Central Theme of the Thesis
The central theme investigated in this theses is acoustic phonons in confined geometries at the nanoscale. In this context, confined geometries refers to the concept that the lateral extend of the investigated structures are, in one or more dimensions, of the same order of magnitude as the acoustic phonon propagation distance.
Using pump-probe spectroscopy, the influence of the geometry, i.e. the boundary con-ditions or the reduced dimensions, on the vibrational properties of the nanostructures was investigated. In addition to the geometry, the influence of the absorption length of the pump and probe light on the obtained phonon spectra was investigated.
The order of topics in this thesis follows an increase in complexity of the investigated sys-tems, in regards to excitation and detection processes, acoustic properties and material combinations, and boundary conditions.
Previous Research
Previous work in the field of phonon spectroscopy has mainly focused on single-layer and bulk systems, on superlattices and embedded cavities, or on specific aspects of phonon propagation, e.g. interface modification.
In the area of nanostructures, prior research either focussed on the MHz regime for beam structures, or on much smaller nanostructures for pump-probe measurements.
Beam structures were mostly investigated in the MHz regime, with excitation mainly by electrical means and only a few reports on optical excitation. Only very little work had been done in the transition regime, i.e. the GHz acoustic properties of beam structures.
For nanostructures, mostly single particles or core-shell particles had been investigated, with a focus on the intrinsic properties, or the interaction with the embedding matrix.
Outline of the Thesis
This thesis builds on and expands previous research by investigating additional material systems, by introducing higher complexities, i.e. two-layered membranes and additional boundary conditions, and by extending the research from membranes to nanostruc-tures.
The thesis can be divided into three main parts: Theoretical considerations and sample fabrication, experimental results of one-dimensionally confined systems, and experimen-tal results of higher-order confinement.
In the first part of the experimental results, the thesis introduces the topic of one-dimensionally confined systems with the example of the least complex case, single-layer membranes with homogenous excitation and detection (silicon membranes), building on the work of Hudert et al. [18] and Bruchhausen et al. [19]. The investigations are then extended to asymmetric excitation and detection (gallium arsenide membranes) and then to two-layered membranes, specifically semiconductor membranes with a thin metal transducer.
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In the second part of the experimental results, higher-dimensional confinement, is dis-cussed on the example of beam structures and disks. The complexity of investigating the acoustic properties of such nm-sized structures is illustrated. It is shown in detail how the acoustic properties of nanomechanical beams can be modified by altering the boundary conditions of the structures. The second part closes with a short overview of investigations of free-standing disk resonators, including a more in-depth discussion of the effect of clamping conditions and an example of how symmetry considerations can reduce the modeling effort.
Summary of Results
The effects of confinement on acoustic phenomena in the GHz range have been inves-tigated in membranes and nanomechanical bridge and disc resonators. The generation and detection processes are compared for systems with increasing complexity. General trends can be observed by careful comparison of the different systems. These will be summarized in the following:
Excitation/Detection Profile The more asymmetric the excitation/detection profile, the more complex is the membrane’s mode spectrum. In particular, with homoge-nous excitation, only odd (breathing) modes can be excited, whereas inhomoge-neous excitation results in both odd and even membrane modes. Under strongly asymmetric excitation, acoustic echoes can be generated which lead to acoustic frequency combs in the frequency domain. The amplitude and shape of the echoes depends on the generation mechanisms and the generation efficiency, as well as on the thermal conductivity of the material.
Layer System The acoustic response of two-layered systems varies with the acoustic mismatch, the interface properties, and the optical absorption of the layers. Ma-terials with high acoustic mismatch and good bonding show similar responses as one-layered systems (e.g. gold-diamond membrane). In the case of poor bond-ing, localized vibrations in the individual layers (e.g. gold film on diamond) and frequency-dependent damping at the interface can be detected.
Degrees of Confinement A reduction of the lateral extend of the investigated struc-ture below the focal spot diameter leads to additional frequencies in the spectrum.
This indicates that additional vibrational modes are generated/detected as a re-sult of the changed boundary conditions, which provide a higher degree of freedom for the acoustic vibrations. A critical examination of the boundary conditions is needed, when simulating the acoustic eigenmodes of the investigated structure, as small deviations can lead to significant changes in the mode spectrum.
