## Simple magneto–optic transition metal models for time–domain simulations |

Optics Express, Vol. 21, Issue 10, pp. 12022-12037 (2013)

http://dx.doi.org/10.1364/OE.21.012022

Acrobat PDF (898 KB)

### Abstract

Efficient modelling of the magneto–optic effects of transition metals such as nickel, cobalt and iron is a topic of growing interest within the nano–optics community. In this paper, we present a general discussion of appropriate material models for the linear dielectric properties for such metals, provide parameter fits and formulate the anisotropic response in terms of auxiliary differential equations suitable for time–domain simulations. We validate both our material models and their implementation by comparing numerical results obtained with the Discontinuous Galerkin time–domain (DGTD) method to analytical results and previously published experimental data.

© 2013 OSA

## 1. Introduction

1. E. Th. Papaioannou, V. Kapaklis, E. Melander, B. Hjörvarsson, S. D. Pappas, P. Patoka, M. Giersig, P. Fumagalli, A. García–Martín, and G. Ctistis, “Surface plasmons and magneto–optic activity in hexagonal Ni anti–dot arrays,” Opt. Express **19**, 23867–23877 (2011) [CrossRef] [PubMed] .

2. E. Melander, E. Östman, J. Keller, J. Schmidt, E. Th. Papaioannou, V. Kapaklis, U. B. Arnalds, B. Caballero, A. García–Martín, J. C. Cuevas, and B. Hjörvarsson, “Influence of the magnetic field on the plasmonic properties of transparent Ni anti-dot arrays,” Appl. Phys. Lett. **101**, 063107 (2012) [CrossRef] .

3. V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia–Martin, J.–M. García–Martín, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Phot. **4**, 107–111 (2010) [CrossRef] .

4. A. García–Martín, G. Armelles, and S. Pereira, “Light transport in photonic crystals composed of magneto–optically active materials,” Phys. Rev. B **71**, 205116 (2005) [CrossRef] .

## 2. Ferromagnetic metals

### 2.1. Magnetic susceptibility

*ω*

^{−2}in the real and

*ω*

^{−3}in the imaginary part), we choose to neglect them in our work. Another possible contribution to the dynamic magnetic susceptibility might be caused by reorientable magnetic dipoles. This can be expected to at least contribute to the static magnetic susceptibility and to have the frequency dependence of the Debye model [6]. Its real part would vanish at high frequencies but the imaginary part would feature an

*ω*

^{−1}–tail. Again, we assume that the relevant time scales are many orders of magnitude larger than the duration of an optical cycle; for the remainder of this manuscript, we thus assume at optical frequencies.

### 2.2. Isotropic dielectric susceptibility

7. V. Korenman, J. L. Murray, and R. E. Prange, “Local-band theory of itinerant ferromagnetism. I. Fermi-liquid theory,” Phys. Rev. B **16**, 4032–4047 (1977) [CrossRef] .

*ω*

_{P}/(2

*π*) and the scattering rate

*γ*

_{D}. Using the well–known connection between a dynamic susceptibility

*χ*(

*ω*) and the corresponding dynamic conductivity this can be equivalently formulated as a differential equation for the polarization current Unfortunately, this model fails to reproduce the losses as can be seen in Fig. 1. The

*ω*

^{−3}–asymptotics of the imaginary part of

*χ*

_{Drude}(

*ω*) is in contradiction to the experimental data, which merely exhibit a

*ω*

^{−1}–behavior. While the Drude model can overall very nicely reproduce the real part of the permittivity of nickel, the losses are grossly under–represented.

*τ*and the background permittivity. A finite number of higher order terms leads to diverging losses at high frequencies. This is unphysical as any susceptibility should vanish at sufficiently short wavelength. Besides that, higher order terms are very likely to create negative losses (i.e. unphysical gain) in some frequency region, leading to numerical instabilities in time–domain simulations. In principle, we could also replace the simple Drude damping

*γ*

_{D}in Eq. (5) with another convolution. This would model a retarded dependence of a scattering mechanism on the movement of electrons (any dependence on the driving field is contained in

*Z*(

*s*)). In the case of QP–phonon scattering, this would model e.g. lattice heating and we assume that no such process is relevant at room temperature and on the time scale of an optical cycle. Even more, to first approximation, it would just renormalize

*ω*

_{P}and

*γ*

_{D}without qualitatively changing anything.

