Phonon Engineering in Isotopically Disordered Silicon Nanowires

: The introduction of stable isotopes in the 13 fabrication of semiconductor nanowires provides an additional 14 degree of freedom to manipulate their basic properties, design 15 an entirely new class of devices, and highlight subtle but 16 important nanoscale and quantum phenomena. With this 17 perspective, we report on phonon engineering in metal-18 catalyzed silicon nanowires with tailor-made isotopic compo-19 sitions grown using isotopically enriched silane precursors 20 28 SiH 4 , 29 SiH 4 , and 30 SiH 4 with purity better than 99.9%. More 21 speci ﬁ cally, isotopically mixed nanowires 28 Si x 30 Si 1 − x with a 22 composition close to the highest mass disorder ( x ∼ 0.5) were 23 investigated. The e ﬀ ect of mass disorder on the phonon 24 behavior was elucidated and compared to that in isotopically pure 29 Si nanowires having a similar reduced mass. We found that 25 the disorder-induced enhancement in phonon scattering in isotopically mixed nanowires is unexpectedly much more signi ﬁ cant 26 than in bulk crystals of close isotopic compositions. This e ﬀ ect is explained by a nonuniform distribution of 28 Si and 30 Si isotopes 27 in the grown isotopically mixed nanowires with local compositions ranging from x = ∼ 0.25 to 0.70. Moreover, we also observed 28 that upon heating phonons in 28 Si x 30 Si 1 − x nanowires behave remarkably di ﬀ erently from those in 29 Si nanowires suggesting a 29 reduced thermal conductivity induced by mass disorder. Using Raman nanothermometry, we found that the thermal conductivity 30 of isotopically mixed 28 Si x 30 Si 1 − x nanowires is ∼ 30% lower than that of isotopically pure 29 Si nanowires in agreement with 31 theoretical predictions. 32

SiNWs. 15 No experiments have, however, been conducted to 82 elucidate these effects.With this perspective, we report in this 83 work the first experimental investigation of the influence of 84 isotope disorder on the phonon behavior in isotopically 85 engineered Si NWs.
The growth of NWs was carried out using the classical gold-(Au) catalyzed vapor phase epitaxy using monoisotopic silane 28 SiH 4 , 29 SiH 4 , and 30 SiH 4 with isotopic purity higher than 99.9%.These precursors were synthesized through the hydrogenation of isotopically enriched SiF 4 . 16The growth f1 conditions are provided in the Supporting Information.Figure f1 1a illustrates the schematics of the two sets of nanowires investigated in this work.The samples consist of isotopically pure 29 Si NWs and isotopically mixed 28 Si x 30 Si 1−x NWs.The former were grown by injecting the monoisotopic 29 SiH 4 precursor, whereas for the latter 28 SiH 4 and 30 SiH 4 were simultaneously introduced in the growth chamber.The control of the content of each isotope in the growing isotopically mixed nanowires was achieved through the control of the partial pressures of the two precursors.The low-magnification scanning electron microscope (SEM) images (taken at a tilt of 60°) of the 28 Si x 30 Si 1−x NWs and 29 Si NWs are displayed in the inset of Figure 1a.The grown NWs are typically 5 μm long with a diameter in the 30−100 nm range.Figure 1b shows the scanning transmission electron microscope (STEM) image of a 29 Si NW.The NW has grown in the [121] direction and the image is taken from [11̅ 1] Si zone axis.The SEM analysis indicates that while the majority of 29 Si NWs have grown along the [111] direction a few have actually grown at an angle of For the isotopically pure NWs the precursor is 29 SiH 4 (purity 99.9%) and for the isotopically mixed NWs, a mixture of 28 SiH 4 (purity 99.99%) and 30 SiH 4 (purity 99.9%) was injected.Crystallization of Si atoms from the supersaturated droplet takes place at the droplet− nanowire interface which becomes the growth front.Inset: Low-magnification SEM images of as grown isotopically mixed 28 Si x 30 Si 1−x NWs and isotopically pure 29 Si NWs, both recorded at a tilt angle of 60°.The scale bars in both the figures denote 1 μm.(b) STEM images of the isotopically pure 29 Si NWs.Left: A single 29   the NW.This is attributed to Au diffusion from the catalyst 114 droplet along the NW sidewalls during the quenching to room 115 temperature after growth interuption. 17,18The high-magnifica-116 tion STEM image of the NW included in Figure 1b and the 117 power spectrum (fast Fourier transform (FFT)) in the inset 118 show that the grown NWs are of the highest crystalline quality.
