Self-assembled fibre optoelectronics with discrete translational symmetry

Fibres with electronic and photonic properties are essential building blocks for functional fabrics with system level attributes. The scalability of thermal fibre drawing approach offers access to large device quantities, while constraining the devices to be translational symmetric. Lifting this symmetry to create discrete devices in fibres will increase their utility. Here, we draw, from a macroscopic preform, fibres that have three parallel internal non-contacting continuous domains; a semiconducting glass between two conductors. We then heat the fibre and generate a capillary fluid instability, resulting in the selective transformation of the cylindrical semiconducting domain into discrete spheres while keeping the conductive domains unchanged. The cylindrical-to-spherical expansion bridges the continuous conducting domains to create ∼104 self-assembled, electrically contacted and entirely packaged discrete spherical devices per metre of fibre. The photodetection and Mie resonance dependent response are measured by illuminating the fibre while connecting its ends to an electrical readout.

(a) was used, while illuminating a single sphere, as shown in Figure 5(a); Inset -IV curve obtained in the dark (black), the dark current of the fibres prior to break up (red)shows no measurable current due to lack of contact between the semiconducting core and the electrodes.  Figure 9 -(a) Experimentally measured responsivity (black) compared with theoretically calculated responsivity (red), as a function of wavelength for a fibre with a spheres radius of 11 μm. The fibre was illuminated by a spherical lens in a single sphere excitation as described in Figure 5(a). (b) The FFT density of the experimental responsivity (black) and the theoretical responsivity (red) as a function of wavelength between two adjacent resonant peaks are described in the upper and the lower graphs, respectively. The location of the first order peak in the experimental and the theoretical results is  peak = 3.9±0.3 nm and  peak = 4.0nm, respectively (marked by an arrow).
Supplementary Figure 10 -(a) Experimentally measured responsivity (black) compared with theoretically calculated responsivity (red), as a function of wavelength for a fibre with a spheres radius of 11 μm. The fibre was illuminated by a cylindrical lens in a multiple sphere excitation as described in Figure 5(b). (b) The FFT density of the experimental responsivity (black) and the theoretical responsivity (red) as a function of wavelength between two adjacent resonant peaks are described in the upper and the lower graphs, respectively. The location of the first order peak in the experimental and the theoretical results is  peak = 4.15±0.3 nm and  peak = 4.2nm, respectively (marked by an arrow).
Supplementary Figure 11 -The FFT density of the experimental responsivity (black) and the theoretical responsivity (red) as a function of wavelength between two adjacent resonant peaks are described in the upper and the lower graphs, respectively, for fibres containing spheres with radius of 5 μm. Multiple excitation scheme. The location of the first order peak in the experimental and the theoretical results is marked by an arrow.

Supplementary Note 1: Timescale for break up and characteristic period
Tomotika model 1 predicts the perturbation growth rate of a Rayleigh-Plateau instability 2,3 in a "cylindrical thread of a viscous liquid surrounded by another viscous fluid": ⁄ , is the radius of the inner fluid, either the semiconducting core or the electrodes; ⁄ , and are the viscosity of the cladding and the inner fibre component, respectively; is the interfacial surface tension between the two liquids. and / are the perturbation growth time and rate respectively, and is the periodicity of the spheres that are present in the fibre after break up. Function has the following form 1 : In these expressions , , and are all functions of expressed in determinantal form as follows:  Additionally, we have measured the radii of the initial semiconducting core and the radius of the resulting spheres and their period for three different initial core diameters. The results are summarized in Supplementary Table 2 and are in agreement with Tomotika 1 model.

Supplementary Note 2 -Additional Results and Data -Sphere Size and Shape
Supplementary figures 2-4 show the fibre structure for different fibre core sizes, which result in spheres radius of , and respectively.
We have found that a normal distribution fits well the distribution of sphere radii, where the stated error is the standard deviation of the distribution. As can be seen in the Supplementary Figures 2-4 This is consistent with the relatively lower responsivity when measuring multiple spheres as described in Supplementary Note 3 (Supplementary Table 6, multiple sphere excitation).
Additionally, as can be seen in Supplementary Figures 2(b), 3(d) and 4(c), the semiconducting particles are very close to spherical. By measuring their diameter in two perpendicular directions allows us to estimate the ellipticity of these particles. The ratio between the minor and major axes yields in a maximum ratio of . Thus, we assume that the particles are very close to spherical.