*m*

_{eff}of the quasi–particles as well as on the partial carrier densities. This holds true for every electronic subsystem, such as spin–up electrons and spin–down electrons. Thus, we find individual plasma frequencies

*τ*

^{↑},

*τ*

^{↓}are quite different, too. At room temperature, however, the Drude damping

*γ*

_{D}is dominated by QP–phonon scattering because the QP life time is very large close to the Fermi surface and magnons are thermally less excited due to their higher energies. Because QP-phonon scattering is not sensitive to spin orientation, we may assume that both subsystems are subject to essentially the same scattering rate. As a result, the conduction electrons are described by two retarded Drude models with a common denominator. These terms can be replaced by a single retarded Drude term with effective plasma frequency and effective relaxation time

## 3. Magneto–optic materials

### 3.1. Insulators

8. P. R. Berman, “Optical Faraday rotation,” Am. J. Phys. **78**, 270–276 (2009) [CrossRef] .

**B**

_{ext}and whose absolute value Ω = |Ω⃗

*|*is the angular cyclotron frequency of the particle with charge −

*e*and mass

*m*.

9. M. König, K. Busch, and J. Niegemann, “The Discontinuous Galerkin time–domain method for Maxwells equations with anisotropic materials,” Phot. Nano. Fund. Appl. **8**, 303–309 (2010) [CrossRef] .

10. J. Alvarez, L. D. Angulo, A. R. Bretones, and S. G. Garcia, “3–D Discontinuous Galerkin time–domain method for anisotropic materials,” IEEE Antenn. Wireless Propag. Lett. **11**, 1182 (2012) [CrossRef] .

### 3.2. Metals

4. A. García–Martín, G. Armelles, and S. Pereira, “Light transport in photonic crystals composed of magneto–optically active materials,” Phys. Rev. B **71**, 205116 (2005) [CrossRef] .

11. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second–harmonic generation at metal surfaces,” Phys. Rev. B **21**, 4389–4402 (1980) [CrossRef] .

*e*, it reads in our notation: Here, −

*e*is the electron charge,

*n*the local electron density,

**v**the corresponding velocity field,

**j**= −

*en*

**v**is the current density and

*p*(

*n*) =

*αn*

^{5/3}is the density–dependent pressure of free fermions with some constant

*α*. For the sake of readability, we suppressed the explicit space–and time–dependence of

*n*,

**v**,

**j**,

**E**and

**H**. From this, the differential equation of the Drude model (which is a linear model) can be recovered by first linearizing the equations. To this end, we introduce equilibrium quantities

*n*

_{0},

**j**

_{0}=

**0**,

**E**

_{0}=

**0**,

**H**

_{0}=

**B**

_{ext}/

*μ*and small deviations

*δn*,

*δ*

**j**=

**j**,

*δ*

**E**=

**E**,

*δ*

**H**. We drop all products of the latter four quantities (i.e. the nonlinear terms) and find where

*p*′(

*n*) = (

*∂p*(

*n*))/(

*∂n*). By averaging over a sufficiently large volume, all non–local effects are smeared out, which is equivalent to dropping the spatial derivatives: This is the Drude model with a Lorentz force added.

*τ*. In principle, there might be a

*τ*–dependent prefactor to the Lorentz force but this is not of practical relevance as it can be absorbed in the cyclotron frequency Ω, which is eventually fitted to experimental data. Finally, higher order terms might be expected to arise from a more elaborate microscopic description.

### 3.3. Susceptibility tensor

*z*–axis. The general solution to this is with

*A*=

*p*

^{(n)}(−i

*ω*) and

*B*= (−i

*ω*)

^{n}^{−1}Ω

*. Obviously, the tensor element*

_{k}*χ*is just the isotropic model. The nontrivial entries of the Lorentz oscillator model with Lorentz force are: The nontrivial entries of the Drude model with Lorentz force are (in accordance with [4

_{zz}4. A. García–Martín, G. Armelles, and S. Pereira, “Light transport in photonic crystals composed of magneto–optically active materials,” Phys. Rev. B **71**, 205116 (2005) [CrossRef] .

*x*– or

*y*–direction, the index subscripts have to be interchanged in the above result. As the material response is linear, an arbitrary magnetic field direction can be handled by superimposing individual magnetic fields along the three Cartesian directions.

## 4. Auxiliary differential equations

9. M. König, K. Busch, and J. Niegemann, “The Discontinuous Galerkin time–domain method for Maxwells equations with anisotropic materials,” Phot. Nano. Fund. Appl. **8**, 303–309 (2010) [CrossRef] .

### 4.1. Drude and Lorentz oscillator models

*m*in Ω⃗ =

*e*

**B**/

*m*is regarded as an effective mass that must be fitted to experimental data.

