Higher Eigenmode Tricks in Multifrequency Atomic Force Microscopy

Tips & Tricks for Optimizing Sensitivity and Indentation Depth

  • Fig. 1: Teflon sample imaged in bimodal AFM using the 2nd ((a) and (b)) and 3rd ((c) and (d)) eigenmodes with varying amplitude.  A1 = 100 nm, setpoint 50%, f1 ≈ 48 kHz, f2 ≈ 286 kHz, f3 ≈ 783 kHz and k1 ≈  2.4 N/m. Fig. 1: Teflon sample imaged in bimodal AFM using the 2nd ((a) and (b)) and 3rd ((c) and (d)) eigenmodes with varying amplitude. A1 = 100 nm, setpoint 50%, f1 ≈ 48 kHz, f2 ≈ 286 kHz, f3 ≈ 783 kHz and k1 ≈ 2.4 N/m.
  • Fig. 1: Teflon sample imaged in bimodal AFM using the 2nd ((a) and (b)) and 3rd ((c) and (d)) eigenmodes with varying amplitude.  A1 = 100 nm, setpoint 50%, f1 ≈ 48 kHz, f2 ≈ 286 kHz, f3 ≈ 783 kHz and k1 ≈  2.4 N/m.
  • Fig. 2: Bimodal AFM measurement of thin Nafion film under varying A2 using the same cantilever as for figure 1.
  • Equation 1

We offer a simple explanation of the effect of the higher eigenmode’s free amplitude on probe sensitivity and indentation depth in bimodal atomic force microscopy (AFM). The explanation is based on a qualitative analysis of the cantilever’s equation of motion and offers a concrete basis for the optimization of the higher eigenmode contrast. The concepts discussed are also applicable to multifrequency AFM techniques involving more than two eigenmodes.

Many AFM methods have been developed since the first dynamic mode, known as amplitude modulation (AM-AFM [1]). Multifrequency AFM (MF-AFM [2]) is based on the excitation of the probe at multiple frequencies, either in a discrete fashion combining multiple sinusoidal excitations, generally corresponding to the active eigenmodes, or by constructing the drive signal as a continuous frequency band [2]. The original MF-AFM method, introduced by Garcia and coworkers [3], is based on the first approach and involves the first two cantilever eigenmodes. Specifically, the fundamental mode is operated using AM-AFM [1], while the second eigenmode is driven in open loop (i.e., with constant excitation amplitude and frequency). The AM-AFM scheme on the fundamental mode provides the topography, while the phase and amplitude of the higher eigenmode offer additional contrast that can be related to the surface material properties [2,3]. The two eigenmodes are generally only weakly coupled, so their operation can often be optimized nearly independently. Despite the extensive literature on MF-AFM, fine-tuning of the higher eigenmode parameters is not always straightforward. This article offers simple tools for this purpose. We assume the user is experienced with the AM-AFM method, so the operation of the first eigenmode is not discussed. MF-AFM methods using more than two eigenmodes [4,5] and combinations of AM-AFM with frequency modulation techniques [6] are not discussed either, but the general concepts are still applicable.

Eigenmode Sensitivity

We explore the sensitivity of an oscillating eigenmode by examining its equation of motion, loosely defining sensitivity as the ability of the oscillation to be perturbed by “small” changes in the tip-sample forces, which are in turn governed by the material properties.

When the oscillation is perturbed, the values of the contrast signals (e.g., phase and amplitude) change [1].
Equation (1) is a modified version of the equation of motion of a damped harmonic oscillator [1], where distances have been normalized by the free oscillation amplitude, time has been normalized by the inverse of the driving angular frequency ω and we have also used the relationship between the free oscillation amplitude Ao, the quality factor Q, the eigenmode force constant k and the driving force amplitude (see equation 1 [4,7]).

Equation 1

z is the normalized tip position, t the normalized time, and Fts the tip-sample force, which depends on the normalized tip-sample distance zts.

