Calculation of the optimal imaging parameters for frequency modulation atomic force microscopy

Applied Surface Science 140 Ž1999. 352–357

Calculation of the optimal imaging parameters for frequency modulation atomic force microscopy Franz J. Giessibl ) , Hartmut Bielefeldt, Stefan Hembacher, Jochen Mannhart UniÕersitat 1, D-86135 ¨ Augsburg, Institute of Physics, Electronic Correlations and Magnetism, Experimentalphysik VI, UniÕersitatsstrasse ¨ Augsburg, Germany Received 1 July 1998; accepted 11 August 1998

Abstract True atomic resolution of conductors and insulators is now routinely obtained in vacuum by frequency modulation atomic force microscopy. So far, the imaging parameters Ži.e., eigenfrequency, stiffness and oscillation amplitude of the cantilever, frequency shift. which result in optimal spatial resolution for a given cantilever and sample have been found empirically. Here, we calculate the optimal set of parameters from first principles as a function of the tip–sample system. The result shows that the either the acquisition rate or the signal-to-noise ratio could be increased by up to two orders of magnitude by using stiffer cantilevers and smaller amplitudes than are in use today. q 1999 Elsevier Science B.V. All rights reserved. PACS: 07.79 Lh; 61.16 Ch; 87.64 Dz; 34.20 Cf Keywords: Atomic force microscopy; Frequency modulation atomic force microscopy; Dynamic force microscopy; Atomic resolution; Tip–sample interaction; Dissipation; Thermal noise

1. Introduction Although Binnig, Quate and Gerber anticipated that the atomic force microscope ŽAFM. they invented in 1985 w1x would ultimately achieve true atomic resolution in vacuum, it took almost 10 years before the Si Ž111.-Ž7 = 7. reconstruction w2–7x and other conducting w8x and insulating surfaces w9,10x were imaged with atomic resolution by AFM. Frequency modulation ŽFM. AFM, originally invented by Albrecht et al. in 1991 w11x for magnetic force

)

Corresponding author. Tel.: q49-821-598-3675; Fax: q49821-598-3652; E-mail: [email protected]

microscopy, is the method which was used in the first demonstration of true atomic resolution in vacuum and is now common practice in vacuum AFM. In this technique, a cantilever ŽCL. with spring constant k and eigenfrequency f 0 is subject to positive feedback such that it oscillates with a constant amplitude A 0 . When the oscillating CL is approached to a sample, its oscillation frequency changes from f 0 to f s f 0 q D f due to the forces between the tip of the CL and the sample. Scanning the CL across the sample Ž x–y plane. and adjusting z such that f is constant yields a map z Ž x, y, D f, k, f 0 , A 0 .. This map provides an atomic picture of the surface if the vertical noise is smaller than the atomic corrugation. Up to now, combinations of D f,

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 3 - 4

F.J. Giessibl et al.r Applied Surface Science 140 (1999) 352–357

353

Table 1 Parameters used for imaging various surfaces by FM-AFM newline k wNrmx

f 0 wkHzx

D f wHzx

A 0 wnmx

kA 0 wnNx

g awfN'm x

Sample

Ref.

Lateral resolution

43.0 17.0 23.5 33.0 30.0 36.0 28.0 37.0 41.0 34.0 10.0 37.0 2.5 2.5

276.0 114.0 153.0 264.0 168.0 160.0 270.0 276.0 172.0 151.0 290.0 276.0 60.0 60.0

y60 y70 y70 y670 y80 y63 y80 y50 y10 y6 y95 y350 y16 y32

40.0 32.0 19.0 4.0 13.0 12.7 15 10.0 16.0 20.0 10.0 1.5 15.0 3.3

1720 544 447 132 390 457 420 370 654 680 100 55.5 37.5 8.25

y74.7 y65.4 y28.2 y21.2 y21.2 y20.3 y15.2 y6.7 y4.8 y3.8 y3.3 y2.7 y1.2 y0.3

SiŽ111. SiŽ111. SiŽ111. SiŽ001. NaClŽ001. InAsŽ110. TiO 2 Ž110. SiŽ111. SiŽ111. InPŽ110. SiŽ111. SiŽ111. KClŽ001. SiŽ111.

w3x w2x w6x w7x w10x w12x w13x w7x w14x w8x w5x w7x w15x w15x

atomic atomic atomic atomic atomic atomic atomic atomic atomic atomic atomic atomic f 3 nm f 0.6 nm Ž x ., 2 nm Ž y .

a

w x g s D fkA3r2 0 rf 0 , see Ref. 16 .

