Comparison of the Experimental Data with Different Damping Models

In Figure 4.4, the values from Table 4.1 are plotted on a double-log scale (blue circles).

Further the calculated damping times for different (bulk) models are shown in grey.

These models are compared to the data and discussed in the following. The models that are used to construct the fit to the experimental data, which is shown as a red solid line, are displayed in red, broken lines [14].

As it can be clearly seen, the observed lifetimes neither scale with the existing theories for bulk phonon lifetimes, nor do they agree with the bulk values. The observed lifetimes in the membranes are significantly shorter than the lifetimes observed by Daly et al. [15]

in bulk silicon.

In order to describe the damping of phonons in thin membranes, it is necessary to con-sider the two contributions of intrinsic and extrinsic damping. In nanoscale systems, and therefore also in nm-thin membranes, the extrinsic damping mechanisms are ex-pected to have a stronger influence on the overall damping compared to the case of bulk material.

Figure 4.4: Lifetimes vs. fundamental frequencies in thin silicon membranes. The blue dots give the measurement data from Table 4.1, the top x-axis indicates the membrane thickness corresponding to the fundamental frequency of the bottom scale. Indicated as well are the bulk values from Daly et al. [15] (red triangles), and the 346 nm membrane value from Bruchhausen et al. [19] (black square). Grey lines show the lifetimes computed using various standard models of phonon damping. The broken red lines reproduces the calculations for boundary scattering and three phonon processes. In solid red is the overall fit (Equation 4.8) to the membranes data shown. Taken from [14].

4 Experimental Results - One Dimensional Confinement

Akhiezer Damping and Herring Model

There are two models which describe contributions to the intrinsic damping in bulk materials. This intrinsic damping cannot be neglected, because even if one considers perfect crystals, or investigates the damping in the best grown or perfect single crys-tals, intrinsic damping is always found. The two models are called Akhiezer relaxation damping and “three phonon relaxation”.

The Akhiezer damping describes the damping of phonons with frequencies above 10 GHz due to the anharmonicity of the lattice, and the influence of the acoustic strain field to the population of wave packets of high frequency phonons [14, 64, 65]. This model does not exhibit a strong frequency dependence, and, in the case of the ultra-thin sil-icon membranes, overestimates the phonon lifetimes by a factor of 100. The Akhiezer contribution is displayed in Figure 4.4 as a thin grey and broken line at the top.

The alternative model, the three phonon scattering model, or, in its generalization as well called Herring Model, considers three phonon interactions or scattering. In this model, the scattering rates are derived from a first order perturbation theory of a har-monic potential. This model is based on the assumption, that for a certain single mode frequencyω, and the average life time of thermal phonons τ_th, the expression ωτ_th 1 is valid. The lifetime of the mode then can be described as a function of temperature T

by: 1

τ =BTⁿω^m, (4.3)

with the three parameters B, n and m being fitted to experimental data [14]. By using the values for bulk silicon, obtained by Daly et al. [15] and Cahill et al. [66], namely B = 2.4×10⁻¹⁹s K⁻¹, n = 1and m= 2, the dashed, grey curve labeled Herring in the upper right of the Figure 4.4 is obtained. This curve greatly overestimates the lifetimes of the membrane modes. This is in so far not surprising, as the influence of the “nano characteristics” of the membrane, i.e the influence of the surface, is completely neglected in these derivations. By fittingB, nand m to the experimental data of the membranes, the value of B = 5.7×10⁻¹⁷s K⁻¹ is obtained. This value is two orders of magnitude larger than the bulk values, which does not appear to be realistic.

Since the Akhiezer model does not reproduce the frequency dependence, and the Herring model yields unrealistic values, it is apparent that the intrinsic damping models are not sufficient to describe the damping in the membranes. Therefore an approach, considering not only the intrinsic damping, is described in the following.

The key objective of this model is to reproduce the frequency dependence of the ex-perimental data from the membranes, and to avoid doubtful parameter assumptions, as the fit of the Herring model yielded. Therefore two contributions to the lifetime are calculated separately, namely the contribution of the intrinsic scattering, using a modi-fied Debye model, and the contribution of the surface roughness scattering. The surface roughness scattering is calculated using an assumption of a specularity parameter, as it is discussed later.

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4.2 Ultra-thin Silicon Membranes

Debye Model for Intrinsic Scattering

Since the before discussed intrinsic models failed to reproduce the experimental data with realistic fitting values, the lifetimes are directly calculated using a Debye approximation.

