Confined Modes in Gold-Diamond Membranes

In the following, a 200 nm and a 470 nm diamond membrane are discussed (nominal values). Both membranes were measured using 3 mW for the probe laser (λ=820 nm), and 30 mW for the pump laser (λ=790 nm). The nominal thicknesses were obtained using SEM measurements [52]. Both membranes are covered with the already mentioned

∼12 nm layer of gold, which acts as acoustical transducer.

The thickness of the gold layer matches the optical absorption length in gold, which re-sults in about 65% absorption of the pump light. This leads to the following assumptions concerning the generation and detection processes: At this thickness of the gold, one can assume a homogenous heating of the gold film due to the absorption profile and the fast hot electron diffusion [36]. The diamond itself will not be excited, as it is transpar-ent at the pump light wavelength [63]. The ratio of 12 nm gold film to 200 nm / 470 nm membrane will lead to an extremely asymmetric excitation profile, eventually leading

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4.4 Diamond Membranes

0 200 400 600 800 1000 1200 -20

(a) Extracted oscillations obtained from two diamond membranes of 200 nm (top) and 470 nm thickness (bottom).

(b) Numerical fast Fourier transform of the data given in (a).

Figure 4.18: Extracted oscillations and corresponding numerical fast Fourier transform from two diamond membranes of different thicknesses, with identical gold coating. The signal of the thin membrane (blue) shows a decay within the measurement window and a spectral composition of up to five harmonics of the fundamental mode. The signal of the thick membrane (red) shows long living oscillations superimposed with pulses at the beginning, and a spectral composition of nine harmonics of the fundamental mode. Here the amplitude modulation due to the acoustic pulses in the time domain can be observed, indicated by the amplitude variations of the modes.

to the excitation of odd and even harmonics of the fundamental mode of the combined, two layer system.

The absorption of the pump pulse in the gold film will lead to a rapid heating of the film. The generated thermal stress within the gold film excites strain pulses, which will travel perpendicular to the membranes surface through the interface into the diamond membrane. Due to the acoustic reflection coefficient of 0.0075 between both materials, the strain pulses will extend almost unattenuated into the diamond. The result is a combined oscillation of the gold and diamond layers.

The detection process is twofold: At the gold layer there will be photoelastic detection of strain, but there will also be a non-negligible contribution of the Fabry-Pérot detection, as the semi-transparency of the gold film allows for the detection of the dynamics inside the diamond layer. Therefore, in the time domain the signal will be a superposition of the sinusoidal of the Fabry-Pérot and a series of pulses detected on the surface of the gold film.

Time Domain Analysis

In Figure 4.18a, the extracted modulations of the time domain reflectivity changes ob-tained from the 470 nm (red) and 200 nm (blue) thick membranes are shown. The original time trace is dominated by a strong electronic onset and a fast thermal decay

4 Experimental Results - One Dimensional Confinement

extracted modulations show two components in the time trace, as it can be clearly seen in the 470 nm case: an almost saw-tooth like shape, decaying very slowly throughout the whole measurement window, and some prominent, pulse-shaped modulations / acoustic echoes in the first half of the time window. In the case of the 200 nm membrane the oscillations damp out within the first 600 ps after excitation (damping time ∼160 ps), and the signal is mostly dominated by the echoes at the beginning. In this case it is not possible to clearly distinguish and extract individual echoes, since the round trip time of a pulse through the membrane, given by

t_RT

2 = d_Au v_l^Au +d_C

v^C_l = 18.8 ps (30 ps for 470 nm),

is similar to the temporal width of the acoustic echoes (∼12 ps), as estimated from the 470 nm membrane.

The echoes seen in these measurements show a single minimum or maximum, depending on the origin of detection. The shape of these echoes originate in the change of the thickness induced by the strain pulse, when arriving either at the front (maximum) or back surface (minimum) of the membrane. When these echoes arrive at the surface, the latter will start to displace, and the integral over the strain becomes non-zero. The alternating shape of the echoes are due to the phase shift of π when the echoes are reflected, i.e. in the gold or diamond layer the maximal strain has a different sign, so that the surface moves into different direction upon arrival of the echoes, and therefore resulting in a different change in optical reflectivity.

