Supplementary Information for Semiconductor Solar Superabsorbers

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1 Supplementary Information for Semiconductor Solar Superabsorbers Yiling Yu, Lujun Huang, Linyou Cao, * Department of Materials Science and Engineering, North Carolina State University, Raleigh NC 7695; Department of Physics, North Carolina State University, Raleigh NC 7695; * To whom correspondence should be addressed. lcao@ncsu.edu This PDF file includes Fig. S-Fig. S3 S. Calculation for the density of leaky modes including Fig. S4-S7 Fig. S8-Fig.S Table S Reference S-S3

2 a Absorption efficiency Si nanoparticle CLMT Mie Wavelength (nm) b Absorption efficiency 3 a-si nanoparticle CLMT Mie Wavelength (nm) c Absorption efficiency Si square NW Wavelength (nm) FDTD CLMT d Absorption efficiency Si SiO CLMT Mie Si/SiO multishell NW Wavelength (nm) Figure S. The general validity of CLMT for the analysis of light absorption in semiconductor nanostructures. Calculated spectral light absorption efficiency using the CLMT model (red dashed lines) and well-established analytical or numerical methods (blue solid lines) for (a) Si nanoparticles in radius of 5 nm, (b) a-si nanoparticles in radius of 5 nm, (c) Si square nanowires in size of nm, and (d) core-multishell Si/SiO nanowires in radius of 3 nm, which equally distributes among the five layers involved. Insets are schematic illustrations for the nanostructures calculated.

a Resonant Wavelength (nm) 4 5 6 7 8 9 (ma) x.5.5.5.5.5 Radiative loss q rad b x.5 Solar absorption (ma).5.5.5.4.3.. Radiative loss q rad 4 6 8 Resonant wavelength λ (nm) Figure S.

3 a Resonant Wavelength (nm) (ma) x Radiative loss q rad b x.5 Solar absorption (ma) Radiative loss q rad Resonant wavelength λ (nm) Figure S. The solar absorption of single leaky modes in D Si structures. (a) Calculated solar absorption of single leaky modes in D Si structures as a function of radiative loss (horizontal axis) and resonant wavelength (vertical axis). (b) The optimal absorption (red line) and associated radiative loss (blue line) as a function of the resonant wavelength.

a Resonant Wavelength (nm) 4 5 6 7 8 9 (ma) x 9.5.5.5.5 Radiative loss q rad b x 9.6.5.5 Solar absorption (ma).5.5 4 6 8 Resonant wavelength λ (nm) Figure S3.

4 a Resonant Wavelength (nm) (ma) x Radiative loss q rad b x Solar absorption (ma) Resonant wavelength λ (nm) Figure S3. The solar absorption of single leaky modes in D CdTe structures. (a) Calculated solar absorption of single leaky modes in D CdTe structures as a function of radiative loss (horizontal axis) and resonant wavelength (vertical axis). (b) The optimal absorption (red line) and associated radiative loss (blue line) as a function of the resonant wavelength Radiative loss q rad

S. Calculation for the density of leaky modes For the convenience of mode identification, most of this modal analysis is performed with D structures, in which the leaky mode can be identified easier.

$we first examine D square structure with a refractive index of n as illustrated in Figure S4. We consider modes in wave vector k space.$

5 S. Calculation for the density of leaky modes For the convenience of mode identification, most of this modal analysis is performed with D structures, in which the leaky mode can be identified easier. However, the principle obtained from D structures can apply to D structures as well. a a Figure S4. Schematic illustration for D square nanowires (left) and for its corresponding mode distribution in k space (right). we first examine D square structure with a refractive index of n as illustrated in Figure S4. We consider modes in wave vector k space. The interspace between neighboring modes in x direction and y direction can be written as Δk x = π/na and Δk y = π/na. The space occupied by one mode isspace Δk x Δk y. For a given space between k and k, /4 π ( k k), the number of modes can be written as π ( k k ) N( λ, λ ) π a n ( ) πan ( ) = = = 4 ΔkxΔky λ λ λ λ (S) Similarly, we also derive the number of modes in D cubic structure as 4π( k k ) 4π 4π N( λ, λ ) a n ( ) Vn ( ) = = = ΔkxΔkyΔkz 3 λ λ 3 λ λ (S) From eqs. (S)-(S), we can find out the density of modes at given wavelength λ as for D, for D,!(" ) = dn # An = (S3) 3 d" "!(" ) = dn d" = 4#Vn3 " 4 (S4) For the case of solar light absorption, we need take into account both transver magenetic (TM) and transverse electric (TE) modes. Therefore, the density of leaky modes for two polarization are:

a Number of modes 8 6 4 Analytical Theoretical circular NW b Number of modes 8 6 4 Analytical Theoretical sphereical NP c Number of modes 8 6 4 3 Radius (nm) Theoretical Numerical d Rectangular NW 4

6 for D,!(" ) = 4# An " 3 (S5) for D,!(" ) = 8#Vn3 " 4 (S6) While derived from square and cubic structures, these equations can apply to nanostructures with arbitrary shapes, as illustrated in Figure S5. a Number of modes Analytical Theoretical circular NW b Number of modes Analytical Theoretical sphereical NP c Number of modes Radius (nm) Theoretical Numerical d Rectangular NW 4 6 Size d (nm) d Number of modes Radius (nm) Theoretical Numerical d Hexagonal NW 3 4 Size d (nm) Figure S5. The number of leaky modes calculated using eqs.(s)-(s) as a function of size, which is referred as theoretical and plotted as dashed lines. Also plotted is the number of leaky modes directly counted from the eigenvalue calculation of leaky modes using the methods we reported previously. -3 We can see that eqs.(s)-(s) can indeed reasonably describe the number of leaky modes in nanostructures with arbitrary shapes. A i, n i A, n A, n Figure S6. Schematic illustration for heterostructures with absorbing and non-asborbing materials.