119 The STEM analysis of the 28 Si x 30 Si 1−x NWs (not shown here) 120 confirms that the two sets of NWs have identical structural and 121 morphological properties.
122 Raman spectroscopy was employed to investigate the 123 vibrational properties of these NWs.To enable the analysis 124 of individual NWs, the as-grown NWs were first transferred 125 onto Au-capped Si to suppress the background signal from the 126 underlying substrate during subsequent Raman analysis.127 Backscattering micro-Raman experiments were carried using 128 two laser lines 488 nm (low power measurements) and 514 nm 129 (high power measurements) at incident power densities in the 130 range of 0.08−17.76mW/μm 2 (see Supporting Information for 131 more details).The average length of the NWs after dispersion 132 on Au was found to be on the order of 2−3 μm.All Raman 133 measurements were performed on single NWs that are in an 134 excellent thermal contact with Au layer (Figure 1c).Clustered 135 and suspended NWs were avoided in this analysis as they heat 136 up faster when exposed to laser beam, which influences their f2 137 Raman modes. 19Figure 2 shows the Si−Si LO phonon spectra 138 of single 28  A first analysis of the data is based on the quasi-harmonic 148 approximation, which is a valid approximation for semi-conductors at room temperature. 20Herein, it is important to minimize the excess local heating of the NWs, which would occur when the incident laser power is sufficiently high.Hence, all calculations involving the quasi-harmonic approximation were carried out on data sets recorded at the lowest incident power density of 0.08 mW/μm 2 at which the local temperature of the NWs is confirmed to be equal to the ambient temperature of 300 K (Supporting Information).A close inspection of the spectra exhibited in Figure 2 shows two separate phonon related effects.First, at all incident laser powers the Raman spectrum for 28 Si x 30 Si 1−x NWs is always broader and red-shifted as compared to the spectrum of 29 Si NWs.Second, regardless of the type of the NW, as the incident power increases, all peaks broaden and redshift.This effect is f3 due to laser-induced heating of the NWs. Figure 3 depicts the evolution of the average peak position and the evolution of the average fwhm with incident power density for both 28  × ⟨m⟩ 30−Si .Note that composition calculated employing the quasi-harmonic approximation is always an average estimate.
In Figure 3a, the shift rate of the average peak position with power density up to 5−6 mW/μm 2 is only slightly higher for 28 Si x 30 Si 1−x NWs as compared to 29 Si NWs (the corresponding slopes of the dotted lines are 0.38 and 0.30 (cm −1 μm 2 )/(mW), respectively).At higher power densities, the behavior of 28 Si x 30 Si 1−x NWs is markedly different from 29 Si NWs with the average phonon frequency of the former undergoing a drastic redshift as compared to the latter (Figure 3d).The evolution of fwhm follows the same trend as the shift in phonon frequency.Below ∼6 mW/μm 2 , the average fwhm of 28 Si x 30 Si 1−x NWs and 29 Si NWs evolve qualitatively similarly with incident power density.At all incident power densities the spectra of 28 Si x 30 Si 1−x NWs are always broader than that of 29 Si NWs.This can be also verified from the respective spectra displayed in Figure 2. At low and mid power ranges, the average fwhm of the former is about 3−3.5 cm −1 broader than the latter.A detailed discussion on the origin of this effect will be presented later in the text.As it can be seen in Figure 3e, at high power regime the spectra of 28 Si x 30 Si 1−x NWs start to broaden much rapidly as compared to 29 Si NWs.The difference in absolute values of the average fwhm between Figure 3, panels b and e, comes simply from the difference in spectral resolution between the two setups (Supporting Information).
