## Transmission line equivalent circuit model applied to a plasmonic grating nanosurface for light trapping |

Optics Express, Vol. 20, Issue S1, pp. A141-A156 (2012)

http://dx.doi.org/10.1364/OE.20.00A141

Acrobat PDF (1198 KB)

### Abstract

In this paper, we show how light absorption in a plasmonic grating nanosurface can be calculated by means of a simple, analytical model based on a transmission line equivalent circuit. The nanosurface is a one-dimensional grating etched into a silver metal film covered by a silicon slab. The transmission line model is specified for both transverse electric and transverse magnetic polarizations of the incident light, and it incorporates the effect of the plasmonic modes diffracted by the ridges of the grating. Under the assumption that the adjacent ridges are weakly interacting in terms of diffracted waves, we show that the approximate, closed form expression for the reflection coefficient at the air-silicon interface can be used to evaluate light absorption of the solar cell. The weak-coupling assumption is valid if the grating structure is not closely packed and the excitation direction is close to normal incidence. Also, we show the utility of the circuit theory for understanding how the peaks in the absorption coefficient are related to the resonances of the equivalent transmission model and how this can help in designing more efficient structures.

© 2011 OSA

## 1. Introduction

1. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Mater. **9**, 205–213 (2010). [CrossRef]

2. A. Tiwari, A. Romeo, D. Bätzner, and H. Zogg, “Flexible CdTe solar cells on polymer films,” Prog. Photovolt: Res. Appl. **9**, 211–215 (2001). [CrossRef]

3. A. Romeo, A. Terheggen, D. Abou-Ras, D. L. Bätzner, F.-J. Haug, M. K. D. Rudmann, and A. N. Tiwari, “Development of thin-film Cu(In,Ga)Se2 and CdTe solar cells,” Prog. Photovolt: Res. Appl. **12**, 93–111 (2004). [CrossRef]

*μm*of material thickness to completely absorb incident sunlight [4, 5

5. V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. **22**, 4794–4808 (2010). [CrossRef] [PubMed]

6. P. N. Saeta, V. E. Ferry, D. Pacifici, J. N. Munday, and H. A. Atwater, “How much can guided modes enhance absorption in thin solar cells?” Opt. Express **17**, 975–20, (2009). [CrossRef]

7. V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. **8**, 4391–4397 (2008). [CrossRef]

3. A. Romeo, A. Terheggen, D. Abou-Ras, D. L. Bätzner, F.-J. Haug, M. K. D. Rudmann, and A. N. Tiwari, “Development of thin-film Cu(In,Ga)Se2 and CdTe solar cells,” Prog. Photovolt: Res. Appl. **12**, 93–111 (2004). [CrossRef]

23. www.cst.com (2011 CST Computer Simulation Technology AG).

_{2}. In Section 4, we present the absorption enhancement with respect to a flat metallic film solar cell. In particular, we demonstrate how the circuit theory allows us to interpret the peaks in the absorption spectrum in terms of an equivalent open circuit condition at the air-Si interface, which can be used as a guideline to design more efficient structures.

## 2. Plasmonic modes and photonic modes

*h*, as shown in Fig. 1(a). The thickness of the Ag film is larger than the skin depth. The optical constants of Ag and Si have been taken from Ref. [24]. When the light strikes the solar cell, part of the energy is trapped inside the Si slab. The number and the nature of the excited modes inside the Si slab can be calculated by means of the spectral domain Green’s function (GF) of the structure. The GF singularities represent the modes supported by the Si slab [22,25

25. A. Polemi, A. Toccafondi, and S. Maci, “High-frequency Green’s function for a semi-infinite array of electric dipoles on a grounded slab. Part I: formulation,” IEEE Trans. Antennas Propag. **49**, 1667–1677 (2001). [CrossRef]

*z*), as shown in Fig. 1(b). The transmission line source is assumed to be fed by an elementary point source placed at the interface between air and Si (

*I*in Fig. 1(b)). This spectral point source (i.e. elementary dipole) represents the expansion of all plane waves impinging on the structure from all possible directions [22], and additionally, it acts as the current generator of the equivalent electric circuit. The model is valid for both TM and TE polarization by simply selecting the correct representation of the modal impedances along the transmission line. In particular, where

