž d 0 . 2 cz
s
Ez
f 0 E 2 Vint Ž d 0 . 2 cz
E z2
.
Ž 6.
A further simplification can be introduced by assuming that the equilibrium point of the oscillation changes only slightly during an approach to the sample surface. Under this condition, the force gradient can be calculated at the support–sample distance d f d 0 , and Eq. Ž6. is modified to: D ffy
f 0 E Fint Ž d . 2 cz
Ez
.
Ž 7.
The last two equations are the origin of the oftenrepeated statement that the frequency shift occurring in dynamic force microscopy is a measure for the force gradient of the tip–sample force w12–14,16,17x. It has, however, to be emphasized that according to the derivation of Eqs. Ž6. and Ž7., this is for general tip–sample forces only true in the limited case of small resonance amplitudes.
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
3.3. Perturbation theory
D f Ž n s 5. s y
Another way to calculate the frequency shift is to use perturbation theory within the Hamilton–Jacobi formalism w18x, as it has been reported by Giessibl w19x. 1 The approach is valid only if the motion of the tip, which oscillates with a changed frequency f s f 0 q D f, is still approximately harmonic. This condition is fulfilled when the tip–sample force is small compared to the cantilever force. Then, d f d 0 , and consequently D s d 0 y A f d y A. Applying the formalism to tip–sample forces of the type: Fn Ž z . s y
Cn zn
,
Ž 8.
the frequency shift can be calculated from the integral w19x: D f Ž n. s
1 f 0 Cn
Ž 9. ndx Ž d q Acos Ž x . . 1 f 0 C n 2p cos Ž x . D f Ž n. s H n d x. 2p c z A 0 Ž D q A Ž 1 q cos Ž x . . . Ž 10 . 2p c z A
H0
For tip–sample forces which can be described as the sum of different inverse power laws Fint Ž z . s Ý nŽ Cn .rŽ z n ., the resulting frequency shift is given by D f s Ý n D f Ž n.. For small values of n, comparatively simple analytic solutions of the integral can be found: D f Ž n s 1. s
f 0 C1 c z A2
D f Ž n s 2. s y D f Ž n s 3. s y
D f Ž n s 4. s y
ž
1y
cz
1
Ž d y A2 .
3 f 0 C3 2 cz 1 f 0 C4 2 cz
/
'd 2 y A2 2
8 cz
Ž d 2 y A2 .
9r2
,
3 f 0 C6 A4 q 12 A2 d 2 q 8 d 4 8 cz
Ž d 2 y A2 .
11r2
Ž 15 . .
Ž 16 . These equations might be especially valuable for the measurement of long-range forces Žvan der Waals, magnetic or electrostatic.. If the oscillation amplitude A is much larger than the nearest tip–sample distance Ž A 4 D ., Eq. Ž10. can be simplified to w19x: D ffy
f0
Cn
'2 p c z A2r3
D ny1r2
I1 Ž n . ,
Ž 17 .
`
I1 Ž n . [
1
Hy` Ž1 q y
2
n
dy
.
p ° s~ 1 P 3 PPP Ž 2 n y 3 . ¢p 2 Ž n y 1. ! ny 1
ns1 n ) 1.
Ž 18 .
It can be shown that the accuracy of this approximation increases not only with larger oscillation amplitudes A, but also with larger values of n.
4. Comparison between the different methods
d
f 0 C2
5 f 0 C5 d Ž 3 A2 q 4 d 2 .
where I1Ž n. is given by:
cos Ž x .
2p
D f Ž n s 6. s y
347
3r2
,
,
d
Ž d 2 y A2 .
5r2
A2 q 4 d 2
Ž d 2 y A2 .
Ž 11 .
7r2
Ž 12 . ,
Ž 13 .
,
Ž 14 .
1 Other perturbation methods have been applied for the case of a driven non-linear oscillator by Boisgard et al. w20x and Sasaki and Tsukada w21x.
In order to judge about the quality and the region of validity of the different analytical methods to calculate the frequency shift in dynamic force microscopy, both the force gradient method and the perturbation theory method are applied to two different model tip–sample forces Žlong-range and shortrange forces, respectively.. The results from the corresponding calculations are than compared with the ‘exact’ results obtained by the numerical solution of the equation of motion Eq. Ž1. w22x. However, it should be pointed out that even though for both the long-range and the short-range forces specific force laws have been assumed in this paper, the qualitative behavior of the different methods is independent from the specific force law and the chosen force parameters.