k, f 0 , and A 0 which do provide true atomic resolution have been found empirically. With currently available CLs, the oscillation amplitude A 0 needs to be up to 100 times the interatomic distances for obtaining atomic resolution Žsee Table 1.. However, the spring constant k and the eigenfrequency f 0 have to be selected from the discrete set of commercially available CLs. Since the spring constant of a CL cannot be freely chosen, it is important to clarify for which set of operating parameters Ž k, f 0 , A 0 . best performance is to be expected. Here, we present a calculation based on first principles for the optimal set of k, f 0 and A 0 as a function of the tip–sample potential Vts Ž x, y, z .. The models for VtsŽ x, y, z . we use here include exponential and inverse-power laws.

tude, it is not straightforward to see whether there is an optimum Žminimum in vertical noise. for a specific set of k, f 0 and A 0 . These optimal parameters will depend on the characteristics of the tip–sample potential. Three major contributors to the vertical noise in z Ž x, y, D f, k, f 0 , A 0 . can be identified Žsee Fig. 1.. Ž1. Frequency noise: a variation d d leads to a variation in the frequency shift according to dŽ D f .

2. Calculation of vertical noise Since the imaging signal in FM-AFM is a frequency shift D f, the best signal-to-noise ratio is expected for large D f and little noise dŽ D f .. However, for a given minimum distance d between the front atom of the CL and the sample both D f and dŽ D f . decrease with amplitude. If D f were independent of A 0 , minimal noise would result for A 0 ™ `. Since the magnitude of D f decreases with ampli-

Fig. 1. Schematic of a frequency modulation atomic force microscope.

F.J. Giessibl et al.r Applied Surface Science 140 (1999) 352–357

354

s ŽEŽ D f ..rŽEd .d d. Thus, the vertical noise caused by frequency noise is: d zfs

dŽ D f .

EŽDf .

Since the noise contributions in z-direction are statistically independent, the total z-noise is: 2

.

Ž 1.

2 Ždz. s

Ed

d fthermal s d Ž D f . thermal s

(

k B TBf 0 kA20 p Q 0

Ž 2.

where k B is the Boltzmann constant, T the absolute temperature, B the detection bandwidth and Q0 the quality factor of the CL w11x. The detection bandwidth B determines the imaging speed. If a section with a width of 100 atoms is to be imaged at an imaging speed of three lines per second, B needs to be at least 2 = 100 = 3rs s 600 Hz. There is also a contribution to frequency noise dŽ D f . instrumental which depends on the quality of the frequency detector. Since these noise sources are uncorrelated, the total frequency noise is given by

(

2

2

d Ž D f . s d Ž D f . thermal q d Ž D f . instrumental .

Ž 3.

The denominator in Eq. Ž1., ŽEŽ D f ..rŽEd ., is calculated further below. Ž2. Amplitude noise: the fluctuation of the total energy of the CL is approximately k B T. Thus, k Ž A 0 q d A 0 . 2r2 f kA20r2 q k B T and the amplitude noise is given by: d A0 f

kA 0

 0 EŽDf .

2

2

q Ž d A 0 . q Ž dZinstr . . .

Ž 5.

Ed

Albrecht et al. w11x have calculated the thermodynamic lower limit of the frequency noise

k BT

dŽ D f .

.

Ž 4.

For typical values of kA 0 ŽTable 1., this contribution is negligible even at room temperature. Ž3. Mechanical noise: the mechanical loop from CL to sample is closed by the xyz scanner and the coarse positioning system. Acoustic noise, building vibrations etc. cause a variation of the distance between CL and sample dZinstr.. Proper design of the microscope and insulation from external vibrations help to bring dZinstr. down to a few picometers.

Since amplitude- and mechanical noise are negligible, in the following, we only consider the noise contribution from frequency noise. 2.1. Calculation of the deriÕatiÕe of the frequency shift 2.1.1. Frequency shift for inÕerse-power forces For tip–sample forces given by Fts Ž q . s

C

Ž 6.

qn

where C is a constant and q is the distance between the center of the front atom of the tip and the plane defined by the centers of the surface atoms, the frequency shift D f s f y f 0 is given by Ref. w16x: D fs A0 s n

f0

C

2k d

Ž 7.

nq1

if A 0 < d. For A 0 4 d the frequency shift is given by: D f l A 0 s Yn

f0

C

kA3r2 0

d

Ž 8.

ny1r2

with Yn [

1

1

`

'2p Hy` Ž 1 q y

2

n

d y.

Ž 9.

.