_{g}*X*= 0 in air,

*X*=

*Si*in silicon, and

*X*=

*Ag*in the silver. In the above equations,

*ξ*=

_{X}*ξ*

_{0}/

*n*represents the characteristic impedance of the material, expressed in terms of the free space characteristic impedance

_{X}*ξ*

_{0}and the refractive index of the medium

*n*. Analogously,

_{X}*k*=

_{X}*n*

_{X}k_{0}represents the wavenumber in the material with

*k*

_{0}being the free space wavenumber. The spectral dependence is taken into account through the transverse wavenumber

*k*represents the component of the incoming light wavevector that is parallel to the interface. This component is required to be constant according to Snell’s laws. At the air-Si interface, the equivalent impedance for both TM and TE polarizations can be calculated as where

_{t}*p*= TM,TE. All the impedances in Eq. (2) are variables of

*k*. From hereinafter, this dependence is considered understood and suppressed for simplicity of notation. The spectral GF associated with the electric field can be now calculated as the equivalent voltage at the interface where the point source is located [22]. This leads to the following implicit expression where the constant

_{t}*I*can be set to 1 without lack of generality. In general,

_{g}*I*is proportional to the dipole momentum

_{g}*Iδ*, where

*I*is current flowing in a dipole of length

*δ*. Since we are looking at the dispersion, the amplitude of the generator is not crucial. By inserting Eq. (2) into Eq. (3), we then obtain where the formulas in Eq. (1) can be used for each impedance to obtain an explicit TM and TE spectral GF expression.

*λ*) is the reflection coefficient at the air-Si interface. The transmission coefficient is neglected because it is negligibly small for these thicknesses. The reflection coefficient can be expressed via the equivalent impedance at the same interface as where

*k*=

_{t}*k*

_{0}sin

*θ*= 0 (see Fig. 1), which implies that the two polarizations are equivalent. The structure, in fact, is totally symmetrical. The agreement between the calculated solution from the transmission line model and the full wave result is excellent. We also want to emphasize the spectral nature of the interface impedance,

*Z*, shown in Fig. 3(b). It is easy to verify that the peaks of absorption occur when the real part of the impedance at the air-Si interface assumes large values and the imaginary part crosses the zero line. This corresponds to an open circuit condition, which implies a peak of the voltage in the equivalent circuit. From the electromagnetic point of view, this behavior reflects a quasi-PMC condition at the interface, justifying a maximum in the tangential electric field, which in turn maximizes penetration inside the Si slab. The peaks shift in the spectrum when the angle of incidence is varied from normal, but the general concepts still hold.

_{int}## 3. Transmission line model of the 1-D grating nanosurface

*w*, height

*r*, and are in a periodic configuration of period

*d*. The Si has height

*h*. The conventional TM and TE polarization reference is taken and defined with respect to the plane crossing the grating (

*xz*plane). The presence of the ridges introduces additional scattering inside the Si slab, which results in an increase of the photonic and plasmonic path length. A qualitative description of the scattering phenomena is depicted in the top of Fig. 5. The cavity modes are primarily dictated by the reflection of the electromagnetic wave on the metal surface or the ridges of the grating. Furthermore, because of the plasmonic nature of Ag, the light impinging on the edges of the ridge creates an accumulation of positive and negative charge resulting in a dipole-like behavior on each ridge, which is only present for TM polarization. If the ridges are not closely packed, this effect is stronger than the coupling between adjacent ridges. For TE polarization, the electric field is oriented along the edge of the ridges and does not undergo diffraction. This behavior is clear from Fig. 5(a) and (b), where the electric field distribution is shown for both polarizations. In the TM case, the dipole-like electric field appears around the ridge of the grating, while in the TE case, the field distribution is affected just by the reflection mechanism.

*Z*. We use an equivalent transmission line model, which is shown in Fig. 6 and 8 for TE and TM polarizations, respectively. Each unit cell of the grating nanosurface is subdivided into two zones. At the interface between Zone 1 and Zone 2, the tangent component of the electric or magnetic field is continuous. Depending on the polarization, we will use this information to connect the two zones according to the circuit theory.

