Surface plasmon-induced enhancement of the magneto-optical Kerr ...

Demidenko et al.

Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B

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Surface plasmon-induced enhancement of the magneto-optical Kerr effect in magnetoplasmonic heterostructures Y. Demidenko,1 D. Makarov,2,* O. G. Schmidt,2 and V. Lozovski1,3 1

V. Lashkariov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Nauki ave. 45, Kyiv 03028, Ukraine 2 Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstraβe 20, D-01069 Dresden, Germany 3 Institute of High Technologies, Taras Shevchenko National University of Kyiv, Volodymyrska str. 64, Kyiv 01601, Ukraine *Corresponding author: d.makarov@ifw‐dresden.de Received April 14, 2011; revised July 6, 2011; accepted July 8, 2011; posted July 8, 2011 (Doc. ID 145958); published August 8, 2011 We developed a self-consistent analytical approach to describe the optical response of a magnetoplasmonic heterostructure upon surface plasmon polariton (SPP) excitation. The approach is based on the effective susceptibility concept in the frame of the Green’s function method and accounts for the local-field effects in the system. The formalism was applied to describe the polar Kerr effect in magnetoplasmonic bilayers consisting of Co and Au on an attenuated total reflection prism. It is demonstrated that the excitation of the SPP in the plasmonic film can lead to an enhancement of the Kerr effect. Analysis performed for the s- and p-polarized incident radiation showed that the enhancement of the Kerr effect is of resonant character, and it is observed when the incident angle of the probing light is close to the angles of the SPP excitation. Thickness-dependent study revealed that the largest enhancement of the Kerr effect is observed for rather thin magnetic layers in the range of couple of nanometers. © 2011 Optical Society of America OCIS codes: 260.2065, 240.5420, 240.6680, 310.5448.

1. INTRODUCTION Active plasmonics is a modern research field where active functionality is added to conventional optical devices. In this respect, composite materials have to be used, enabling one to alter optical properties by external means, including temperature as well as magnetic and electric fields [1–4]. By combining magnetic (i.e., Co) and plasmonic (i.e., Au) materials, the so-called magnetoplasmonic composite can be realized [4–6]. The characteristic feature of the magnetoplasmonic systems is the possibility of resonant interaction between the plasmonic and magnetic subsystems [4,7–9]. This interaction can, in part, lead to an enhancement of the magneto-optical effects, i.e., Kerr effect. Magneto-optical effects constitute the change of the polarization or/and attenuation of an incoming light while interacting with magnetized media. There is a number of experimental and theoretical works devoted to study the properties of magnetoplasmonic systems [10–13]. Thus, for example, the enhancement of the magneto-optical polar Kerr effect in Fe–Cu bilayers was demonstrated to be caused by the plasma resonance effect of the free charge carriers in Cu [14]. Besides the academic interest behind such a plasmon-induced modification of the magneto-optical response, the possibility to enhance the Kerr effect is crucial, especially for the investigation of ultrathin magnetic films, when small signals have to be measured experimentally [15]. However, issues with high losses in magnetic layers have to be addressed, as they cause an overdamping of plasmon resonance and thus a possible decrease of the magnetoplasmonic resonance enhancement. Therefore, the optimization of the structural properties of 0740-3224/11/092115-08$15.00/0

magnetoplasmonic systems, i.e., the thickness of the layers, is of great importance. The theoretical study of the properties of magnetoplasmonic systems is performed by numerical methods mainly. For example, in the work by García-Martín et al. [16], the numerical solution of Maxwell’s equations was applied to examine a magnetoplasmonic system consisting of an array of cylindrical nanoparticles. However, it is advantageous to have a simple and straightforward tool that allows the estimation of the magneto-optical properties of complex systems, consisting of magnetic and plasmonic components. In this case, analytical approaches have to be put forward. In this respect, the s-matrix propagation method was applied by Bonod et al. [17] to treat analytically the structure properties to optimize the surface-plasmon enhancement of the magneto-optical effects. Particularly, it was shown that the most effective enhancer of the magneto-optical effects is the high-quality surface plasmons for the transverse excitation geometry and may be the overdamped surface plasmons for the polar and longitudinal ones [17]. We put forth the analytical approach [6,8] based on the Green’s function method [18–20] to treat the magnetooptical properties in composite magnetic/nonmagnetic nanoparticles. The approach accounts for the local-field effects and, thus, allows treating correctly the influence of the system shape and dimensions on its magneto-optical response. In addition to the localized surface plasmons characteristic of nano-objects, propagating surface plasmon polaritons (SPPs) are of great application relevance [21] in spectroscopy [22], microscopy [23], and sensorics [24–26], to name a few. There are studies on the magnetoplasmonic effects in systems with © 2011 Optical Society of America

