Conference-NC-AFM 2010-Pressure Dependence of the Q-factor of Cantilevers Used for NC-AFM

We implement a test bed allowing the characterisation of cantilevers used in non-contact atomic force microscopy under controlled atmospheric conditions prior to using the cantilevers in a force microscope. The effective Q-factor of two types of cantilevers is measured as a function of the ambient pressure varied from 10−8 mbar to normal pressure. The Q-factor is found to be almost constant up to a pressure in the range of 10−2 to 10−1 mbar and then decreases by about three ordes of magnitude when increasing the pressure further to normal pressure. The pressure dependence of the effective Q-factor is approximated by analytical models based on the intrinsic damping of the cantilever, mounting losses and pressure dependent terms accounting for damping in the molecular flow and viscous regimes. The experimental data can be well described by the models in both regimes. [DOI: 10.1380/ejssnt.2011.30]


I. INTRODUCTION
Dynamic scanning force microscopy operated in the non-contact mode (NC-AFM) has long been a domain of measurements under conditions of the ultra-high vacuum (UHV) as in most NC-AFM implementations, imaging is based on the detection of small frequency shifts in the oscillation of a high-Q cantilever [1].Oscillating the cantilever in a gas of higher pressure, specifically normal pressure or in a fluid reduces the Q-factor dramatically and, consequently, the force gradient detection sensitivity is strongly diminished [2].However, it has been shown that the physical detection limit for NC-AFM imaging in air and liquids is low enough to allow for atomic resolution imaging and atomic and molecular structures can be detected provided the opto-electronic detection system is optimized to achieve its technical limits [3][4][5].The quality of NC-AFM images obtained in air can further be enhanced by applying advanced experimental techniques such as robust feedback loops [6], Q-control [7] or bimodal detection [8].Molecular resolution in air has been demonstrated [9] and the relevance of the Q-factor for such measurements has been clearly pointed out [10].
The effective Q-factor [11] of a cantilever oscillating in air has been investigated for a variety of cantilever geometries [12][13][14][15] and it has been shown that the effective Q-factor in air can be increased by optimising the geometry [16].However, to obtain a higher Q factor for NC-AFM measurements in a gas atmosphere, it may be more straightforward to reduce the ambient pressure and it is therefore of interest to determine the precise dependence of the Q-factor on the pressure of the ambient atmosphere.In this contribution we investigate the Q-factor of cantilevers oscillating in a gas atmosphere where the ambient pressure is varied from 10 −8 mbar to normal pressure.We compare our measurements to models predicting the Q-factor based on the intrinsic damping of the cantilever, mounting losses and pressure dependent terms accounting for damping in the molecular flow and viscous regimes.
The quality factor Q of a damped oscillating system is generally defined as the ratio of the energy W stored in the oscillating system to the energy ∆W dissipated per cycle [12]: There are several damping mechanisms contributing to the dissipation of energy in the oscillation of a cantilever.The reciprocal of the effective Q-factor 1/Q eff describes the total damping which is determined by the intrinsic damping 1/Q 0 of the cantilever, the damping 1/Q mount of the cantilever fixation in the AFM system and the air damping 1/Q air which needs to be considered when experiments are not performed under UHV conditions.
In a recent publication [11], we have extensively discussed most of the factors influencing the effective Qfactor of a cantilever and here we focus on the effect of the ambient gas atmosphere.
The pressure dependence of the Q-factor of silicon cantilevers has been investigated experimentally by Bianco et al. [14] as well as by Blom et al. [12] for certain ranges of the ambient pressure.It is possible to distinguish between pressure regions that are dominated by different damping mechanisms namely the molecular flow region and the viscous flow region [12].To distinguish these regions, we use the Knudsen number where λ is the mean free path of the gas molecules and w the width of the gas layer in motion, which here is the cantilever width [13].The mean free path of the gas molecules is given by where p is the gas pressure, n the number density, and d = 3.7 × 10 −10 m the diameter of the involved molecules [17].
The viscous flow regime is defined as K n < 0.01, the free molecular flow regime is defined as K n > 10 while the range 10 > K n > 0.01 is referred to as the transition e-Journal of Surface Science and Nanotechnology Volume 9 (2011) regime where molecular as well as viscous damping contribute [13].For a typical cantilever width of 30 µm, K n equals 10 at a pressure of p = 0.23 mbar, while K n equals 0.01 at a pressure of p = 230 mbar.For the molecular region, the pressure-dependent Q-factor is calculated applying a model derived by Christian [14,18]: where M is the mass of the gas molecules, R the gas constant, T the absolute temperature and p the pressure.Cantilever properties enter via the cantilever width w, thickness t and eigenfrequency f 0 = ω 0 /2π.For the viscous damping regime the model of Hosaka et al. is used [19].
with η being the dynamic viscosity of the gas.
To illustrate the importance of different contributions to the cantilever damping, we compile respective results in Table I.Using the known material properties of silicon cantilevers and typical dimensions, we calculate Q 0 according to a procedure described in detail in [11].These contributions are compared to the contribution Q air caused by viscous air damping at 10 3 mbar.
From this table, it can clearly be seen that the damping due to the ambient gas exceeds all other effects by orders of magnitude and, therefore, Q air is clearly the dominating contribution to the effective Q-factor.

