IOP Publishing doi:10.1088/1742-6596/61/1/188
Journal of Physics: Conference Series 61 (2007) 949–954 International Conference on Nanoscience and Technology (ICN&T 2006)
Noise in Frequency Modulation-Dynamic Force Microscopy J´ erˆ ome Polesel-Maris1 and S´ ebastien Gauthier2 1 2
CSEM SA, rue Jaquet-Droz 1, Case Postale, CH-2002 Neuchˆ atel, Switzerland GNS-CEMES-CNRS, 29 rue Jeanne Marvig, F-31055 Toulouse, France
E-mail:
[email protected] Abstract. Analytical expressions giving the noise power spectral densities of the signals of interest in frequency modulation-dynamic force microscopy are derived. They are validated by comparison with numerical simulations performed with a realistic virtual force microscope. Thermal noise as well as detection noise are considered. A tip-substrate interaction incorporating a van der Waals and a Morse potential is considered as an illustrative example.
1. Introduction Frequency Modulation-Atomic Force Microscopy (FM-AFM) is a rather complex technique. Its complete analysis is difficult mainly because it includes non-linearities, which can be classified in two types: 1) the tip-substrate interaction is non-linear [1] because the oscillating amplitudes are large compared to its range; 2) the control system of the microscope includes itself nonlinearities, in the form of multipliers, which make the analysis by usual linear control theory methods impossible. Nevertheless, a good understanding of the noise of the instrument is an important goal. It allows to precise the limits to the accuracy of the measurement set by the different kind of noise and what are the optimal settings to approach them. It can also help to avoid mesurement artefacts and to compare different instrumental choices. A virtual AFM was recently shown to provide a realistic numerical model of a real FM-AFM, including noise [2], but no analytical model has yet been proposed. The aim of this paper is to set analytical expressions for the noise in FM-AFM and to validate them by comparison with results produced by the virtual AFM. This paper is a summary of a longer one, which is in preparation. For the detailled structure of the virtual AFM, we refer to [2]. 2. Noise analysis of the FM-AFM oscillator We briefly recall the structure of the control system of a FM-AFM illustrated in figure 1. The cantilever C is maintained oscillating at its resonance frequency ωc by a positive feedback loop which is constituted by an Automatic Gain Control (AGC) device and a Phase-Locked Loop (PLL)-based frequency demodulator. When properly set, the PLL outputs a signal exc(t) at the resonant frequency ωc with a phase shift of π2 relative to the oscillation of the cantilever z(t). The amplitude is measured by the Synchronous Demodulator (SD) of the AGC and compared with a pre-set value A∗ . The resulting error signal is passed through a Proportional-Integral © 2007 IOP Publishing Ltd
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C F(t) δF(t)
+
e(t)
z(t)
PIadc
SD
+
d(t)
δz(t)
D(t) +
+
+
+
PLL exc(t) w(t)
∆f *
A*
PI
Figure 1. Scheme of the complete control system. controller (PI) giving w(t) called the damping signal, because it is proportional to the tipsubstrate energy dissipation when the above mentionned phase shift of π2 is realized [3]. The Proportional gain P and Integral gain I allow to set the stability and the precision of the AGC regulation loop. When used in the constant frequency detuning imaging mode, the Automatic Distance Control loop (ADC) adjusts the tip-substrate distance D(t) in order to maintain the frequency detuning ∆f (t) at a pre-set value ∆f ∗ . The error signal ∆f ∗ − ∆f (t), after being passed through a Proportional-Integral controller (PIadc ), acts on the tip-substrate distance D(t) via a piezoelectric transducer. We consider only two noise sources, which are known to dominate in FM-AFM [5, 6]. (i) The thermal equilibrium noise of the cantilever, of two-sided Power Spectral Density (PSD) 2 −1 BT Sf = 4kk ωc Q (in N .Hz ) where k and Q are the stiffness and quality factor of the cantilever, kB the Boltzmann constant and T the temperature. Due to the equipartition of energy, the power of thermal noise should be shared equally between amplitude and phase noise. Thus, Sf = Sfa + Sfφ , with Sfa = Sfφ = 12 Sf . (ii) The noise of the deflection sensor Sz . The amount and type of noise depends on the specific method used and of its detailled implementation. In the absence of more precise specifications, we will also consider that Sza = Szφ = 12 Sz . 