5 Experimental Results - Higher Order Confinement
5.3 Nanomechanical Beam Structures
5.3.1 Length, Width and Thickness Dependence of Beam Modes
This section shows preliminary results of two examples of how pump-probe spectroscopy can be used in the investigation of the vibrational properties of free-standing nanome-chanical resonators, and how it is possible to identify individual beam modes in the GHz-frequency range. By using the thickness and length dependance of the frequency shift of the individual modes, it is possible to identify the nature of the detected modes when comparing them to finite element simulations. The results presented in this section are discussed in more detail in the master’s thesis of Moritz Merklein [161].
First, the thickness and length dependance of the lowest detected mode is used to il-lustrate that this mode can be identified as a higher order of a string-like resonance.
Following this argumentation, the length and width dependance of the three lowest (in frequency) detected modes is discussed. Here it will be shown that it is possible to identify and simulate the shape of the modes, showing the different origins within the beam structure.
The beams which are discussed in the following are all fabricated from stoichiometric silicon nitride of either 57 nm, 107 nm or 157 nm thickness, all with a top layer of 20 nm gold. Two widths (450 nm, 400 nm) are discussed in order to illustrate the width depen-dance of the vibrational spectra of these beams. In addition to the width variation, also the lengths of the beams are changed, from 1-5 µm, here a small shift in mode frequency is also apparent.
Figure 5.6 shows the numerical FFTs of 400 nm wide beams of varying length and thick-ness. The complete spectra, Figure 5.6a, is taken from a 107 nm thick beam, showing clearly distinguishable peaks at 2 GHz, 6 GHz and 13 GHz, as well as an additional peak at 45 GHz. The 45 GHz correspond to the vibration of the gold film, which are discussed in detail in Section 5.3.4.1 on Page 136. The FFT of beams with 57 nm and 157 nm thickness show the same pattern, with slightly shifted frequencies, but are omitted here for clarity. The data indicates that the peak position changes slightly with beam length.
In order to illustrate this behavior a normalized zoom into the frequency range below 6 GHz is shown in Figure 5.6b. This graph now shows the first mode obtained from the measurements of the 57 nm (broken line) and 157 nm (solid line) thick beams, where the 107 nm data was omitted for clarity. The curves are normalized in order to illustrate the behavior more clearly. For both set of curves the peak position shifts with increasing beam length, and the overall frequency increases with increasing thickness. The data
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Figure 5.6: Numerical Fourier Transform obtained from measurements of 400 nm wide silicon nitride beams with 20 nm gold layer on top and different lengths. (a) Complete spectrum with clearly distinguishable peaks at 2 GHz, 6 GHz, 13 GHz and 45 GHz. The nature of these modes is explained in the text and in Figure 5.7. (b) Length dependance of the normalized lowest frequency mode obtained from beams of 57 nm and 157 nm thickness. For both set of curves the peak position shifts with increasing beam length, and the overall frequency increases with increasing thickness.
for the frequency shift, especially for the ∼2 GHz values, are shown for qualitative rea-soning only, as the shift is smaller than the measurement uncertainties of 0.8 GHz from the width of the Fourier window.
The dependance of the frequency shift to the nitride thickness and to the beam length indicate that the shape of this mode is a vibration of the whole beam, with a modulation similar to a vibrating string mode. These assumptions find even more support when considering the mode shape, as it is shown in Figure 5.7, left.
Through comparison with finite element simulations it is possible to identify the nature of the three modes: The simulated mode shapes are illustrated in Figure 5.7, right. The color code shows the estimated displacement amplitude, where blue corresponds to zero amplitude and red to a large displacement amplitude. The mode at 2 GHz resembles a “classical” beam mode, or in other words a higher harmonic (n=5) of a string mode, when compared to the Euler-Bernoulli beam theory. The modes with higher frequencies are primarily located in the upper part of the beam, i.e. the gold film. The 6 GHz mode shows a modulation located mainly at the edges of the beam, while the 13 GHz mode shows the highest modulation at the center of the beams width. Both modes show a high frequency modulation along the beams long axis, which illustrates the length dependance of these modes.
Remarkably, the observed length dependance here hints towards a reversed behavior, i.e. the mode frequencies tend to increase with increasing beam length. For the 2 GHz mode, this behavior is well below the measurement uncertainty (±0.2 GHz), and even
5 Experimental Results - Higher Order Confinement
for the higher frequencies only a small change of ∼0.8 GHz is observed. The physical reason behind this behavior could not be found, although very good agreement with the modeling was achieved. Additionally a correlation between the beams width and the frequency can be seen: wider beams (450 nm) tend to show a slightly lower frequency than narrow beams (400 nm), see Figure 5.7 as well. This phenomenon agrees with the classical Euler-Bernoulli theory that shorter length lead to higher stiffness and thus to higher frequencies.
Interesting to note is that the mode frequencies shift only by 0.2 GHz with respect to length changes of several µm, yet width changes of 50 nm lead to a change in frequency of 0.5 GHz. The small length dependance is probably due to the fact that the beam length is larger than the focal spot size, i.e. the beam can be assumed to approximate an infinitely extended beam, similar to the membranes.
Limitations of the Presented Results
To note is that the here presented mode shapes are preliminary results and need further confirmation, as the complete interaction between the suggested modes and the detection is not completely integrated in the simulations. The overall strain distribution in the beam plays an additional important role in the detection process. This will be discussed in the following section in more detail.
A second other limitation comes from the frequency resolution of the experimental set-up of 800 MHz. The simulations provide a number of similarly shaped modes within a very small frequency range, the actual measurement might as well be a superposition of these modes. In order to follow the evolution of these modes, more measurements with beam variations in much smaller step sizes would be necessary to identify and estimate any trend. Further detailed simulations would be necessary to distinguish different contributions of multiple modes and to eliminate uncertainties due to mode coupling, crossing, anti-crossing or similar phenomena, see for example Reference [150].
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~12 GHz ~6 GHz ~2 GHz
0
+
displacement (arb. units) 1234502468101214
Fre que ncy (G Hz)
length (µm)
Experiment: 450nm 400nmSimulation: 450nm 400nm Figure5.7:Lengthdependanceofbeammodes:Simulationandexperiment:Left:Lengthdependanceofthethreelowestfrequencies oftwo450nmand400nmbeamsof107nmNitridethicknessand20nmgoldlayer.Thethreemodesshowaclearlengthdependance, wherethemodesofthe400nmbeamareslightlyhigherinfrequencycomparedtothe450nmbeam.Thesimulatedfrequenciesshow excellentagreementwiththeexperimentalresults.Right:FiniteElementSimulationofthethreesetsofmodesshownontheleft. Shownisthemodeshapeforthe3µmlongand450nmwidebeam,inordertoillustratethe(suggested)natureofthemode.Thecolor codeshowsthesurfacedisplacement,exaggeratedforbettervisibility.Herebluedenotesnodisplacementandredlargedisplacement. Theshapeofthemodesandthelimitationsofthispredictionarediscussedinthetext.
5 Experimental Results - Higher Order Confinement
Figure 5.8: Left: SEM pictures of a free-standing (top) and a soft-landed beam (bottom).
The lateral dimensions of the structures investigated are given in the top picture. The beams themselves consist of a layer of 107 nm Si3N4 and a top layer of 20 nm gold, at a width of 177 nm. The bottom picture is a close-up to illustrate the soft-landed character of the beam.
Right: Schematics of the soft-landing process. Here, the boundary conditions of the vibrations of the beams are indicated (a) for the free-standing case (BC1) and (c) for the soft-landed case (BC1 and BC2).