Theory and application of a novel co-resonant cantilever sensor

2.1 System modelling

As mentioned above, a cantilever can be described by a harmonic oscillator model with different numerical values for each of its eigenfrequencies [8]. This model can also be applied to the co-resonantly coupled system, resulting in a coupled harmonic oscillator model as depicted in figure 1a. Furthermore, the external interaction can be included as an additional spring k3 attached to the nanocantilever. k3 represents an external force gradient and the derivation of this coherence can be found elsewhere, for example in [15].

In order to evaluate the sensor’s behaviour, the amplitude response curves, especially of the microcantilever, are of interest because this is the quantity which is evaluated experimentally. These can be derived by solving the system of differential equations which describes the coupled harmonic oscillator model. These evaluations are very complex and it is therefore favorable to employ electro-mechanical analogies in order to use the capabilities of circuit analysis tools. The details of the circuit modelling are discussed in [16]. Here it suffices to state that the analogies force F≡ current I and velocity v≡ voltage U are used which preserve the structure of the mechanical model [17], [18], [19]. The resulting circuit model depicted in figure 1b is used for all subsequent simulations.

Figure 1

(a) Mechanical model of a coupled harmonic oscillator driven by a periodic force applied at the microcantilever and (b) corresponding electric circuit model. d1,2, k1,2 and m1,2 are the beams’ damping constant, spring constant and effective mass, respectively. The external interaction is modelled by an additional spring k3.

In the following discussion of the effects of the co-resonant coupling, the exemplary values listed in table 1 will be used for micro- and nanocantilever. They are comparable to values observed for actual sensors. Furthermore, we will only discuss the amplitude response curves for the microcantilever as these are the ones which are evaluated experimentally. However, please note that the same simulations and considerations are possible for the nanocantilever and can be found in [16].

In the following, it will be necessary to distinguish between the properties of the individual subsystems and the co-resonantly coupled system. Therefore, the former will be noted by the indexes 1,2 and the properties of the coupled system by indexes a,b.

Figure 2

Amplitude response curve of the microcantilever calculated for the values from table 1 with nanocantilever eigenfrequency of (a) 200 kHz and (b) 102 kHz (2 % frequency deviation).

2.2 Effect of frequency matching

To study the effect of frequency matching, the model from figure 1 is considered without the additional spring k3. In the first case, we assume an unmatched frequency state, where the eigenfrequency of the microcantilever is f1 = 100 kHz and that of the nanocantilever f2 = 200 kHz. The corresponding amplitude response curve for the microcantilever is depicted in figure 2a and numerical values are listed in table 2. It exhibits one clear resonance peak very close to the eigenfrequency of the microcantilever. However, since it is a coupled system, a second resonance peak exists very close to the eigenfrequency of the nanocantilever. The amplitude of the second peak is too small to be visible in the microcantilever’s amplitude response curve due to the vastly different parameters of the subsystems.

This changes significantly in case of the eigenfrequency matching, where two clear resonance peaks are evident in the microcantilever’s amplitude response curve (figure 2b). Furthermore, the resonance frequencies of the coupled system are clearly distinct from the individual beams’ eigenfrequencies, indicating that they belong to the coupled system as a whole.

2.3 Effect of an interaction

Adding an interaction to the system, i. e. considering the complete model from figure 1, allows to study the influence of the co-resonance on the sensitivity of the system. The sensitivity S in this case can be defined as the frequency shift of the resonance peak Δf induced by an external force gradient Δk=k3, hence S=Δf/k3, and is dependent on the spring constant according to equation (2).

As an example, an interaction of k3=10−5 N/m was used in a simulation which is two orders of magnitude smaller than the smallest spring constant of the system. Figure 3 depicts the corresponding amplitude response curves for the microcantilever at 2 % frequency deviation between micro- and nanocantilever. It can clearly be seen that the frequency shift induced by k3 is different for the two resonance peaks: the left-hand side peak with the higher amplitude shows a smaller frequency shift of Δfa = 144 Hz compared to the peak with the smaller amplitude on the right Δfb = 396 Hz.

Figure 3

Amplitude response curve for the microcantilever in case for 2 % frequency deviation between the eigenfrequencies of micro- and nanocantilever with and without an interaction k3. The frequency shift induced by the interaction is indicated for both resonance peaks.