Since both membranes are fabricated on the same substrate, both membranes have an identical gold coating, allowing for the comparison of these echoes. The only difference of both membranes is the diamond thickness. This indicates that the damping might be dominated by interface/boundary scattering at the gold-diamond interface, as the echoes damp out in the 200 nm membrane within∼450 ps, while in the thick membrane they can be found until∼900 ps. The factor of two in lifetime could account for twice the scattering events at this interface, when neglecting the scattering at the air-interfaces, yet the origin cannot be determined accurately. The damping of the fundamental mode of the 470 nm membrane can unfortunately not be used in a direct comparison, as it does not decay within the measurement window, and thus might be partly altered by off-resonance subharmonic driving.

Frequency Domain Analysis and Contributions to the Detection Process

By numerical fast Fourier transformation of the time domain traces, the mode spectra of the extracted oscillations can be obtained, see Figure 4.18b. The spectra are dominated by the fundamental dilatational mode of the two-layer membranes, 14.46 GHz for the 470 nm and 26.5 GHz for the 200 nm membrane. Both spectra show the typical series of equidistant peaks of the confined modes inside the membranes, all at integer multiples of the fundamental mode. The mode spacing of the 200 nm membrane is almost twice as

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4.4 Diamond Membranes large as the mode spacing of the 470 nm membrane, as expected from the theory. Higher order modes can be observed up to 160 GHz, in the case of the 470 nm membrane up to the ninth harmonic. The linear dependence of these modes can be seen in Figure 4.19.

The small deviation of the mode position from the linear dependence follows a sinusoidal behavior, and can be explained by a first order perturbation theory, as described in Reference [46], as this in not a single layer membrane.

0 2 4 6 8 10

Figure 4.19:Plot of the mode frequency over mode number for two diamond mem-branes. For both membranes the linear dependence of the modes is clearly seen.

The small deviation from the linear de-pendance can be explained by a first oder perturbation theory.

Two major features of interest can be seen in the spectra obtained from both mem-branes. The 470 nm membrane shows an alternating Fourier amplitude for the peaks of mode order 2≤n≤7 ,while the 200 nm membrane shows double peaks at all mode numbers (1 - 3).

The alternating behavior in the mode spectrum of the 470 nm membrane can be ex-plained by the two contributions to the detection process. The modes with odd mode number (1, 3, . . . ) are detected predominantly by the Fabry-Pérot process, the even modes are detected by the photoelastic effect. A more distinct analysis of these contri-butions is given in Section 4.5 on the example of the silicon membranes with aluminum transducer. In short, the detected signal is a superposition of both contributions, both with opposite sign. The Fabry-Pérot contribution originates mainly from the movement of the diamond layer, i.e. the motion of the diamond surfaces. Since the Fabry-Pérot detection is not sensitive to the strain pulses inside the membrane, i.e. the integral over the strain inside the membrane is zero, the additional contribution to the detection is from the photoelastic effect in the gold layer. Here the strain pulses, which can be seen in the beginning of the time traces, are detected. The detection of the pulses results in the additional existence of the even modes in the Fourier spectrum, with the overall modulation of a pulse, i.e. a maximum at the fundamental frequency of the gold film [46].

The superposition of both contributions results in the observed amplitude modulation of the mode spectra in the frequency domain.

The spectral maximum of the fundamental resonance of the gold film can be calculated via f = _4d^vl = 67GHz, in good agreement with the experimental result. The use of the closed-pipe resonance is in contrast to the previously calculated result of the gold film resonance, where, in the unpolished case of the substrate, the open-pipe resonance

4 Experimental Results - One Dimensional Confinement

f = _2d^vl was assumed. Two arguments support the validity of the two approaches of the calculation of the resonance. For one, the open/closed-pipe resonances derived in the theoretical framework are truly valid for large impedance mismatches only, which cannot be assumed in the case of gold and diamond. In this case, a transition between both cases might occur, or the correct resonance case might as well be dependent on certain surface/interface conditions. A theoretical description of this transition regime was not available in the course of this thesis. This leads to the second argument: The unpolished areas (open pipe resonance) might have some residual surface contamination or unknown surface layer properties, which might influence the acoustic matching. The surface properties depend on the fabrication process and final treatment of the fabrica-tion process [80], and are subject to the unknown prior treatment of the manufacturer.