_{int}### 3.1. Transverse electric excitation

*k*=

_{t}*k*

_{0}sin

*θ*and

*θ*is the angle of incidence of the incoming light. We write the transmission line expression again for convenience: For Zone 2, we calculate an equivalent impedance

*h*–

*r*instead of

*h*. For TE polarization, at the junction between Zone 1 and Zone 2, the tangent electric field (tangent to the plane separating the two zones) is always continuous for any angle of incidence (see Fig 6). From a circuit theory point of view, this means that the electric potential difference at the nodes of the two impedances is constant, and the connection between the two loads is then a shunt connection. When dealing with shunt connections, it is easier to express the loads in terms of admittances

*f*

_{1}=

*w/d*and

*f*

_{2}= (

*d – w*)/

*d*. Thus, the equivalent interface impedance to be used in Eq. (7) can be written as and the absorption can be calculated using Eq. (6). The result for normal incidence (

*θ*= 0) is shown in Fig. 7(a) (red line) along with the full wave CST simulation of the grating nanosurface. The agreement is good overall except for some small discrepancies, which are likely due to higher order effects that are not considered in the model.

### 3.2. Transverse magnetic excitation

*Z*, as the series of a capacitance and a resistance (Si has losses) with

_{C}*C*=

*Re*(

*ε*)

_{Si}*w*and

*R*=

_{C}*Im*(

*ε*)

_{Si}*w*. Then, the impedance

*Z*and

_{C}*f*

_{1}=

*w/d*and

*f*

_{2}= (

*d*–

*w*)

*/d*. Thus, the equivalent interface impedance to be used in Eq. (7) can be written as and then the absorption can be calculated through Eq. (6). The result for normal incidence (

*θ*= 0) is shown in Fig. 7(b) (blue line) and compared with a full wave CST simulation. Despite the degree of approximation of the equivalent impedances calculated above, which only account for the largest first order effects, the results show that the global physical mechanism is captured by the model. Also, we show the absorption for the same structure without accounting for the impedance

*Z*in the model (dotted line). This result is important because it demonstrates how the plasmonic nature amplifies the diffraction at the ridges of the grating. The calculated absorption without

_{C}*Z*still agrees with the full wave solution, except in the range of the spectrum where the SPP is strongly excited (see dispersion diagram in Fig. 2).

_{C}### 3.3. Inclusion of an antireflective coating

*h*=

_{AR}*λ*/(4

*n*cos

_{AR}*θ*), where

*n*is the refractive index of the material used as AR coating. We use here TiO

_{AR}_{2}with refractive index

*n*= 2.5 [5

_{AR}5. V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. **22**, 4794–4808 (2010). [CrossRef] [PubMed]

*h*= 60

_{AR}*nm*. We apply the transmission line equivalent circuit model for both TE and TM polarizations, as shown in Fig. 9(b). The equivalent impedance utilized in Section 3.1 for TE and Section 3.2 for TM needs to be transformed by an extra piece of transmission line of length

*h*and characterized by a characteristic impedance as in Eq. (1), where

_{AR}*X*=

*AR*. The new

*Z*impedance is then calculated at the air-AR coating interface using the usual transmission line formula as in Eq. (8) for TE and in Eq. (10) for TM. This result is then plugged into Eq. (7) to evaluate the absorption coefficient analytically via Eq. (6). The calculated result is compared with a CST simulation in Fig. 10 for both polarizations. The agreement is extremely good throughout the optical spectrum.