The redshift in peak position and broadening of Raman spectra with increasing power densities are indicative of how the NWs are reacting to laser-induced heating.From Figure 3, it can be seen that the behavior of the two types of NWs only show slight differences at low power, but at high power 28   calculated the phonon frequency for Nat Si at 0 K, 21 ω 0 Nat−Si = 529 cm −1 (using only the three phonon process).By taking into account the change in the reduced mass, we calculate ω 0 Iso−mix (for 28 Si x 30 Si 1−x NWs) and ω 0 29−Si (for 29 Si NWs) to be 519.80 and 520.81 cm −1 , respectively.C was estimated from the data recorded at the lowest laser power density (0.08 mW/μm 2 ) corresponding to a temperature of 300 K. Next, ∏(T) was calculated from Figure 3a,b for different incident power densities and the NW local temperature was then estimated as displayed in Figure 3c,f.The plot reveals that 28 Si x 30 Si 1−x NWs are getting slightly more heated up as compared to 29 Si NWs in the low power density regime (<6 mW/cm 2 ), the temperature of the former is higher by ∼10−15 K at 1.94 mW/ μm 2 and ∼25−30 K at 5.88 mW/μm 2 as compared to the latter.However, at the highest power of 11.76 mW/μm 2 the difference in temperatures is quite significant.Indeed, the temperature of 28 Si x 30 Si 1−x NWs is almost 120 K higher than that of 29 Si NWs.At low power regime, the rate of increase of temperature with increasing laser power, ΔT/ΔP, is ∼1.30times higher for 28 Si x 30 Si 1−x NWs than 29 Si NWs and becomes ∼6 times larger in the high power regime (6−12 mW/μm 2 ).
To evaluate the change in thermal conductivity between the two sets of NWs, which led to the result described above, we used Raman nanothermometry 22,23 in conjunction with a heat transport model.This model draws its basis from the assumptions that the region of the NW exposed to laser acts as the heat source and that the major portion of the generated heat is dissipated by conduction along the NW growth axis and at the NW-Au interface (Supporting Information Figure S3).Details of the model are provided in the Supporting Information.Around 300 K, based on the rate of increase of temperature, we estimated the ratio of thermal conductivities of 29   Si NWs to 28 Si x 30 Si 1−x NWs κ Si−29 /κ Iso−Mix to be ∼1.30.This means there is a ∼30% decrease in κ Iso−Mix as compared to κ Si−29 .Interestingly, this value is close to the theoretical prediction of 30% reduction in case of 28 Si 0.5 30 Si 0.5 bulk alloys as compared to Nat Si at room temperature, 24 but it is slightly higher than ∼20% reduction predicted for 28 Si 0.5 29 Si 0.5 NW as compared to 28 Si NW at 300 K. 14 It is also noteworthy that the observed mass disorder-induced change in Si NW thermal conductivity is lower the 50% reduction demonstrated for 12 C 0.5 13 C 0.5 graphene as compared to purified 13 C graphene. 25r Si NWs, Yang and co-workers predicted that at room temperature a much significant reduction in thermal conductivity up to ∼70% can be achieved when a 28 Si NW is mixed at 50% with 42 Si atoms. 14However, 42 Si being radioactive with a half-life of ∼13 ms cannot obviously be implemented for any practical purpose.Note that the ratio of thermal conductivity was specifically calculated in the low power regime because the fluctuations in the measured peak position of different 28 Si x 30 Si 1−x NWs at high power regime are very large (Supporting Information Figure S2:Ab) thus making the estimation of the temperature of 28 Si x 30 Si 1−x NWs fraught with large uncertainties.
Although our experimental data are consistent with early theoretical predictions, isotope mixing alone cannot explain all the observed differences in phonon properties between isotopically disordered and pure NWs.In Figure 3b, the fwhm of 28 Si x 30 Si 1−x NWs is consistently larger by almost 3−3.5 cm −1 at all incident power compared to the 29 the following equation 21

346
(3) 347 Similar to eq 1 we have neglected the four phonon 348 interaction and normalized the data for Δ Anhrm Iso−Mix Bulk at T = 6 349 K to find the constant "A".ω 0 for the bulk 28 Si 0.50 30 Si 0.50 sample was calculated by the same approach used to calculate ω 0 Iso−Mix and ω 0 29−Si .We found Δ Anhrm Iso−Mix Bulk (T = 300 K) = 1.36 cm −1 .