In addition to these general patterns, each topic covered additional aspects:
Silicon Membranes This section presents the first pump-probe measurements on sub-10 nm thick silicon membranes. The measurements of thicknesses ranging from 7 nm to 200 nm allowed to investigate the damping times over the corresponding
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frequency range from 500 GHz to 20 GHz, respectively. The results allow for a deeper understanding of the damping of acoustic phonons in silicon, especially in thin membranes. At these thicknesses, the interface/interface roughness plays a major role in the damping, and it is shown that (bulk) damping models are insufficient to explain the observed damping times. A new model including the surface roughness is suggested, improving the understanding of damping in such confined systems.
Gallium Arsenide Membranes This section illustrates the variability of the experimen-tal set-up by combining temperature-dependent measurements and exploiting the possibilities of the wavelength tuneability of the ASOPS system. The wavelength-dependent measurements at low temperatures illustrate the different contributions to the electronic response of the membranes. The large variation of the oscillation amplitude in these measurements demonstrates the high sensitivity of this tech-nique to small thickness variations in such a membrane. Furthermore, the high sensitivity of this measurement method was exploited to determine the width of a 3.5 kHz-wide resonance curve with a center frequency of 12.8 GHz, a sensitivity which had not been reported before, with the technique of subharmonic resonant driving. The measurements reveal a monoatomar flatness of the membrane which allows to determine the intrinsic acoustic damping without surface roughness con-tributions.
Diamond Membranes In this section, the first measurements of GHz resonances in diamond membranes are reported. The membranes were cut using a focused ion beam, providing several membranes of different thicknesses on one sample. The combination of gold as a transducer and a diamond membrane resulted in a two-layer membrane, which acted, due to the excellent acoustic mismatch, similar to an acoustic one-layer system. The investigations further show how the surface treatment, i.e. the polishing of the surface, influenced the adhesion of the acoustic transducer. Open questions remain with regard to the observed acoustic spectra, in particular the occurrence of unaccounted double peaks in the frequency response of the membrane system.
Metal-Semiconductor Membranes This section demonstrates on various material com-binations how the choice of acoustic transducer affects the acoustic response of the investigated system. The results from different material combinations illustrate the differences in acoustic generation and the different contributions of the gener-ation processes. Using a thin aluminum film as acoustic transducer on a silicon membrane, a frequency comb spanning up to 300 GHz was detected. The frequen-cies correspond to a series of acoustic echoes in the time domain. The shape of these echoes can be explained by the different contributions of the deformation potential and thermoelastic generation. Further, it is illustrated how the acoustic transducer can be used in metrology to probe the interface quality, or how the same acoustic response can be achieved with different material combinations. One example shows how minor variations in sample preparation, i.e. the native oxide
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layer accumulated when transferring the sample from the etch to the evaporation chamber, can influence the acoustic transport through the interface, across the oxide layer, and how such a minor difference can be detected using picosecond ultrasonics.
Nanomechanical Beam Structures The section starts with some preliminary results illustrating the transition from membranes to beams – the membrane mode di-minishes, while beam modes appear, when the width decreases below the fo-cal spot diameter. Also, preliminary results show first GHz-range, pump-probe-characterization of two-layered beams, indicating a cross-section and length-depen-dance of the beam modes of free-standing beams. The mode shapes are identified with the help of finite element method (FEM) simulations. The section’s main focus is the influence of the boundary conditions on the vibrational model spec-trum of silicon nitride beam structures. Specifically, it is shown how changes in boundary conditions can be used to taylor the vibrational properties of the beam.
By soft-landing the beam on the substrate, the bottom of the beam was kept fixed, adding a third boundary condition. This change in BCs resulted in a shift of the mode spectrum towards higher frequencies, shifting the fundamental mode from the MHz regime into the GHz regime. The frequency of the fundamental mode is the highest reported fundamental resonance frequency (12 GHz) of such a µm sized beam. Similar frequencies are usually reported for structure sizes in the sub-µm range. The last part of the section shows the spatially resolved probing of vibra-tional modes in a µm-sized structure, demonstrating the different origins of the individual modes within the structure.