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
348
The frequency shift D f s f y f 0 has been plotted throughout the paper as a function of the distance d y A in order to make the data comparable with experimentally obtained results as well as to make frequency shift data calculated for different oscillation amplitudes A comparable. The distance d y A is thereby not exactly equal to the distance D s d 0 y A Žsee Fig. 1; Eq. Ž3. for the definitions of D and d 0 , respectively.. Nevertheless, since the difference d y d 0 is very small Žand far below the resolution limit of scanning force microscopes in most cases., the approximation D f d y A is valid with good precision, and the Ž d–A.-axis can be interpreted as an axis showing the distance between tip and sample at the point of closest approach. To describe the cantilever properties, a spring constant of c z s 40 Nrm and an eigenfrequency of f 0 s 170 kHz have been used. 4.1. Long-range forces As a model for a long-range force between the tip and the sample, we chose the van der Waals force between a tip with radius R and the sample. This force is often described by w16,19,23x: FvdW Ž z . s y
AH R 6 z2
,
Ž 19 .
where A H represents the Hamaker constant. Typical ˚ and A H s 0.1 aJ w19,23x. parameters are R s 120 A Retardation effects which would affect the interac˚ in a real tion at distances larger than about 100 A physical system w23x are neglected for simplicity. A comparison between the different calculation methods applied to this force law using the values of R and A H given above is displayed in Fig. 3. The results obtained numerically from Eq. Ž1. are marked by symbols for two different amplitudes. A comparison with the frequency shift obtained from Eq. Ž12. Žsolid line. shows that the accuracy of the perturbation theory is better than 3% for all amplitudes and tip–sample distances. Contrary to this result, the interpretation of the frequency shift as a force gradient is only satisfac˚ for a small tory for distances greater than 30 A ˚ Ž amplitude of 10 A dashed–dotted line in Fig. 2a, calculation performed using Eq. Ž7... If the ampli˚ the force gradient method tude is increased to 100 A,
Fig. 3. Comparison between the different methods to calculate the ˚ Ža. resonance frequency shift for the oscillation amplitudes 10 A ˚ Žb. at the example of the van der Waals force ŽEq. and 100 A Ž19... The exact result obtained numerically from Eq. Ž1. is marked by black circles in Ža. and by squares in Žb.. The deviation between these results and Eq. Ž12. Žsolid lines. is small and within an accuracy of better than 3%. The analytical approximations obtained from Eq. Ž17. are plotted by dashed lines and give good agreement only if the nearest tip–sample distance is small compared to the amplitude. The calculation using the force gradient method Eq. Ž7. Ždashed–dotted lines. is only useful in the limit of small amplitudes and large tip–sample distances Žsee text..
fails for all distances up to 30 nm ŽFig. 2b.. This result agrees with the assumptions used in the derivation of the frequency shift formula Eq. Ž7.. If the variation of the tip–sample force is too large within one oscillation period of the tip, the approximation of Fint Ž z . by the first two terms of the Taylor expansion is inappropriate, and Eq. Ž5. consequently fails. Therefore, the validity of the formula depends on the amplitude as well as on the variation of the tip–sample force at the equilibrium point. It is interesting to note that Eq. Ž7. can also be obtained from the perturbation theory ŽEq. Ž12.. using d 4 A: y
f 0 C2 2 3r2
czŽ d 2 y A .
fy
f 0 C2 cz d
3
sy
f 0 E F2 Ž z . 2 cz
Ez
.
Ž 20 .
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
Eq. Ž17. is another limit of the perturbation theory. According to its derivation, it only provides useful results if D < A. The same result can be obtained directly from Eq. Ž12.: f 0 C2 f 0 C2 y sy 3r2 2 2 3r2 2 czŽ d y A . c z Ž D q 2 DA . fy
f 0 C2 2'2 c z D 3r2A3r2
.
349
approximation of such forces with the first two terms of a Taylor series has a very small range of validity, limiting the applicability of the force gradient method to very small amplitudes. As demonstrated in Fig. 4, the oscillation amplitude A must be as small as 0.1 ˚ to obtain sufficient agreement between the results A
Ž 21 .
Consequently, the calculation of the frequency shift ˚ only if works even for large amplitudes like 100 A ˚ the nearest tip–sample distance is smaller than 10 A Ždashed line in Fig. 3b.. To summarize this section, we have found at the example of a van der Waals force ŽEq. Ž19.. that in the case of long-range tip–sample forces Žwhich are usually described by inverse power laws as given in Eq. Ž8. with small n, i.e., n F 6., the frequency shift is satisfactorily calculated with Eqs. Ž11. – Ž16.. The force gradient method ŽEq. Ž7.. and the approximation Eq. Ž17., however, showed an application range which is unacceptably small for a useful interpretation of experimental data. 4.2. Short-range forces Short-range forces between the front atoms of the tip and nearest sample surface atoms are often described by the force of a Lennard–Jones potential w16,19x: 12 E0 r 0 13 r0 7 FL J s y , Ž 22 . r0 z z
žž / ž / /
where E0 represents the binding energy and r 0 the equilibrium distance; typical parameters are E0 s 1 ˚ Introducing these values in Eq. eV and r 0 s 4 A. Ž22., it is found that the tip approaches the sample surface without any hysteresis or instability effect, since the condition for a ‘jump-to-contact’ w19x:
E 2 Vint Ez
s y
2 max
E Fint Ez
and c z A q w Fint x max - 0,
) cz max
Ž 23 .
is not fulfilled. It is the nature of short-range forces that their decay length is very small. Consequently, a linear
Fig. 4. Comparison between the exact results for the frequency shift obtained from the equation of motion Žindicated by the symbols. using the short-range force ŽEq. Ž22.., and the corresponding approximations calculated according to Eq. Ž5. Žsolid lines., Eq. Ž6. Ždashed lines., and Eq. Ž7. Ždashed–dotted lines. as ˚ Žb. a function of the oscillation amplitude A: Ža. As 0.1 A, ˚ and Žc. As10 A. ˚ Whereas at As 0.1 A, ˚ the solid and As1 A, the dotted line are very close together and show good agreement with the exact results, all lines differ significantly at larger amplitudes from the numerically obtained data. In Žc., this deviation is most prominent: all three lines are indistinguishable and approximately identical to D f f 0 kHz on this scale.
