Corrigendum: Observation of strongly enhanced photoluminescence from inverted cone-shaped silicon nanostructures

Silicon nanowires (SiNWs) attached to a wafer substrate are converted to inversely tapered silicon nanocones (SiNCs). After excitation with visible light, individual SiNCs show a 200-fold enhanced integral band-to-band luminescence as compared to a straight SiNW reference. Furthermore, the reverse taper is responsible for multifold emission peaks in addition to the relatively broad near-infrared (NIR) luminescence spectrum. A thorough numerical mode analysis reveals that unlike a SiNW the inverted SiNC sustains a multitude of leaky whispering gallery modes. The modes are unique to this geometry and they are characterized by a relatively high quality factor (Q ~ 1300) and a low mode volume (0.2 < (λ/neff)3 < 4). In addition they show a vertical out coupling of the optically excited NIR luminescence with a numerical aperture as low as 0.22. Estimated Purcell factors Fp ∝ Q/Vm of these modes can explain the enhanced luminescence in individual emission peaks as compared to the SiNW reference. Investigating the relation between the SiNC geometry and the mode formation leads to simple design rules that permit to control the number and wavelength of the hosted modes and therefore the luminescent emission peaks.


Supplementary information S2:
The PL intensity from C1, C2, C3, and C4 was baseline corrected between 1000 and 1200nm (black line) and fitted (red line) by a sum of multiple Voigt peak profiles where each profile is given by and Here, and are a Gaussian and Lorentzian distribution with the widths and , respectively. The spectral peak width Δ is then given by the approximation Δ = 1.1 + �0.9 2 + 8 ln 2 21 . Accordingly the Qfactors were calculated as where is the spectral position of the peak maximum.

Supplementary information S3:
For the mode analysis, broadband dipole pulses (850-1250nm) polarized in x-and z-direction ( , ) were excited in the maxima of the pump laser absorption in a SiNC with the geometry of C4 and a SiNW with geometry NW1 (compare Fig. 1c). In a and b it can be seen that within the SiNW attached to the wafer, the pulse energy (proportional to E 2 ) decays much faster i.e. it is by far more optically 'leaky' than the SiNC, which is able to retain more optical energy over a longer time span. c and d show a Fourier transformation of the optical power emitted through the top facet of the SiNW and SiNC for t>600fs. While the spectra for the SiNC for different excitation still shows a strong emission in a multitude of sharp peaks, the emission of the SiNW is much weaker and only a few shallow peaks are visible in the spectrum. This is in good agreement to the PL spectra in Fig. 2a where in contrast to the SiNW the emission of the SiNC shows strong additional peaks.
wavelength [nm] Supplementary information S4: The figure shows the good agreement between the peak positions in the experimental emission spectrum of C4 and the positions of radiative energy maxima extracted from the numerical simulations. However, only about 63% of the peaks found experimentally are confirmed numerically. This can be explained by the fact that slight deviations in the geometry of the real SiNC C4 lead to the occurrence of additional modes and/or peak shifts that are not found by the simulations based on the ideal geometry of an inverted cone with dimensions as given in S1. Supplementary information S5: Figure 3c in the manuscript displays the xy cross sectional energy density for the WGMs at 1153nm, 1119nm, 1176nm (2x), and 1104nm. The exemplary chosen modes typically represent the branches HE61, HE81, HE101, HE81a, and HE101a identified in Fig. 3b. In HEij, the indices i and j correspond to the number of azimuthal and radial nodes, respectively 2 . With the additional index a, we distinguish the two appearance forms of HE81 and HE101.

Supplementary information S6:
The NA is estimated by a numerical analysis 3 of the far field radiation intensity of the modes in Fig. 3. We excite the SiNC C4 with a broadband dipole ( , as described in the main text) and monitor the radial distribution of the radiative energy in the z direction at a distance of z1=20nm and z2=200nm above the top facet (see the right scheme below). The graph shows the normalized radial distribution of the radiative intensity from the 922nm mode for the two distances. For simplified analysis, we use the double distance of the outer maxima, i.e. 80nm, as the broadening of the light cone between the distance of z1=20nm and z2=200nm away from the top facet. We apply

Supplementary information S7:
The temporal change of carriers in a unit volume of a solid state emitter is given by where is the generation rate and the sum of all carrier recombination processes. We rewrite as with representing the portion of the absorbed excitation intensity , generating elementary charges in the volume of a SiNW or SiNC. can be expressed as the sum of all radiative and non-radiative recombination processes = + + (8) where is the spontaneous (radiative) emission rate, is the sum of no non-radiative recombination processes (and carrier leakage) and is the stimulated radiative emission. Assuming stationary conditions / = 0 and a low photon density leading to ≅ 0 the combination of (6), (7), and (8) leads to 4 = + (9).
Using (9)  Here, ℎ / is the energy of a photon with wavelength and is absorbed fraction of the photon flux injected by the pumping laser. To find the radiative emission emitted in direction of the analyzer, an out-coupling efficiency can be introduced, and accordingly 4 = ℎ . (13) Supplementary information S8: (10) can be rewritten as in which is the non-radiative recombination lifetime and is the spontaneous emission lifetime. We calculate for C4 an NW1 according to the approximation for SiNWs 5, where is the surface recombination velocity, is the bulk lifetime and the diameter of the SiNW. This approximation is valid for < 2 where is the diffusion constant of carriers in Si. Since = 5 • 10 2 for an oxidized Si surface 7 and 2 ≅ 7 • 10 3 with = 36 2 and ≅ 500 , using (8) is justified in the presented case. The bulk lifetime of carriers in crystalline Si (n-type, 1-5Ωcm) can range between ≅ 10 −4 − 10 −108,9 . For a similar bulk material quality it is strongly decreasing at high injection levels (Auger-effect), so the value of in (8) for a low injection is determined by the (in this case) very low surface recombination lifetime = 4 / ≅ 10 −8 , where in contrast for a very high injection it will be dominated the bulk Auger-recombination. Since C4 and NW have similar surface properties (SiO2passivation) and bulk material quality and their volume and surface/volume ratio is roughly the same, we can expect their to be comparable for the same injection conditions. These are in fact given for the compared experimental spectra (Fig. 2a,b) that both have been acquired under excitation with an 1.28mW CW laser at 660nm, for which both structures absorb about 40% of the light incident at the top facet (Fig.  1c).
can be calculated as 10 where the radiative recombination probability = 1.1 • 10 −14 3 and is the photo generated carrier density under optical pumping. Accordingly, is dependent on intrinsic material properties and carrier injection, and therefore (as described above for ) will be comparable for C4 and NW1 under the given experimental conditions. This means that if a further Purcell enhanced emission can be neglected, has about the same dimension for C4 and NW1.

Supplementary information S9:
The mode volume of all modes determined in C4 (Fig. 3b)  We find 0.01µ 3 < < 0.05µ 3 intuitively a higher mode volume for a higher orbit of the leaky WGMs in the structure. For the mode at λ=1027nm we find ≅ 0.01µ 3 (see below a visualization of the torus containing the ½ of the mode optical energy).