Reasons for Use of Anodic Nanotube Arrays in Hybrid and Dye Sensitized Solar Cells and Photo-Electrochemical Cells

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1 Supplemental Information: Optical Properties of Titania Dioxide Nanotube Arrays Reasons for Use of Anodic Nanotube Arrays in Hybrid and Dye Sensitized Solar Cells and Photo-Electrochemical Cells Titanium dioxide nanotube arrays have a geometry that provides a very high aspect ratio in which the active microscopic surface area of a nanotube array sample typically are from 5 to 5 times the macroscopic sample surface area. 1-4 This property combined with the high index of refraction of titanium dioxide material (n ave.5) and the vertical tubular structure makes TiO nanotube arrays excellent candidate structures for solar cells and hydrogen production by photo-electrochemical water-splitting for the following reasons: a) For water splitting, it provides a very large interface between the titanium dioxide (absorber layer/hole conducting layer) and water. 4-7 For hybrid solar cells or Dye Sensitized Solar Cells (DSSC s) it provides a large interface for charge separation so that photon absorption happens either right at or close to the interface for charge separation. This enables cheaper materials (like polymers) with short exciton or free carrier diffusion lengths to still have good efficiencies (the maority of electron-hole pairs will be separated 1-6, 8-1 before they recombine). b) For wavelengths at or smaller than the tube diameters it is proposed that the vertically aligned tubular structure with the high index of refraction of the tube walls (TiO with n ave.5) may enhance light trapping and absorption by acting as a waveguide or through photonic crystal or scattering effects. All these conectures will be examined in this work by presenting theoretical models for these phenomena and comparing it with measured transmission and reflection data for free-standing nanotube arrays. c) For wavelengths long compared to the tube diameters, the nanotube array provides an increasing mean index of refraction (real and imaginary parts) along the tube lengths, resulting in absorption of a high percentage of the incident light, as will be shown in agreement with previous work 13. (For water-splitting PEC cells light absorption is only by the TiO nanotubes themselves and hence primarily only UV light with energy above the TiO bandgap is absorbed. For solar cells the absorption spectrum needs to be expanded into the visible. This is typically done either by adding a highly absorbing dye to the nanotube surface (DSSC cells) or filling the nanotubes with a light absorbing material (for instance a polymer in organic-inorganic cells).

2 Differences Between Annealed and Non-Annealed Anodic TiO Nanotube Arrays The morphology, optical properties (mostly diffuse reflection) 5, and crystallinity between nonannealed (amorphous) and annealed (polycrystalline anatase) has been well documented by multiple studies. 5, 6, 11, 1, 18- These studies found that the basic geometry remains the same as long at the annealing temperature is not too high (< approx.. 5 to 6 C) so as to induce the phase change from anatase to rutile which destroys the nanotube structure. The main difference between the polycrystalline and amorphous nanotubes are that the polycrystalline nanotubes have significantly better conductivity (the reason making them more desirable for solar cells). Furthermore, ust as in the bulk the polycrystalline material have a sharper more well defined band edge around the usual anatase value of 3. ev. Furthermore, annealing removes most of the residual impurities from the anodization process (for instance fluoride) which result in less visible range impurity absorption. On the other hand, if a reducing atmosphere (like vacuum or H ) is used annealing can introduce deep traps and associated mid-band absorption mostly attributed to oxygen vacancies. Additional Experimental Details The experimental set-up for the integrating sphere (a) and single fixed angle measurements (b & c) are shown in figure S1 below.

3 (a) (b) (c) Figure S1. Illustration of the integrating sphere spectrometer as used to measure total reflectance. For transmission the sample is placed at the sphere entrance.(b) Illustration of the normal incidence optical measurements, I o is the incident beam intensity, I T is the transmitted beam intensity and I r is the backward reflected beam intensity. Figure (c) Illustration of the coaxial fiber optic bundle used to measure specular reflectance at normal incidence.