350
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
Fig. 5. Comparison between the exact results for the frequency shift obtained from the equation of motion Žindicated by the symbols for three different oscillation amplitudes. using the short-range force ŽEq. Ž22.., and the corresponding approximations calculated according to Eq. Ž17. Žsolid lines.. The exact results are well-reproduced.
found by the numerical solution of the equation of motion Žindicated by the symbols. and the results obtained using Eq. Ž5. Žsolid lines. or Eq. Ž6. Ždashed lines.. However, it should be noted that using a cantilever with c z s 40 Nrm as assumed throughout ˚ is at this paper, an oscillation amplitude of 0.1 A room temperature already realized solely by thermal activation w24x. Moreover, since the equilibrium point d 0 differs close to the sample surface significantly from the support–sample distance d if small amplitudes are applied, Eq. Ž7. Ždashed–dotted lines. is misleading even for such a small value of A. With increasing amplitudes, the force gradient method gets completely useless Žsee Fig. 4b and c.. Contrary to the situation described above, the frequency shift caused by a short-range force of the type Eq. Ž8. with large values of n Ž n G 7. can be calculated in the region close to the sample surface Ž A 4 D . satisfactorily using Eq. Ž17.. This issue is shown in Fig. 5; the result obtained by the numerical solution of the equation of motion is well-reproduced.
5. Summary We presented a study on different numerical and analytical methods to calculate the frequency shift of a cantilever oscillated with its resonance frequency near a sample surface. A comparison between the exact results obtained numerically by the solution of
the equation of motion and various analytical approximations demonstrates that the often used interpretation of the frequency shift as a measure for the force gradient is only true in the limit of very small oscillation amplitudes Ž A ™ 0.. A much better approximation of the frequency shift can be realized by means of perturbation theory within the Hamilton– Jacobi formalism which leads to a set of comparatively simple analytic formulas for long-range as well as for short-range interaction forces based on inverse power laws.
Acknowledgements We are indebted to W. Allers, A. Schwarz, and B. Gotsmann for helpful discussions. Financial support from the Deutsche Forschungsgemeinschaft ŽGrant No. WI 1277r2-2. and the Graduiertenkolleg ‘Physik nanostrukturierter Festkorper’ is gratefully acknowl¨ edged.
References w1x G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Phys. Rev. Lett. 49 Ž1982. 57. w2x For a review on STM see, e.g., R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications, Cambridge University Press, Cambridge, 1994. w3x J. Tersoff, D.R. Hamann, Phys. Rev. Lett. 50 Ž1983. 1998. w4x J. Tersoff, D.R. Hamann, Phys. Rev. B 31 Ž1985. 805. w5x G. Binnig, C.F. Quate, C. Gerber, Phys. Rev. Lett. 56 Ž1986. 930. w6x S. Ciraci, A. Baratoff, I.P. Batra, Phys. Rev. B 41 Ž1990. 4728. w7x C.F. Quate, Surf. Sci. 299r300 Ž1994. 980. w8x E. Tekman, S. Ciraci, J. Phys. C: Condens. Matter 3 Ž1991. 2613. w9x F.J. Giessibl, Science 267 Ž1995. 68. w10x H. Ueyama, M. Ohta, S. Sugawara, S. Morita, Jpn. J. Appl. Phys. 34 Ž1995. L1086. w11x M. Bammerlin, R. Luthi, E. Meyer, A. Baratoff, J. Lu, ¨ ¨ M. Guggisberg, C. Gerber, L. Howald, H.-J. Guntherodt, Probe ¨ Microsc. 1 Ž1997. 3. w12x Y. Martin, C.C. Williams, H.K. Wickramasinghe, J. Appl. Phys. 61 Ž1987. 4723. w13x R. Erlandsson, G.M. McClelland, C.M. Mate, S. Chiang, J. Vac. Sci. Technol. A 6 Ž1988. 266. w14x T.R. Albrecht, P. Grutter, D. Horne, D. Rugar, J. Appl. Phys. ¨ 69 Ž1991. 668. w15x B. Gotsmann, D. Kruger, H. Fuchs, Europhys. Lett. 39 ¨ Ž1997. 153; 41 Ž1998. 583.