The difference in spectral resolution between our setup and the In comparison, the fwhm of isotopically mixed 28 Si x 30 Si 1−x 358 NWs, after correcting for the spectral resolution, consists of 359 three contributions Iso Mix NW Anhrm Iso Mix NW Isotope Iso Mix NW Surface Iso Mix NW 360 (5)

Δ Surface
Iso−Mix NW is the broadening due to surface scattering of 362 phonons.This contribution is peculiar to NWs but absent for 363 bulk materials.Similarly for the fwhm of the isotopically pure   29 Si bulk crystals at different incident laser (488 nm) power densities: 1.94, 0.97, 0.17, and 0.08 mW/μm 2 .(b) Evolution of average peak position and (c) evolution of average fwhm with incident laser power density for both 28 Si 0.6 30 Si 0.4 and 29 Si bulk samples extracted from the corresponding Voigt fits.In all of the three figures the data points for the 28 Si 0.6 30 Si 0.4 bulk sample are shown in empty blue squares and that of 29 Si bulk sample are shown in filled black circles.The averaging was done over measurements on four different spots on each sample.In (a), the red and the green curves correspond the Voigt fit of the respective raw data.In (b,c), the error bars in both the figures are smaller than the data symbols used.The dotted lines in both the figures are guide to the eye.
for both types of NWs investigated in this work as they have comparable diameters.It is therefore reasonable to conclude that the surface induced broadening of Raman spectra is the same for both types of NWs, that is Δ Surface Iso−Mix NW ≈ Δ Surface Si−29 NW = 0.88 cm −1 .Now, the left-hand side (LHS) of eq 5 at the lowest incident laser power density is 5.5 cm −1 , whereas the right-hand side (RHS) after summing up Δ Anhrm Iso−Mix NW , Δ Isotope Iso−Mix NW , and Δ Surface Iso−Mix NW equates to 2.3 cm −1 .The fact that the equality of LHS and RHS in eq 5 does not hold suggests that there must be some other source of spectral broadening that has not been considered in eq 5. We rule out the possibilities of phonons scattering at crystallographic defects because the two sets of NWs are of high crystalline quality.This suggests that the excess broadening is induced by effects other than those listed above.
To verify the calculations above, we performed a series of control experiments on isotopically mixed and isotopically pure bulk samples.These isotopically engineered bulk crystals were grown by floating zone technique.The secondary ion mass spectrometer analysis (not shown here) estimated that the isotopically mixed bulk sample is composed of about 60% of 28   Si and 40% of 30 Si.The spectra of both 29 Si and 28  Si 0.4 sample is slightly smaller than that of 29 Si bulk sample.Consequently, the spectra of the former are blueshifted at all incident power as compared to the latter.The evolution of the average peak position and the average fwhm with incident laser power densities for both bulk samples are shown in Figure 4, panels band c, respectively.The data displayed in Figure 4b,c were averaged over measurements on four different spots on each sample.The 28 Si 0.6 30 Si 0.4 and 29 Si bulk sample have peaks at 514.9 and 512.3 cm −1 , respectively.Unlike the case of Si NWs, these phonon frequencies do not change significantly with increasing laser power density.This is an expected behavior because the effect of laser heating is ineffective in bulk samples, which have higher thermal conductivities as compared to NWs.The average fwhm, nearly 3 cm −1 smaller than those measured for NWs, also shows a very limited increase with laser power density that is almost identical for both bulk samples.It is worth noting that the Si−Si mode of the 28 Si 0.6 30 Si 0.4 bulk sample is broader only by 0.4 cm −1 at the lowest incident laser power than the Si−Si mode of 29 Si bulk sample, which is significantly less than the 3.2 cm −1 difference found between the modes of the two sets of NWs.The difference of 0.4 cm −1 between the average fwhm of the two bulk samples at the lowest laser power density is not entirely coming from isotope scattering effect.Indeed, the average mass of 28 Si 0.6 30 Si 0.4 being smaller than the average mass of 29 Si, the anharmonic scattering of phonons, which scales inversely with average mass, is larger in the 28 Si 0.6 30 Si 0.4 bulk sample as compared to that of the 29 Si bulk sample at a fixed temperature.The contribution of this excess anharmonic phonon scattering to the Raman line width is hidden within the difference of 0.4 cm −1 between the average fwhm of the two bulk samples.A plausible explanation of the observed broadening is the nonuniform mixing of 28 Si and 30 Si isotopes during the VLS growth of 28 Si x 30 Si 1−x NWs.Indeed, the excess broadening of the Raman spectra for the 28  kept the same as that of an isotopically pure 29 Si NW.Peak 1 (red) is at 508.77 cm −1 , peak 2 (green) is at 512.07 cm −1 , and peak 3 (black) is at 515.33 cm −1 .The local compositions corresponding to these three peaks are 26.9, 45.8, and 65.3%, respectively.The estimated uncertainty from the spectral resolution of our Raman setup is about 7%.
