Physical interpretation of frequency-modulation atomic force microscopy

Physical interpretation of frequency-modulation atomic force microscopy

Franz J. Giessibl and Hartmut Bielefeldt

Universita¨t Augsburg, Institut fu¨r Physik, Experimentalphysik 6, D-86135 Augsburg, Germany 共Received 24 March 1999兲

Frequency modulation atomic force microscopy is a method for imaging the surface of metals, semiconduc- tors and insulators in ultrahigh vacuum with true atomic resolution. The imaging signal in this technique is the frequency shift ⌬f of an oscillating cantilever with eigenfrequency f₀, spring constant k and amplitude A, which is subject to tip-sample forces Fts. Here, we present analytical results of⌬f ( f0,k,A) for several basic classes of Fts. With these results, a method to calculate images is derived and demonstrated with an example.

The scanning tunneling microscope 共STM兲 has provided spectacular images of conducting surfaces on the atomic scale.¹ After the invention of atomic force microscopy 共AFM兲by Binnig, Quate, and Gerber in 1986,²the possibil- ity of imaging insulating surfaces with atomic resolution in real space seemed to be very close. True atomic resolution with static AFM on inert samples has been reported in 1993.³ However, imaging reactive surfaces like Si共111兲in ultrahigh vacuum by static AFM has proven to be impractical because of chemical bonding between tip and sample and wear on the atomic scale.⁴ The application of frequency modulation AFM 共FM-AFM兲 共Ref. 5兲with large amplitudes in noncon- tact mode has helped to overcome problems with tip-sample bonding and finally, the 7⫻7 reconstruction of Si 共111兲 could be resolved also by AFM.^6–7 The initial experiments have been refined and the quality of FM-AFM images reaches that of the STM.^8–10 Other semiconductors,¹⁰ ionic crystals,¹¹ metal oxides,¹² metals,^13,14organic monolayers,¹⁵ a film of xenon physisorbed on graphite,¹⁶ and deliberately created defects on CaF₂共111兲 共Ref. 17兲 have been imaged with atomic resolution. The first and second international workshop on noncontact AFM共NC-AFM98 in Osaka, Japan and NC-AFM99 in Pontresina, Switzerland兲 have helped to establish FM-AFM as a powerful experimental technique.

While the number of surfaces which are imaged by FM- AFM with atomic resolution is growing rapidly, open ques- tions in the quantitative interpretation of the FM-AFM re- main. The imaging signal in FM-AFM is the shift in resonance frequency ⌬f of an oscillating cantilever 共CL兲. Here, we derive for the first time analytical solutions of⌬f for three classes of tip-sample potential V_ts: inverse power-, power- and exponential potentials. With this relations, we establish a powerful method for the calculation of FM-AFM images.

In FM-AFM, a CL with a very high-Q factor (⬇10⁵), eigenfrequency f₀ and spring constant k is subject to con- trolled positive feedback such that it oscillates with a con- stant amplitude A. When this CL is brought close to a sample, its frequency changes from f₀ to f⫽f₀⫹⌬f . This frequency change ⌬f is used to create an image z_{CL– b}(x,y ,⌬f ) by scanning the CL in the x⫺y plane while adjusting the z position of the base of the CL z_{CL– b} such that

⌬f stays constant 共see Fig. 1兲. Typical parameters are ⌬f

⬇⫺100 Hz k⬇20 N/m, f₀⬇200 kHz and A⬇10 nm—see Table I in Ref. 18 for an overview.