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨ w16x I.Yu. Sokolov, G.S. Henderson, F.J. Wicks, Surf. Sci. 381 Ž1997. L558. w17x R. Perez, M.C. Payne, I. Stich, K. Terakura, Phys. Rev. Lett. 78 Ž1997. 678. w18x H. Goldstein, Classical Mechanics, 2nd edn., AddisonWesley, 1980. w19x F.-J. Giessibl, Phys. Rev. B 56 Ž1997. 16010. w20x R. Boisgard, D. Michel, J.P. Aime, ´ Surf. Sci. 401 Ž1998. 199.
351
w21x N. Sasaki, M. Tsukada, Jpn. Appl. Phys. 37 Ž1998. L533. w22x W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd edn., Cambridge University Press, London, 1992. w23x J. Israelachvili, Intermolecular and Surface Forces, 2nd edn., Academic Press, London, 1992. w24x E. Meyer, H. Heinzelmann, in: R. Wiesendanger, H.-J. ŽEds.., Scanning Tunneling Microscopy II, Guntherodt ¨ Springer, Heidelberg, 1992.
Calculation of the frequency shift in dynamic force microscopy ) H. Holscher , U.D. Schwarz, R. Wiesendanger ¨ Institute of Applied Physics and Microstructure Research Center, UniÕersity of Hamburg, Jungiusstrasse 11, D-20355, Hamburg, Germany Received 22 August 1998; accepted 1 September 1998
Abstract A theoretical study of the quality and the range of validity of different numerical and analytical methods to calculate the frequency shift in dynamic force microscopy is presented. By comparison with exact results obtained by the numerical solution of the equation of motion, it is demonstrated that the commonly used interpretation of the frequency shift as a measure for the force gradient of the tip–sample interaction force is only valid for very small oscillation amplitudes and leads to misinterpretations in most practical cases. Perturbation theory, however, allows the derivation of useful analytic approximations. q 1999 Elsevier Science B.V. All rights reserved. PACS: 61.16.Ch; 07.79.Lh; 02.60.Cb Keywords: Atomic force microscopy; Frequency modulation force microscopy; Dynamic force microscopy; Tip–sample interaction; Frequency shift; Oscillating cantilever
1. Introduction In 1981, the invention of the scanning tunneling microscope ŽSTM. by Binnig et al. w1x enabled the direct observation and manipulation of individual surface atoms w2x. In spite of this experimental success, the question of the interpretation of the observed atomic-scale contrast remained, which was answered 1983 by the finding of Tersoff and Hamann w3,4x that the tunneling current between tip and sample is in first approximation proportional to the local density of states of the surface at the position of the tip. Contrary to the STM, the scanning force microscope ŽSFM. w5x, invented in 1986, is believed to
)
Corresponding author. Tel.: q49-40-4123-6201; Fax: q4940-4123-5311; E-mail: [email protected]
image in the so-called contact mode the ‘topography’ of the surface Ži.e., the total density of states of the surface atoms. w6x. It therefore produces results complementary to STM investigations and has the advantage that it is not restricted to conducting samples. Theoretical considerations suggest, however, that a single atom at the tip end is not stable at loading forces which are practicable in contact mode under ambient conditions or in ultrahigh vacuum ŽG 10y9 N. w7x. On the other hand, blunt tips prevent the observation of point-like defects, although the periodicity of the atomic lattice can still be resolved w8x. This limitation was overcome in 1995 by Giessibl w9x who obtained atomic resolution in the so-called non-contact or dynamic mode of the scanning force microscope. In this mode, the observation of atomic defects on semiconductors w10x as well as on insula-
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 5 5 2 - 2
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
tors w11x could be realized. The measurement principle is thereby based on the fact that the interaction force between tip and sample causes a shift of the resonance frequency of the cantilever which can be detected and used for image formation. This frequency shift is usually interpreted in terms of simple models w12–14x and associated with the gradient of the tip–sample interaction force. Thus, it is said that the dynamic force microscopy measures the force gradient, whereas contact mode force microscopy is directly sensitive to the force. In this paper, we will discuss and compare different methods which have been used in the past for the calculation of the frequency shift. As a general result, it can be concluded that an interpretation of the frequency shift in terms of force gradients is unjustified in most cases, and physical arguments based on this interpretation are consequently misleading.
2. The principle of dynamic force microscopy The principle of dynamic force microscopy and the notations used in this article are shown in Fig. 1. A cantilever is vibrated with a fixed resonance amplitude A at its resonance frequency f. This resonance frequency is different from the eigenfrequency f 0 of the free cantilever due to the influence of the interaction force between the tip and sample and
Fig. 1. Scheme of the notations used in this article. The cantilever oscillates with a resonance amplitude A. During one cycle, the tip approaches the sample to the nearest tip–sample distance D. The distance between the tip and the sample at the point where the cantilever in undeflected is called the support–sample distance d.