_{int}## 4. Absorption enhancement

*A*(

_{flat}*λ*) is the absorption for the flat metal film and

*A*(

_{grating}*λ*) is for the grating nanosurface. The absorption in each case can be calculated from Eq. (6) and has been shown previously in Figs. 3 and 7 for TE and TM polarizations, respectively. The gain in absorption is shown in Fig. 11(a) for normal incidence. Similar figures have been presented before; however, we want to interpret this result according to our transmission line model. To better understand the nature of the peaks in

*G*(

*λ*), we also show the TE and TM equivalent impedance at the air-Si interface compared with the flat film solar cell. In particular, in Fig. 11(b) and (c), the real and imaginary part of

*Z*is plotted. As described in Section 2, the peaks of absorption occur when the equivalent interface impedance acts like an open circuit, or equivalently the interface acts like a PMC surface. For the grating nanosurface, whenever the equivalent impedance shows a peak in its real part and a crossing of the zero line in its imaginary part, the absorption is maximum. Since we are examining the ratio between the grating absorption and the flat case absorption, the gain shows a substantial increase only when the open circuit condition between the two cases does not overlap. In fact, when it overlaps, the effect is effectively equivalent and leads to an overall cancellation. Also, when the open circuit condition holds only for the flat case, the gain trends toward a minimum. From the plot of

_{int}*G*(

*λ*) in Fig. 11(a), it is clear that the grating nanosurface has a better performance for the TM polarization. The TM curve shows two main peaks around 750

*nm*and 920

*nm*, while the TE curve shows only one main peak around 900

*nm*. In addition, the TM absorption enhancement peaks are broader in bandwidth. This occurs because the open circuit TE resonances are only due to cavity modes resonances, which are sharp and narrowband. The TM resonances occur as a combination of cavity and plasmonic modes, which improve the gain in both amplitude and bandwidth. As a consequence, better performance is attained when the nanosurface is designed to combine cavity and plasmonic modes to increase the number of open circuit conditions and possibly to increase the bandwidth of the resonances.

## 5. Conclusions

## Acknowledgments

## References and links

1. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Mater. |

2. | A. Tiwari, A. Romeo, D. Bätzner, and H. Zogg, “Flexible CdTe solar cells on polymer films,” Prog. Photovolt: Res. Appl. |

3. | A. Romeo, A. Terheggen, D. Abou-Ras, D. L. Bätzner, F.-J. Haug, M. K. D. Rudmann, and A. N. Tiwari, “Development of thin-film Cu(In,Ga)Se2 and CdTe solar cells,” Prog. Photovolt: Res. Appl. |

4. | W. Koch, A. Endrös, D. Franke, C. Häbler, J. P. Kaleis, and H.-J. Möller, “Bulk crystal growth and wavering for PV,” |

5. | V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. |

6. | P. N. Saeta, V. E. Ferry, D. Pacifici, J. N. Munday, and H. A. Atwater, “How much can guided modes enhance absorption in thin solar cells?” Opt. Express |

7. | V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. |

8. | M. Green, Third generation photovoltaics: advanced solar energy conversion, Springer series in photonics (Springer, 2003). URL http://books.google.com/books?id=TimK46htCMoC. |

9. | D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Opt. Express |

10. | R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with broadband absorption enhancements,” Adv. Mater. |

11. | A. Polyakov, S. Cabrini, S. Dhuey, B. Harteneck, P. J. Schuck, and H. A. Padmore, “Plasmonic light trapping in nanostructured metal surfaces,” Appl. Phys. Lett. |

12. | C. Haase and H. Stiebig, “Optical properties of thin-film silicon solar cells with grating couplers,” Prog. Photovolt: Res. Appl. |

13. | K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

14. | T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” Electron. Commun. |

15. | N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures for Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron Devices |

16. | J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun. |

17. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A |

18. | T. Tamir and S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. |

19. | A. Chutinan and S. John, “Light trapping and absorption optimization in certain thin-film photonic crystal architectures,” Phys. Rev. A |

20. | Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express |

21. | I. T. A. Luque and A. Marti, “Light intensity enhancement by diffracting structures in solar cells,” J. Appl. Phys. |

22. | L. Felsen and N. Markuvitz, |

23. | www.cst.com (2011 CST Computer Simulation Technology AG). |

24. | E. Palik, |

25. | A. Polemi, A. Toccafondi, and S. Maci, “High-frequency Green’s function for a semi-infinite array of electric dipoles on a grounded slab. Part I: formulation,” IEEE Trans. Antennas Propag. |

26. | B. Davies, “Locating the zeros of an analytic function,” J. Comput. Phys. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(160.4760) Materials : Optical properties

