Summary and Outlook

In this section, it was demonstrated that ultra-thin silicon membranes are interesting candidates to study lifetimes of high frequency acoustical phonons in acoustical cavities.

By using different thicknesses of thin membranes, it was possible to study the frequency dependence of the damping times of the first oder dilatational modes, without any influence from a substrate or potential acoustic transducers.

A comparison of the experimental data obtained from membranes with the data ob-tained from bulk samples, and a comparison with the corresponding theories for lifetime calculations showed, that the nanoscale nature of the membranes is of such great influ-ence, that there is neither agreement between the experimental results obtained from the membranes with the bulk theory, nor with the experimental data of Daly et al. [15], for membrane thicknesses below 200 nm. To overcome this discrepancy, a twofold approach for the lifetime calculations was presented. This approach introduced two models, which consider not only internal damping, but also the influence of phonon scattering at the membranes surfaces, including surface roughness.

4 Experimental Results - One Dimensional Confinement

Process Frequency Dependence Temperature Dependence

phonon-phonon scattering (intrinsic) ≤ω² Yes

interface-/surface roughness ω²−ω⁴ No

Table 4.2: Contributions to the damping of phonon lifetimes. Contrary to the surface rough-ness, the intrinsic damping is temperature dependent. This temperature dependence can be used to separate intrinsic from surface roughness mediated damping. Adapted from [69].

For frequencies in the range above 100 GHz, the investigated damping times were found to be dominated by external damping mechanism, namely interface or surface rough-ness. The boundary or surface roughness can be modeled by a wavelength dependent specularity parameter p. Assuming that the roughness is the dominant contribution to the damping, the boundary scattering model yields excellent agreement between the lifetimes predicted by the model and the experimental results. In the lower frequency range, below 100 GHz, this boundary scattering model overestimates the lifetimes, and the damping is assumed to be dominated by internal damping processes, which can be described by phonon-phonon interaction processes.

The combination of the two models yields a satisfactory agreement with the experi-mental data obtained from the membranes. Further enhancements to the model should concentrate on an improved modeling of the internal contributions, especially for the transition regime between bulk samples and membranes of thickness above 100 nm. Fi-nally, the transition between both regimes, i.e. dominating boundary or dominating intrinsic scattering, needs further evaluation, in order to better understand the influence of the confinement onto the lifetimes of the phonon modes.

Temperature depended measurements could shed further light onto the individual con-tributions, particularly in the transition regime. In Table 4.2, the temperature and frequency dependence of the two contributions, i.e. internal and surface roughness, to the damping are given. As the surface roughness scattering is assumed to be not temperature dependent, opposing to the intrinsic scattering, temperature dependent measurements could help to further understand the individual contributions.

Additionally, it should be possible to modify the surface roughness in a controlled way, see for example the work of Klingele [59]. In this thesis, two cases of rough and extremely smooth membrane surfaces were investigated, and a significant difference in the lifetime of the confined mode was found. This technique could be used to further improve and validate the surface roughness model, and might as well help to improve the modeling of the transition regime.

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4.3 GaAs Membranes

4.3 GaAs Membranes

Complementing the investigations of silicon membranes, GaAs membranes were fabri-cated as a second model system of single-layer and crystalline semiconductor membranes.

GaAs is especially interesting to study, because of its optical properties in the near in-frared region at around 800 nm. At room temperature the optical absorption length of light around 800 nm is a factor of 10 shorter in GaAs (direct band gap) compared to the absorption length in silicon (indirect band gap) [70]. The influence of this shorter absorption length on the excitation and detection of confined acoustic modes will be demonstrated in the first part of this section in comparison to the measurements taken at the silicon membranes.

Furthermore, the change in optical absorption of GaAs is strongly temperature depen-dent in the spectral region of 800 nm, as this is the region close to the band gap transition energy of GaAs. This dependance can be utilized to tune the absorption length in one material system. Here, and in the following, the transition at the Γ-point is referred to, when speaking of the band gap. This temperature dependence of the band gap energy Eg of GaAs is displayed in Figure 4.6 for the temperature range from 4 K to 300 K.

At room temperature, the band gap energy takes the value of 1.424 eV, equivalent to 870 nm, while at temperatures below 10 K the band gap reaches its maximum of 1.519 eV or 817 nm, neglecting excitonic effects.

0 50 100 150 200 250 300

(a) Temperature dependence of the band gap energyE_g of GaAs.

(b) Band diagram of GaAs at 300 K.

Taken from [71].

Figure 4.6:Temperature dependence of the band gap energy Egof GaAs. The energy equivalent of the central wavelength, to which the pump and probe laser can be tuned, is marked in the grey shaded area. This is the range in which one can tune the laser to be on- or off-resonance with the band gap energy.

Optical investigations with the ASOPS system in the temperature range of 4 K to 200 K are performed using an optical cryostat in combination with the microscope set-up. By cooling the membranes to temperatures below 170 K, the band gap energy of GaAs shifts to values above 1.47 eV. This is equivalent to the photon energy below 840 nm, which is then in the tuning range of the Ti:sapphire lasers. This combination of tuning range and

4 Experimental Results - One Dimensional Confinement

(a)Optical picture of the membrane including pump and probe spots.

membrane edge AlGaAs Buffer Layer

10 µm

(b) SEM picture taken with 1.40 k magnifica-tion, taken at the center of the membrane.

Figure 4.7: Images of the investigated GaAs membrane.

the excitation and detection of the confined acoustical modes inside such a membrane, as one can tune the wavelength with respect to the transition energy Eg.

The temperature dependent transition energies, as displayed in Figure 4.6, are calculated using the Varshni equation [48]:

E_g =E₀− αT²

T +β, (4.9)

where E_g denotes the band gap energy, E₀ the energy of the band gap at 0 K, and α,β are material specific constants.

In comparison to gallium arsenide, the band gap of silicon is around 1.1 eV at room temperature, far off the tuning range of the lasers. In terms of the optical absorption length of 800 nm light in the membrane, the absorption length in gallium arsenide is at room temperature around 830 nm [72], compared to the∼8 µm in silicon. This leads to a more asymmetric excitation profile compared to the silicon case, which can lead to an excitation of odd and even harmonics of the fundamental confined acoustical mode in gallium arsenide.

In the following, measurements taken at room temperature (RT) and at nominally 4 K (LT) are discussed. The GaAs membranes investigated at low temperatures were fabri-cated using the back-etching process described in Chapter 3.2.1. The RT measurements were performed at membranes using the focused ion beam approach. The sample dis-cussed in the LT case was fabricated in the context of the bachelor thesis of Scheel [73], and preliminary measurements are discussed there.

In Figure 4.7 an optical image³, taken with the CCD camera of the measurement set-up,

3This picture was digitally edited to remove dust spots originating from the optics of the camera.

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4.3 GaAs Membranes and a SEM picture of the center of the membrane are shown. For the optical image, the membrane was illuminated from the backside, showing the semi-transparent membrane in color and the (thick) opaque substrate in black. The SEM image gives an idea of the flatness of the membrane within the resolution of the SEM in this magnification (1.40 k).

Additional close-up SEM images (not shown) of the membranes showed no detectable surface roughness, i.e. residuals of the etching processes, within the ultimate resolution limit of the SEM. Surface roughness measurements using an atomic force microscope (AFM) could not be performed, as the deep etching through the substrate (>500µm) and the small width of the membrane did not allow to place an AFM cantilever on the membrane.