**j**

_{L}, it features a secondary auxiliary field

**p**

_{L}. The frequency–domain equation of motion for

**j**

_{L}including the Lorentz force is: Introducing a secondary auxiliary field we find The time–domain ADEs are obtained by Fourier transforming the last two equations: Again, the parameter

*m*in the expression for Ω⃗ must be fitted to experimental data for each Lorentz oscillator individually and should not be regarded as the effective mass of an actually existing particle. Negative values for

*m*appear to be all but uncommon.

### 4.2. Retarded Drude model

## 5. Parameter fits

13. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B **9**, 5056–5070 (1974) [CrossRef] .

14. Š. Višňovský, V. Pařízek, M. Nývlt, P. Kielar, V. Prosser, and R. Krishnan, “Magneto–optical Kerr spectra of nickel,” J. Magn. Magn. Mater. **127**, 135–139 (1993) [CrossRef] .

15. Š. Višňovský, “Magneto–optical Ellipsometry,” Czech. J. Phys. B **36**, 625–650 (1986) [CrossRef] .

13. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B **9**, 5056–5070 (1974) [CrossRef] .

14. Š. Višňovský, V. Pařízek, M. Nývlt, P. Kielar, V. Prosser, and R. Krishnan, “Magneto–optical Kerr spectra of nickel,” J. Magn. Magn. Mater. **127**, 135–139 (1993) [CrossRef] .

14. Š. Višňovský, V. Pařízek, M. Nývlt, P. Kielar, V. Prosser, and R. Krishnan, “Magneto–optical Kerr spectra of nickel,” J. Magn. Magn. Mater. **127**, 135–139 (1993) [CrossRef] .

## 6. Numerical results

### 6.1. Halfspace problem

15. Š. Višňovský, “Magneto–optical Ellipsometry,” Czech. J. Phys. B **36**, 625–650 (1986) [CrossRef] .

*h*

_{max}= 100nm, the metallic parts into tetrahedra with

*h*

_{max}= 50nm; on each tetrahedron the field components were expanded into Lagrange polynomials [5]. We varied the polynomial order to show convergence.

### 6.2. Experimental data

1. E. Th. Papaioannou, V. Kapaklis, E. Melander, B. Hjörvarsson, S. D. Pappas, P. Patoka, M. Giersig, P. Fumagalli, A. García–Martín, and G. Ctistis, “Surface plasmons and magneto–optic activity in hexagonal Ni anti–dot arrays,” Opt. Express **19**, 23867–23877 (2011) [CrossRef] [PubMed] .

*h*

_{max}= 100nm in air and

*h*

_{max}= 50nm in the silicon substrate and the metal. The expansion basis consisted of fifth order Lagrange polynomials.

*ε*= 11.9. We believe that all these modifications do not dramatically spoil our results (Fig. 5). This has to be compared to the experimental and numerical data in Fig. 3(b) of [1

1. E. Th. Papaioannou, V. Kapaklis, E. Melander, B. Hjörvarsson, S. D. Pappas, P. Patoka, M. Giersig, P. Fumagalli, A. García–Martín, and G. Ctistis, “Surface plasmons and magneto–optic activity in hexagonal Ni anti–dot arrays,” Opt. Express **19**, 23867–23877 (2011) [CrossRef] [PubMed] .

## 7. Summary

## Acknowledgments

## References and links

1. | E. Th. Papaioannou, V. Kapaklis, E. Melander, B. Hjörvarsson, S. D. Pappas, P. Patoka, M. Giersig, P. Fumagalli, A. García–Martín, and G. Ctistis, “Surface plasmons and magneto–optic activity in hexagonal Ni anti–dot arrays,” Opt. Express |

2. | E. Melander, E. Östman, J. Keller, J. Schmidt, E. Th. Papaioannou, V. Kapaklis, U. B. Arnalds, B. Caballero, A. García–Martín, J. C. Cuevas, and B. Hjörvarsson, “Influence of the magnetic field on the plasmonic properties of transparent Ni anti-dot arrays,” Appl. Phys. Lett. |

3. | V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia–Martin, J.–M. García–Martín, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Phot. |

4. | A. García–Martín, G. Armelles, and S. Pereira, “Light transport in photonic crystals composed of magneto–optically active materials,” Phys. Rev. B |

5. | K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser & Photon. Rev. |

6. | M. Y. Koledintseva, K. N. Rozanov, A. Orlandi, and J. L. Drewniak, “Extraction of Lorentzian and Debye parameters of dielectric and magnetic dispersive materials for FDTD modeling,” J. Electr. Eng.–Slovak |

7. | V. Korenman, J. L. Murray, and R. E. Prange, “Local-band theory of itinerant ferromagnetism. I. Fermi-liquid theory,” Phys. Rev. B |