The surface properties of the membranes were altered in the fabrication process through polishing with the focused ion beam, which then might have lead to an improved acoustic matching or better adhesion, leading to the closed pipe resonances.

The origin of the observed double-peak feature in the FFT of the 200 nm thick mem-brane could not be determined in the context of this thesis. It could not be explained by the numerical simulations of the spectra, therefore the following hypotheses of a detection-based origin of the splitting is suggested [V. Gusev, priv. comm.]: “It could be possible that the splitting of this resonance peak is a single resonance, but the different mechanisms of detection lead in their superposition to a splitting of this resonance. This splitting could be due to an anti-phase detection of the two contributions with slightly different resonance width, then the narrow resonance could lead to the dip in the broad resonance. In order to validate this hypothesis, it is necessary to account correctly for the relative amplitudes and phases of the Fabry-Pérot and the photoelastic detection.”

The phases of the detection were not implemented in the numerical simulations, so this hypothesis could not be validated.

4.4.3 Summary

The diamond membranes with metal transducers were the first example in this thesis of a membrane system with very asymmetric excitation and detection. They were chosen, because of the almost identical acoustic impedance of both materials, and therefore, from the acoustics point of view, simulating almost perfect a one-layered membrane.

These membranes show a similar acoustic behavior as the GaAs membranes, but with an even more asymmetric excitation and detection. The excitation profile was, due to the excitation of the gold film only, strongly asymmetric, therefore launching strain pulses of very short duration (∼12 ps) into the membrane. The obtained time traces show a superposition of these strain pulses / acoustic echoes and the saw-tooth like shape of the membranes thickness oscillations, i.e. the first oder dilatational mode and its higher harmonics, very similar to the room temperature measurements of the GaAs membranes.

In the frequency domain, the spectra shows the corresponding superposition of Fabry-Pérot contributions to the detection in the diamond, and the photoelastic contribution

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4.4 Diamond Membranes from the detection in the gold film, in form of equidistant spaced modes of multiple integers of the fundamental mode. The amplitude variations of the modes follow the general _n¹2 dependance of the Fabry-Pérot contribution, but modulated with the spectral response of the acoustic echoes, mainly visible for the higher harmonics. The mode spacing follows the known linear dependence of a single layer membrane, with small sinusoidal deviations from the linear theory, which can be explained by a first oder perturbation theory. Compared to the GaAs membrane, the photoelastic contribution to the detection leads to a stronger modification of the spectral amplitudes of the modes in the FFT.

The prior not exactly known longitudinal sound velocity of the sample was obtained by the measurement of the Brillouin Oscillations. The sound velocity and the determined layer thicknesses correspond very well to the nominal values, demonstrating that this method is reliable to obtain material parameters in a non-destructive way from otherwise transparent materials like diamond, i.e. showing how a transparent material can be probed with help of a thin metallic coating.

The use of the two different membrane thicknesses with identical coating, i.e. the fabrica-tion on the same chip, shows a possible way to investigate the influence of the individual contributions to the damping or scattering mechanisms of the high frequency acoustic pulses, e.g. the influence of the interface between the metal coating and the semicon-ductor. This will be further illustrated in the next section as well. The advantage of the fabrication of different membrane thicknesses from one substrate is the fact that the material properties and the coating thickness are identical, and that there is no variation in properties due to fabrication uncertainties. This allows for a systematic comparison of the results, e.g. as shown here for the thickness determination of the gold film and the diamond sound velocity.

The lack of diamond membranes with sufficiently small thickness inhibited a further investigation of the damping properties of the combined oscillatory mode properties, as a direct examination of the damping times was not possible for the large thickness.

Interesting aspects of such studies could be the influence of the surface properties on the lifetime. Especially for the membranes fabricated with focused ion beam this influence could be interesting, as the ion beam fabrication is known to leave an amorphous layer at the cut area, as well as residual implanted ions in the vicinity of the cuts [52]. These aspects seemed to play no further role in the investigations of these samples, but with reduced membrane thickness they might gain importance.

4 Experimental Results - One Dimensional Confinement