From Membranes to Beams

5.2 From Membranes to Beams

The confinement of acoustic modes inside a membrane was discussed in the previous chapter. For all previously presented results in this thesis, the lateral dimensions of the membranes were at least one order of magnitude larger than the membrane thickness, which allowed to ignore the influence of the in-plane extension of the membranes, thus treating this problem as purely one-dimensional.

From the experimental geometry, i.e. the geometry of the set-up, primarily acoustic modes are detected, which are traveling perpendicular to the membrane’s surface, or perpendicular to the lateral extension of the membrane, as illustrated in Figure 5.3 (left).

Given the assumption of an infinitely extended membrane, the properties of the acoustic modes depend (in first order) only on the membrane’s thickness, and the material param-eter of the membrane, see Equation 2.44. Therefore, the following question consequently arises:

How do the vibrational properties of the membrane change when the lateral dimensions of the membrane are reduced into the micro or nanometer range?

This question is illustrated on the left of Figure 5.2. The sketch shows a membrane (black lines) of thickness h, where the lateral extent is indicated for three different cases (A,B,C) by the red lines. For better understanding, a SEM image is shown above the sketch, where a series of membranes (or beams) can be seen. The red circle illustrates the size of the laser focal spot. Case A describes a membrane which is larger than the focal spot, a case which is discussed in the previous chapter. If the membrane’s width w is reduced to a value which is in the order of the size of the focal spot (B), effects of the confinement of the membrane or plate are expected to be detectable in the experiments.

When the width is comparable to the thickness of the structure, these structures are, with decreasing width, more generally referred to as plates, beams or bars. The third case (C) shows a bar where the width is of similar size as the height of the bar. The ultimate limit is shown in the right of Figure 5.2, where a thin beam of almost squared cross-section (w∼150 nm ,h∼110 nm) of 2.5 µm length is shown.

With respect to the measurement geometry and to the limitations from the set-up, e.g.

focal spot size and frequency resolution, the above mentioned complex question can be reduced to the twofold question: What happens to the vibrational properties, when the lateral extension of the membrane is reduced to

a) the order of the lateral extension of the laser’s focal spot, or b) the order of the thickness of the membrane?

In this thesis, it was only possible to touch on this topic due to its complexity. Therefore only a short overview over the first results is presented in the following. A similar ques-tion was discussed elsewhere for macroscopic beams [147–149]. A more recent theoretical discussion of the question also included the effect of localized edge modes and their cou-pling to propagating membrane/beam modes [150, 151]. The effect of such edge-modes

5 Experimental Results - Higher Order Confinement

Figure 5.2: Illustration of the concept of changing membrane width. Left: The sketch shows a membrane of thickness h and in blue the intensity distribution of the laser’s focal spot.

Three cases of membrane width w are illustrated by red lines. Case A is the case in which the membrane is wider than the focal spot, as seen for example in the previous chapter, where the system could be treated as a one-dimensional confined system. Case B and C illustrate the effect of the lateral confinement, i.e. a narrow membrane/bar or a wide beam (B) or a thin beam (C). Here differences in the vibrational properties compared to case A are expected. All three cases are further shown in the SEM picture above, which shows a series of bars and beams cut from a free-standing membrane. Right: SEM image of a beam structure of 2.5 µm long beam of∼150 nm width, illustrating the limit of case C. The red circle indicates the extension of the∼2µm diameter of the laser focal spot.

is also of concern when the width of the plate is smaller than the diameter of the focal spot, as for example in case B. Here, the beam is placed at the slope of the intensity distribution, small changes in the width of the beam can lead to large changes in the reflected intensity (red arrows). This results in a large contribution of such an edge-mode in the obtained experimental spectra, especially when compared with a thickness oscillation alone.

Mode Conversion

The first and primary question here is: Can membrane modes be excited and/or detected with this setup when the ratio of membrane thickness h to width w approaches 1, or more general _w^h ≤ 1, as illustrated in Figure 5.2. In the case of a beam or membrane with a metal transducer, a plane wave is excited in the top layer, as discussed in the previous chapter. This plane wave then travels into the underlying second layer of the membrane/beam. Therefore the general assumption should be that such a fundamental breathing mode can be excited in both structures. This assumption is illustrated in Figure 5.3 for a membrane (left) and for a plate or beam (right). Apart from the excitation mechanisms, the vibrational properties are mainly influenced by the geometry of the structure, i.e. if the sides can move in-plane of the membrane’s lateral extend.

