Supplementary to:
Periodicity in bimodal atomic force microscopy
Chia-Yun Lai1,Ψ , Victor Barcons2, Ψ , Sergio Santos1,*, Matteo Chiesa1
1
Laboratory for Energy and NanoScience (LENS), Institute Center for Future Energy (iFES),
Masdar Institute of Science and Technology, Abu Dhabi, UAE
2
Departament de Disseny i Programació de Sistemes Electrònics, UPC - Universitat
Politècnica de Catalunya, Av. Bases, 61, 08242 Manresa (Barcelona), Spain
We note that the expression in (9) here reproduced as (S1) can be computed in terms of
experimental observables and is equivalent to others proposed in the literature1, 2.
 ETn Q( m ) 1

 k ( m ) An2 

2
n  tan 1  2n

VnQ ( m )




(S1)
Nevertheless an important remark is made here in terms of anharmonicity/harmonicity and
fundamental period T. Namely, the expressions for V(m)n (6) and ET(m)n (8) are to be
computed over a fundamental period T as given by (2). Furthermore, the amplitudes An and
phases ϕn are to be averaged over the fundamental period T. The validity of these statements
was verified via numerical integration of (3) and resulted in agreement between numertical
integration and expressions to the significant digits shown in Table I.
1
Fig. 2 in the main text is here expanded to a higher value of r, i.e. r=6.33 (Figs. 1Sa and 1Sb)
for which T≈1,400 μs.
Details on Table I
It is particularly interesting to note the variations in the sign of the virial Vq in (6) with
variations in T as parameterized by r. That is, whether ϕq lies above or below 90 degrees
depends on Vq, thus the choice of r affects where ϕq lies. Furthermore, the sign of the energy
transfer from and to other frequencies from the monitored frequency q, i.e. ETq, also depends
on r. All values in Table I, except the phase shift ϕq which is given in degrees, are given in
eV. Finally note that Edis=0.00 throughout since the interaction is conservative.
We note that when the fundamental theory of multifrequency AFM was developed1, peaks at
lower frequencies than fD1 were already observed but were assumed to be too small to cause
significant perturbation.
2
FIG. S1 Examples of the frequency spectrum (amplitude/frequency) for an interacting tip
according to cantilever anharmonicity/harmonicity as parameterized by r=q/p. The panels on
the right correspond to zooms of the panels on the left.
At this point we note that the frequency spectrum plots in Fig. 2 and S1 have been obtained
by employing the standard FFT algorithm with sufficient resolution to detect the fundamental
frequency and its higher harmonics. The values in Table I were computed from the numerical
values obtained directly from the FFT. Nevertheless, in standard AFM equipment lock-inamplifiers are typically employed. Standard expressions in AFM are commonly derived by
manipulating the equation of motion (3) and integrating over a period T(1) given by
3
T(1) 
2
(1)
(S2)
Figure S2
For completeness, we also show in Fig. S2 the FFTs obtained for the free cantilever. The data
corresponds to the waveforms in Fig. 3 and the corresponding FFTs in Fig. 4. The results
show that for the free cantilever, that is, when the cantilever is not interacting with the nonlinear tip-sample force Fts, the only frequencies present in the spectrum are those that are
externally excited, i.e. fD1 and fD2, or in terms of the normalized values 1 and r. The
frequencies have been normalized as before with fD1. Note however that some frequencies,
other than those near 1 and r, might be present even for the free cantilever (red circles in Fig.
S2). These are a consequence of the optical and electronic system of the instrument and do
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not have physical relevance. These were present throughout in the experiments both when the
cantilever was free and when it was engaged.
FIG S2. Experimental FFTs obtained for a free cantilever for a) r=5.9, b) r=6 and c) r=6.2.
The data corresponds to the waveforms in Fig. 3 for the free cantilever.
Finally, more information on the phenomena reported here can be found at arXiv.org
http://arxiv.org/abs/1407.3165.
1.
2.
J. R. Lozano and R. Garcia, Physical Review Letters 100, 076102-076105 (2008).
S. D. Solares, Beilstein Journal of Nanotechnology, 298–307 (2014).
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