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propagating SPP [27–30]. In part, the two first works [27,28] were devoted to the theoretical description in the frame of Green’s dyadic technique of the magneto-optical effects in metallic multilayers under the condition of total internal reflection. The numerical simulations, which account for the variation of the angle of incidence at a fixed wavelength and the variation of the wavelength at a fixed angle of incidence, were discussed. In the present work, we developed a self-consistent analytical approach to describe the optical response of a magnetoplasmonic planar heterostructure on the excitation of the SPP. The peculiarity of this approach is in the possibility to treat the composite with a magnetic layer possessing out-of-plane easy axis of magnetization. The developed formalism was applied to describe the polar Kerr effect in magnetoplasmonic bilayers consisting of Co and Au on an attenuated total reflection (ATR) prism. The peculiarities of the resonant interaction between plasmonic and magnetic components is the focus of the discussion. The article is organized as follows: in Section 2, we give general remarks on the model and outline system classes that can be treated using the proposed theoretical approach. Section 3 deals with the Green’s functions for the model of choice. Application of the developed approach to consider polar Kerr effect in magnetoplasmonic composite is discussed in Section 4. Numerical tabulation of the analytical expression of the polar Kerr effect in Co–Au bilayers and discussion of the obtained results is summarized in the Section 5.

2. MODEL We consider a magnetoplasmonic composite of the total thickness h (h ¼ hp þ hm ), which consists of a plasmonic (thickness hp ) and a ferromagnetic (thickness hm ) layer. From now on, characteristics related to magnetic and plasmonic films will be indicated with the subscripts “m” and “p,” respectively. The composite is located on an ATR prism with dielectric constant εpr [31], as shown in Fig. 1. The proposed approach is limited to the consideration of magneto-optical effects in magnetoplasmonic composites with ultrathin magnetic layers. In the following, it is assumed that the magnetic film possesses the easy axis of magnetization oriented perpendicular to the film plane (along the z axis, Fig. 1). Furthermore, we restrict our consideration to the case when

Demidenko et al.

the external magnetic field is applied along the z axis as well (Fig. 1). Optical response of the magnetoplasmonic composite will be treated considering the dielectric functions of the layers. Thus, the dielectric functions of the plasmonic part can be presented in the form ↔ðPÞ

ε

↔

ðωÞ ¼ εðPÞ ðωÞU ;

εðPÞ ðωÞ ¼ 1 −

ω2pp ; ωðω þ iγ p Þ

ð1Þ

↔

where U is unit dyadic and ωpp and γ p are the plasma frequency and the damping constant, respectively. The dispersion properties of the magnetic film can be described in the frame of the Drude model for magnetized plasma [32] 0 1 1 iQðω; BÞ 0 ↔ðMÞ B C ε ðω; BÞ ¼ εd ðωÞ@ −iQðω; BÞ 1 0 A; ð2Þ 0 0 1 with εd ðωÞ ¼ 1 −

ω2pm ; ωðω þ iγ m Þ

and Qðω; BÞ ¼ ωc =ω:

ð3Þ

Here, ωc ¼ e · B=m
is the cyclotron frequency (e denotes the charge of an electron, B is the strength of the magnetic field acting on the system, and m
is the effective mass of an electron).

3. GREEN’S FUNCTIONS OF THE SYSTEM In order to describe the magneto-optical response of a composite consisting of thin magnetic layers, the pseudovacuum Green’s function method will be applied [20]. In this approach, the Green’s function of the medium with two interfaces is considered first [19]. Then, the so-called k-z representation is used, where the Fourier transformation in the film plane (coinciding with the XOY plane of the Cartesian coordinate system) is performed. The latter is justified as the system is assumed to be homogeneous in the plane. The Green’s function depends on the plane where the source of the field is located (z0 ) and the plane where the field is observed (z). Therefore, several Green’s functions differing by location of the source and observation planes should be considered— ↔ab

