Two-Dimensional Phononic Crystals: Disorder Matters

The design and fabrication of phononic crystals (PnCs) hold the key to control the propagation of heat and sound at the nanoscale. However, there is a lack of experimental studies addressing the impact of order/disorder on the phononic properties of PnCs. Here, we present a comparative investigation of the influence of disorder on the hypersonic and thermal properties of two-dimensional PnCs. PnCs of ordered and disordered lattices are fabricated of circular holes with equal filling fractions in free-standing Si membranes. Ultrafast pump and probe spectroscopy (asynchronous optical sampling) and Raman thermometry based on a novel two-laser approach are used to study the phononic properties in the gigahertz (GHz) and terahertz (THz) regime, respectively. Finite element method simulations of the phonon dispersion relation and three-dimensional displacement fields furthermore enable the unique identification of the different hypersonic vibrations. The increase of surface roughness and the introduction of short-range disorder are shown to modify the phonon dispersion and phonon coherence in the hypersonic (GHz) range without affecting the room-temperature thermal conductivity. On the basis of these findings, we suggest a criteria for predicting phonon coherence as a function of roughness and disorder.

contactless technique of 2-laser Raman thermometry, here applied to PnCs for the first time.
2D phononic crystals were fabricated of free-standing silicon membranes (Norcada Inc.) using electron beam lithography and reactive ion etching to generate ordered and disordered hole patterns with equal filling fractions ( Fig. 1(a-c)) [47]. The disorder was introduced by random displacements of the holes in x and y direction within the unit cell of the PnC lattice. The hole positions of the disordered PnC were defined by p = p 0 ± ǫ · s, where p is the displaced hole position along the two in-plane axes, p 0 is the ordered lattice position, ǫ is a random number between 0 and 1 and s is the maximum displacement which was set to 45 nm. The level of disorder in percentage of the period a = 300 nm is than quantified by n = s/a · 100% = 15%. Figs The coherent acoustic phonon dynamics of the ordered and disordered PnCs with the same filling fractions are investigated using femtosecond pump-probe reflectivity measurements using asynchronous optical sampling [48][49][50] (see Fig. 1(g) and section Methods for details). The optical excitation of acoustic vibrations arises from the electronic and thermal stresses induced by the pump pulse which are determined by the generated electron-hole pair density and the temperature-induced lattice deformation, respectively [49,51,52]. The excited out-of-plane (dilatational) oscillations change the optical cavity thickness of the membrane which leads to a modulation of the probed reflectivity by the Fabry-Perot effect [49,53,54]. The blue shaded areas indicate the frequency ranges in which coherent acoustic phonons are detected. It should be noted that only phonon modes with amplitudes greater than the noise level are displayed in the fits, thus, the highest phonon frequency is not a strict limit for phonon coherence as indicated by the gradient in the blue shaded range. Using this experimental approach, we can directly obtain the complete zone-center (q = 0) coherent phonon spectrum from the GHz to the THz regime. In the case of the bare membrane ( Fig. 2(d)), the different harmonics in the vibrational spectrum appear as equidistant peaks as a consequence of the confinement of the acoustic modes [13,49,55]. The lowest frequency peak at 16.8 GHz thereby corresponds to the first order symmetric (S 1 ) mode and higher frequency modes up to the 9th harmonic at 151 GHz are clearly visible. The absence of even harmonics can be understood taking into account that those modes have only in-plane displacement at the Γ point so that no modulation of the optical cavity thickness occurs [54].
Using the value of the longitudinal sound velocity for Si [001] of v L = 8433 m/s [56], the thickness of the membrane d determines the frequencies of the observed modes f n = nv L /2d, where f n is the frequency of the n-th harmonic (n = 1, 3, 5, ...) of the symmetric (dilatational) mode [49]. In addition, a weak signal at 23.8 GHz is observed (0.03 of the amplitude of S 1 ).
Using FEM modeling, we identify this peak as the symmetric S 2 mode which might be visible due to excitation of propagating in-plane modes with small but nonzero wave numbers and therefore nonzero out-of-plane displacement [57].