The frequency shift ⌬f as a function of F_ts⫽⫺⳵^Vts/⳵^z has been calculated by first order perturbation theory using the Hamilton-Jacobi approach¹⁹

⌬f共x_{CL– b}, f₀,k,A兲⫽⫺ f₀² kA²

冕

0

1/f₀

F_ts共x_tip兲q

⬘

共t兲dt 共1兲

with the CL deflection q

⬘

^{(t)⫽A cos(2}␲^f0t) and x_tip

⫽关x_{CL– b}⫹const.,y_{CL– b},z_{CL– b}⫹q

⬘

^(t)兴^T. This result has been confirmed by other approaches.^20–25First-order pertur- bation theory yields excellent results because max兩V_ts兩 ⱗ10 eV, while E⫽0.5kA²⬇5 keV. A Fourier approach q

⬘

^(t)⫽兺m⬁⫽0a_mcos(2␲^mft) 共Refs. 20–22兲 shows that both the dc deflection (a₀⫽A⌬f / f₀) and the higher orders are very small 关兩a_m兩⭐2A/(1⫺m²)兩⌬f / f₀兩 for m⭓2 with

⌬f / f₀ⱗ10^⫺3兴 compared to the fundamental amplitude a₁

⫽A. At the lower turnaround point of the tip x_tip^ltpthe precise value of the deflection of the CL is given by the condition of constant energy E⫽0.5kq_ltp

⬘

²⫹V_ts(x_tip^ltp)⫽0.5kA², thus q_ltp

⬘

⬇

⫺A⫹A•V_ts(x_tip^ltp)/2E. Since兩V_ts兩is small compared to E, the FM-AFM image z_{CL– b}(x, y ,⌬f ) is approximately equal to the map described by x_tip^ltp.

The scaling properties of ⌬f are such that it is useful to introduce a ‘‘normalized frequency shift’’¹⁹

␥共z,A兲ªkA^3/2

f₀ ⌬f共z, f₀,k,A兲. 共2兲 Substituting z

⬘

⫽A关1⫹cos(2␲^f0t)兴in Eq.共1兲we find

FIG. 1. Schematic of an oscillating cantilever共CL兲with tip and front atom共fa兲at its lower turnaround point共ltp兲close to a sample.

PHYSICAL REVIEW B VOLUME 61, NUMBER 15 15 APRIL 2000-I

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␥共x_tip^ltp,A兲⫽ 1

&␲

冕

0

2AF_ts共x_tip^ltp⫹z

⬘

^ez兲

冑

^z

_⬘

1⫺z

⬘

冑

^1⫺A2A^z

^⬘

dz

⬘

^. 共3兲

In typical experiments, A is very large compared to the range of F_tsand the second factor in the integral is close to unity in regions where F_ts is nonvanishing. Therefore, the ‘‘large- amplitude’’ normalized frequency shift

␥lA共x_tip^ltp兲⫽

冕

0

⬁F_ts共x_tip^ltp⫹z

⬘

^ez兲

␲

冑

^2z

⬘

^dz

⬘

^共4兲 is a good approximation for ␥ for any class of tip-sample forces, provided that the range of the forces is small com- pared to A.

The interaction of a macroscopic tip of an AFM with a sample is a complicated many-body problem, which can be solved with molecular dynamics calculations.²⁶ For a quali- tative analysis, quite realistic model forces can be con- structed from a chemical contribution modeling the interac- tion of the front atom 共fa兲with the sample F_faplus the van der Waals 共vdW兲 contribution of the rest of the tip F_background.¹⁹Since␥is linear in F_tsit is practical to expand F_ts in terms of basic types: 共a兲 inverse-power forces, 共b兲 power forces, and共c兲exponential forces and superimpose the individual solutions for ␥^.

共a兲 F_ts(z)⫽C/zⁿ 共Lennard-Jones potential, vdW forces, electrostatic and magnetic forces²⁷兲. Insertion in Eq.共3兲and comparing the result with the integral representation of the hypergeometric function F_c^a,b(␵) 共Ref. 28兲yields:

␥共z,A兲⫽C

冑

^A

zⁿ

冋

^F1n,0.5

冉

^⫺z2A

冊

^⫺F2n,1.5

冉

^⫺z2A

冊册

^{. 共5兲}

For large amplitudes (A/zⰇ1) the transformation formula F_c^a,b(␵)⫽(1⫺␵)^⫺bF_c^b,c⫺a(␵/(␵⫺1)) is useful 共15.5.9. in Ref. 28兲and

