## Dispersion, Wave Propagation and Efficiency Analysis of Nanowire Solar Cells

Optics Express, Vol. 17, Issue 12, pp. 10399-10410 (2009)

http://dx.doi.org/10.1364/OE.17.010399

Acrobat PDF (1304 KB)

### Abstract

We analyze the electromagnetic properties of InP/InAs nanowire solar cells for different geometries. We address both eigenvalue calculations to determine the wave propagation as well as source problems to simulate direct perpendicular illumination by three-dimensional finite element calculations. We demonstrate the validity of a 2D waveguide modal analysis as a method of estimating the results of the computationally far more demanding 3D analysis. The resulting data is employed in a detailed balance analysis in order to determine the optimum set of bandgap energies for a single-junction and dual-junction cell as well as the corresponding efficiency limit. The efficiency of the nanowire design can approach the efficiency of conventional thin-film designs despite the low volume fill-factor.

© 2009 Optical Society of America

## 1. Introduction

1. L. Tsakalakos, “Nanostructures for photovoltaics,” Mater. Sci. Eng. R **62(6)**, 175–189 (2008).
[CrossRef]

2. W. Shockley and H. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. **32**, 510 (1961).
[CrossRef]

3. B. Tian, T. Kempa, and C. Lieber, “Single nanowire photovoltaics,” Chem. Soc. Rev. **38(1)**, 16–24 (2009).
[CrossRef]

## 2. Geometry

*xy*-plane with a unit cell

*U*of square footprint. It is embedded in an infinite homogeneous medium of refractive index

*n*

_{0}. We shall refer to the distance between the centers two adjacent nanowires as the array period

*a*

_{0}, the cylindrical nanowires have a diameter of

*d*

_{0}, optionally they are embedded in a coaxial cylindrical waveguide of the diameter

*d*

_{WG}.

*h*

_{0}. Each unit cell is assumed to be illuminated by a plane wave through its top boundary

*∂U*

_{top}. The interface between free space and the plane of all top boundaries is located by definition at

*z*=0. These conventions are illustrated in Fig. 1.

*h*

_{0}=2 µm and

*n*

_{0}=1. The passive waveguide consists of silicon dioxide with a diameter of

*d*

_{WG}=200 nm (referred to as “with oxide”) or

*d*

_{WG}=0 nm (referred to as “without oxide”). Furthermore we conduct two series of simulations assuming the nanowires consist of InP and InAs, respectively. We use the dielectric parameters measured for thin films and bulk provided in [4].

## 3. Methods

### 3.1. Optical Field Calculation

_{0}of a harmonic wave in terms of its free-space wave-vector k0 or free space wavelength λ

_{0}. In our calculation we are employing the finite element method and the underlying variational formulation is equal to [5, 7

7. F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. **39(4)**, 341–352 (2007).
[CrossRef]

#### 3.1.1. 3D Finite Element Simulation

6. J. Nedelec, “Mixed finite elements in R 3,” Numerische Mathematik **35(3)**, 315–341 (1980).
[CrossRef]

*z*-oriented Poynting vector S

_{inc}. We assume that reflection can occur at the interface between free space and the nanowire array, therefore the total power flux is assumed to be the superposition of power fluxes of the incident and scattered fields.

_{sc}(λ

_{0}) has components in -

*z*-direction. The simulation domain in -

*z*-direction is terminated by perfectly matched layers (PML) sufficiently far away from the array structure. The materials composing the structure are represented by their respective wavelength-dependent complex refractive index. We perform simulations over a wavelength range of 300 nm≤λ

_{0}≤2000 nm in order to cover the relevant parts of the solar spectrum. The resulting local spectral power dissipation

*p*(x,λ

_{0}) in a lossy medium serves as the basis for the subsequent detailed balance calculation.

*V*⊆

*U*with respect to the power incident upon the unit cell

*U*as relative power dissipation

*η*(

_{p}**x**,λ

_{0}). In this context, note that the unit cell does not contain any sources.

**z**̂ being the unit direction vector along the

*z*-axis. Note that the denominator represents the integral over the entire top boundary of the unit cell while the volume

*V*may be any volume within

*U*. In the scope of this publication,

*V*comprises the entire semiconductor nanowire. Furthermore we refer to the relative power dissipation of the entire unit cell as total relative power absorption

*η*

_{abs}(

*λ*

_{0}). The transmittivity of the surrounding medium-to-nanowire-interface

*T*is defined as the ratio of the power flux into the array compared to the power flux in an infinite volume of the surrounding medium.

**S**

_{sc}(

*λ*

_{0}), the value of

*T*(

*λ*

_{0}) is within 0≤

*T*(

*λ*

_{0})≤1. Finally we want to introduce the convention that the power

*P*(

*z*,

*λ*

_{0}) penetrating a surface within the unit cell is calculated as

*P*

_{inc}defined in analogy as an integral over

*S*

_{inc}(

**x**,

*λ*

_{0}).