2.1. Analytical formulation We start from the simplest situation, where the cantilever is oscillating at a large distance from the substrate. The tip-substrate interaction is then negligible and the ADC is opened. We indicate only the starting point and the result of the calculation. Details will be published elsewhere. The displacement z(t) of the tip can be modelized as an amplitude and phase modulated carrier [4]: z(t) = Ac [1 + m(t)] sin[ωc t + φ(t)]. With m(t) = m0 cos(Ωt + θa ), φ(t) = φ0 sin(Ωt + θφ ) and assuming small maximum deviations φ0 and m0 , we get: z(t) ≈ Ac sin(ωc t) + δza {sin[(ωc + Ω)t + θa ] + sin[(ωc − Ω)t − θa ]} +δzφ {sin[(ωc + Ω)t + θφ ] − sin[(ωc − Ω)t − θφ ]} with δza = Ac m20 and δzφ = Ac φ20 . The crossterm in φ0 m0 1 is neglected. Starting from this expression for z(t), the damping signal w(t) at the output of the AGC can be calculated as well as the excitation signal at the output of the PLL (figure 1). Then the excitation force F (t) = exc(t).w(t) + δF (t) is applied to the cantilever through the transfer function: z(ω) 1 Cl(ω) = = ω2 −ω2 c F (ω) k( 2 + jω ) ωc
Qωc
Near resonance, ω = ωc + Ω, with Ω ωc and Cl(ωc + Ω) = C(Ω)
1 j . k(− 2Ω +Q ) ω
Neglecting the
c
second order noise terms - in δza δzφ - one finally gets P SDw (Ω), the PSD of the damping signal, and P SD∆f (Ω), the PSD of the frequency detuning signal: P SDw (Ω) =
2 2Fagc (Ω) 2 φ 4 (|C(Ω)| Sf + Sza ) 1 − 2jC(Ω)Fagc (Ω)
(1)
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Figure 2. Simulated (grey lines) and calculated (black lines) (a) P SDw (ν) and (b) P SD∆f (ν). ωc fc = 2π = 270 kHz, k = 30 N.m−1 , Q = 45000, A∗ = 20 nm, P = 0.01 N.m−1 , I = 0.5 −1 −1 N.m .s , τD = 2 ms, Kv = 5280 rad.s−1 , τ1 = 55 µs, τ2 = 160 µs, Sf = 3.14 10−30 N2 .Hz−1 (T = 300 K). ν2 P SD∆f (Ω) = 4 2 Ac
2 H(Ω) 2 (|C(Ω)| Sfa + Szφ ) 1 − j k C(Ω)H(Ω)
(2)
Q
1 I with the transfer function of the AGC Fagc (Ω) = − 1+jΩτ (P + jΩ ) and the transfer function of D Kν the PLL H(Ω) = jΩ(1+jΩτ1 )(1+jΩτ2 )+Kν . τD is the time constant of the lowpass filter of the SD, P (resp. I) is the gain of the proportional (resp. integral) branch of the P-I controller of the AGC. Kν is the loop gain of the PLL, τ1 and τ2 the time constants of the second order lowpass filter of the PLL [2]. The noise of the damping signal depends on the thermal phase noise and the detection amplitude noise while the frequency detuning noise depends on the thermal amplitude noise and the detection phase noise. This results from the fact that, for noise or small displacements from equilibrium, the two feedback loops of the control system work independently.
2.2. Numerical simulations Figure 2 shows P SDw (ν) and P SD∆f (ν) obtained by numerical simulation compared with their analytical expressions (1) and (2) for different amounts of detection noise. When the detection noise is small (Sz = 0 or 2 10−28 m2 .Hz−1 ), the spectra are close to their thermal limit in the measuring bandwidth, indicated by horizontal dotted lines. In this situation, the integrated noise 1 scales as ν 2 . The dissipation noise thermal limit is given by lim [P SDw (Ω, Sz = 0)]Ω→0 = 4Sfφ . This value agrees with that given in [7], within a numerical factor of the order of unity and with [8], within a factor of 2. For the parameters of figure 2, it gives, when expressed in energy per oscillation cycle 4Sfφ π 2 A2c = 10−6 (eV/cycle)2 .Hz−1 , resulting in a variance of 2 10−4 (eV/cycle)2 when integrating from - 100 Hz to + 100 Hz. The dissipation sensitivity is then 14 meV/cycle, comparable to the best experimental data obtained so far [9]. The frequency detuning noise thermal limit is given by: lim [P SD∆f (Ω, Sz = 0)]Ω→0 =
fc kB T 2πkA2c Q
This two-sided PSD is in agreement with the one-sided expression calculated by Albrecht et al [10]. For the parameters of figure 2, it amounts to 3.2 10−7 Hz2 .Hz−1 , resulting in a variance of 6.4 10−4 Hz2 . The frequency detuning sensitivity is then of the order of 0.025 Hz. When the detection noise dominates (figure 2b with Sz = 2 10−24 m2 .Hz−1 ), P SD∆f (ν) increases as ν 2 3 from 10 Hz to 1 kHz leading to an integrated noise scaling as ν 2 as explained by Giessibl [5].