345
Fig. 2. Graph illustrating the reason for the frequency shift occurring in dynamic force microscopy Žsee text..
changes while changing the resonance amplitude A or the distance d between the sample and the tip if the cantilever is undeflected. Since this distance is varied in the experiment by moving the position of the cantilever support relative to the sample surface, d will be called the support–sample distance in the following. Additionally, the tip–sample distance at the point of closest approach is denoted as D. The reason for the shift of the cantilever resonance frequency can easily be understood by looking at the potentials plotted in Fig. 2. If the cantilever is far away from the sample surface, the tip moves in a parabolic potential Ždotted line., and its oscillation is harmonic. In such a case, the tip motion is sinusoidal and the resonance frequency is given by the eigenfrequency f 0 of the cantilever. If, however, the support–sample distance d is reduced, the potential which determines the tip oscillation is modified and given by an effective potential Žsolid line. represented by the sum of the parabolic potential and the tip–sample interaction potential Ždashed line.. This effective potential differs from the original parabolic potential and shows an asymmetric shape, contrary to the symmetric shape of the parabolic potential. Consequently, the resulting tip oscillation becomes inharmonic, and the resonance frequency of the oscillation now depends on the oscillation amplitude A. Since the effective potential experienced by the tip changes with the support–sample distance d, the frequency shift D f depends on both parameters: D f Ž d, A..
346
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
3. Calculation of the frequency shifts
microscopy, the interaction force Fint is usually expanded into a Taylor series:
The aim of this chapter is to review three different methods which can be used for the calculation of the frequency shifts occurring in dynamic force microscopy, namely solving the equation of motion, the force gradient method and the calculation of the frequency shifts using perturbation theory.
at the equilibrium point d 0 which is given by the root of:
3.1. Equation of motion
c z Ž z y d . s Fint Ž z . .
An exact way to calculate the frequency shift is to solve the equation of motion of the tip in its potential and to determine the deviation of the actual resonance frequency from the eigenfrequency of the cantilever. Setting up the correct equation of motion is, however, not just straightforward since the damping mechanisms involved are difficult to consider exactly and small inharmonicities of the free cantilever oscillation might occur. Additionally, a correct simulation of the tip movement has to include the properties of the microscope feedback which drives the cantilever support w15x. Nevertheless, the basic principle of the frequency modulation technique w14x which is commonly used in dynamic force microscopy operated in vacuum includes the compensation of the cantilever damping by the driving mechanism in order to measure the resonance frequency f independently from the actual damping. Therefore, it is under most circumstances sufficient to neglect the above mentioned issues and to calculate the path of the tip from the conservative equation of motion: m z z¨ q c z Ž z y d . s Fint Ž z . ,
Ž 1.
where m z s c zrŽ2p f 0 . 2 is the effective mass and c z the spring constant of the cantilever; the term Fint Ž z . s yŽEVint Ž z ..rŽE z . describes the interaction force between tip and sample. The amplitude A is controlled through the initial conditions z 0 [ z Ž t s 0., Õ˙ 0 [ 0 s z˙Ž t s 0.. 3.2. Force gradients A widespread method to handle a non-linear equation of motion like Eq. Ž1. is to simplify the non-linear term Fint and to solve the resulting equation of motion analytically. In the case of dynamic force
Fint Ž z . s Fint Ž d 0 . q
E Fint Ž d 0 . Ez
Ž z y d0 . q . . . , Ž 2.
Ž 3.
Fint Ž z . can be approximated within the range of the oscillation Ž d 0 y A F z F d 0 q A. by the first two terms of the Taylor series if adequate oscillation amplitudes are applied. For general interaction forces Fint Ž z ., such oscillation amplitudes are normally very small. Eq. Ž1. can than be written as: m z z¨ q c z Ž z y d . s Fint Ž d 0 . q
E Fint Ž d 0 . Ez
Ž z y d0 . . Ž 4.
The solution of this differential equation is given by z Ž t . s d 0 q AcosŽ2p Ž f 0 q D f . t ., where: D f s f0
ž(
1y
1 E Fint Ž d 0 . cz
Ez
/
y1 .
Ž 5.
Using '1 y x f 1 y xr2, Eq. Ž5. reduces to: D ffy
f 0 E Fint Ž d 0 . 2 cz
s
Ez
f 0 E 2 Vint Ž d 0 . 2 cz
E z2
.
Ž 6.
A further simplification can be introduced by assuming that the equilibrium point of the oscillation changes only slightly during an approach to the sample surface. Under this condition, the force gradient can be calculated at the support–sample distance d f d 0 , and Eq. Ž6. is modified to: D ffy
f 0 E Fint Ž d . 2 cz
Ez
.
Ž 7.
The last two equations are the origin of the oftenrepeated statement that the frequency shift occurring in dynamic force microscopy is a measure for the force gradient of the tip–sample force w12–14,16,17x. It has, however, to be emphasized that according to the derivation of Eqs. Ž6. and Ž7., this is for general tip–sample forces only true in the limited case of small resonance amplitudes.