Following the discussion of the acoustic phonon dynamics of the non-patterned membrane, we now focus on the modification of the frequency spectrum in ordered and disordered PnCs by hole patterning of the original membrane. The time resolved pump-probe reflectivity spectra for the ordered and disordered PnCs are displayed in Fig. 2 In order to explain the observed differences between the unpatterned membrane and the ordered PnC, we calculate the acoustic phonon dispersion relation and the 3-dimensional displacement fields for the zone-center modes up to 55 GHz by means of FEM modeling as described in Ref. [12]. In principle, the reflectivity of the membrane is modulated by the induced mechanical modes which result in non-zero average thickness variations. Here the model assumes a predominant role of the optical cavity thickness mechanism with negligible contribution of the photoelastic effect (PE) [49]. The pump-induced variation of the membrane thickness ∆d is of the order of a picometer and directly proportional to the measured relative change of the reflectivity ∆R/R 0 . In the case of the PnCs the mechanical modes are complex and ∆d is position dependent. We correlate the amplitudes A i of the modes ω i with their corresponding average change of thickness |∆d| calculated over the whole FEM unit cell using the formula: where u i (z) are the out-of-plane displacement components, and S is the free surface area.
The displacement fields of all the FEM solutions are normalized in such a way that all the modes store the same elastic energy and are populated according to the Planck distribution at high temperature.
The acoustic phonon dispersion relations for the membranes before and after hole patterning are displayed in Fig. 3(a) and 3(b), respectively. For the unpatterned membrane, the first three symmetric modes S 1 , S 2 , and S 3 are precisely reproduced by the FEM simulations regarding both amplitude and frequency ( Fig. 2(g) and Fig. 3(a)). The decreasing amplitude of the higher harmonics can be accurately described by Eq. (1) which in the case of the unpatterned membranes simplifies to a 1/ω 2 relation (Fig. 2(d)) [49]. The fact that the mode amplitudes obey this relation indicates that the phonon coherence is not destroyed by, e.g., surface roughness up to at least 150 GHz. In the case of the ordered PnCs, the strong modifications of the phonon dispersion relation in Fig. 3(b) compared to the bare membrane ( Fig. 3(a)) arises from band folding and band splitting when the Bragg condition is satisfied [12].  Up to this point, we have limited our discussion to the hypersonic (GHz) frequency range of the phonon spectrum. Taking into account that no coherent phonon modes in the ordered and disordered PnCs could be observed at frequencies above 55 and 20 GHz, respectively, we use a different approach to investigate the influence of order and disorder on the thermal properties: Two laser Raman thermometry (2LRT) [58]. The main advantage of this technique with respect to, for example, electrical measurements or time-domain thermoreflectance (TDTR), is given by its contactless nature avoiding the introduction of additional thermal interface resistances. Thus, the thermal conductivity of the PnCs can be directly obtained from the measurements without additional modeling. A spatially fixed heating laser generates a localized steady-state thermal excitation, whereas a low power probe laser measures the spatially-resolved temperature profile with sub-micrometer resolution through the temperature dependent Raman frequency of the optical phonons in the material. Fig. 4(a) displays the temperature profiles for ordered and disordered PnCs and the unpatterned Si membrane obtained by 2LRT with the experimental arrangement shown schematically in Fig. 1(h). Applying Fourier's law in 2-dimensions for a thermally isotropic medium leads to a temperature field T (r) with: where (r 0 , T 0 ) is an arbitrary point in the temperature field, P abs is the absorbed power, f = 0.528 is a correction factor for the missing material due to the holes in the PnCs with a filling fraction of φ = 0.267 [25], d is the thickness of the PnC membranes, and κ 0 is the thermal conductivity. Here, κ 0 can be treated as temperature independent since the temperature range is sufficiently small (≈ 50 K). We recall that for the case of bulk Si the thermal conductivity changes by about 15% in the range from 350 K to 400 K [59].
This variation represents only an upper (bulk) limit since the temperature dependence is typically reduced as boundary scattering increases. Fig. 4(b) displays the thermal decays in logarithmic scale according to Eq. (2), thus, the slope of the thermal decay is directly related to κ 0 . The purely linear decay observed in this graph validates the temperature independent treatment of κ. A deviation from this linear relation is expected in cases where κ = κ(T ) as discussed in Ref. [58]. Based on these measurements, we obtain the same value for the thermal conductivity κ 0 = 14 ± 2 Wm −1 K −1 for the ordered and disordered PnCs compared to κ 0 = 80 ± 3 Wm −1 K −1 in case of the unpatterned membrane. Considering the correction factor for the material loss in the holes of the PnCs, we would obtain an effective thermal conductivity of the PnCs in the absence of size effects of f κ 0 = 46 ± 3 Wm −1 K −1 .