To put that in perspective, the resulting frequency shifts are compared to those of the individual subsystems. The individual microcantilever has a frequency shift of Δf1 = 0.5 Hz, while the individual nanocantilever shows a frequency shift of Δf2 = 509 Hz. These values indicate that the coupled system’s relevant sensor properties lie in between those of the individual beams. Furthermore, the resonance peak with the smaller amplitude is stronger influenced by the nanocantilever’s properties, i. e. it exhibits a bigger frequency shift, while the peak with the larger amplitude is more dominated by the microcantilever’s properties.

These considerations lead to effective parameters (spring constant, quality factor) in describing the behaviour of the two resonance peaks of the co-resonantly coupled system. Numerical values for the spring constants are ka = 0.0043 N/m and kb = 0.0013 N/m for the left and right peak, respectively. A comparison to k1 = 1 N/m for the microcantilever and k2 = 0.001 N/m for the nanocantilever demonstrates the effect of the co-resonant coupling on the spring constant and ultimately the sensitivity of the coupled system. It shows that a large fraction of the nanocantilever’s low stiffness becomes accessible within the coupled state. Further analysis of these effective parameters, e. g. their dependence on the degree of frequency matching and the individual beams’ parameters, is the subject of an ongoing study.

So far, we can conclude that the co-resonant coupling of a micro- and a nanocantilever gives access to a large portion of the nanocantilever’s high sensitivity while at the same time preserving the ease of oscillation detection. Furthermore, the co-resonance leads to the necessity of effective parameters for describing the two resonance peaks of the coupled system.

3.1 Sensor geometry and fabrication

The co-resonant sensor geometry employed for cantilever magnetometry as well as a sketch of the resulting measurement setup are shown in figure 4a and d. A carbon nanotube (CNT) was used as nanocantilever due to its low spring constant and very good mechanical stability. The CNTs were grown by aerosol-assisted chemical vapor deposition [22], separated by ultrasonication and dispersed onto a TEM-grid. An individual CNT was transferred to the free end of a commercially available tipless silicon microcantilever (Nanosensors TL-CONT, NanoWorld AG) by micromanipulation and attached by electron beam-assisted material deposition (amorphous carbon), to achieve a mechanical coupling in series.

The eigenfrequencies of both beams were determined with a custom-made vibration stage inside a scanning electron microscope (SEM) by sweeping the excitation frequency and searching for the maximum oscillation amplitude (example in figure 4b). The microcantilever’s eigenfrequency was measured before the attachment of the carbon nanotube and was assumed to only change minimally due to the coupling with the very light CNT. The nanocantilever’s individual eigenfrequency was determined by ‘de-coupling’ the system, i. e. by holding the microcantilever in place with the micromanipulator and therefore only driving the nanocantilever oscillation. Matching of the eigenfrequencies was achieved by electron beam-assisted material deposition at the free end of the nanocantilever. The nanocantilever’s oscillation was observed in the scanning electron microscope to ensure closely matched frequencies.

The sample was another carbon nanotube filled with few individual Co₂FeGa Heusler nanoparticles, which was placed at the free end of the nanocantilever by micromanipulation (see figure 4c). Heusler compounds are a class of intermetallic materials which feature very favorable transport and magnetic properties, making them candidates for prospective applications in spintronics, energy technology and thermoelectrics [23]. Detailed information about the fabrication process and the sample can be found in [24], [25], [26].

3.2 Results

Detailed results have been published in [24] and therefore only the key points will be summarized here.

All experiments were conducted under high vacuum and at room temperature with standard magnetic force microscope equipment which allowed for sensor excitation with a piezo-actor and oscillation detection via laser-deflection. The external magnetic field was applied through an electromagnet with variable field strength depending on the electric current. The frequency shift for both resonance peaks of the coupled system was tracked with a phase-locked loop.

The frequency shift in dependence of the external magnetic field shows a hysteretic behaviour which can be expected for a ferromagnetic sample. However, when tracking the frequency shift of the smaller resonance peak (right-hand side peak fb in figure 4d), distinct jumps in the frequency shift signal were observed. It is known from other magnetometry experiments that such jumps indicate the magnetization reversal of a magnetic sample [21]. In the aforementioned experiments we observed three clear jumps which we associate with the magnetization reversal of the biggest Heusler nanoparticles of our sample. It is unlikely that these jumps originated from thermal fluctuations or noise as they occurred reproducibly when the experiment was repeated at different times and on several days. Furthermore, the derived magnetic switching fields agree well with values reported in another study for ensembles of Heusler-filled carbon nanotubes [25], [26].