Pertinent values of Yn are Y1 s 1r '2 , Y2 f 0.36, Y3 f 0.27, Y4 f 0.22, Y7 f 0.16 and Y13 f 0.11 w16x. For functions g s Ž d . s cdy3 r2 and g l Ž A 0 . s y3 r2 A0 a function g Ž d, A 0 . s

1 ym 3 m r2

c

d

q A30m r2

1rm

Ž 10 .

fulfills lim A 0 r d ™ 0 g Ž d, A 0 . s g s Ž d . and lim A 0 r d ™` g Ž d, A 0 . s g l Ž A 0 .. The simplest case m s 1 yields

F.J. Giessibl et al.r Applied Surface Science 140 (1999) 352–357

an approximation for D f which is sufficiently precise for intermediate amplitudes 1 Yn f 0 C D fs ny 1r2 2Y d n 3r2 k A3r2 d 0 q n Yn f 0 C 1 s . Ž 11 . 2Yn nq1 k 3r2 ny1r2 A0 d q d n The variation of the frequency shift with distance is given by the derivative:

ž

/

EŽD f . Ed sy =

Yn f 0 C k ny 3r2 q 2Yn Ž 1 q 1rn . d n Ž n y 1r2. A3r2 0 d

ž

ny1r2 A3r2 q 0 d

2Yn n

2

d nq1

.

/

2.1.2. Frequency shift for exponential forces For exponential forces

Ž 13 .

the frequency shift for small amplitudes A 0 < l is: f0 D fs A0 s F eyd r l Ž 14 . 2kl 0 and the frequency shift for large amplitudes A 0 4 l w16x: D fl A0 s

f 0'l F0 eyd r l kA3r2 0

'2p

.

with the derivative

EŽD f .

1

D f. Ž 17 . Ed l With this calculation ŽEŽ D f ..rŽEd . all the elements of the z-noise are complete. Finally, we have to take into account that we cannot choose the stiffness of the CL k independently from the amplitude A 0 . Forces between tip and sample in vacuum are in general attractive before the CL makes contact. Therefore, there is a distance region where the CL is unstable and snaps uncontrolled to the surface Ž ‘ jump-to-contact’ . , unless either k ) max ŽyŽE 2 Vts .rŽEd 2 .. s k tsmax w17x or kA 0 ) maxŽyFts . s Ftsmax w16x. These two conditions can be merged to kss

sy

Ftsmax

Ž 18 .

A 0 q Ftsmaxrk tsmax

with a ‘safety factor’ s ) 1.

Ž 12 .

Fts Ž q . s F0 eyq r l

355

2.2. Effect of nonconserÕatiÕe Vt s The tip–sample potential has so far been treated as nondissipative. However, Erlandsson et al. w4x, Bammerlin et al. w10x, Sugawara et al. w14x, Guethner w5x and others have found significant damping of the CL when it oscillates close to the surface. A general model in analogy to friction is that the dissipative force component is given by dq Fdiss Ž q . s ym Fts Ž q .

Ž 15 .

Again, we can create an approximation for all amplitudes f0 D fs F0 eyd r l Ž 16 . p A 0 3r2 2kl 1q 2 l

ž ( ž / /

1 For amplitudes in the order of the decay length Ž A 0 s d, l., the error of the all-amplitude approximation is 7%, 2%, 4% and 90% for exponential forces and inverse-power forces ns13, 7 and 1, respectively. A precise formula for ns1 is given in H. Holscher et al. w18x. ¨

dt dq

dq

r

.

dt

Ž 19 .

dt The parameter r determines the type of ‘friction’: r s 0 corresponds to velocity-independent friction, r s 1 is commonly used for treating damped oscillators and r s 2 describes friction in fluids. Unfortunately, little is known about the nature of the dissipation process in FM-AFM. For the simplest case r s 0, the energy loss per oscillation cycle D Ets is given by: d

D Ets s y

H

dq2 A 0

dq2 A 0

m Fts Ž q . d q q

H

d

m Fts Ž q . d q.

Ž 20 .

F.J. Giessibl et al.r Applied Surface Science 140 (1999) 352–357

356

3. Results and conclusion The z-noise is thus given by

d zfs

d

k Yn Ftsmax

(

=

ž

ny

1 2

2

3r2

A0

žž /

q

d

A0

2Yn n

2

/

3r2

/ž /

q 2Yn Ž 1 q 1rn .

d

k B TB

Ž 23 .

p QkA20 f 0

for inverse-power forces and d zfs

Fig. 2. Calculated noise in sample topography Žperpendicular to surface. as a function of amplitude for exponential and inversepower force models.