The importance of each term in equation (1) is determined by its order of magnitude. We focus on the last term of the right hand side, which contains the tip-sample force Fts. We see that for constant force Fts, the magnitude of this term (and hence its ability to perturb the eigenmode oscillation) increases if the product kAo is decreased, since kAo is in the denominator, and vice-versa. During an experiment the user has control of both quantities: k can be changed by selecting a different eigenmode (higher modes have increasingly higher k [2]), while Ao can be adjusted by changing the drive amplitude. If the user seeks greater sensitivity, then the product kAo should be decreased, and vice-versa. Note also that when the cantilever is made more sensitive, and thus more likely to be perturbed by external forces, it is less capable of indenting the sample deeply. This is because greater indentation leads to larger tip-sample forces and thus greater perturbation of the oscillation. Therefore, along with increased sensitivity, indentation depth is also reduced. Conversely, when the product kAo is large, the cantilever is less sensitive and more difficult to perturb, and thus penetrates deeper into the sample. In attempting to analyze results corresponding to deeper indentation alongside with results corresponding to shallower indentation, one must realize that a direct comparison between the two cases is not always possible, since deeper subsurface regions of the sample may exhibit different material properties and features that can influence the overall contrast. Thus, changes in the contrast as a result of adjustments of the product kAo may be partly due to subsurface features that become visible or invisible, when indentation depth is increased or decreased, respectively, in addition to changes in probe sensitivity.

Experimental Example

Figure 1 (Teflon tape sample) illustrates the effect of changing the amplitude of the 2nd ((a) and (b)) and 3rd ((c) and (d)) eigenmodes, which in turn changes the eigenmode’s kAo product. In each case, as Ao is increased, the range of the phase contrast becomes smaller and moves further away from 90º, which corresponds to the highest sensitivity. One can also note that the phase ranges for the 2nd eigenmode are closer to 90º (more sensitive) than the ranges for the 3rd eigenmode (less sensitive), which is as expected, since the latter has a larger force constant, and hence a larger product kAo for a given value of Ao. Figure 2 (edge of a spin coated Nafion film on a silicon oxide substrate) illustrates the increase in indentation depth as the free amplitude is increased. Greater indentation leads to a smaller measured value of the film height, since the polymer film is compressible but the substrate is highly incompressible.


We have provided qualitative insight into one of the key factors governing the sensitivity of the cantilever eigenmodes in Multifrequency AFM, namely the free oscillation amplitude, which is useful for controlling indentation depth and for optimizing the material property contrast range during imaging.


[1] Ricardo Garcia, Ruben Perez: Dynamic atomic force microscopy methods, Surface Science Report 47: 197–301 (2002) doi: 10.1016/S0167-5729(02)00077-8

[2] Ricardo Garcia, Elena T. Herruzo: The emergence of multifrequency force microscopy, Nature Nanotechnology, 7: 217-226 (2012) doi: 10.1038/nnano.2012.38

[3] Tomás R. Rodrı́guez, Ricardo Garcia: Compositional mapping of surfaces in atomic force microscopy by excitation of the second normal mode of the microcantilever, Applied Physics Letters, 84: 449-451 (2004) doi: 10.1063/1.1642273

[4] Daniel Ebeling, Babak Eslami, Santiago De Jesus Solares: Visualizing the subsurface of soft matter: simultaneous topographical imaging, depth modulation, and compositional mapping with triple frequency atomic force microscopy, ACS Nano, 7: 10387–10396 (2013) doi: 10.1021/nn404845q

[5] Santiago D. Solares, Sangmin An, Christian J. Long: Multi-frequency tapping-mode atomic force microscopy beyond three eigenmodes in ambient air, Beilstein Journal of Nanotechnology, 5: 1637-1648 (2014) doi: 10.3762/bjnano.5.175

[6] Gaurav Chawla, Santiago D. Solares: Mapping of conservative and dissipative interactions in bimodal atomic force microscopy using open-loop and phase-locked-loop control of the higher eigenmode, Applied Physics Letters, 99: 074103 (2011) doi: 10.1063/1.3626847

[7] Babak Eslami, Daniel Ebeling, Santiago D. Solares: Trade-offs in sensitivity and sampling depth in bimodal atomic force microscopy and comparison to the trimodal case, Beilstein Journal of Nanotechnology, 5: 1144–1151 (2014) doi: 10.3762/bjnano.5.125

Miead Nikfarjam1, Enrique A. López-Guerra2, Santiago D. Solares2, Babak Eslami1 

1 University of Maryland, Department of Mechanical Engineering,, USA
2 The George Washington University, Department of Mechanical and Aerospace Engineering, Washington, USA, www.solaresspmlab.com

Dr. Babak Eslami

University of Maryland
Department of Mechanical Engineering
College Park, MD, United States

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