Simulation of the Mode Spectrum

The aim of this project was the tailoring of the vibrational mode spectra of such beams, and the (selective) modification and suppression of certain modes. The modification employed in this thesis was the release of the beam structure from the substrate, and the subsequent removal of the pedestals of the beam structure, as described in Section 3.2.4 and illustrated in Figure 5.8, right. By soft-landing the beam directly onto the substrate, a second boundary condition (BC2) – a fixed bottom – was applied to the beam. This added boundary condition strongly affects the vibrational mode spectra. The effect of this modulation was simulated using the finite element simulation software Comsol¹. In Figure 5.9, a dispersion relation like picture for the first five vibrational modes,

1As the structures investigated in this section are not analytically solvable due to their complex-ity, Comsol, a finite element simulation software, was used. The simulations were performed by E. Barrettto of the group of Elke Scheer, University of Konstanz. Figures 5.9, 5.10, 5.11 and 5.15 were provided by E. Barretto.

5 Experimental Results - Higher Order Confinement

Figure 5.10: Shapes of the modes given in Figure 5.9 for the free-standing (blue) and soft-landed (red) beams. The figure shows the displacement along the beam, where the direction of the displacement and the relative magnitude are indicated using arrows, and a cross-section view at the center of the beam (Color code: blue – small displacement, red – large displacement).

Simulated withCOMSOL.

simulated for free- and soft-landed beams, is shown². This dispersion relation was con-structed from calculations of the distinct eigenmodes of the beam with their respective higher harmonics, and is strictly correct only for the discrete values given at the y-axis.

The connecting lines can be seen as “an guide to the eye” only. The calculations were performed by running an eigenfrequency analysis over the considered geometry with the respective boundary conditions. The resulting frequencies were then assigned to the cor-responding mode and mode number. The modes of the free-standing beam are shown in blue, with frequencies of the fundamental (n=1) mode order ranging from a few hundred MHz (modes A, B, C) to several GHz (modes D and E). The modes of the soft-landed beam are given in red, starting at frequencies for the n = 1 order at 5 GHz (mode F) and higher (modes G to J).

The additional boundary condition shifts the complete mode spectrums lower bound, starting at values of some hundred MHz, towards starting values just above 5 GHz, completely suppressing the modes of the free-standing beam and allowing only for a new set of modes F to J. These modes do have a certain similarity to the modes of the free-standing beam, yet one has to keep in mind that there is no direct link between these modes, i.e. no transition from free to soft-landed modes can be assumed.

2Due to the complex structure of the beam and the nature of these simulations, the dispersion curves cannot be scaled to other geometries

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5.3 Nanomechanical Beam Structures

A B C

E D

(a) free-standing beam

F G H

J I

(b) soft-landed beam

Figure 5.11: Strain distribution of the modes of the free-standing and soft-landed beams as simulated by Comsol. The strain magnitude is given by the color code. Due to symmetry considerations only modes with a symmetric strain distribution along the beam contribute to the detected signal.(Color code: white = no strain, increasing color intensity = increasing strain)

5 Experimental Results - Higher Order Confinement

In Figure 5.10 the corresponding displacements of the modes from Figure 5.9 are dis-played for the mode of order n = 1. The arrows indicate the direction and relative magnitude of the displacement of the beam, also the corresponding cross section dis-placement at the center of the beam is given for each mode (color code: blue to red indicates increasing displacement).

Detection Process and Symmetry Considerations

The complete detection mechanism of the vibrational modes is not yet fully understood, as for this the light-matter interaction of the beam would have to be implemented into the Comsol model. Yet the dominant contribution to the detection is presumed to be the strain distribution along the beam structure, as it is shown in Figure 5.11. As the focal spot has the size of the beam, it will average over the total strain distribution.

Therefore only modes with a relatively uniform symmetric strain distribution along the beam (same sign) are expected to contribute significantly to the time dependent change in reflectivity. In the case of anti-symmetric strain distributions the contribution to the reflectivity change is expected to average out over the length of the beam. From the volumetric strain distribution, see Figure 5.11, and the symmetry of the modes, it follows that only the modes B and E are suitable to contribute to the signal of the free-standing beam, whereas for the soft-landed beam mainly the modes H and J are expected to contribute significantly to the signal. To calculate the detected strain distribution, the volumetric strain was integrated over the whole structures (top) surface and was then normalized by the maximum strain amplitude. The resulting strain is later compared to the experimental results.

5.3.4 Experimental Results: Free-Standing and Soft-Landed Beams

Figure 5.12 shows the time domain reflectivity changes ∆R/R obtained at the free-standing beam (blue curve) and at the soft-landed beam (red curve). The dynamical behavior at both beams follows the following description: At zero time delay between pump and probe a sharp and pronounced rise in ∆R/R is observed, followed by a superposition of an ∼43 GHz oscillation and several lower frequency oscillations. The sharp rise is due to the absorption of the pump pulse in the gold layer, which results in a rapid heating of the electron gas. The heated electron gas induces a strong change in the dielectric function and thus in∆R/R [38]. As the electron gas thermalizes with the lattice in the gold film, an impulsive thermal stress is generated therein [38]. This thermal stress acts as the source for the observed set of vibrational modes.

By removing the electronic background the vibrational contributions can be extracted.

The above described dynamics are in principle identical in both systems, except the amplitude and the life-time of the 43 GHz oscillation of the soft-landed beam cannot

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5.3 Nanomechanical Beam Structures

Δ R/R x 10

-10 -8 -6

-4

free standing

0 200 400 600 800 1000 1200

-2 -1 0

Δ R/R x 10

time (ps)