The reduction in the thermal conductivity of the PnCs down to 18% of the value of the unpatterned membrane, about a six-fold reduction, is in accordance with other recent studies of the thermal conductivity of PnCs with comparable dimensions [16,19,23,[25][26][27].
This drastic reduction cannot be solely explained by the mass loss due to hole patterning.
Instead, two aspects need to be considered: (i) diffuse boundary scattering and (ii) phonon coherence. A decrease of thermal conductivity by diffuse boundary scattering is expected due to the increase of the surface area caused by the introduction of holes as well as the hole wall roughness of about 7 nm due to the patterning process (c.f. Figs. 1(e) and 1(f)).
We address the issue of phonon coherence by plotting in Fig. 4(c) the phonon frequencies of the measured coherent acoustic phonons (Fig. 2) as function of the characteristic size R limiting in each case the measured phonon frequency range, i.e. surface roughness, hole wall roughness, and average lattice site displacement. Next, we plot the corresponding phonon frequencies for selected specularity parameters p as function of R: where the specularity parameter p expresses a quantitative measure for the percentage of specular scattering at a normal surface with given roughness (1 for purely specular scattering and 0 for purely diffusive scattering). The dependence of p as function of the phonon wavelength is displayed for given values of R in SI Fig. 5. Attempting to derive a general criteria for the non-coherent phonon regime (fully diffusive phonon scattering), we extrapolate the specularity parameter towards 0 in SI Fig. 5. The corresponding phonon wavelength is given in good approximation by the line for p = 0.01 in Fig. 4(c). By plotting the phonon wavelength λ ph as function of roughness for this specularity parameter, we obtain λ ph ≤ 10R as simple criteria for the non-coherent phonon regime. Following this approach, we obtain frequency limits of 800 GHz, 115 GHz, and 36 GHz for surface roughness values of 1 nm, 7 nm, and 22.5 nm, respectively. It is important to note that the specularity parameter as introduced by Ziman [60] only considers the surface roughness for a normal incidence wave, not wall roughness or lattice site displacement. However, despite the different types of roughness in our phononic crystals, the computed values reproduce the general tendency of the measured decreasing high frequency limit of coherent phonons in our phononic crystals. In fact, our experimental data suggests that phonon coherence is already affected by roughness corresponding to a specularity parameter between 0.3 and 0.5 (c.f. Fig. 4(c)). Using the more conservative value of p = 0.5, we find in a rough approximation that λ ph > 25R constitutes a realistic criteria for the coherent phonon regime. Consequently, we suggest that disorder, quantified by the average hole displacement from the periodic lattice sites, can also be considered as a type of roughness for long wavelength phonons as can be seen when plotting the measured coherent phonon frequencies of the disordered PnCs for a roughness R = 25 nm in  Two-laser Raman thermometry: Thermal conductivity measurements were conducted using two-laser Raman thermometry, a novel technique recently developed to investigate the thermal properties of suspended membranes [58]. A spatially fixed heating laser generates a localized steady-state thermal excitation, whereas a low power probe laser measures the spatially-resolved temperature profile with sub-micrometer resolution through the temperature dependent Raman frequency of the optical phonons in the material. Both lasers were focused on the PnCs using 50× microscope objectives with numerical apertures of NA = 0.55. The power of the heating laser with a wavelength of λ heat = 405 nm was set to 1 mW and the power of the probe laser with a wavelength of λ probe = 488 nm to 0.1 mW in order to avoid local heating by the probe laser while measuring the temperature field. The absorbed power is measured for each sample as the difference between incident and transmitted plus reflected light intensities probed by a calibrated system based on a non-polarizing cube beam splitter. The measurements were performed at ambient pressure which introduces heat losses through convective cooling. This effect accounts for about 30% of the thermal conductivity in our samples, i.e., the measured values for the thermal conductivity of the PnCs were κ 0 = 21 ± 2 Wm −1 K −1 . After correcting for the heat transport due to convective cooling we obtained the reported value of κ 0 = 14 ± 2 Wm −1 K −1 for the ordered and disordered PnCs. A detailed discussion of the influence of convective cooling on the experimental values obtained for the thermal conductivity in Si PnCs is being published elsewhere [61].

Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: Description of the finite element modeling simulations; femtosecond pump-probe reflectivity spectra for a bare membrane, an ordered PnC, and a disordered PnC; specularity parameter as function of phonon wavelength for selected roughness values.