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Figure 3. Simulated and calculated (a) P SDw (ν) and (b) P SD∆f (ν) for P = 0.01 N.m−1 , I = 0.5 N.m−1 .s−1 and P = 0.1 N.m−1 , I = 10 N.m−1 .s−1 with Sz = 2 10−26 m2 .Hz−1 . Figure 3 shows how the settings of the AGC influence the PSDs. As expected, P SD∆f (ν) is not affected by these settings, since the two feedback loops work independently. For P = 0.1 N.m−1 , I = 10 N.m−1 .s−1 , a peak appears on P SDw (ν). It corresponds to a tendancy of the AGC loop to oscillate (“ringing”) at the peak frequency. For P = 0.01 N.m−1 , I = 0.5 N.m−1 .s−1 , the loop operates near critical damping. The agreement between analytical and simulated spectra is quite good, showing that the approximations involved in the derivation of expressions (1) and (2) are valid for the considered parameters. 3. Introducing the tip-substrate interaction We consider in the following the tip-substrate interaction potential as a sum of a van der Waals and a Morse contribution. An approximate analytical expressions can be derived for the frequency detuning considering the tip trajectory as sinusoidal, a very good approximation due to the high Q of the cantilever. Following equation (29) in reference [5], one writes the following integral for the frequency detuning: f2 ∆f = c 2 k Ac
1/fc 0
Ftip−sample [z(t)] z(t) dt
with the cantilever oscillation z(t) = Ac sin(2πfc t) and the force Ftip−sample between the tip and the sample such as: Ftip−sample [z(t)] = −
H R1 − 2 Eb κ(1 − e−κ(D+z(t)−σ) )e−κ(D+z(t)−σ) 6 (D + z(t))2
D is the distance between the tip at rest and the surface. The other parameters are explained in figure 4. f (D) obtained with the virtual AFM is compared in figure 4 with this analytical expression, showing a very good agreement. Let us consider an amplitude fluctuation δAc and a distance fluctuation δD. Due to the nonlinear character of the interaction, these fluctuations will induce a frequency fluctuation: δ∆f =
∂∆f ∂∆f δD = α(Ac , D)δAc + β(Ac , D)δD δAc + ∂Ac ∂D
∂∆f with α = ∂∆f ∂Ac and β = ∂D . These couplings can be taken into account by considering the expression of z(t) = Ac [1 + m(t)] sin[ωc t + φ(t)]. The amplitude fluctuation is given by δAc (t) = Ac m(t) = Ac m0 cos(Ωt + θa ). The distance fluctuation is imposed as δD(t) =
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Figure 4. Circles: f (D) obtained with the virtual AFM with the following parameters: P = 0.1 N.m−1 , I = 10 N.m−1 .s−1 , A∗ = 20 nm and without noise. Interaction parameters, for the van der Waals contribution: Hamaker constant H = 1 eV, tip radius R1 = 10 nm, for the Morse contribution: depth of the potential minimum Eb = 2.15 eV, equilibrium atomic bond distance σ = 0.235 nm and decay of the potential κ = 15.5 nm−1 . Continuous line: analytical expression. Dotted lines: −α(D). Dashed lines: β(D). −α and β are distinct but not distinguishable at the scale of the figure. δD0 cos(Ωt + θd ). The couplings α and β generate a new phase modulation contribution φint (t) such that φ(t) = φ0 sin(Ωt + θφ ) + φint (t) with: φint (t) =
αAc m0 δ∆f βδD0 = cos(Ωt + θa ) + cos(Ωt + θd ) ν ν ν
Then: int int {cos[(ωc +Ω)t+θa ]+cos[(ωc −Ω)t−θa ]}+δzD {cos[(ωc +Ω)t+θd ]+cos[(ωc −Ω)t−θd ]} z(t) = z0 (t)+δzA c 2
int = αAc m0 = αAc δz and δz int = βAc δD . Starting from this expression of z(t), the where δzA a 0 D 2ν ν 2ν c calculation proceeds as in section 2.1. Expression (1) for P SDw (ν) stays unchanged, because it does not depend on the position phase noise while expression (2) for P SD∆f (ν) becomes:
2 H 1 − j k CH + βFadc H Q 2 2 αA C c φ 2 × |C|2 Sfa + Szφ + (Sf + |Fagc | Sza ) 1 − 2jCFagc ν
ν2 P SD∆f (ν) = 4 2 Ac
Iadc jΩ the transfer function of the ADC. Due to φ (Sfa , Sf , Sza , Szφ ) contribute now to the frequency noise.
with Fadc (Ω) = Padc +
the interaction, the four
noise sources distance D is given by:
Finally, the PSD of the
4 ν2 P SDD (ν) = |Fadc | P SD∆f (ν) = 2 2 β Ac 2
2 βFadc H [noise source terms] 1 − j k CH + βFadc H Q
The results displayed in figure 5 show good agreement between simulated and calculated data. It is seen that the frequency noise scales approximately as α2 : it is dominated by the non-linear contribution. It increases when the tip is closer to the substrate, except at very short distances, when α2 begins to decrease due to the repulsive interaction (figure 4). ∆f shows a more complex behaviour due to the 1/β 2 dependance when βFadc is kept constant as in figure 5.
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Figure 5. (a) P SD∆f (ν), (b) P SDD (ν) with βPadc = 0.2 and βIadc = 400. 4. Conclusions The analytical expressions for the Power Spectral Densities of the different signals produced by FM-AFM presented here open many possibilities. We plan to use them to compare and to optimize different instrumental choices, for instance for which kind of measurement a tuning fork is a better choice than a cantilever. Acknowledgments This work was partly funded by the NMP Program of the European Community for RTD activities, under the STREP project NANOMAN (Contract n NMP4-CT-2003-505660). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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