#### 3.1.2. 2D Waveguide Simulation

*z*-direction. We are thus modeling the nanowire array as an infinite array of infinitely extended nanowires. The validity of this assumption is not evident in any way because the nanowires have a length of

*h*

_{0}which is in the order of a few wavelengths. For the geometries we investigated, however, direct comparisons between the 2D and the 3D calculation revealed good matching. We employ a separation of variables ansatz by inserting

*z*-coordinate other than a scaling with exp(-

*jβ z*) leading to the reduction of the problem dimensionality. Hereby, we require the permittivity and the permeability to be isotropic with respect to the

*z*-axis. This limitation is intrinsic to the ansatz in Eq. (7) and does not limit the generality of the method in case of isotropic materials or at negligible anisotropy along the

*z*-direction.

*xy*-plane) in terms of first order curl-conforming Nedelec basis functions, the electric field in the

*z*-direction is approximated by linear node-based shape functions [8

8. M. Hano, “Finite-element analysis of dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. **32(10)**, 1275–1279 (1984).
[CrossRef]

*n*

_{eff,m}(

*λ*

_{0})=

*λ*

_{0}

*β*(

_{m}*λ*

_{0})/2

*π*of a mode

*m*and as the corresponding field profile

**E**

*(*

_{m}*x,y*), respectively. The relation

*β*(

_{m}*λ*

_{0}) is the well-known dispersion relation of the waveguide structure with the frequency being represented by

*λ*

_{0}. For reasons of clarity we shall omit to mention the dependence of

*n*

_{eff,m}(

*λ*

_{0}) on the wavelength

*λ*

_{0}.

^{2}, the size and symmetry of the arrangement leads to a virtually infinite number of modes with high degeneracy of each mode. We can safely reduce the total number of modes by analyzing the modal behavior of a unit cell of the structure, thereby imposing spatial periodicity of the electromagnetic mode. This reduction leads to four distinct symmetry expansions of the field. Assuming an excitation of the cell by perpendicularly incident plane waves, only a field expansion with one Dirichlet and one Neumann boundary condition on two opposing edges of the unit cell remains to be considered.

*P*propagating in the mode

_{m}*m*can be calculated using the Beer-Lambert law

*m*is defined as [9

9. R. Orobtchouk, “On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling,” Springer Series in Optical Sciences **119**, 263–290 (2006).
[CrossRef]

*m*. This integral represents the correlation of the incident wave and the modal fields. The modal transmittivity

*t*is equal to

_{m}*d*

_{0}at

*a*

_{0}=600 nm and

*λ*

_{0}=300 nm is provided in Fig. 3. The figure illustrates two characteristic values of the modes: the modal penetration depth defined as the distance in

*z*-direction leading to a decrease of intensity by 1/

*e*and the modal transmittivity

*t*as introduced in Eq. (10). The graphs indicate that for thin diameters the coupling into a weakly absorbing mode with most energy confined in the free space between the wires (mode no. 3 in Fig. 2) is dominant. This observation can be explained by the fact that the modal electric field distribution of mode no. 3 is fairly uniform throughout the entire unit cell and correlates well with the incident plane wave.

_{m}*d*

_{0}=180 nm is minuscule. The variation of the direction of the modal electric field distribution also reduces the coupling efficiency of mode no. 2 which exhibits — in contrast to the well-coupling mode no. 3 — a change of the direction of the electric field within the unit cell. The associated zero-crossing of mode no. 2 is indicated by the blue strip through its field profile in Fig. 2. Note, the coupling into mode no. 3 decreases with increasing diameter as the free space between the nanowires (to which the mode is confined) is reduced.

*P*(

_{m}*z*=0) of the mode

*m*at the top interface of the solar cell to be equal to

#### 3.2. Detailed Balance Calculation

2. W. Shockley and H. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. **32**, 510 (1961).
[CrossRef]

*hν*>

*E*generate electron-hole pairs with an internal quantum efficiency

_{g}*η*≤1. The local spectral generation rate reads as

_{eqe}*V*of constant

*E*

_{g}*λ*being the wavelength with photon energy of

_{g}*E*. Assuming radiative recombination only the corresponding IV-characteristic can be derived to be [10]

_{g}2. W. Shockley and H. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. **32**, 510 (1961).
[CrossRef]

## 4. Discussion

*η*

_{abs}(

*λ*

_{0}) to the value calculated using the waveguide approach

*η*

_{abs,2D}(

*λ*

_{0}). The results are in good agreement as illustrated in Fig. 4. Two designs of conventional thin-film InAs and InP solar cells were added for comparison illustrating the implicit anti-reflective effect of the nanowire array. The hypothetical 2 layer anti-reflective coating with

*n*

_{1}=1.6 and

*n*

_{2}=2.3 was obtained using the genetic algorithm discussed in [11

11. M. Schubert, F. Mont, S. Chhajed, D. Poxson, J. Kim, and E. Schubert, “Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm,” Opt. Express **16(8)**, 5290–5298 (2008).
[CrossRef]

*z*-direction.

*d*

_{0}approaching the diameter of the waveguide

*d*

_{WG}. The structure also exhibits local maxima of the total relative power absorption that are caused by the dispersion characteristic of the structure. In contrast, the dielectric properties of bulk do not exhibit this phenomenon.