4 Figure S. A fiber optic bundle identical to the one used in the normal incidence back reflection measurements. For the normal incidence directly back reflected or nominal specular reflection measurements a fiber bundle probe as shown in figure S was used. The core fiber collects the returning light and send it to the spectrometer whilst the surrounding six fibers is connected to the combined deuterium and a tungsten-halogen lamp and serve as the light source. The fiber diameters are ~ 1 m so that the expanding beam spot of the specularly reflected returning light from any of the source fibers is significantly overlaps with all or most of the probe fiber. Discussion of Perturbations or Rings Along the Nanotube Walls The semi-periodic rings shown in Figure 6a & b along the nanotube walls have been widely observed. 19, 1-5 Some authors associate the rings with fluctuations in the anodization current during growth. 19, 1-6 It may be that the nanotube-growth system typically is naturally oscillatory. The severity of these perturbations and smoothness of the nanotubes can be increased by changing and optimizing the fabrication conditions. Li et al. 1 went the opposite way and intentionally varied or pulsed the anodization current. By doing this they fabricated nanotube arrays with very well ordered periodic indentations along their length. When the length scales of these indentations were chosen to correspond to visible light a photonic crystal that strongly diffracts visible light was obtained. Index of Refraction of Amorphous TiO The index of refraction of TiO in the UV, visible and NIR regions is in Figure S (measured by Filmetrics 7 ). Since the density, stoichiometry, degree of crystalline order and sometimes purity of amorphous TiO can vary there is some variation in the reported spectral plots of n and for amorphous TiO. However, the measurement by Filmetrics shown in Figure S is very consistent 18, 8-3 with other published measurements.

5 Refractive Index TiO Index of Refraction n Wavelength (nm) Figure S3. The real and imaginary refractive index of pure bulk amorphous TiO. 7 Critical Angle for Total Internal Reflection of Amorphous TiO The large refractive index of amorphous TiO leads to a relatively large critical angle for total internal reflection calculated and shown in Figure S3. Also shown is the percentage of randomly orientated radiation that will escape when incident on an amorphous TiO /air interface. The closed end of the nanotube array or barrier layer is an approximately 5 nm thick film of amorphous TiO. Except for the fact that some radiation may penetrate all the way through the barrier layer the plot in Figure S4 can be used as a rough estimate for the closed side critical angle.

6 Escapes Probability (%) Critical Angle (deg) Critical Angle For Total Internal Reflection of Amorphous TiO Wavelength (nm) Figure S4. Escape probability (%) for light incident on the closed of the nanotube array at a random angle (see text for assumptions) and the critical angle for total internal reflection of amorphous TiO calculated using the data in Figure S. As a result the critical angle for total internal reflection ( c ) at the closed edge of the nanotube array could be as low as 5 degrees across the visible range and hence more than 7% of visible light that incident on this interface from the inside at a random angle will be totally internally

7 Back Reflection (%) reflected (see Figure S4). Back Reflection Wavelength (nm) Figure S5. The probability (%) that that light back-scattered onto the closed nanotube surface from the inside at a random angle will escape. (Calculated using the critical angle in Figure S3) Appendix A Estimates of the effective refractive index Throughout the text we use the concept of a mean or effective refractive index to estimate the expected magnitude of various effects. Here we will present a formula for making these estimates for a nanotube array. In particular we will estimate the mean refraction indices for the open and closed sides, n1 and n. On the open side, we represent the nanotube structure as a close-packed triangular lattice of tubes of outer radius R and inner radius R 1. The lattice constant is equal to R, and hence, the unit cell area is equal to 1/ (3) R. Then, if the index of refraction of TiO is denoted by n, the mean index of refraction is given by 1/ ( R R1 ) n [(3) R ( R R1 )] ( R R1 )( n 1) n1 1. (A1) 1/ 1/ (3) R (3) R

8 On the closed end, we have a triangular lattice of closes tubes of radius R and lattice constant R, for which the mean index of refraction is n 1/ R n R R 1 n 1/ 1/ [(3) ] ( 1). (A) (3) R (3) If we take for instance a wavelength such that n 3 and an array such that R /.9 1 R, we obtain n and n.81. Substituting these values in Eqs. (5) and (7), we obtain for the reflection coefficients for the open and closed ends.11 and.6, respectively, in the long tube limit. Thus, for light incident on the open end, the standard eikonal approach predict that about 98% of the incident radiation will not be lost to back reflection and either be absorbed or transmitted. It is clear from the data in figure 4b that reflectance is much larger than the eikonal model predicted (for example at 8 nm we have ~ % predicted vs. 5% measured for the open side and ~3% predicted and 3% measured). As stated in the text and explored in section 4b and 4c there is clearly additional mechanisms at work. At short wavelengths, for example 4 nm, the deviation from the eikonal predictions for reflection is even starker with the open side back-reflecting more than the closed side. Appendix B - Detailed derivation of the scattering model The scattering due to disorder in the nanotube array can be treated using a method that follows the method used in Jackson's book 31 on electromagnetic theory to treat scattering by density fluctuations. (Jackson's treatment in turn is likely based on the work of Einstein and 3, 33 Smoluchowski to treat critical opalescence.) We begin with Maxwell's equations in a medium