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Plasmonics

**History**

Original Manuscript: October 28, 2011

Revised Manuscript: December 13, 2011

Manuscript Accepted: December 16, 2011

Published: January 2, 2012

**Citation**

Alessia Polemi and Kevin L. Shuford, "Transmission line equivalent circuit model applied to a plasmonic grating nanosurface for light trapping," Opt. Express **20**, A141-A156 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-S1-A141

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### References

- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Mater.9, 205–213 (2010). [CrossRef]
- A. Tiwari, A. Romeo, D. Bätzner, and H. Zogg, “Flexible CdTe solar cells on polymer films,” Prog. Photovolt: Res. Appl.9, 211–215 (2001). [CrossRef]
- A. Romeo, A. Terheggen, D. Abou-Ras, D. L. Bätzner, F.-J. Haug, M. K. D. Rudmann, and A. N. Tiwari, “Development of thin-film Cu(In,Ga)Se2 and CdTe solar cells,” Prog. Photovolt: Res. Appl.12, 93–111 (2004). [CrossRef]
- W. Koch, A. Endrös, D. Franke, C. Häbler, J. P. Kaleis, and H.-J. Möller, “Bulk crystal growth and wavering for PV,” Handbook of Photovoltaic Science and Engineering, pp. 205–255 (John Wiley2003).
- V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010). [CrossRef] [PubMed]
- P. N. Saeta, V. E. Ferry, D. Pacifici, J. N. Munday, and H. A. Atwater, “How much can guided modes enhance absorption in thin solar cells?” Opt. Express17, 975–20, (2009). [CrossRef]
- V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett.8, 4391–4397 (2008). [CrossRef]
- M. Green, Third generation photovoltaics: advanced solar energy conversion, Springer series in photonics (Springer, 2003). URL http://books.google.com/books?id=TimK46htCMoC .
- D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Opt. Express19, 95–107 (2011). [CrossRef]
- R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with broadband absorption enhancements,” Adv. Mater.21, 3504–3509 (2009). [CrossRef]
- A. Polyakov, S. Cabrini, S. Dhuey, B. Harteneck, P. J. Schuck, and H. A. Padmore, “Plasmonic light trapping in nanostructured metal surfaces,” Appl. Phys. Lett.98, 104–107 (2011). [CrossRef]
- C. Haase and H. Stiebig, “Optical properties of thin-film silicon solar cells with grating couplers,” Prog. Photovolt: Res. Appl.14, 629–641 (2006). [CrossRef]
- K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966). [CrossRef]
- T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” Electron. Commun.31, 116–120 (1977).
- N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures for Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron Devices54, 1926–1933 (2007). [CrossRef]
- J. Chen, Q. Wang, and H. Li, “Microstructured design of metallic diffraction gratings for light trapping in thin-film silicon solar cells,” Opt. Commun.283, 5236–5244 (2010). [CrossRef]
- M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A71, 811–818 (1981). [CrossRef]
- T. Tamir and S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol.14, 914–927 (1996). [CrossRef]
- A. Chutinan and S. John, “Light trapping and absorption optimization in certain thin-film photonic crystal architectures,” Phys. Rev. A78, 023,825 (2008). [CrossRef]
- Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express18, 366–380 (2010). [CrossRef]
- I. T. A. Luque and A. Marti, “Light intensity enhancement by diffracting structures in solar cells,” J. Appl. Phys.104, 502–034, (2008).
- L. Felsen and N. Markuvitz, Radiation and scattering of waves, 1st ed. (Prentice-Hall, Englewood Cliffs, NJ, 1973).
- www.cst.com (2011 CST Computer Simulation Technology AG).
- E. Palik, Handbook of optical constants of solids, 1st ed. (Academic Press, Orlando, 1985).
- A. Polemi, A. Toccafondi, and S. Maci, “High-frequency Green’s function for a semi-infinite array of electric dipoles on a grounded slab. Part I: formulation,” IEEE Trans. Antennas Propag.49, 1667–1677 (2001). [CrossRef]
- B. Davies, “Locating the zeros of an analytic function,” J. Comput. Phys.66, 36–49 (1986). [CrossRef]

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