Supplementary Figure S1: typical assembly scheme for fabrication of Si core preform cm long 8 mm / 2 mm, and 15 cm section of 2 mm rod are

Transcription

1 Supplementary Figure S1: typical assembly scheme for fabrication of Si core preform cm long 8 mm / 2 mm, and 15 cm section of 2 mm rod are sleeved and fused with a thick rod to seal the bottom cm long Si sod is inserted into the pocket. The bottom is vacuum fused to ensure hermetic closure cm long 2 mm rod is inserted to close the pocket and the structure is vacuum fused on the top. 1

2 Supplementary Figure S2: typical assembly scheme for fabrication of preform for fibre redraw cm long 12 mm / 6 mm, 6mm / 1mm tubes and 15 cm section of 1mm rod are sleeved and fused with a thick rod to seal the bottom cm long fiber section is inserted into the pocket. The bottom is vacuum fused to ensure hermetic closure cm long 1mm rod is inserted to close the pocket and the structure is vacuum fused on the top. 2

3 Supplementary Figure S3: side view of the fibre used for torch-scaling process with OD = 280 µm and D = 4 µm (left) and the side view of the fibre after the triple-iteration scaling (right). This is a typical result of the fibre high-tension low-ratio downscaling using a hydrogen torch. The scaling process here is applied to a fibre with outer diameter of the silica cladding of OD = 280 µm and a core diameter of D = 4 µm and consists of 3 scaling redraw iterations. The first redraw is conducted using a heating flame resulting from a hydrogen flow of 0.23 l/sec, second - of 0.21 l/sec, third - of 0.19 l/sec. Feed speed v f and draw speed v d were chosen to be 0.05 mm/s and 0.25 mm/s, respectively. The scaling factor of the fibre cross section resulting from one iteration is n = v d v f = 5. Beginning with OD = 280 µm after a triple-iteration process we should end with OD = OD = 25 µm. We estimate geometrically from those pictures, given OD=280 μm and D=4 μm for the original fibre, that the resulting OD'=(24±1) µm, assuming that the cross sectional geometry of the fibre is preserved, which is a reasonable assumption for the high tension draw process. The resulting core size is expected to be D'=(340±15) nm. 3

4 Supplementary Figure S4: Viscosity of silica μ SiO 2 as a function of temperature between Si melting and boiling points. In comparison, μ Si decreases only of 16% over the same temperature range, from 7.8x10-4 to 6.6x10-4 Pa.s. Under uniform heating, capillary breakup of a long in-silica-fibre Si core is predicted to occur with a regular spatial period λ T entirely set by the core diameter D, the Si/silica interfacial tension γ and the viscosity contrast μsi μ SiO 2 between Si (viscosity μsi ) and silica (viscosity μ SiO 2) at the chosen temperature T (42). In theory, the range of temperatures for which breakup can be induced is comprised between Si melting (1,414 o C) and boiling (2,357 o C) points. While μ Si varies very little over that range (15), μ SiO 2 drop by several order of magnitudes (41), and the viscosity contrast thus highly depends on T. Note that the drop of μ SiO 2 makes breakup impractical for temperatures above 2,200 o C, beyond which silica itself is too liquid. 4

5 Supplementary Figure S5: Viscosity ratio μsi (solid blue line left axis) and resulting dominant μ SiO 2 period λ T as a function of uniform heating temperature. λ T is displayed relative to its minimal possible value, λ min = πd (solid red line right axis), and the ratio is thus independent of the core size a. Despite its large variations with T, μsi μ SiO 2 remains very small, which means that the dominant wavelength λ T is always tens to hundreds of times larger than the smallest physically achievable breakup period, λ min = πd. Since the final sphere size is tied to the breakup period via volume conservation, the naturally dominant long breakup periods under uniform heating make this method unsuitable to yield submicron spheres. 5

6 Supplementary Figure S6: approximated temperature profile used for the simulations, with origin x = 0 at the point of liquefaction of Si (T=1,414 o C). 6