Interestingly, Raman spectra recorded along the growth axis of individual 28 Si x 30 Si 1−x NWs show that neither the peak position nor the fwhm of Si−Si mode vary along the nanowire growth axis (Figure 5b).This suggests that the isotopic content is uniform along the growth axis and thus the inferred nonuniformity of the isotopic content seems to be associated with the radial distribution of the two isotopes.To verify this intriguing observation, the nanowires investigated by Raman were also analyzed using atom probe tomography (APT), which is the only technique capable of providing the threedimensional (3D) distribution of different isotopes in a nanoscale structure with a near atomic resolution.Details of the APT analysis will be reported elsewhere.Figure 5c displays the radial profiles of 28 Si and 30 Si isotopes across the diameter of an isotopically mixed nanowire.The average isotopic compostion as estimated from APT 28 Si 0.41 30 Si 0.59 which is close to the content obtained from Raman analysis ( 28 Si 0.47 30 Si 0.53 ).Importantly, we note that, as predicted from Raman spectra, APT analysis also confirms that the radial distribution of the two isotopes is not uniform, whereas their profiles along the growth axis (not shown here) remain unchanged also in agreement with Raman data (Figure 5b).Moreover, APT profiles demonstrate that the two isotopes are distributed in three different regions (Figure 5c): (1) Near the surface where 28 Si ( 30 Si) content is higher (lower) than its average content in the entire nanowire (region I).The width of this region is about 26.3% of the nanowire diameter.
(2) At the core of the nanowire where 30 Si ( 28 Si) content is higher (lower) than its average content in the entire nanowire (region III).The width of this region is about 34.3% of the nanowire diameter.
(3) A transition region between the two regions I and III where the content of 30 Si ( 28 Si) increases (decreases) monotonically inward from nanowire surface to its core.The width of this region is about 39.4% of the nanowire diameter.
The average isotopic composition of each region is x (I) = 0.50 ± 0.01 ( 28 Si 0.5 30 Si 0.5 ); x (II) = 0.35 ± 0.01 ( 28 Si 0.35 30 Si 0.65 ); and x (III) = 0.75 ± 0.01 ( 28 Si 0.25 30 Si 0.75 ).Clearly, APT analysis confirms Raman-based observations reported above.At the same time, the 3D atom-by-atom distribution of each isotope within a single nanowire also raises fundamental questions about the basic mechanisms and dynamics of the VLS growth.Addressing these very important questions extends beyond the main focus of this Letter.
In summary, we have demonstrated the growth of isotopically mixed Si NWs via the VLS process using isotopically enriched silane precursors 28 SiH 4 , 29 SiH 4 , and 30 SiH 4 .Using Raman spectroscopy, the vibrational properties of these NWs were investigated and compared to that of isotopically pure 29 Si NWs having a close reduced mass.The outcome of the comparative study indicates that there is an enhanced phonon scattering in isotopically mixed NWs, which manifests itself at two interrelated levels.First, the measured Raman spectra of the 28 Si x 30 Si 1−x NWs were found to react to laser power quite differently from those of 29 Si NWs.The redshift in peak position and broadening of Raman spectra are more significant for the former as compared to the latter with the local

Figure 1 .
Figure1.(a) A schematic illustration of the VLS growth of the isotopically engineered Si NWs.Vapor phase precursors are supplied to Au−Si eutectic droplet.For the isotopically pure NWs the precursor is29 SiH 4 (purity 99.9%) and for the isotopically mixed NWs, a mixture of28 SiH 4 (purity 99.99%) and30 SiH 4 (purity 99.9%) was injected.Crystallization of Si atoms from the supersaturated droplet takes place at the droplet− nanowire interface which becomes the growth front.Inset: Low-magnification SEM images of as grown isotopically mixed28 Si x 30 Si 1−x NWs and isotopically pure 29 Si NWs, both recorded at a tilt angle of 60°.The scale bars in both the figures denote 1 μm.(b) STEM images of the isotopically pure 29 Si NWs.Left: A single 29 Si NW.The NW have grown along the [121] direction and the image taken from the [11̅ 1] Si zone axis.The scale bar in the figure is 200 nm.Middle: STEM image of the NW sidewalls showing gold decoration on the facets.The scale bar in the figure is 20 nm.Right: High-magnification STEM image (taken from the region marked by the red box in the middle image) and the power spectrum (FFT) in the inset shows the high crystalline quality of the NW.The scale bar in the figure corresponds to 1 nm.(c) SEM image of a single 29 Si NW after sonication and dispersion atop Au capped Si substrate.The scale bar denotes 1 μm.