*n*

_{NW}, the surrounding refractive index

*n*

_{0}and the volume fill-factor

*δ*. The 2D waveguide approximation discussed in this paper, however, can be interpreted as more sophisticated effective medium approach where the modes which depend on the exact geometry constitute spatially superimposed effective media. This aspect also manifests itself in the validity of the Beer-Lambert law on a mode-by-mode basis.

*n*

_{eff}}≈1. This mode is referred to as mode no. 3 in Fig. 2 and Fig. 3. The nanowire structure therefore acts as an anti-reflective coating by itself.

12. S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, “Dielectric functions and electronic structure of InAsP films on InP,” Appl. Phys. Lett. **91**, 041,917 (2007).
[CrossRef]

*πd*

^{2}

_{0}/4

*a*

^{2}

_{0}≈0.1 is sufficient to yield seventy percent of the conventional thin-film solar cell efficiency. The resulting efficiency limits indicate that micro-concentration is occurring as the efficiencies the nanowire cells exceed the efficiency of the thin-film cell weighted with the relative nanowire area. Due to the long penetration depth at high wavelengths, we observe an increase of the optimum bandgap energies for a dual-junction design exceeding the bounds of the InAs/InP material system. In order to achieve high efficiency within the given material system, it is imperative to increase absorption as the efficiency drop is more pronounced in case the optimum bandgap energy lies outside of the material system.

## 5. Conclusion and Outlook

## Acknowledgements

## References and links

1. | L. Tsakalakos, “Nanostructures for photovoltaics,” Mater. Sci. Eng. R |

2. | W. Shockley and H. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. Appl. Phys. |

3. | B. Tian, T. Kempa, and C. Lieber, “Single nanowire photovoltaics,” Chem. Soc. Rev. |

4. | E. Palik, |

5. | J. Jianming, “The finite element method in electromagnetics,” Wiley & Sons (1993). |

6. | J. Nedelec, “Mixed finite elements in R 3,” Numerische Mathematik |

7. | F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, “Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver,” Opt. Quantum Electron. |

8. | M. Hano, “Finite-element analysis of dielectric-loaded waveguides,” IEEE Trans. Microwave Theory Tech. |

9. | R. Orobtchouk, “On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling,” Springer Series in Optical Sciences |

10. | G. Létay and A. Bett, “EtaOpt-a program for calculating limiting efficiency and optimum bandgap structure for multi-bandgap solar cells and TPV cells,” Spectrum |

11. | M. Schubert, F. Mont, S. Chhajed, D. Poxson, J. Kim, and E. Schubert, “Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm,” Opt. Express |

12. | S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, “Dielectric functions and electronic structure of InAsP films on InP,” Appl. Phys. Lett. |

**OCIS Codes**

(350.6050) Other areas of optics : Solar energy

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Solar Energy

**History**

Original Manuscript: April 29, 2009

Revised Manuscript: May 27, 2009

Manuscript Accepted: June 3, 2009

Published: June 5, 2009

**Citation**

J. Kupec and B. Witzigmann, "Dispersion, Wave Propagation and Efficiency Analysis of Nanowire Solar Cells," Opt. Express **17**, 10399-10410 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10399

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### References

- L. Tsakalakos, "Nanostructures for photovoltaics," Mater. Sci. Eng. R 62(6), 175-189 (2008). [CrossRef]
- W. Shockley and H. Queisser, "Detailed Balance Limit of Efficiency of p-n Junction Solar Cells," J. Appl. Phys. 32, 510 (1961). [CrossRef]
- B. Tian, T. Kempa, and C. Lieber, "Single nanowire photovoltaics," Chem. Soc. Rev. 38(1), 16-24 (2009). [CrossRef]
- E. Palik, Handbook of optical constants of solids (Academic Press, 1985).
- J. Jianming, "The finite element method in electromagnetics,"Wiley & Sons (1993).
- Q1. J. Nedelec, "Mixed finite elements in R 3," Numerische Mathematik 35(3), 315-341 (1980). [CrossRef]
- F. R¨omer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39(4), 341-352 (2007). [CrossRef]
- M. Hano, "Finite-element analysis of dielectric-loaded waveguides," IEEE Trans. Microwave Theory Tech. 32(10), 1275-1279 (1984). [CrossRef]
- Q2. R. Orobtchouk, "On Chip Optical Waveguide Interconnect: the Problem of the In/Out Coupling," Springer Series in Optical Sciences 119, 263-290 (2006). [CrossRef]
- Q3. G. L’etay and A. Bett,"EtaOpt-a program for calculating limiting efficiency and optimum bandgap structure for multi-bandgap solar cells and TPV cells," Spectrum 20 (2001).
- M. Schubert, F. Mont, S. Chhajed, D. Poxson, J. Kim, and E. Schubert, "Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm," Opt. Express 16(8), 5290-5298 (2008). [CrossRef]
- S. Choi, C. Palmstrøm, Y. Kim, D. Aspnes, H. Kim, and Y. Chang, "Dielectric functions and electronic structure of InAsP films on InP," Appl. Phys. Lett. 91, 041,917 (2007). [CrossRef]

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