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
3.3. Perturbation theory
D f Ž n s 5. s y
Another way to calculate the frequency shift is to use perturbation theory within the Hamilton–Jacobi formalism w18x, as it has been reported by Giessibl w19x. 1 The approach is valid only if the motion of the tip, which oscillates with a changed frequency f s f 0 q D f, is still approximately harmonic. This condition is fulfilled when the tip–sample force is small compared to the cantilever force. Then, d f d 0 , and consequently D s d 0 y A f d y A. Applying the formalism to tip–sample forces of the type: Fn Ž z . s y
Cn zn
,
Ž 8.
the frequency shift can be calculated from the integral w19x: D f Ž n. s
1 f 0 Cn
Ž 9. ndx Ž d q Acos Ž x . . 1 f 0 C n 2p cos Ž x . D f Ž n. s H n d x. 2p c z A 0 Ž D q A Ž 1 q cos Ž x . . . Ž 10 . 2p c z A
H0
For tip–sample forces which can be described as the sum of different inverse power laws Fint Ž z . s Ý nŽ Cn .rŽ z n ., the resulting frequency shift is given by D f s Ý n D f Ž n.. For small values of n, comparatively simple analytic solutions of the integral can be found: D f Ž n s 1. s
f 0 C1 c z A2
D f Ž n s 2. s y D f Ž n s 3. s y
D f Ž n s 4. s y
ž
1y
cz
1
Ž d y A2 .
3 f 0 C3 2 cz 1 f 0 C4 2 cz
/
'd 2 y A2 2
8 cz
Ž d 2 y A2 .
9r2
,
3 f 0 C6 A4 q 12 A2 d 2 q 8 d 4 8 cz
Ž d 2 y A2 .
11r2
Ž 15 . .
Ž 16 . These equations might be especially valuable for the measurement of long-range forces Žvan der Waals, magnetic or electrostatic.. If the oscillation amplitude A is much larger than the nearest tip–sample distance Ž A 4 D ., Eq. Ž10. can be simplified to w19x: D ffy
f0
Cn
'2 p c z A2r3
D ny1r2
I1 Ž n . ,
Ž 17 .
`
I1 Ž n . [
1
Hy` Ž1 q y
2
n
dy
.
p ° s~ 1 P 3 PPP Ž 2 n y 3 . ¢p 2 Ž n y 1. ! ny 1
ns1 n ) 1.
Ž 18 .
It can be shown that the accuracy of this approximation increases not only with larger oscillation amplitudes A, but also with larger values of n.
4. Comparison between the different methods
d
f 0 C2
5 f 0 C5 d Ž 3 A2 q 4 d 2 .
where I1Ž n. is given by:
cos Ž x .
2p
D f Ž n s 6. s y
347
3r2
,
,
d
Ž d 2 y A2 .
5r2
A2 q 4 d 2
Ž d 2 y A2 .
Ž 11 .
7r2
Ž 12 . ,
Ž 13 .
,
Ž 14 .
1 Other perturbation methods have been applied for the case of a driven non-linear oscillator by Boisgard et al. w20x and Sasaki and Tsukada w21x.
In order to judge about the quality and the region of validity of the different analytical methods to calculate the frequency shift in dynamic force microscopy, both the force gradient method and the perturbation theory method are applied to two different model tip–sample forces Žlong-range and shortrange forces, respectively.. The results from the corresponding calculations are than compared with the ‘exact’ results obtained by the numerical solution of the equation of motion Eq. Ž1. w22x. However, it should be pointed out that even though for both the long-range and the short-range forces specific force laws have been assumed in this paper, the qualitative behavior of the different methods is independent from the specific force law and the chosen force parameters.
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
348
The frequency shift D f s f y f 0 has been plotted throughout the paper as a function of the distance d y A in order to make the data comparable with experimentally obtained results as well as to make frequency shift data calculated for different oscillation amplitudes A comparable. The distance d y A is thereby not exactly equal to the distance D s d 0 y A Žsee Fig. 1; Eq. Ž3. for the definitions of D and d 0 , respectively.. Nevertheless, since the difference d y d 0 is very small Žand far below the resolution limit of scanning force microscopes in most cases., the approximation D f d y A is valid with good precision, and the Ž d–A.-axis can be interpreted as an axis showing the distance between tip and sample at the point of closest approach. To describe the cantilever properties, a spring constant of c z s 40 Nrm and an eigenfrequency of f 0 s 170 kHz have been used. 4.1. Long-range forces As a model for a long-range force between the tip and the sample, we chose the van der Waals force between a tip with radius R and the sample. This force is often described by w16,19,23x: FvdW Ž z . s y
AH R 6 z2
,
Ž 19 .