In Figure 5.3 the effect of a free or fixed side of a beam on the propagation of a plane wave is illustrated. In the case of fixed sides, the wave travels through the membrane and is reflected at the bottom. As the sides are fixed in the in-plane direction, only 122

5.2 From Membranes to Beams

fixed sides plane wave

free sides plane wave

Figure 5.3: Illustration of the effect of fixed and free sides of a membrane or beam. A plane wave is excited in the top layer of the membrane/beam and propagates into the bottom layer.

Left: Here the case of an infinitely extended membrane is illustrated. For an excited plane wave, at a certain place of the membrane, the “sides” can be considered to be fixed, as the infinite extent of the membrane prohibits free movement in the direction of the membranes extend. Therefore this is a purely one-dimensional problem, and the wave travels perpendicular to the membrane’s lateral extension, and is reflected at the bottom. Right: If the plane wave is excited in a beam, or a membrane where the lateral extensions are in the order of the membrane’s thickness, the sides are free, i.e. the side of the beam can extent perpendicular to the traveling direction of the plane wave. This allows for mode conversion at every point of the side of the beam the wave is passing by. The result are reflected waves travelling into the beam. The resulting superposition of these waves is more complex than in the membrane case, and is not analytically solvable. A set of four fundamental types of resulting modes is shown in Figure 5.4.

movement along the propagating direction is allowed. In the case of free sides, such as seen in a bar or beam, the side wall can extend outwards, perpendicular to the propagation direction. This allows for mode conversion at this boundary, and waves start to travel into the beam, leading to a more complex mode spectrum. The result of this mode conversion is illustrated in Figure 5.4, where four fundamental types of modes of a beam are illustrated (cross-section), ignoring the extension of the beam along the z-direction (the beam’s long axis). The longitudinal mode is the analogue of the membranes fundamental breathing/dilatational mode, and consequently turns into this mode in the limit of an infinitely extended membrane. If one considers as well the extension of the beam in the z-direction, i.e. the full three-dimensional problem, it is useful to use finite element methods to search for the fundamental beam modes.

bending torsional

longitudinal

y x

Figure 5.4: Four fundamental types of beam or plate modes. Shown is the cross section of a beam (x,y-direction). The beam is assumed to be infinitely long in the z-direction. The longitudinal mode is the analogue to the membrane’s fundamental breathing mode. Adopted

5 Experimental Results - Higher Order Confinement

Experimental Indications of Mode Conversion

The experimental data shown in Figure 5.5 illustrates the effect of the above discussed influence of the confinement. Compared are two numerical fast Fourier transform (FFT) of measurements from a 57 nm thick silicon nitride membrane with a 20 nm thick gold transducer, and a 800 nm wide and ∼2.5 µm long beam, made from the same material system. The FFT shows the membrane’s fundamental breathing mode (∼28 GHz) and the first harmonic thereof. For clarity, the graph scales only to 80 GHz, as in the fre-quency range above 80 GHz only membrane modes (∼90 GHz, ∼121 GHz) are present.

Deviations from the purely linear dependance can be explained by a first order pertur-bation theory, as described in Reference [46]. The membrane modes cannot be seen in the FFT of the beam. Instead, a series of low frequency modes below 20 GHz and a single mode at 45 GHz are present. The type of these beam modes are discussed in the following sections. This result indicates that the lateral confinement has significant in-fluence on the fundamental dilatational (membrane) mode already at a width to height ratio of 10 : 1.

A more detailed analysis of this transition from membrane to beam modes is not shown, as this is beyond the scope of this thesis. To note is that this topic very complex, and the investigations to fully understand and model the vibrational properties of this transition depend on a variety of parameters, which need to be carefully controlled. For example it is not possible to ignore the beam length when discussing the width dependance of these beam modes. This will be shown in the next section.

0 20 40 60 80

Figure 5.5: Comparison of the experimental results of a membrane and a wide beam, made from silicon nitride with a top layer of ∼20 nm gold. Shown on the left are the numerical Fourier Transform of measurements from a membrane, with lateral extent larger than the focal spot, and of a wide beam with a length larger than the focal spot and a width of 800 nm. The geometry is illustrated at the right. The FFT shows the membrane’s fundamental mode and the first harmonic thereof (blue). The beam does not show the membrane’s fundamental mode, but a series of low frequency modes below 20 GHz (red). This indicates that at a ratio of ∼10 : 1 (width to height) the confinement shows significant influence on the vibrational spectrum.

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