7 Supplementary Figure S7: Time to achieve dominance τ(λ, v f ) (solid blue line left axis) for v f = 10 μm.s -1, the minimum is reached for λ d = 73 μm. For comparison, the inverse of the growth rate under uniform heating at T=1,770 o C is given (solid red line right axis), and the minimum is not reached in the plot window, thus λ T > 900 μm. 7

8 Supplementary Figure S8: Predicted λ d as a function of feed speed v f (red triangles) and experimental data (black squares, identical to Figure 2c). The discrepancy shows that the real σ(λ, x) must differ from the one derived here in a way that advantages the longer wavelengths. 8

9 Supplementary Figure S9: optical microscope frames in a transmission mode of a 323 nm core fibre. The optical frames I, II, and III are taken 3mm one from another to assure the macroscopic uniformity of the sample. The spheres released from the same samples were inspected around the same location where those images are taken and were used for statistical characterization of the resulting sphere size distribution in Supplementary Fig. S10. Breakup conditions are: hydrogen flow 0.17 l/m, oxygen flow 0.08 l/m, feed speed 10 µm/sec. Unlike the uniform-heating induced breakup, the gradual liquefaction approach yields sphere of very regular dimensions. 9

10 Supplementary Figure S10: left - Statistical analysis of the sphere size distribution, where a Gaussian distribution (red solid line) is fitted to the measured distribution in sphere sizes (gray histogram and cumulative green dotted line). The statistical analysis on that sample reveals that the diameter distribution fits to a normal distribution centered at (460 ± 24) nm; right - Aberration-free optical images of the fibre-embedded chain of spheres immersed in silica RI-matched liquid taken in transmission (top) and reflection (bottom) illumination modes. The fibre-embedded sphere chain immersed in a silica RI-matched liquid (50350, Cargille Laboratories) and observed using a confocal microscope both in transmission and reflection mode. 10

11 Supplementary Figure S11: optical microscope frames in a transmission mode of a 4 µm core fibre breaking up into 21 µm spheres (top) and the same fibre scaled by factor of 5 breaking up into 6 µm spheres. Breakup conditions are: hydrogen flow 0.3 l/m, oxygen flow 0.1 l/m, feed speed 20 µm/sec. 11

12 Supplementary Figure S12: largest spheres produced: the frame shows optical microscope snapshot in a reflection mode of a 100 µm core fibre breaking up into 250 µm spheres. Breakup conditions are: hydrogen flow 0.47 l/m, oxygen flow 0.31 l/m, feed speed 10 µm/sec. The Supplementary Fig. S9, S11, S12 prove the scalability in size of the sphere fabrication process over 3 orders of magnitude between the scales of hundreds of nanometers to hundreds of microns. 12

13 Supplementary Figure S13: Electron diffraction patterns of three different sphere cross-sections, showing representative multicrystalline grain structure. The dashed circles depict the aperture sizes corresponding to the respective diffraction patterns. Arrows point at some of the observable grain boundaries. 13

14 Supplementary Figure S14: EDX analysis of the sphere shown in Figure 4b confirming the absence of oxide formation at grain boundaries within the core of the sphere. EDX line scans are overlaid onto a dark field (DF) STEM micrograph (Scale bar 100 nm). 14

15 Supplementary Figure S15: Optical microscope images of a dual-core fibre, one core is p-type Si, and the other core is n-type Si. 15

16 Supplementary Figure S16: Current-voltage characteristics (semilog) of the pn-diode (black) and of its constituent p-type Si (red) and n-type Si (blue) spheres. Current was capped at 3 ma to prevent possible damages to the structure, limiting the range of voltages applied to the most conductive n-type sphere. 16