Besides the magnetic switching fields, an average magnetic moment, anisotropy field and particle aspect ratio (based on the demagnetization factors) could be derived by fitting the measured frequency shift data. A very good agreement is found between the values for both resonance peaks of the coupled system as well as with few other published experiments.

Consequently, these experiments indicate the immense potential of the co-resonantly coupled approach in cantilever magnetometry. We were, for the first time, able to observe magnetization reversal of individual Heusler nanoparticles at room temperature and with standard laser-deflection detection. Furthermore, we demonstrated that both resonance peaks of the coupled system strongly benefit from the high sensitivity of the nanocantilever.

4.1 Experimental details

The sensor is fabricated by micromanipulation, focused ion beam and scanning electron beam techniques. First, the oscillation of the microcantilever is measured with a vibration stage to determine its eigenfrequencies and identify the position of the nodal point (which can also be calculated). There, a pillar is grown by electron beam-assisted deposition of amorphous carbon. Finally, a FeCNT is attached to the end of the spacer element by micromanipulation and electron beam-assisted material deposition, with both long axes aligned. The eigenfrequency of the nanocantilever is determined again on the above-mentioned vibration stage in the SEM by holding the microcantilever with the micromanipulator, thereby decoupling the system. Frequency matching is achieved by material deposition at the free end of the FeCNT as for the cantilever magnetometry sensors [28].

The FeCNT is filled with a single magnetic domain iron core [31] so that effectively only the magnetic pole at the FeCNT’s free end interacts with the sample’s magnetic stray field and therefore constitutes a very defined monopole-like magnetic interaction [28].

By the co-resonant coupling, a system with effective parameters is created like in the case of cantilever magnetometry. Hence, the effective spring constant of the coupled system is drastically reduced compared to the individual microcantilever, resulting in a strongly increased sensitivity as demonstrated by the experimental results for a Co/Pt multilayer sample [29].

For these measurements, the co-resonantly coupled sensor was first excited at the first flexural mode of the microcantilever, performing a measurement of the perpendicular magnetic stray field gradient. In a second step, the measurement was repeated for the in-plane direction by driving the system at the co-resonantly enhanced second flexural mode of the microcantilever. Please note that this is one of the two resonance frequencies of the coupled system which are close to the second eigenfrequency of the microcantilever.

Figure 6

Line section of lateral operation mode frequency shift data measured on a Pt/Co multilayer sample with a bidirectional magnetic force microscopy sensor equipped with an FeCNT as interaction tip. The calculated lateral data is based on data of the same position measured with the same sensor in perpendicular operation mode assuming a rigid, i. e. not co-resonantly enhanced, carbon nanotube. Additionally this calculated data was multiplied by a factor of 50 to match the magnitude of the measured lateral data, thereby demonstrating the co-resonant signal enhancement.

4.2 Results and discussion

The CoPt multilayer sample is a well-known system in terms of its magnetic domain structure and therefore it is possible to use the measured perpendicular, i. e. out-of-plane, stray field map to analytically calculate a theoretical in-plane stray field map with a Fourier transform-based approach. This calculation assumes a rigid carbon nanotube, i. e. no co-resonant signal enhancement. Details about these calculations can be found in [28], [32], [33].

The calculated image can be compared to the measured in-plane data obtained with the co-resonantly enhanced sensor. A line section for such a comparison is depicted in figure 6. Please note that a Fourier filter was applied to the calculated data for increased clarity as the perpendicular, i. e. out-of-plane, experimental data and, therefore, the derived in-plane data had a low signal-to-noise ratio due to experimental limitations. The in-plane experimental data was used as measured.

Both line sections, for calculation and measurement, agree very well in terms of the curve shape. However, the calculated data had to be multiplied by a factor of approximately 50 to reach the same magnitude as the measured data. This is a direct consequence of the co-resonant coupling and demonstrates its immense potential for signal enhancement in this technique. Further experiments indicated an even higher signal enhancement by three to four orders of magnitude.