Nanomechanical Disk Resonators Nanomechanical disk resonators are presented as an example of three-dimensionally confined structures. Due to the rotation sym-metry, it was possible to reduce the model to a two dimensional model. In 2D, the simulations are comparable to a simple Euler-Bernoulli cantilever. Differences be-tween the Euler-Bernoulli theory, the FEM modeling and the experimental results are explained by the slight deviations in the assumed clamping conditions. It is shown that the rigidly attached center of the disk is needed to explain all modes from experimental results, and that the vibrations of the free standing part of the disk extend into the upper part of the center of the disk. In addition, a good agree-ment between layer thickness increase and increase of frequency of fundaagree-mental mode was found. Finally, some indications were found, that the fundamental vi-bration mode might be independent of the disk size itself, and presumably depend only on the length of the undercut of the structure.
Methodological Contributions
In addition to the pump-probe measurements, the thesis made a number of methodolog-ical contributions. Specifmethodolog-ically, several new methodologies, covering sample preparation, improvements in the experimental setup and finite element modeling have been devel-oped during the course of the thesis and will be summarized in the following.
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Sample Preparation A major amount of work in the context of this thesis dealt with the optimization of the sample fabrication processes, which, apart from the fab-rication of the diamond membranes, was newly developed and optimized for the research questions under investigation. Compared to research in the MHz regime, which is relatively robust to (sub-) nm variations in structure size, in GHz acous-tics, small deviations in the sample preparation procedure can lead to significant changes in the acoustic response of the system. In the context of this thesis, ac-cordingly, high demands on sample quality, reliability, and accuracy called for an optimization of the sample fabrication process. Examples of this sensitivity can be found throughout the thesis, for example in
• Silicon Membranes: influence of surface roughness
• Gallium Arsenide Membranes: resonance curve (thickness variation) and sub-harmonic driving: detection of atomar flatness of the membrane
• Metal-Semiconductor Membranes:
– Layer thicknesses
– Diamond: differences between polished and unpolished surfaces – Metrology: acoustic pulses.
The effect of such nm-size variations in the structure’s dimensions, interface bond-ing or surface roughness played an even greater role in the investigations of the beam and disk structures, compared to the membranes.
Limits of the Experimental Set-Up The thesis demonstrates the limits of the experi-mental set-up, in terms of lateral resolution when scanning the beam structure, as well as in terms of frequency resolution when determining the resonance width of the GaAs membrane. The lateral resolution was diffraction-limited, while for the resonance, the length of the laser cavity was stabilized to mechanical fluctuations below 30 nm, at a cavity length of 30 cm. The method was pushed to its ultimate limits with wavelength-dependent measurements at cryogenic temperatures (close to 4 K) with diffraction-limited lateral resolution.
FEM Modeling of Nanostructures The thesis demonstrates how the nature of acous-tic vibrations in nanostructures can be identified by comparing experimental re-sults with FEM modeling, taking into account boundary conditions and symmetry considerations. Specifically, the discussion of the disks resonators illustrates how symmetry considerations can be used to reduce the modeling effort, and how care-ful consideration of the choice of boundary conditions can lead to an improved match between the simulations and the experimental results.
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Conclusion
In conclusion, the thesis demonstrates how the ASOPS pump-probe technique can be used for investigations of the properties of acoustic phonons in nanostructures. The the-sis gives a novel overview over the properties of acoustic phonons in confined geometries, covering a wide range of material combinations. Further, it illustrates the transition to higher confinements as well as beam structures using pump-probe, which had not been done before. It demonstrates the dependency of the vibrational properties of nanostruc-tures on the confinement and boundary conditions, the absorption profile of pump and probe light, and the material’s acoustic properties. It provides a general discussion and comparison of all industry-relevant semiconductor membranes in one work, with a partic-ular focus on materials interesting for silicon-based IC integration, and chemically inert materials for sensors in harsh environments, such as diamond, silicon nitride, and gold.
Furthermore, the methodological contributions provide a base for further investigations in this research area.