␥lA共z兲⫽

冑

2¹␲

⌫

冉

^n⫺12

冊

⌫共n兲 C

z^n⫺1/2 共6兲 where⌫(n) is the Gamma function.²⁸

共b兲 F_ts(z)⫽C(⫺z)^m for z⬍0 and F_ts(z)⫽0 for z⬎0 共Hertzian contact forces with m⫽3/2 for a spherical tip on a flat surface and adhesion forces with m⫽0兲 共Ref. 27兲

␥⫽C兩z兩^m⫹1/2⌫共m⫹1兲

冑

²␲

冋

^{F⌫共m0.5,0.5⫹m1.5⫹}

^冉

^3/2⫺2Az兲

^冊

^{⫹2Az F⌫共m0.5,1.5⫹m2.5⫹}

^冉

^5/2⫺2Az兲

^冊 册

^.

共7兲 For large amplitudes

␥lA共z兲⫽ ⌫共m⫹1兲

冑

²␲⌫共m⫹3/2兲C共⫺z兲^m⫹1/2 for z⬍0. 共8兲 共c兲F_ts(z)⫽F₀e^⫺␬z共Morse potential兲

␥⫽F₀e^⫺␬z

冑

^A关M₁^0.5共⫺2␬^A兲⫺M₂^1.5共⫺2␬^A兲兴, 共9兲 where M_b^a(␵) is the Kummer function.²⁸ For large ampli- tudes (␬^AⰇ1) we can use the asymptotic expression of M_b^a(␵) 共13.5.1. in Ref. 28兲and find

␥lA共z兲⫽F₀e^⫺z␬

冑

²␲␬ ^{. 共10兲} On conductive samples, simultaneous STM and FM-AFM operation is possible. Since the bandwidth of the tunneling current (I_t) preamplifier is i.g. much smaller than f₀ of the CL, the measured I_t is a time-average. With I_t(z)⫽I₀e^⫺␬tz we find

具^It共z,A兲典^⫽I0e^⫺␬tzM₁^1/2共⫺2␬tA兲. 共11兲 When ␬tAⰇ1, 具^It(z,A)典^/It(z,0)⬇1/

冑

²␲␬tA yielding a re- duction to 1/35 for typical parameters 共A⫽10 nm and ␬t

⬇20 nm^⫺1for metals兲.

Figure 2 shows the ratio ␥^(z,A)/␥lA(z) as a function of A/z and A␬, respectively. In a typical experiment, A/z and A␬ are between 10 and 1000 during imaging where␥lA(z) and ␥(z,A) are almost identical. As A/z becomes smaller, the sensitivity to chemical 共short range兲 forces increases. In addition to a better signal-to-noise ratio¹⁸ the use of smaller amplitudes makes sharp tips less important since the sensi- tivity to the macroscopic vdW forces decreases.

Equation 共4兲 shows that␥lA increases with the range of F_tsand␥has the dimension of N

冑

m. Table I shows␥lAand F_ts

冑

␭ with range ␭for three basic types of forces laws. For exponential forces F_ts(z)⫽F₀e^⫺␬z the range ␭ defined by

FIG. 2. Ratio between normalized frequency shift␥and large-amplitude approximation␥lAas a function of A/z and A␬, respectively.

TABLE I. Three basic types of tip-sample potentials and corre- sponding normalized frequency shift␥.