Supplementary Note 3: Electrical behavior model of spherical semiconductor fibres
Our goal is to develop a model for the electric behavior of fibres that contain continuous electrodes and semiconducting spheres. We will do it in several steps.
1. Resistance of a single sphere: Since a resistance of a perfect sphere is infinite due to two singular contact points, we approximate this problem, by introducing a truncated sphere.
Assuming the sphere has a homogeneous conductivity , an infinitesimal unit of the sphere has a resistance of: , where z is the axis pointed between the two truncated caps, and A(z) is the area that is defined by z, which is normal to the area of the sphere.
Integrating with respect to z gives the resistance of a sphere -: Where R is the sphere radius, is the conductivity of the semiconductor, is the distance between the truncated cap and the center of the sphere, and ⁄ is the contact factor, which is a function of the contact between the electrodes and the sphere.
2. The next step is to consider a fibre system which contains only one sphere that is connected to continuous electrodes. Here we assume that conductors have much smaller resistance than the semiconducting sphere, as it is in our case, and that there is no junction resistance between the electrodes and the sphere, at the contact.
The carrier generation rate 6 is given by Supplementary Equation 3.

( ) ( )
Where n is the carrier density in the semiconductor, n dark is the carrier density in the semiconductor without illumination, is the carrier lifetime, is the photon flux and is the quantum efficiency. For the case of As 2 Se 5 which is an amorphous arsenic selenium semiconductor, the electric transport is governed by holes 7 .
Quantum efficiency of photo-generation and collection for a spherical particle is given by ( ) The resistance of the illuminated sphere is given by Supplementary equation 6 and the resistance of the spheres in the dark is: Since these spheres are all connected in parallel, the total resistance of the system is: The current in the system is thus: Where V is the applied voltage.
The dark current which is present when the fibre is not illuminated ( ) is given by Supplementary Equation 11.

( ) ( )
As can be seen in Figure 4(b) in the main text, the dark current is not linear with V. The reason for this nonlinear behavior is the dependence of the mobility of holes on the field, which is described by the Poole Frenkel effect 7,9 . The dependence of the mobility on the field is exponential as shown in Supplementary equation 12: where is the density of states in the valence band, is the density of the trapping centers, is the carriers' microscopic mobility, is the thermal activation energy at zero field, and is the Poole Frenkel constant. There are no exact values in the literature of As 2 Se 5 for and , but for a very similar and commonly used material -As 2 Se 3 , , and as reported in 7 .
The dark current has thus the following form: where A and B are constants which were used as parameters to fit the expression in Supplementary   Equation 13 to the experimental data. The fit results are summarized in Supplementary Table 3. On the other hand, the values of the pre-factor A change considerably between the experiments; this could not be explained just by the difference in sphere radius. These differences are most probably affected by the contact area between the sphere and the electrodes (mathematically described by factor), which has a significant influence on the pre-factor A.

According to the values of B in Supplementary
The responsivity of the fibre, when only one sphere is illuminated is given by Supplementary equation 14: where P is the power of the illumination that is incident on the sphere/s, and . h is Where is the measured current, is the total power supplied by the laser, and is a geometrical factor that takes into account the relative power that was incident on the spheres.
Assuming that the laser beam flux is a two dimensional Gaussian, it is possible to calculate the factor by the following expression: ; . The geometric corrections are summarized in Supplementary Table   4.
The values of the measured responsivity such as shown in Figure 5 Table 5.
The values of were calculated as described in Supplementary Note 4.
As a comparison between the experiments and theory, an average of the responsivity was taken in a light wavelength range of 825-850nm. Supplementary table 6 summarizes the measured responsivities and expected responsivity from theory, for one measurement from each type.
From Supplementary Table 6, we notice that the measured responsivity is very similar to the theoretical prediction, taking into to account the fact that the calculation of the theoretical responsivity was a first order model. The variance between the responsivity of the results obtained for multiple-sphere and single-sphere excitations could be attributed to the number of connected spheres. From these values it is most probable that in the multiple sphere experiment the majority of the spheres were connected in specimens with spheres of 11μm radius, and ~35% of the spheres were disconnected in specimens with 5μm radius. It is reasonable that as the sphere radius decreases, the effect of electrode fluctuations and sphere size dispersion will adversely affect the contact between the electrodes and the spheres, increasing the number of misconnections, as supported by the electrical measurements.

Supplementary Note 4 -Resonances theory
Mie theory 8 describes the interaction of electromagnetic radiation with spherical particles. The ultimate result of the theory is given by the scattering, extinction and absorption factors. We focus on the absorption as it will directly affect the absorption of the light in the semiconductor which could be measured through the response of the photodetecting spheres in the fibres.
We have measured the refractive index and the extinction coefficient of As 2 Se 5 by spectrophotometry using an evaporated film (thickness of ) of As 2 Se 5 and employing Swanepoel method to calculate the refractive index of the material 11 . The results are shown in Supplementary Figure 7.
Using the values of the refractive index and extinction coefficient we obtain for sphere radii of , and as shown in Supplementary Figure 8.