17 Supplementary Note 1. Building a predictive model for sphere size scaling with feed speed Due to the complexity of the dynamics of Si core reshaping, constructing a predictive model for the final sphere size as a function of the selected feed speed v f is mathematically involved, but a framework can be developed. As mentioned in caption of Supplementary Fig. S5, the final sphere size is tied to the observed dominant breakup period λ d via volume conservation. With an analogous treatment as in the isothermal regime, an instability growth rate can be associated with each possibly observable breakup period λ. However, because the temperature is not uniform, and instead is a function of T(x) of the position x within the flame, these growth rates are now both spectrally and spatially dependent, which writes σ(λ, x). If we define the time t = 0 as the time when the Si core tip has been liquefied at the liquefaction front position x = 0 then between the time t and t + dt the amplitude of a perturbation of wavelength λ grows by a factor e σ(λ,x λ(t,v f ))dt, where x λ t, v f is the position of the portion of molten silicon considered, at the time t and feed speed v f. Since breakup occurs from the tip and one sphere at a time, only the section of length λ including the tip is relevant, and thus x λ t, v f should be chosen anywhere between the tip position v f t and exactly one period λ behind, at x = v f t λ. This range of possible choice for the representative position of that section of molten Si can be seen as a fit parameter for the model. We can then define a breakup period dominance criterion M as the logarithm of the perturbation amplification necessary to achieve for a wavelength to establish dominance and set the observed breakup period. Mathematically, this write: τ(λ,v f ) τ 0 (λ,v f ) σ(λ, x λ (t, v f ))dt = M (S1) where τ(λ, v f ) is the time at which the considered breakup period λ would establish dominance at feed speed v f, provided no other period already has, and τ 0 (λ, v f ) is the time after which growth of the considered molten Si section is physically permitted. Indeed, for such growth to be possible, at least a length λ of Si must have been liquefied, and thus τ 0 λ, v f = λ. Assuming that the Si core initially v f displays small imperfections at all wavelengths, which can grow due to capillary forces, M would typically lie in the 5-10 range, yielding amplifications of these initial perturbations of a factor e M ranging from a few hundreds to tens of thousands. Resolving the dominance criterion equation above and obtaining τ(λ, v f ) for every possible λ and for a given feed speed v f would allow us to predict the dominant breakup period λ d, as it is simply the one for which τ(λ, v f ) is minimal. However, doing so requires using growth rates σ(λ, x) uniquely adapted to our particular setup. Determining an accurate analytical expression of σ(λ, x) is beyond the scope of this paper and will be the object of future, dedicated work. However, for a given temperature profile T(x), we can derive an estimate of σ(λ, x) based on the growth rate expression given in (15) for uniform-heating induced breakup of infinite cylinders. Clearly, our system differs widely from the conditions of applicability of this expression, but insight can be gained from the discrepancy found 17

18 between the predicted results using this method and the experimental results displayed in Figure 2, obtained with a 4μm Si core (a=2 μm). An approximated temperature profile T(x) is generated for the simulations, as displayed in Supplementary Fig. S6. Only the part of the flame where the temperature rises is modeled, to reflect the experimental observation that spheres detach from the Si core before reaching the middle point of the flame. The interfacial tension γ at the Si/silica interface is considered equal to 1.5 J.m -2, independently of the temperature. As mentioned above, this model offers some freedom in the choice of representative position for a section of molten Si, x λ t, v f. A natural choice is simply the average position of such section, which writes x λ t, v f = v f t λ. Attempts at resolving the dominance equation with an expression of 2 σ(λ, x λ (t, v f )) derived from (11) - using for each location the corresponding temperature and materials viscosities - and intuitive values of M in the 5-10 range would yield no solution at most feed speeds. This means that growth rates derived this way have values that are too low to induce significant perturbation amplification by the time the Si core tip reaches the middle of the flame. In other words, growth rate values typical to infinite cylinder uniform breakup are slower than their gradual liquefaction breakup counterparts. One way to assess how much faster the growth rates in the gradual breakup setup are is to resolve the equation with a smaller value of M. Doing so, for M=1, yields a solution τ(λ, v f ) for most pairs (λ, v f ) that are of practical interest. Supplementary Figure S7 displays the evolution of τ(λ, v f ) with λ for v f = 10 μm.s -1. The dominant breakup period λ d (v f ) is simply the one where the displayed curve is at is minimum. For comparison, the inverse of the growth rate as defined in (15) at the maximum temperature in our profile (T=1,770 o C) is also displayed as a function of breakup period. We can readily see that the minimum is reached for a much longer wavelength in the case of uniform heating, and this illustrates precisely what the gradual heating technique is about: giving an advantage to shorter wavelengths. Supplementary Figure S8 displays the predicted λ d (v f ) using this approach, alongside the experimental results. The predicted values are within a factor 5 of the observed result, but it is apparent that this framework consistently predicts that short breakup periods should dominate, in disagreement with experimental observations. As a conclusion, we can therefore infer that the real growth rates σ(λ, x) that should be used are at not only faster than their counterpart in the uniform heating setup, but also their distribution is more favorable to dominance of longer wavelengths. 18