8. | P. R. Berman, “Optical Faraday rotation,” Am. J. Phys. |

9. | M. König, K. Busch, and J. Niegemann, “The Discontinuous Galerkin time–domain method for Maxwells equations with anisotropic materials,” Phot. Nano. Fund. Appl. |

10. | J. Alvarez, L. D. Angulo, A. R. Bretones, and S. G. Garcia, “3–D Discontinuous Galerkin time–domain method for anisotropic materials,” IEEE Antenn. Wireless Propag. Lett. |

11. | J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second–harmonic generation at metal surfaces,” Phys. Rev. B |

12. | R. E. Wyatt, |

13. | P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B |

14. | Š. Višňovský, V. Pařízek, M. Nývlt, P. Kielar, V. Prosser, and R. Krishnan, “Magneto–optical Kerr spectra of nickel,” J. Magn. Magn. Mater. |

15. | Š. Višňovský, “Magneto–optical Ellipsometry,” Czech. J. Phys. B |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(160.3820) Materials : Magneto-optical materials

(260.3910) Physical optics : Metal optics

**ToC Category:**

Materials

**History**

Original Manuscript: February 19, 2013

Revised Manuscript: March 16, 2013

Manuscript Accepted: March 20, 2013

Published: May 10, 2013

**Citation**

Christian Wolff, Rogelio Rodríguez-Oliveros, and Kurt Busch, "Simple magneto–optic transition metal models for time–domain simulations," Opt. Express **21**, 12022-12037 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12022

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### References

- E. Th. Papaioannou, V. Kapaklis, E. Melander, B. Hjörvarsson, S. D. Pappas, P. Patoka, M. Giersig, P. Fumagalli, A. García–Martín, G. Ctistis, “Surface plasmons and magneto–optic activity in hexagonal Ni anti–dot arrays,” Opt. Express 19, 23867–23877 (2011). [CrossRef] [PubMed]
- E. Melander, E. Östman, J. Keller, J. Schmidt, E. Th. Papaioannou, V. Kapaklis, U. B. Arnalds, B. Caballero, A. García–Martín, J. C. Cuevas, B. Hjörvarsson, “Influence of the magnetic field on the plasmonic properties of transparent Ni anti-dot arrays,” Appl. Phys. Lett. 101, 063107 (2012). [CrossRef]
- V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia–Martin, J.–M. García–Martín, T. Thomay, A. Leitenstorfer, R. Bratschitsch, “Active magneto-plasmonics in hybrid metal-ferromagnet structures,” Nat. Phot. 4, 107–111 (2010). [CrossRef]
- A. García–Martín, G. Armelles, S. Pereira, “Light transport in photonic crystals composed of magneto–optically active materials,” Phys. Rev. B 71, 205116 (2005). [CrossRef]
- K. Busch, M. König, J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser & Photon. Rev. 5, 773–809 (2011).
- M. Y. Koledintseva, K. N. Rozanov, A. Orlandi, J. L. Drewniak, “Extraction of Lorentzian and Debye parameters of dielectric and magnetic dispersive materials for FDTD modeling,” J. Electr. Eng.–Slovak 53, 97–100 (2002).
- V. Korenman, J. L. Murray, R. E. Prange, “Local-band theory of itinerant ferromagnetism. I. Fermi-liquid theory,” Phys. Rev. B 16, 4032–4047 (1977). [CrossRef]
- P. R. Berman, “Optical Faraday rotation,” Am. J. Phys. 78, 270–276 (2009). [CrossRef]
- M. König, K. Busch, J. Niegemann, “The Discontinuous Galerkin time–domain method for Maxwells equations with anisotropic materials,” Phot. Nano. Fund. Appl. 8, 303–309 (2010). [CrossRef]
- J. Alvarez, L. D. Angulo, A. R. Bretones, S. G. Garcia, “3–D Discontinuous Galerkin time–domain method for anisotropic materials,” IEEE Antenn. Wireless Propag. Lett. 11, 1182 (2012). [CrossRef]
- J. E. Sipe, V. C. Y. So, M. Fukui, G. I. Stegeman, “Analysis of second–harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980). [CrossRef]
- R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, 2005).
- P. B. Johnson, R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974). [CrossRef]
- Š. Višňovský, V. Pařízek, M. Nývlt, P. Kielar, V. Prosser, R. Krishnan, “Magneto–optical Kerr spectra of nickel,” J. Magn. Magn. Mater. 127, 135–139 (1993). [CrossRef]
- Š. Višňovský, “Magneto–optical Ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986). [CrossRef]

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