7 We can calculate the number of leaky modes in heterogeneous structures that consist of multiple materials as illustrated Figure S6. for D, for D, N ( λ, λ) π (/ λ / λ) ni Ai i = (S7) 4 N( λ, λ ) = π (/ λ / λ ) nv (S8) i i 3 i The subscript i here denote different parts of the structure. n i is the refractive index of the ith part, and A i, V i is the corresponding area or volume of the ith part. The validity of these equations is confirmed in Figure S7a, which shows a good consistence between the number of leaky modes calculated using eq. S7 and that counted from eigenvalue calculation -3. We believe that only those leaky modes mainly related with the absorbing materials may substantially contribute to light absorption. Number of leaky modes Numerical Theoretical Si ZnO Radius (nm) Figure S7. The number of leaky modes for a ZnO/Si/ZnO nanowire, with a constant size ratio of. 4,.36, and.4 from inner to outside, calculated using eqs.(s7) as function of the radius, referred as theoretical and plotted as dashed blue line. Also plotted is the number of leaky modes directly counted from the eigenvalue calculation of leaky modes using COMSOL, referred as numerical and plotted as red circle. Inset is a schematic illustration of the NW.

8 a Solar absorption (ma) x Volume (µm3) b Volume V (µm3).4. CdTe v L L Period L (µm) Figure S8. Solar superabsorption in single nanostructures and an array of nanostructures. (a) The maximal solar absorption of single D nanostructures in which CdTe is included as the absorbing materials as a function of the volume of CdTe materials. The calculation result includes an estimated % error as indicated by the shaded areas. The inset is a schematic illustration for the nanostructure, whose irregular shape is intentionally chosen to illustrate that the structure may have any arbitrary shapes. (b) Solar superabsorption limit of an array of D nanostructures. The minimum volume of CdTe materials necessary to absorb >9% of the solar radiation above the band gap is plotted as a function of the period of the array. The inset is a schematic illustration for the nanostructure array.

9 Table S. Calcuated eigenvalue and radiative loss for the leaky modes in different structures

10 Absorption efficiency Wavelength (nm) Figure S9. The robustness of the design. The calculated spectral absorption efficiency for the designed nanosstructure arrays with a little bit difference in geometrical features. The structure is the one studied in the main text. The detailed geometrical features of the three structures are listed in the following table. Structure side height a-si SiC ZnO SiO Period Efficiency. 8nm 38nm nm 3nm 3nm 5nm 54nm.69mA/cm 9.8%. 9nm 39nm nm nm 4nm 4nm 54nm.58mA/cm 9.3% 3. 7nm 37nm nm 4nm nm nm 48nm.35mA/cm 89.3%

11 .4 CdTe Volume (µm 3 ) Period (µm) Figure S. The relationship between the designed structure (the red dot) and the predicted minimum volume for CdTe. The structure is shown in Figure 5d inset in the main text.

12 a Absorption efficiency b air 6 nm Lambertian limit 4 nm 3 nm substrate mirror SiO 7nm ZnO 3nm SiC 3nm CIGS 3nm Our design Wavelength (nm).3 Volume (µm 3 ) Period (µm) Figure S. Design of solar superabsorbers for CIGS. (a) The calculated spectral absorption efficiency of a designed structure including 3nm thick CIGS. The Lambertian limit for CIGS with an efficient thickness of 5.5 nm is also given (black). The inset shows the geometry of the designed structure. (b) The relationship between the designed structure (the red dot) and the predicted minimum volume for CIGS.

13 Solar Absorption (ma/cm ) 5 5 θ 55 nm ZnO 8 nm substrate mirror 38 nm SiO 5 nm ZnO 3 nm SiC 3 nm a-si 5 nm Incident angle θ ( o ) Figure S The dependence of the solar absorption on incident angle. The dimension of the structure modeled is given as shown. The definition of the incident angle θ is also schematically illustrated in the figure. References S S S3 Huang, L. J., Yu, Y. L. & Cao, L. Y. General Modal Properties of Optical Resonances in Subwavelength Nonspherical Dielectric Structures. Nano Letters 3, , (3). Yu, Y. & Cao, L. Coupled leaky mode theory for light absorption in D, D, and D semiconductor nanostructures. Opt Express, , (). Yu, Y. & Cao, L. The phase shift of light scattering at sub-wavelength dielectric structures. Opt. Express, , (3).

Semiconductor Solar Superabsorbers

Semiconductor Solar Superabsorbers Yiling Yu 2, Lujun Huang 1, Linyou Cao 1,2 * 1 Department of Materials Science and Engineering, North Carolina State University, Raleigh NC 27695; 2 Department of Physics,