19 Supplementary Note 2. Identification and of the PN junction and IV-curve analysis When inducing simultaneous breakup of parallel Si cores containing dopants of different natures and concentration, physical contact between the formed spheres can be achieved as is illustrated and demonstrated in Figure 5c. However, because of the chances of partial mixing of the liquid Si as well as possible inter-sphere dopant diffusion during the cooling down or on the contrary, the possibility that poor bonding of the two spheres at their shared interface might prevent an electrical contact to be formed, electrical measurements are required to properly establish the existence of a pn junction. For practical reasons due to the atypical geometries of the globular structures we fabricate, ion implantation or other techniques of heavy-doping of the Si outer surface (such as spin-on-dopant) are not readily available to us. For this reason, in this proof of concept we have chosen to use N- and P-Si with the highest concentrations of dopants available to us, in order to enable direct electrical probing using tungsten needles. Ideally, degenerate Si elements would be employed to that effect, namely n-type Si with an electron donors concentration N D in excess of 2.86x10 19 cm -3 and p-type- Si with an electron acceptors concentration N A in excess of 3.1x10 19 cm -3, but such high concentrations were not available in the 2mmdiameter rods format used in the initial step of our process. While the n-type Si used in this experiment was close to degenerate level (N D =1.6x10 19 cm -3 ), the p-type Si had a dopant concentration nearly one order of magnitude below optimum (N A =6.9x10 18 cm -3 ), which calls for a verification that contacts with external tungsten needles are not themselves rectifying. This is done by testing, for the pn junction disclosed in this work, that current can flow both ways through each sphere when probed individually, with satisfactory levels of conduction. Supplementary Fig. S16 displays the current-voltage characteristics obtained when both probes were sitting on either sphere, as well as the IV curve obtained when applying a voltage across the interface between the spheres. When testing the conduction through each sphere individually, the external probe tips were between 10 and 30 microns apart, while both types of Si have a rather small resistivity (ρ n = 4x10-3 Ω.cm for the n- type Si and ρ p =10-2 Ω.cm for p-type Si, as specified by the supplier). This yields expected resistances of the conduction paths within Si that, although dependent on the exact positions of the probes, are in the order of 1 to 10 Ω (for n-type Si and p-type Si respectively). The data displayed in Supplementary Figure S16 is however consistent with series resistances comprised between 180 Ω and 300 Ω for the n-type Si sphere and between 0.65 kω and 8 kω for the p-type Si sphere. Note that we give ranges to account for the small non-linearity of these characteristics, a consequence of the dopants concentrations being below the degenerate levels - as mentioned above. The conclusion that we can draw from this analysis is that for both the n-type Si and the p-type Si used in this study, it is the contact properties between the external probes and the spheres - rather than the distance between the probe-tips itself - that sets the dominant series resistance. We can therefore readily compare the different curves shown in Supplementary Fig. S16, regardless of the small differences in probe tips locations between measurements. While at first we could try to account for the effective blocking of the reverse current across the junction (black squares) by invoking the presence of a highly resistant path (with a resistance superior to 10 MΩ) between the spheres, this hypothesis does not hold when considering the much smaller overall resistance (down to a few kω) measured in the forward regime. The electrical behavior of the structure we have fabricated is clearly rectifying, a property that 19