II. EXPERIMENTAL
The oscillation behavior of cantilevers is investigated using a test setup that has been described in detail elsewhere [11].In brief, cantilevers are mounted with a clamp fixation so that they can easily be removed for further use.The test setup is housed in a compact vacuum chamber equipped with a turbomolecular pump and an ion getter pump.A combined pirani/cold-cathode vacuum gauge (PKR 251, Pfeiffer Vacuum, Asslar, Germany) allows us to measure the pressure from normal to UHV conditions.The pressure in the chamber can be controlled by backfilling with nitrogen using a metering valve.Cantilevers to be examined are mounted on a stage capable of holding up to 12 specimens.The cantilevers are aligned by milled recesses and fixed with copper-beryllium springs allowing for their reuse in a force microscope after their characterization.The cantilever mounting stage is equipped with a linear drive to select one of the twelve cantilevers for the measurement.Precise alignment in the optical path is facilitated by an X−Y positioning table.The base plate of the mounting stage is glued to a piezo ceramic plate with gold electrodes that is excited to vibration by applying an appropriate AC voltage.
To measure the Q-factor, we use a sine wave generator to excite the cantilever holder and sweep the frequency in a range centered on the resonance frequency f 0 of the cantilever.A Lock-In amplifier records the deflection signal as a function of the excitation frequency.The eigenfrequency f 0 and the Q-factor are obtained from the frequency spectrum by a least squares fit of Eq. ( 7) to the data.
where the cantilever is assumed to be a damped harmonic oscillator excited at the frequency f exc with the amplitude A exc [20].For this measurement it is important to adjust the sweep speed to the time constants involved in the measurement.For high-Q 75 kHz cantilevers, the amplitude time constant τ Q = Q/(πf 0 ) is typically τ Q = 0.7 s under UHV conditions defining the upper limit [21].The time constant of the Lock-In detector recording the cantilever oscillation signal is set to τ LI = 0.3 s while the sweep time is typically t sweep = 200 s.For a precision measurement of Q, the sweep range is chosen to be symmetric about the resonance frequency with start and stop set to frequencies where the oscillation amplitude is about 0.1 times the onresonance maximum.By this adjustment, any distortion of the recorded resonance curve and corruption of the determined Q-factor resulting from an inappropriate choice of the effective measurement time constant and the sweep speed can be excluded.

III. RESULTS AND CONCLUSIONS
In a set of experiments, we measure the Q-factor as a function of the ambient pressure.First, we compare results obtained under UHV conditions to those obtained at normal pressure.Increasing the pressure by 10 orders   FIG.2: Q-factor of a 75 kHz cantilever and a 300 kHz cantilever as a function of ambient pressure.Starting at UHV conditions, the vacuum chamber is back filled with nitrogen until normal pressure is reached.The solid lines show predictions for the Q-factor assuming 1/Qmount = 0.The dashed lines show predictions additionally assuming Qmount = 282,000 for the 75 kHz cantilever and Qmount = 316,000 for the 300 kHz cantilever.
of magnitude starting from UHV conditions results in a slight shift in the resonance frequency but a dramatic reduction in the Q-factor as demonstrated by the example resonance curves shown in Fig. 1.
This figure shows a narrow resonance curve for a 300 kHz cantilever recorded under UHV conditions and a broad resonance curve for the same cantilever recorded after backfilling the chamber to a pressure of 100 mbar.In Table II, the quality factors Q air eff of different cantilevers measured at normal pressure are compared to the corre-sponding values Q vac eff obtained in vacuum.A major conclusion to be drawn from this table is that there is no correlation of Q-factors measured in air to the respective UHV values.Even within a group of the same type of cantilevers, the order of increasing Q-factors in air does not correspond to the values in UHV what is most obvious for the 75 kHz cantilevers.Therefore, it is not possible to obtain any estimate for a Q-factor effective in UHV from a measurement performed in air.
Figure 2 shows the Q-factors of a 75 kHz and a 300 kHz cantilever as a function of the ambient pressure.
Up to a pressure of 10 −4 mbar, the measured Q-factors remain virtually unchanged compared to UHV conditions.In this region, no viscous damping occurs but collisions of the cantilever with the increasing number of residual gas molecules results in a slight reduction of the Q-factor.A dramatic reduction of Q is observed for pressures above 10 −2 mbar for the 75 kHz cantilever and above 10 −1 mbar for the 300 kHz cantilever.The measured data is compared to predictions derived from Eq. ( 2) without the influence of Q mount (solid line) using known dimensions (Table I) and material properties [22] of the cantilever without any fitting.Additional plot lines show predictions considering Q mount (dashed line) as derived by a procedure described in [11].These results confirm that for a measurement performed under UHV conditions Q mount is important for 75 kHz cantilevers but of minor relevance for 300 kHz cantilevers.When the pressure approaches normal conditions, Q eff is completely controlled by Q air .For both types of cantilevers, we find good agreement in the free molecular flow and viscous flow regimes, respectively.In the transition regime, however, neither model can well describe the pressure dependence of the Q-factor.
In summary, we measured the Q-factor of two NC-AFM cantilevers differing in geometry over a range of 11 decades of ambient pressure.For both types of cantilevers, the pressure dependence can be well described by theoretical models in the free molecular flow and viscous flow regimes.This insight is of practical use for measurements performed at elevated pressure as the methodology introduced here allows a prediction of the Q-factor enhancement when reducing the ambient pressure from normal pressure to a lower value.

TABLE I :
Contributions to the Q-factor of NC-AFM cantilevers calculated from the models described in the text for typical cantilever dimensions (l = 225 µm, w = 28 µm, t = 3 µm for 75 kHz cantilevers and l = 125 µm, w = 30 µm t = 3.6 µm for 300 kHz cantilevers).

TABLE II :
Q-factor Q vac eff and resonance frequency f0 of cantilevers measured in UHV compared to values measured in air.Q vac eff is determined at a pressure in the range from 10 −7 to 10 −8 mbar and Q air eff is measured under normal pressure conditions.