Figure1.(a) A schematic illustration of the VLS growth of the isotopically engineered Si NWs.Vapor phase precursors are supplied to Au−Si eutectic droplet.For the isotopically pure NWs the precursor is29 SiH 4 (purity 99.9%) and for the isotopically mixed NWs, a mixture of28 SiH 4 (purity 99.99%) and30 SiH 4 (purity 99.9%) was injected.Crystallization of Si atoms from the supersaturated droplet takes place at the droplet− nanowire interface which becomes the growth front.Inset: Low-magnification SEM images of as grown isotopically mixed28 Si x 30 Si 1−x NWs and isotopically pure 29 Si NWs, both recorded at a tilt angle of 60°.The scale bars in both the figures denote 1 μm.(b) STEM images of the isotopically pure 29 Si NWs.Left: A single 29 Si NW.The NW have grown along the [121] direction and the image taken from the [11̅ 1] Si zone axis.The scale bar in the figure is 200 nm.Middle: STEM image of the NW sidewalls showing gold decoration on the facets.The scale bar in the figure is 20 nm.Right: High-magnification STEM image (taken from the region marked by the red box in the middle image) and the power spectrum (FFT) in the inset shows the high crystalline quality of the NW.The scale bar in the figure corresponds to 1 nm.(c) SEM image of a single 29 Si NW after sonication and dispersion atop Au capped Si substrate.The scale bar denotes 1 μm.

Si x 30 Si 1−x and 29
Si NWs.In Figure 3a,b are displayed the data recorded at low laser power densities averaged over a large number (>10 of single NWs).High-power measurements are given in Figure 3d,e.The peak position and fwhm of 4−5 individual 28 Si x 30 Si 1−x NWs and 29 Si NWs, as extracted from the Voigt fit of the raw data, at both low and high power levels are displayed in Figure S2 (Supporting Information).Interestingly, both the broadening and the redshift are found to be more pronounced for 28 Si x 30 Si 1−x NWs.For instance, the average peak position within the laser power range investigated varies by about 4 cm −1 for 28 Si x 30 Si 1−x NWs as compared to ∼1 cm −1 for 29 Si NWs.This indicates that the two types of NWs react differently to local heating induced by laser.In the following, we elucidate the origin of the remarkable changes in Raman spectra as a function of the NW isotopic content.According to the virtual crystal approximations (VCA), a simple harmonic analysis predicts that the energy of a phonon mode is inversely proportional to the square root of the average isotopic mass 1 −ω phonon ∝ (1/⟨m⟩) 1/2 .Here m is the average isotopic mass given by ⟨m⟩ = ∑ i c i m i , with c i being the fractional composition of an isotope of mass m i .Using the ratio of the average peak position at the lowest incident power density of 0.08 mW/μm 2 (Figure 3a) and the value of ⟨m⟩ 29−Si , we computed ⟨m⟩ Iso−mix = 29.05amu.Thus, the corresponding fractional composition of 28 Si in the isotopically mixed NWs is x = 0.47 ± 0.07 calculated from the known values of ⟨m⟩ 28−Si and ⟨m⟩ 30−Si in the identity: ⟨m⟩ Iso−mix = x × ⟨m⟩ 28−Si + (1 − x)

Figure 2 .
Figure 2. Si−Si LO normalized phonon spectra of 28 Si x 30 Si 1−x NW and Si x30 Si 1−x NWs are much more affected than29 Si NWs.A convenient way to quantify this heating effect is to extract the NW local temperature.Herein, an estimate of the effective local 226 temperature is made from the shift in average peak position 227 with the incident laser power.The peak position of a NW, 228 Ω(T) at a temperature "T", is given by 21 Ω(T) = ω 0 + ∏(T), 229 where ω 0 is the peak position at 0 K and ∏(T) is the shift of 230 peak position at T, given by 1) 232 where "C" and "D" are constants.The first term is related to 233 three-phonon anharmonic interaction and the second term 234 represents the four phonon interaction.The probability of the 235 latter being small, we can reasonably neglect it to be left with 236 the first term in the right-hand side of (1).Balkanski et al.

Figure 3 .
