Determination of the Longitudinal Sound Velocity of Bulk Diamond and of the Thickness of the Gold FilmDiamond and of the Thickness of the Gold Film

The elastic properties of synthetically prepared (bulk) diamond vary in the range from 14×10⁵ cm/s to 18×10⁵cm/s for the longitudinal sound velocity [79], depending on the growth quality, impurities and the (poly-) crystallinity. This parameter enters directly into the relation between the frequency of the confined dilatational modes of the mem-brane, and the membranes thickness, therefore it is useful to reduce any uncertainties from this parameter for the use in the following calculations. The longitudinal sound velocity can be experimentally estimated in semi-transparent samples by means of the so-called “time resolved Brillouin scattering” via:

f =v_l· 2n

λ_probe, (4.14)

4 Experimental Results - One Dimensional Confinement

wherenis the real part of the refractive index of the material (here diamond) andλ_probe the center wavelength of the probe pulse. Equation 4.14 can be directly calculated from the sensitivity function approach described in Section 2.4.1. The origin of this frequency can be understood by the following picture: The pump pulse excites strain inside the gold film, which travels in form of a strain pulse into the substrate. One fraction of the probe light is reflected at the sample’s surface, while the other fraction is reflected at the strain pulse inside the diamond. Both reflected fractions interfere, when they are detected. The phase difference between both fractions is directly related to the path difference accumulated from the fraction that travels into the diamond towards the strain pulse and back. As the strain pulse moves further into the substrate, this distance grows, thus altering the interference condition, and forming a dynamic optical Fabry-Pérot cavity. The change in reflection is therefore modulated as [see Reference 13, and Section 2.4.1]:

The corresponding measurement is displayed in Figure 4.17a, the corresponding numeri-cal fast Fourier transform of the oscillations, after removal of the background, is given in Figure 4.17b. The measurements were performed using 3 mW probe power (λ= 820nm) and 30 mW pump power (λ= 790nm).

The signal is dominated by a strong onset, which can be attributed to the excitation of free electronic carriers in the gold film, as the pump pulse is absorbed. High frequency oscillations (∆R/R∼ 10⁻⁶) are modulated on the decay of the electronic and thermal background, they can be extracted by means of a moving average. These oscillations are the exspected time-resolved Brillouin oscillations.

This method is only possible to use, because of the semi-transparency of a 12 nm thin gold film. At larger gold film thicknesses the film becomes opaque and this method cannot be used. This method is limited in opaque materials to the absorption depth of the probe light, which, depending on the absorption depth, results in high uncertainties, as only a few cycles of the oscillation can be measured and fitted.

Takingn = 2.4, λ_probe = 820nm and f = 102.14GHz from the FFT, the sound velocity for this sample can be calculated, giving vl = 17448.91m/s, which is in excellent agree-ment with the typical literature values of 17.52×10⁵ cm/s along the [100] direction [63].

This sound velocity is used in any further calculations concerning this sample.

A second parameter, that can be estimated through these measurements, is the thickness of the gold film. The thickness obtained from these measurements can be compared with the nominal values, obtained from the quartz crystal of the sputtering machine. The nominal thickness of the gold film, given by the sputtering machine, was 14±1 nm [52].

By looking at the first 200 ps of the signal given in Figure 4.17, a high frequency os-cillation of only a few cycles can be seen. This osos-cillation can be identified with the thickness oscillations of the gold film. The frequency of these oscillations is 131.7 GHz, as obtained from the FFT, shown in Figure 4.17c. The corresponding thickness of the

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4.4 Diamond Membranes

(a)Time resolved reflectivity changes obtained from bulk diamond with a 12 nm thick gold transducer.

(b)Oscillations extracted from(a). The inset shows the two contributions from the gold film vibration, and the Brillouin oscillation inside the diamond.

(c) Numerical fast Fourier transform of the gold film oscillations seen in (b). The chosen time window was from start up to 75 ps.

0 50 100 150 200

(d) Numerical fast Fourier transform of the Brillouin oscillation from(a). Here a time win-dow from 75 ps to 1200 ps was chosen.

Figure 4.17: Experimental results obtained from the gold film on bulk diamond. The time domain data (a) is dominated by the sharp and pronounced electronic onset. The inset shows the modulation upon the electronic and thermal background. The extracted oscillations (b) can be divided into two parts, as shown in the inset, first the vibrations of the gold film only, and later the Brillouin oscillation, which can be seen up to 500 ps. In (c) the numerical FFT of the extracted oscillations depicts the frequency of the thickness oscillation of the gold film, in (d) the numerical FFT of the Brillouin oscillation is shown.

4 Experimental Results - One Dimensional Confinement

gold film can be calculated via Equation 2.44:

d= v_l

2f = 12.3nm, (4.16)

which agrees well with the nominal thickness. Equation 4.16, which describes the open pipe oscillations, or the free-standing layer, can be used for two reasons: First, the acous-tic impedance of gold (ZAu= 62.5 GPa s/m) is larger than the impedance of diamond (Zdia= 61.6 GPa s/m), see also Section 2.5. Second, this measurement was performed at a place of the sample where no polishing took place during the sample preparation.

These areas can be distinguished by the light and dark areas, as illustrated in Fig-ure 3.7. The polishing results in a presumably better adhesion between the gold and the diamond, as no such oscillations could be found in the polished areas. This can be seen as a hint towards an acoustic decoupling of the film from the diamond in the unpolished case, probably due to residuals from the fabrication process on the diamond surface, or roughness effects. This has not been further investigated.

The acoustic mismatch of ZAu/Zdia= 0.93, being close to the value of one, i.e. perfect matching, makes gold a perfect optical transducer for diamond in the near infrared, as the reflection coefficient between both layers is:

|r|=

This very small reflection coefficient explains, why the gold film oscillations damp out on such a short time scale, and the acoustic energy is transferred almost unattenuated into the diamond. The beating of the signal around 75 ps can be explained by a superposition of the gold film thickness oscillation and the beginning of the Brillouin oscillations.