Basic type V_ts(z) Fts冑^Vts/Fts ␥lA

Inv.-power n⬎1, z⬎0

1 n⫺1

C z^n⫺1

1

冑

n⫺1 C z^n⫺0.5

⌫

冉

^n⫺12

冊

冑

2␲⌫共n兲 C z^n⫺0.5 Power

m⭓0, z⬍0

C共⫺z兲^m⫹1 m⫹1

C共⫺z兲^m⫹0.5

冑

m⫹1

⌫共m⫹1兲C共⫺z兲^m⫹0.5

冑

2␲⌫共m⫹1.5兲 Exponential F₀e^⫺␬z

␬

F₀e^⫺␬z

冑

␬

F₀e^⫺␬z

冑

2␲␬

PRB 61 BRIEF REPORTS 9969

V(z⫹␭)/V(z)⫽1/e is independent of z with ␭⫽V(z)/F(z)

⫽1/␬. For inverse power and power forces, an analog ex- pression␭powerªV(z)/F(z) depends on z, but it shows ana- log scaling properties, i.e., V(z⫹␭power)/V(z)⬇1/e for ex- ponents n⬎1 共⫽1/e for n→⬁兲. Table I shows ␥lA and F_ts

冑

^Vts/F_ts. It is interesting to note that F_ts

冑

^Vts/F_ts/␥lA

⬇

冑

²␲ ^for兩n兩ⲏ2 and F_ts

冑

^Vts/F_ts/␥lA⫽

冑

²␲ for exponen- tial forces and power forces with 兩n兩→⬁. Since the fre- quency shift is a linear function of F_ts关Eq.共1兲兴, the total␥^is a linear combination of the contributions of the basic types F_tsⁱ

␥lA共x_tip^ltp兲⬇ 1

冑

²␲

兺

_i ^{Ftsi 共xtipltp兲}

^冑

^{Vtsi 共xtipltp兲/Ftsi 共xtipltp兲, 共12兲}

where V_tsⁱ is a basic type. Typical chemical forces are⬇⫺1 nN with a range of 0.1 nm and vdW forces are⬇⫺1 nN with a ‘‘range’’ of 1 nm, thus experimental␥

⬘

s are expected to be in the order of⫺10 fN

冑

^m.

To illustrate how Eq. 共12兲 can be used to calculate the FM-AFM image z(x,y ,␥), we consider a relatively simple system: an adsorbed layer of xenon. Xenon forms a closed packed film with next-neighbor distance ␴Xe⬇0.43 nm on graphite, thus the unit vectors of the xenon surface lattice are given by a₁⫽(␴^{,0,0)T and a}2⫽(␴^/2,␴)/2,0)^T. Allers et al.¹⁶have succeeded in imaging such a layer by FM-AFM:

the image shows the expected closed packed structure with a corrugation of 25 pm and atoms appearing high in the image, i.e. no contrast inversion. The following imaging parameters have been used:⌬f⫽⫺92 Hz, k⫽35 N/m, f₀⫽160 kHz and A⫽9.4 nm, i.e., at ␥⫽⫺18 fN

冑

^m.

The interaction of two xenon atoms can be modeled by a Lennard Jones potential

␾LJ共d兲⫽⫺E_bond

冋

²

冉

^␴d

冊

^6⫺

冉

^␴d

冊

¹²

册

^共13兲

with E_bond⫽0.02 eV and ␴⫽0.433 nm.²⁹ Since the tip was made of silicon, we assume that the front atom of the tip is

FIG. 3.共Color兲 共a兲Normalized frequency shift␥lA(x⫽0,y ,z) from y⫽⫺2␴^{to 2}␴⁽␴⫽0.43 nm). Xenon atoms are situated at y⫽0,⫾)␴. The contour lines of constant␥are cross sections of the corresponding FM-AFM image z(x,y ,␥). Maximal corrugation is obtained for␥⬇⫺18 fN冑m共green area兲.共b兲 Arrangement of Xenon atoms on the surface,共c兲top view of␥lAalong red track in共b兲showing a maximal corrugation of⬃5 pm.