20 can only be explained by the presence of a pn junction at the interface between the two spheres. This also indicates that the Si/silica mixture shells surrounding the spheres upon formation (as seen in Figure 4c), while apparently acting as barriers to dopants diffusion and Si mixing, do not prevent electrical conduction. The largest effect of external contacts non-idealities on the measured behavior of the junction can be found where the characteristics of the overall diode structure comes closest to that of one of its constituent part. We see in Supplementary Fig. S16 that this is in the forward regime, where the contribution of the external contact to the P-doped sphere to the overall series resistance is the less negligible. From this analysis we can expect that this global series resistance contains a term that varies by a few kω over the range of voltages of interest. While the previous demonstration proves that we have discovered a new approach to the fabrication of functional Si devices, the displayed IV-curve parts from the ideal diode case in ways that we can readily account for. The most obvious non-ideality is the slow-rising current that indicates the presence of a sizable series resistance in the circuit. From our previous considerations, we expect this resistance to be the sum of a constant term (the internal resistance of the junction) and a term that varies of a few kω with voltage, due to the non-ideal external contact to the p-type Si sphere, these two terms being of comparable importance. However, a series resistance defined this way fails to properly account for the slow rise of current with forward voltage, and cannot account for the second most striking non-ideality of the structure, the growing reverse current with absolute voltage. These trends are the result of a different common phenomenon in pn structures: the trap-assisted generation and recombination of carriers within the depletion region of the diode. As shown all throughout this study, it is indeed expected that the interface between the two spheres is populated with a large concentration of crystalline defects that can act as carrier traps. While modeling this effect rigorously is rather involved, we can derive an analytical expression for it that accounts for most of our observations, using a standard derivation based on a few simplifying approximations and the Shockley-Read-Hall model (29). These consist in noting that the net carrier recombination rate U trap associated with trap-assisted mechanisms is dominated by traps located in the middle of the bandgap, and that within the depletion region of a pn junction, the product of the carriers concentrations is set by the applied bias V, following: np = n i 2 exp ( qv kt ) (S2) where n and p are the concentrations in electrons and holes respectively, n i the electron concentration in intrinsic Si at the considered temperature T, with q the charge of an electron and k the Boltzmann constant. U trap can then be written in the simplified form: U trap (n, p) = n i 2 (exp qv kt 1) τ h0 n+τ e0 p+(τ h0 +τ e0 )n i (S3) where n 0 and p 0 are the electron and hole concentrations under zero bias (at equilibrium) while τ e0 and τ h0 are their lifetime, respectively. An approximation of the total current flowing through the junction can then be obtained by integrating the maximum value of U trap over the entire depletion region. This value is obtained when τ h0 n = τ e0 p and yields: 20

21 I(V) = C V depqn i τ h0 τ e0 (exp qv 1) (S4) 2kT where V dep is the volume of the depleted region inside the junction and C is a dimensionless constant that depends on the interface geometry. This derivation accounts for both effects mentioned above. Indeed, where an ideal diode has a saturation current that does not depend on the bias, here the prefactor in the expression I(V) is proportional to V dep, a depleted volume that grows with the reverse voltage applied to the junction, accounting for the current increase with the negative voltage in Supplementary Figure S16. It also explains the slow forward current rise with V, since it calls for a growth in exp qv instead of the 2kT exp qv expected in a defect-less diode. kt Supplementary References [41] Y. Sato, Y. Kameda, T. Nagasawa, T. Sakamoto, S. Moriguchi, T. Yamamura, Y. Waseda, Viscosity of molten silicon and the factors affecting measurement. Journal of Crystal Growth 249, (2003). [42] R. H. Doremus, Viscosity of silica. Journal of Applied physics 92, (2002) 21