where A H represents the Hamaker constant. Typical ˚ and A H s 0.1 aJ w19,23x. parameters are R s 120 A Retardation effects which would affect the interac˚ in a real tion at distances larger than about 100 A physical system w23x are neglected for simplicity. A comparison between the different calculation methods applied to this force law using the values of R and A H given above is displayed in Fig. 3. The results obtained numerically from Eq. Ž1. are marked by symbols for two different amplitudes. A comparison with the frequency shift obtained from Eq. Ž12. Žsolid line. shows that the accuracy of the perturbation theory is better than 3% for all amplitudes and tip–sample distances. Contrary to this result, the interpretation of the frequency shift as a force gradient is only satisfac˚ for a small tory for distances greater than 30 A ˚ Ž amplitude of 10 A dashed–dotted line in Fig. 2a, calculation performed using Eq. Ž7... If the ampli˚ the force gradient method tude is increased to 100 A,
Fig. 3. Comparison between the different methods to calculate the ˚ Ža. resonance frequency shift for the oscillation amplitudes 10 A ˚ Žb. at the example of the van der Waals force ŽEq. and 100 A Ž19... The exact result obtained numerically from Eq. Ž1. is marked by black circles in Ža. and by squares in Žb.. The deviation between these results and Eq. Ž12. Žsolid lines. is small and within an accuracy of better than 3%. The analytical approximations obtained from Eq. Ž17. are plotted by dashed lines and give good agreement only if the nearest tip–sample distance is small compared to the amplitude. The calculation using the force gradient method Eq. Ž7. Ždashed–dotted lines. is only useful in the limit of small amplitudes and large tip–sample distances Žsee text..
fails for all distances up to 30 nm ŽFig. 2b.. This result agrees with the assumptions used in the derivation of the frequency shift formula Eq. Ž7.. If the variation of the tip–sample force is too large within one oscillation period of the tip, the approximation of Fint Ž z . by the first two terms of the Taylor expansion is inappropriate, and Eq. Ž5. consequently fails. Therefore, the validity of the formula depends on the amplitude as well as on the variation of the tip–sample force at the equilibrium point. It is interesting to note that Eq. Ž7. can also be obtained from the perturbation theory ŽEq. Ž12.. using d 4 A: y
f 0 C2 2 3r2
czŽ d 2 y A .
fy
f 0 C2 cz d
3
sy
f 0 E F2 Ž z . 2 cz
Ez
.
Ž 20 .
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
Eq. Ž17. is another limit of the perturbation theory. According to its derivation, it only provides useful results if D < A. The same result can be obtained directly from Eq. Ž12.: f 0 C2 f 0 C2 y sy 3r2 2 2 3r2 2 czŽ d y A . c z Ž D q 2 DA . fy
f 0 C2 2'2 c z D 3r2A3r2
.
349
approximation of such forces with the first two terms of a Taylor series has a very small range of validity, limiting the applicability of the force gradient method to very small amplitudes. As demonstrated in Fig. 4, the oscillation amplitude A must be as small as 0.1 ˚ to obtain sufficient agreement between the results A
Ž 21 .
Consequently, the calculation of the frequency shift ˚ only if works even for large amplitudes like 100 A ˚ the nearest tip–sample distance is smaller than 10 A Ždashed line in Fig. 3b.. To summarize this section, we have found at the example of a van der Waals force ŽEq. Ž19.. that in the case of long-range tip–sample forces Žwhich are usually described by inverse power laws as given in Eq. Ž8. with small n, i.e., n F 6., the frequency shift is satisfactorily calculated with Eqs. Ž11. – Ž16.. The force gradient method ŽEq. Ž7.. and the approximation Eq. Ž17., however, showed an application range which is unacceptably small for a useful interpretation of experimental data. 4.2. Short-range forces Short-range forces between the front atoms of the tip and nearest sample surface atoms are often described by the force of a Lennard–Jones potential w16,19x: 12 E0 r 0 13 r0 7 FL J s y , Ž 22 . r0 z z
žž / ž / /
where E0 represents the binding energy and r 0 the equilibrium distance; typical parameters are E0 s 1 ˚ Introducing these values in Eq. eV and r 0 s 4 A. Ž22., it is found that the tip approaches the sample surface without any hysteresis or instability effect, since the condition for a ‘jump-to-contact’ w19x:
E 2 Vint Ez
s y
2 max
E Fint Ez
and c z A q w Fint x max - 0,
) cz max
Ž 23 .
is not fulfilled. It is the nature of short-range forces that their decay length is very small. Consequently, a linear
Fig. 4. Comparison between the exact results for the frequency shift obtained from the equation of motion Žindicated by the symbols. using the short-range force ŽEq. Ž22.., and the corresponding approximations calculated according to Eq. Ž5. Žsolid lines., Eq. Ž6. Ždashed lines., and Eq. Ž7. Ždashed–dotted lines. as ˚ Žb. a function of the oscillation amplitude A: Ža. As 0.1 A, ˚ and Žc. As10 A. ˚ Whereas at As 0.1 A, ˚ the solid and As1 A, the dotted line are very close together and show good agreement with the exact results, all lines differ significantly at larger amplitudes from the numerically obtained data. In Žc., this deviation is most prominent: all three lines are indistinguishable and approximately identical to D f f 0 kHz on this scale.