Since

Ž 21 .

E s D ECL q D Ets

Q0 1 q 2 Q 0 D EtsrkA20

.

2

l

3r2

ž ( ž / /( 1q

k B TB

p QkA20 f 0

(

This energy loss diminishes the quality factor Q0 of the CL. Since Q0 s ErD ECL , where E s kA20r2 is the total energy in the CL and D ECL is the energy loss due to internal friction in the CL, the effective Q-value is given by: Qs

A0

for exponential forces. In both cases, the noise is proportional to 1rFtsmax , i.e., the greater the maximum attractive force, the easier it is to obtain true atomic resolution. Also, the noise is proportional to 1r f 0 , i.e., the higher the frequency, the lower the noise. The most interesting implications of Eqs. Ž23. and Ž24. are the dependence of noise with amplitude. The stiffness of the CL is calculated with Eq. Ž18.. The safety factor s is set to 100 Žmost authors use s 4 1 see column ‘kA 0 ’ in Table 1.. Fig. 2 shows a plot of d z f for inverse-power forces with n s 1, 7 and 13 and exponential forces with Ftsmax s 3 nN, d s 250 pm and l s a Bohr s 52.9 pm as a function of A 0rd. The corresponding force models are listed in Table 2. The properties of the CL in use are Q0 s 10 000, f 0 s 100 kHz, T s 300 K, the frequency detector is set to B s 1 kHz and its electronic noise is dŽ D f . instrumental s 0.1 Hz. For Fts

ts

D Ets s 2 m Vts Ž d q 2 A 0 . y Vts Ž d . .

Ftsmax

p

Ž 24 .

HF d q s V , ts

2 k l2

Ž 22 .

The negative effect of dissipation on the Q factor is partially compensated with using large amplitudes. Bammerlin et al. w10x have measured the dissipation for a silicon tip interacting with potassium chloride. Analysis of their data yields m f 0.05.

Table 2 Models used for tip–sample potential Force model Inverse-power, n ) 1 Inverse-power, n s 1 Exponential

Vts Ž q . C

1

ny1 qn y1 C lnŽ q . F0 leyq r l

FtsŽ q .

Ftsma x

k tsmax

D Ets

Crq n

Crd n

nFtsmax rd

m

Ftsma x rd Ftsmax rl

m Ftsmax d lnŽ1 q 2 A 0rd . m FtsmaxlŽ1 y expy2 A 0 r l .

Crq F0 eyq r l

Crd F0 eyd r l

Ftsma x d ny1

1y

1

Ž 1q2 A 0 r d . n y 1

F.J. Giessibl et al.r Applied Surface Science 140 (1999) 352–357

357

and sample needs to be done for improving the performance of FM-atomic force microscopy. Also, Eqs. Ž23. and Ž24. suggest that using CLs with higher f 0 results in less noise. However, if the dissipative part of the tip–sample force depends on the relative velocity between tip and sample Ž r G 1 in Eq. Ž19.., using CLs with higher frequencies might result in poorer resolution, since the dissipated energy will be proportional to f 0 . Acknowledgements We thank Lukas Howald for discussions. This work was supported by BMBF Grant 13N6918r1. Fig. 3. Calculated noise in sample topography as a function of amplitude for exponential and inverse-power force models with dissipation.

s Crq, the optimum is reached for A 0rd f 1. The corresponding spring constant is k s 500 Nrm. For forces with shorter ranges Ž n s 7, 13 and exponential., the lowest noise is obtained for even smaller amplitudes. The corresponding optimal spring constants range up to 3000 Nrm. Fig. 3 shows the noise for nonconservative potentials. For each cycle, an energy D Ets is dissipated with d Ets s 2 m Ž VtsŽ d q 2 A 0 . y VtsŽ d .. with m s 0.05. As expected, larger amplitudes are required for minimum noise. The minimal noise is less pronounced than in the case without damping. The corresponding optimal stiffness is between 30 and 200 Nrm. In summary, we have shown why CLs with k f 10 Nrm require amplitudes in the order of 10 nm. If CLs with k f 300 Nrm were available, B could be increased by 100 or the vertical noise would drop by a factor of 10. Since the z-noise level is already comparable to the other noise sources d A 0 and dZinstr., the major practical advantage is an increase in bandwidth which would allow faster scanning. By comparing Figs. 2 and 3 it is clear that dissipation is crucial for determining the optimal operating amplitude—zero dissipation would favor stiff cantilevers and strong dissipation requires large amplitudes and softer cantilevers. More work on the investigation of the dissipation process between tip

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