Outlook
This thesis covers fundamental research on a broad variety of material systems and topics. It illustrates with various examples a number of interesting findings in regard to the aspects of acoustic phonons in confined geometries. From these findings many new questions arise, especially when considering the obtained results of all topics.
Temperature-dependent measurements of the damping in silicon could help to distin-guish the contributions of intrinsic and extrinsic damping to the overall damping. In particular, the transition from room temperature to cryogenic temperatures could be interesting. Such a measurement would be interesting for the GaAs membranes as well, where the resonance or subharmonic resonance driving method could be applied concur-rently in order to determine the exact line-width of the modes. The frequency doubling of the pump and/or probe laser could also be interesting, allowing to further investigate the absorption length dependance of measurements with 400 nm light. Here, silicon is expected to show a similar behavior as the GaAs membranes, with an enhanced photoe-lastic contribution to the detection. GaAs should show even shorter absorption length as well: similarities to the metal films are expected [74].
The results of the silicon membranes with aluminum transducer open up a way to use frequency combs in metrology. The excited frequency comb can be used to investigate acoustic damping over large frequency ranges at the same time, and to investigate the frequency-dependent influence of interface resistance and acoustic mismatch. In such measurements, the interface roughness would be the same for all frequencies, an advan-tage over membranes of different thickness from different samples. The distinct acoustic echoes allow for a direct estimation of the amplitude damping and investigations of the damping time over the frequency range up to 800 GHz, by determining the spectral am-plitudes for each pulse at a time, as illustrated by Grossmann et al. [46]. These data could, in combination with the measurements of the thin silicon membranes, shine light
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on the frequency-dependent damping mechanisms. Temperature-dependent measure-ments could also help to distinguish between the contributions of intrinsic and extrinsic damping.
The fabrication of the GaAs membranes and the diamond membranes, and several at-tempts to fabricate nanostructures with the focused ion beam, illustrate a challenge when fabricating such structures with a focused ion beam: When cutting material with the ion beam, Ga+-ions are implanted in the material in the vicinity of the cut, which alters the structural properties of the sample. In the case of diamond, a polishing of the surface led to an increase in adhesion between the gold and diamond, suggesting that a careful surface treatment with ion beams, or ion implantation as suggested by Tas et al. [85], opens up a broad range of possibilities to modify the surface’s influence on the damping of the confined modes. The influence of chemical modification of the sur-face (wet-etching) was already shown by Klingele [59], representing another option of influencing the surface properties.
The above presented suggestions are also interesting for higher degrees of confinement.
From a theoretical point of view, it would be desirable to fabricate single layer nanos-tructures from silicon and gallium arsenide, as less complicated mode spectra are ex-pected [151]. This would even allow for analytical estimates of the fundamental vi-brational modes. Also, a comparison between homogenous excitation/detection (silicon beams, red light) and inhomogeneous excitation/detection (blue light) could give more insight into the vibrational properties of such structures. The fabrication of these kinds of structures is the more challenging aspect of such investigations, as the fabrication processes would first need to be established. Cutting beam structures from membranes would also allow to investigate in the transition regime between membranes and beams.
The fabrication process presented in this thesis is limited by the height of the sacrificial layer, only allowing to fabricate beams of a maximum width of ∼800 nm. This limita-tion could be overcome, when cutting beams directly from membranes. Investigalimita-tion in the transition regime would allow to further investigate the questions arising from Reference [176, 177], where the influence of mode crossing and edge-modes are discussed theoretically.
Further, it could be interesting to fabricate beams with defects which would allow to investigate the possibility of further tailoring of the vibrational properties of such beams.
Defects could be designed and integrated in the fabrication process, or subsequently injected using (focused) ion beam.
The measurement of the lateral distribution of the modes of the beam structure also demonstrates a very interesting possibility of the experimental set-up. Damping mecha-nisms, or the dissipation of energy, remain a challenging question in the field of
The measurement of the lateral distribution of the modes of the beam structure also demonstrates a very interesting possibility of the experimental set-up. Damping mecha-nisms, or the dissipation of energy, remain a challenging question in the field of