Figure 3. (a,b) Measurements using 488 nm laser at low incident power density;(d,e) measurements using 532 nm laser at high incident power density.In all figures, the empty blue squares correspond to the isotopically mixed 28 Si x 30 Si 1−x NWs and the filled black circles represent the isotopically pure29 Si NWs.(a,d) Evolution of average peak position with incident laser power density for both28 Si x 30 Si 1−x NWs and 29 Si NWs.(b,e) Evolution of the average fwhm with incident laser power density for both 28 Si x 30 Si 1−x NWs and 29 Si NWs.In panels a and b, the averaging was done over measurements on more than 10 single NWs, and in panels d and e the averaging was done over measurements on 7 single NWs.The error bars in panels a, b, d, and e are double the standard deviation of the peak position and fwhm from respective average values.(c,f) Plots of the effective local temperature of the NWs extracted from the shift in average peak position in panels a and d, respectively.The error bars represent the uncertainty in the calculated temperature due to the standard deviation of the measured peak position.The dotted lines in panels a−f are guides to the eye.
take Δ Anhrm Iso−Mix Bulk | T = Δ Anhrm Iso−Mix NW | T and Δ Isotope Iso−Mix Bulk = Δ Isotope Iso−Mix NW , because the anharmonic scattering and isotope scattering of phonons depend on the temperature and isotopic composition, but not on the size of the material as long as confinement effects are unimportant.Thus, Δ Isotope Iso−Mix NW = 0.065 cm −1 and at the lowest incident laser power density, corresponding to a temperature of about 300 K, Δ Anhrm Iso−Mix NW is 1.36 cm −1 .We can now relate Δ Anhrm Si−29 NW to Δ Anhrm Iso−Mix NW through the inverse mass relation at a fixed temperature Δ Anhrm | T ∝ 1/⟨m⟩, giving Δ Anhrm Si−29 NW (T = 300 K) × Δ Anhrm Iso−Mix NW (T = 300 K) × (⟨m⟩ Iso−Mix /⟨m⟩ Si−29 ) ≈ 1.36 cm −1 .Putting the value of the fwhm of 29 Si NWs at the lowest incident power and Δ Anhrm Si−29 NW in eq 6 we deduce the contribution of surface scattering to the NWs 29 Si− 29 Si Raman peak broadening: Δ Surface Si−29 NW 0.88 cm −1 .Note that the contribution of surface scattering of phonons, which is diameter dependent, is the same

Figure 4 .
Figure 4. (a) Si−Si LO normalized phonon spectra of28 Si 0.630 Si 0.4 and29 Si bulk crystals at different incident laser (488 nm) power densities: 1.94, 0.97, 0.17, and 0.08 mW/μm 2 .(b) Evolution of average peak position and (c) evolution of average fwhm with incident laser power density for both Si x 30 Si 1−x NWs probably originates from the overlap of several narrower peaks corresponding to different regions within a NW with different isotopic content.f5 For instance, in Figure 5a the Raman spectrum of a single 28 Si x 30 Si 1−x NW recorded at the lowest laser power is 445 deconvoluted in three different peaks corresponding to a 28 Si-446 rich area, a 30 Si-rich area, and a transition zone.Because the 447 broadening due to isotopic scattering of phonons at 300 K is 448 only 0.065 cm −1 the fwhm of each of the three peaks has been 449

Figure 5 .
Figure 5. (a) The spectrum of a single 28 Si x 30 Si 1−x NW at an incident power density of 0.08 mW/μm 2 , data points shown in empty blue squares and the cumulative Voigt Fit (pink) has been simulated using the convolution of three different spectrum (red, green, and black) each representing different isotopic composition (details in text) within the NW; (b) peak position and fwhm profiles measured along the growth axis of single 28 Si x 30 Si 1−x nanowires.Each data point is an average over a few measurements on different nanowires.The horizontal dashed lines indicate the average values; (c) APT radial profile of 28 Si (red) and 30 Si (blue) isotopes across the diameter of an isotopically mixed nanowire.The offset in x-axis reflects the thickness of the Ni protective layer deposited around the nanowire to prevent any damage that may occur during FIB processing.
110 ∼19.5°with respect to the [111] direction corresponding to 111 the [121] crystallographic direction.It is noticeable that the 112 NW surface is decorated with Au clusters mainly near the tip of 113 DOI: 10.1021/acs.nanolett.5b00708Nano Lett.XXXX, XXX, XXX−XXX B