Here, the intrinsic scattering is described by anharmonic processes. These processes define the phonon lifetime, which are in first approximation three phonon processes: i) collision: two phonons scatter to form a third, and ii) decay: one phonon decays into two phonons [67]. The general derivation of this model is given in Reference [14], and in its supplementary material. An extended discussion of the assumptions made for this model, and the general validity can be found there as well. For the detected purely longitudinal modes in the membrane, the lifetime can be expressed as [14, 68]:

τ_3−ph⁻¹ (ωL) = ~v_l one adjustable parameter, namely the mode-averaged Grüneisen parameter γ, where s is the mode polarization, n the Bose-Einstein distribution and v the phonon average group velocity.

When two types of phonon polarizations are considered, which are labeled l for longi-tudinal and t for transverse polarization, then two types of the three-phonon processes can be identified. These processes describe the scattering of two longitudinally polar-ized phonons, l+l →l, and the scattering of a longitudinally polarized phonon with a transversely polarized phonon, l +t → l. The first type of these processes dominates most of the scattering, which influences the intrinsic lifetime [14].

By fitting Equation 4.4 to the experimental data, a value ofγ = 1.08is obtained for the reduced Grüneisen parameter. The results are displayed in Figure 4.4 by the broken red line, labeled τ3−ph. Obviously, this model does not reproduce the frequency dependence of the high frequencies as well, but delivers improved results for membrane thicknesses above 200 nm compared to the previously discussed intrinsic damping models.

Surface Roughness

From the paragraphs above it is evident that the models, which consider intrinsic damp-ing only, do not reproduce the frequency dependence in the range above 100 GHz. There-fore it is necessary to take the effects of the confinement, i.e. the two membrane’s sur-faces, into account. This is done by a model which considers the effect of the surface roughness onto the lifetimes of the modes.

The model, which is considered as surface roughness scattering, uses a single parameter only, namely a specularity parameter p. This parameter p describes the flatness of the surface, in form of a deviation from an ideal flat surface. This concept is illustrated in Figure 4.5.

4 Experimental Results - One Dimensional Confinement

To describe the influence of the surface roughness, in this model it is considered that a wave undergoes a phase change φ when it is reflected at the surface of the membrane.

This phase change is related to the thickness deviationy(x) of the surface from an ideal surface plane, marked by the thick black lines in Figure 4.5, and is described by [14]:

φ(x) = 4π λ y(x).

The wavelength-dependent specularity pthen becomes:

p(λ) = exp(−πφ²)

= exp(−16π³η²/λ²). (4.5) Here, η describes the root mean square deviation of the height of the surface from the reference plane. This expression is valid for roughness features, whose width is smaller than 4πη. Looking at Equation 4.5, one can see that the specularity depends on the ratio of feature height and wavelength. This roughness effect increases therefore with decreasing wavelength. Also apparent from Figure 4.5 is the increasing role of the surface roughness, as the membrane thicknessd₀ decreases.

x z

y(x) d₀

∼ η

Figure 4.5: Surface roughness variation y(x) from a reference plane, i.e. the ideally flat membrane surface (thick black line). This deviation is used to model the flatness of the silicon membrane’s surface. The feature size of the roughness is estimated byη, being the root mean square value of the deviation from the reference plane. d₀ describes the nominal membrane thickness.

Under the consideration of multiple reflections, the mean free path in the membrane can be expressed with the help of a characteristic dimension of the structure, i.e. the unperturbed membrane thickness d0, as:

Λ = 1 +p

1−pΛ₀. (4.6)

Here isΛ₀ the mean free path for a membrane with an ideal flat surface, orp= 0. Using the mean free path Λ and the longitudinal sound velocity v_l, the lifetime due to the

64

4.2 Ultra-thin Silicon Membranes

surface roughness can be expressed as [14]:

τ_b = Λ

For a value η = 0.5nm, the result of Equation 4.7 is plotted in Figure 4.4, and labeled with τ_b. Very good agreement between the experimental data and this fit is obtained for membrane thicknesses below 50 nm, or frequencies above 100 GHz, indicating that in this region indeed the surface roughness has strong influence on the lifetime.

In order to fit the experimental data of the membranes over the whole frequency range, the two damping models, the Debye model and the surface roughness model, are com-bined. This can be done by using Matthiesen’s rule for the combination of lifetimes:

1

and by combining the results of Equation 4.4 and Equation 4.7. The fit to the experi-mental data, using a total lifetime τ_T, is shown in Figure 4.4 by the solid red line.

The lifetimes of the combined models result in a satisfying fit to the experimental data, being in excellent agreement in the regime above 100 GHz, and being in a reasonable good agreement with the overall trend of lifetimes over the whole frequency range.