9970 BRIEF REPORTS PRB 61

also silicon. Krupp has shown, that the Hamaker constant of two materials is given by the geometric average between the individual Hamaker constants.³⁰With further assuming, that the equilibrium distance in a silicon-xenon bond is given by the mean value between the bulk nearest neighbor distance in Si (␴Si⫽0.235 nm) and Xe, we can create a LJ model poten- tial for interaction of Si and Xe ␾LJ

Si⫺Xe

with E_bond^Si⫺Xe

⫽0.047 eV and␴Si⫺Xe⫽0.33 nm. F_ts is approximated by:

F_ts共x_fa^ltp兲⫽⫺ C z_fa^ltp⫹␴Si

⫹ ⳵

⳵^zn,m

兺

⫽⫺⬁

⬁

␾LJ Si⫺Xe

共兩x_fa^ltp⫺a_mⁿ兩兲, 共14兲 where C is a constant 共the tip is assumed to be conical or pyramidal, thus the long-range component is given by

⫺C/z, see Refs. 19 and 31兲 and a_mⁿ⫽na₁⫹ma₂. For the calculation of ␥lA(x,y ,z), the attractive part ␾LJ⫺att⬀d^⫺6 and repulsive part ␾LJ⫺rep⬀d^⫺12 have to be treated separately.³² Figure 3共a兲 shows ␥lA(x⫽0,y ,z) from y

⫽⫺2␴^{to y}⫽2␴. The contour lines correspond to cross sec- tions of the image z(x,y⫽0,␥⫽const.). Stable operation of the microscope is only possible in a z range where ⳵␥^/⳵^z

⬍0. The maximal corrugation occurs at ␥optimal

⬇⫺18 fN

冑

^{m and is} ⬇5 pm 关Fig. 3共c兲兴. It is noted that the absolute value of ␥optimal depends strongly on C 共here C

⫽8.6⫻10^⫺19J兲 which is a function of the macroscopic tip shape. However, the maximal corrugation varies little with C and is quite insensitive to the parameters we have calculated for the front atom-sample potential ␾LJ

Si⫺Xe. The deviation between the corrugations in theory and experiment is prob- ably due to elastic sample deformation. We assume that the tip is rigid 共i.e., x_fa⫺x_tipis constant—the force constant be-

tween next neighbors in Si is 170 N/m兲, but the vdW bonds in the Xe are weak (⳵²␾LJ/⳵^d2⬇1 N/m). We believe that the corrugation is strongly enhanced because the Xe atoms are pulled out of the surface. A similar effect has been ob- served in STM where the theoretical corrugation for Al共111兲 was ⬃1 pm while the experimental values were⬃50 pm.³³

In summary, we have found a physical interpretation of large amplitude FM-AFM: the images are a map z(x, y ,␥

⫽const.) where ␥^{(x,y ,z)}⫽兰0⬁F_ts(x,y ,z⫹z

⬘

^)/

冑

²␲^2z

⬘

^dz

⬘

^. When F_ts is known in terms of basic 共i.e., power, inverse power, exponential兲 force types F_tsⁱ(z), ␥

⫽兺iF_tsⁱ (z)

冑

␭i/2␲ ^where ␭i⫽V_tsⁱ (z)/F_tsⁱ (z). Our analytic results for various basic types of tip-sample forces establish the validity of the large-amplitude approximation for the in- terpretation of images and allow a quantitative analysis of

⌬f (z) curves for larger z values 共i.e., 0⬍A/z⬍100兲. We have further found an analytic result for the dependence of the tunneling current as a function of amplitude. Calculations with molecular dynamics, which currently yield F_ts(x, y ,z) 共Ref. 26兲 can be extended to compute the observables in FM-AFM, namely the experimental FM-AFM images z(x,y ,␥^{) with Eq.}共4兲and␥(x,y ,z,A) curves with Eq.共3兲on specific spots x,y on the sample 共e.g. adatom sites and cor- nerhole centers兲. Comparing the experimental results with these calculations, the force vs distance characteristics of chemical bonds between front atom and samples can be di- rectly derived.

We thank A. Baratoff, P. van Dongen, U. Du¨rig, S. Hem- bacher, U. Mair, and J. Mannhart for discussions and W.

Allers et al. for the preprint of Ref. 16. This work is sup- ported by the BMBF共Project No. 13N6918/1兲.

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