9 H c 1 D t (B1a) E c 1 B t (B1b) B (B1c) D (B1d) with the electric field E D / K( r), where D is the displacement field and Kr ( ) is the dielectric constant at the point r in the sample. The vectors H and B are the magnetic field and magnetic induction, respectively. Substituting for E in Eq. (B1b) and taking the curl of that equation gives 1 ( H) [ D / K( r)] c. (B) t Substituting for H using Eq. (B1a), gives [ D / K( r)] c D t (B3) which can be re-written as D D ( K / c) {[( K / K( r) 1] D}, t (B4)

10 where K is the local average of Kr ( ).Expanding the double curl on the left hand side and substituting for D using Eq. (B1d), we obtain D D ( K / c) {[( K / K( r) 1] D} t (B5) which can be written as ik r r ' 1 3 e D D (4 ) d r ' ' ' {[( K / K( r ') 1] D( r ')}. (B6) r r' For r much larger than the length scale of the inhomogeneities in the sample, Eq. (B6) can be approximated by ikr e D D Asc, (B7) r where D is the displacement field of the incident wane and 1 3 ikrˆ r A (4 ) ' ' sc d r e ' ' {[( K / K( r ') 1] D( r ')}. (B8) The scattering amplitude is given by f ˆ A / D (B9) scat sc where ˆ is the unit vector in the direction of the polarization of the scattered wave. In the Born approximation calculation of the extinction coefficient due to disorder scattering for the radiation in Jackson's book 31, the differential cross-section is determined from the scattering amplitude f scat using

11 f scat (B1) and fscat is given in the Born approximation for electric dipole scattering by substituting D for D on the right hand side of Eq. (B6) which to lowest order in K gives k K( r) (B11) iqr 3 f ˆ ˆ scat e d r 4 K where ˆ and ˆ are the unit vectors giving the final and initial polarization directions and q is the difference between the final and initial wavevectors, K is the average dielectric constant and k / and K( r) K( r) K, where Kr ( ) is the dielectric constant at point r in the sample and is the wavelength of the incident wave inside the array (i.e., / n). K represents the local mean value of Kr ( ) and Kr ( ) represents deviations of Kr ( ) from its local mean value. In the spirit of the eikonel approximation, K varies with r but over a length scale large (i.e., comparable to the nanotube length) compared to that of the disorder in the array. Let us break the array up into a collection of identical cells, each having a volume v, whose dimensions are chosen to be of the order of the length scale of the disorder. The integral in Eq. (B11) can then be replaced by an integral over the th cell followed by a sum over, giving f scat kv 4 K iqr ˆ ˆ e, (B1) K where K is. (B13) 1 iqr 3 v e K( r) d r th cell

12 When is much larger than the size of a cell iq r e Jackson's book, the extinction coefficient is given by can be replaced by unity. From the treatment in V 4 kv (4 ) V tot 1 (4 ) V d fscat d( ˆ ˆ ) K e K K iq( rr' ) ', ' (B14) where V is the sample volume and... is an average over all configurations of the nanotubes in the array. The integral over solid angle is an integral over all directions of the wavevector of the scattered light, along with a sum over all polarizations for each wavevector. Here we have assumed that the cells are large enough so that K and K ' are uncorrelated. Then, K K K (B15) ', ' and Eq. (B14) becomes 4 kv d( ˆ ˆ ) N K (4 ) K 1, (B16) where N is the number of cells in the array. If the wavelength is large compared to the dimensions of the cell, we may integrate Eq. (B16) over, which gives 4 kv 1 N K. (B17) 1 K If the wavelength is smaller than one of the dimensions of the cell (most likely the direction along the nanotube axis), the expression 4 kv ( ˆ ˆ ) N K (4 ) K (B18) 1

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