350
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨
Fig. 5. Comparison between the exact results for the frequency shift obtained from the equation of motion Žindicated by the symbols for three different oscillation amplitudes. using the short-range force ŽEq. Ž22.., and the corresponding approximations calculated according to Eq. Ž17. Žsolid lines.. The exact results are well-reproduced.
found by the numerical solution of the equation of motion Žindicated by the symbols. and the results obtained using Eq. Ž5. Žsolid lines. or Eq. Ž6. Ždashed lines.. However, it should be noted that using a cantilever with c z s 40 Nrm as assumed throughout ˚ is at this paper, an oscillation amplitude of 0.1 A room temperature already realized solely by thermal activation w24x. Moreover, since the equilibrium point d 0 differs close to the sample surface significantly from the support–sample distance d if small amplitudes are applied, Eq. Ž7. Ždashed–dotted lines. is misleading even for such a small value of A. With increasing amplitudes, the force gradient method gets completely useless Žsee Fig. 4b and c.. Contrary to the situation described above, the frequency shift caused by a short-range force of the type Eq. Ž8. with large values of n Ž n G 7. can be calculated in the region close to the sample surface Ž A 4 D . satisfactorily using Eq. Ž17.. This issue is shown in Fig. 5; the result obtained by the numerical solution of the equation of motion is well-reproduced.
5. Summary We presented a study on different numerical and analytical methods to calculate the frequency shift of a cantilever oscillated with its resonance frequency near a sample surface. A comparison between the exact results obtained numerically by the solution of
the equation of motion and various analytical approximations demonstrates that the often used interpretation of the frequency shift as a measure for the force gradient is only true in the limit of very small oscillation amplitudes Ž A ™ 0.. A much better approximation of the frequency shift can be realized by means of perturbation theory within the Hamilton– Jacobi formalism which leads to a set of comparatively simple analytic formulas for long-range as well as for short-range interaction forces based on inverse power laws.
Acknowledgements We are indebted to W. Allers, A. Schwarz, and B. Gotsmann for helpful discussions. Financial support from the Deutsche Forschungsgemeinschaft ŽGrant No. WI 1277r2-2. and the Graduiertenkolleg ‘Physik nanostrukturierter Festkorper’ is gratefully acknowl¨ edged.
References w1x G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Phys. Rev. Lett. 49 Ž1982. 57. w2x For a review on STM see, e.g., R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy: Methods and Applications, Cambridge University Press, Cambridge, 1994. w3x J. Tersoff, D.R. Hamann, Phys. Rev. Lett. 50 Ž1983. 1998. w4x J. Tersoff, D.R. Hamann, Phys. Rev. B 31 Ž1985. 805. w5x G. Binnig, C.F. Quate, C. Gerber, Phys. Rev. Lett. 56 Ž1986. 930. w6x S. Ciraci, A. Baratoff, I.P. Batra, Phys. Rev. B 41 Ž1990. 4728. w7x C.F. Quate, Surf. Sci. 299r300 Ž1994. 980. w8x E. Tekman, S. Ciraci, J. Phys. C: Condens. Matter 3 Ž1991. 2613. w9x F.J. Giessibl, Science 267 Ž1995. 68. w10x H. Ueyama, M. Ohta, S. Sugawara, S. Morita, Jpn. J. Appl. Phys. 34 Ž1995. L1086. w11x M. Bammerlin, R. Luthi, E. Meyer, A. Baratoff, J. Lu, ¨ ¨ M. Guggisberg, C. Gerber, L. Howald, H.-J. Guntherodt, Probe ¨ Microsc. 1 Ž1997. 3. w12x Y. Martin, C.C. Williams, H.K. Wickramasinghe, J. Appl. Phys. 61 Ž1987. 4723. w13x R. Erlandsson, G.M. McClelland, C.M. Mate, S. Chiang, J. Vac. Sci. Technol. A 6 Ž1988. 266. w14x T.R. Albrecht, P. Grutter, D. Horne, D. Rugar, J. Appl. Phys. ¨ 69 Ž1991. 668. w15x B. Gotsmann, D. Kruger, H. Fuchs, Europhys. Lett. 39 ¨ Ž1997. 153; 41 Ž1998. 583.
H. Holscher et al.r Applied Surface Science 140 (1999) 344–351 ¨ w16x I.Yu. Sokolov, G.S. Henderson, F.J. Wicks, Surf. Sci. 381 Ž1997. L558. w17x R. Perez, M.C. Payne, I. Stich, K. Terakura, Phys. Rev. Lett. 78 Ž1997. 678. w18x H. Goldstein, Classical Mechanics, 2nd edn., AddisonWesley, 1980. w19x F.-J. Giessibl, Phys. Rev. B 56 Ž1997. 16010. w20x R. Boisgard, D. Michel, J.P. Aime, ´ Surf. Sci. 401 Ž1998. 199.
351
w21x N. Sasaki, M. Tsukada, Jpn. Appl. Phys. 37 Ž1998. L533. w22x W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd edn., Cambridge University Press, London, 1992. w23x J. Israelachvili, Intermolecular and Surface Forces, 2nd edn., Academic Press, London, 1992. w24x E. Meyer, H. Heinzelmann, in: R. Wiesendanger, H.-J. ŽEds.., Scanning Tunneling Microscopy II, Guntherodt ¨ Springer, Heidelberg, 1992.