Gð0Þ ðk; z; z0 Þ, with a and b corresponding to < or/and >. The > sign indicates that the plane is located above the interface “plasmonic film–external medium,” and the < sign means the plane is located below this interface. The following Green’s functions will be used. ↔>>

Fig. 1. (Color online) Sketch revealing the excitation geometry of plasmons in a magnetoplasmonic composite using an ATR prism. Total thickness of the composite is h (h ¼ hm þ hp ), which consists of a plasmonic (thickness hp ) and a ferromagnetic (thickness hm ) layer. The external magnetic field, B, is applied along the OZ axis. The incident electromagnetic wave, Eð0Þ , is shown together with the reflected, EðRÞ , and the transmitted, EðTÞ , ones.

i. The Green’s function Gð0Þ ðk; z; z0 Þ describes the propagation of light in the k-z space of pseudovacuum (thin film on a substrate), when both the excitation plane (z0 ) and the observation plane (z) are in the upper semispace (z > 0, Fig. 2). ↔< iii. The Green’s function GðmÞ ðk; ω; z; z0 Þ accounts for the presence of the thin magnetic layer. Based on the theory

Demidenko et al.

Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B

(a)

4. POLAR KERR EFFECT

z'

Let the polarization of the incident light be β and the polarization of the reflected wave be α. Then, the reflection coefficients, r αβ , can be introduced. The polar Kerr effect (rotation of the polarization plane of the reflected wave) in the geometry under consideration (Fig. 1) can be described by the ratio of the reflection coefficients [33,34]:

↔

z

>> G(0) (k, z, z′)

z 0

x

↔

>< G(0) (k, z, z ')

z ↔

< ðk∥ ; ω; z; lm Þ are exponents, describing the dependence of the Green’s function on the z coordinate. Then, the Kerr effect can be described as follows: ΘSK ðθ; ωÞ ¼ ðξm ðωÞQðωÞÞ2

jΔðSÞ ðθ; ωÞj2 ΦPS ðθ; ωÞ −2ξm ðωÞReb3 ðθÞ ; e jΔðPÞ ðθ; ωÞj2 ΦSS ðθ; ωÞ ð17Þ

with ðmÞ ðθ; ωÞΔðSÞ ðθ; ωÞ ΦSS ðθ; ωÞ ¼ jI < ð0Þyy ðθ; ωÞe

ð18Þ

ΦPS ðθ; ωÞ ¼ jðF ð0Þxx ðθ; ωÞΨðmÞxy ðθ; ωÞ >< 2 þ F ð0Þxz ðθ; ωÞΨðmÞzy ðθ; ωÞÞF ð0Þyy ðθ; ωÞj

þ jðF ð0Þzx ðθ; ωÞΨðmÞxy ðθ; ωÞ >< 2 þ F ð0Þzz ðθ; ωÞΨðmÞzy ðθ; ωÞÞF ð0Þyy ðθ; ωÞj :

Here, b3 ðθÞ ¼

ð19Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εpr sin2 θ − 1; DðmÞ ðθ; ωÞ is the pole part and

ΨðmÞij ðθ; ωÞ is the tensor part of the linear response function X ðmÞij ðθ; ωÞ ¼ ΨðmÞij ðθ; ωÞ=DðmÞ ðθ; ωÞ [see Eq. (A10)]. Similar to Eq. (18), the expression for the Kerr effect can be written for the case when the incident light is p polarized: jΔðPÞ ðθ; ωÞj2 ΦSP ðθ; ωÞ −2ξm ðωÞReb3 ðθÞ ΘPK ðθ; ωÞ ¼ ðQðωÞξm ðωÞÞ2 ðSÞ ; e jΔ ðθ; ωÞj2 ΦPP ðθ; ωÞ ð20Þ with >< ΦSP ðθ; ωÞ ¼ jF ð0Þyy ðθ; ωÞfΨðmÞyi ðθ; ωÞF ð0Þix ðθ; ωÞ cos θ 2 þ ΨðmÞyi ðθ; ωÞF >< ð0Þiz ðθ; ωÞ sin θgj ;

ðmÞ ðθ; ωÞðΔðpÞ ðθ; ωÞÞ2 þ I < − ξm ðωÞF ð0Þxi ðθ; ωÞΨðmÞin ðθ; ωÞ½F ð0Þnx ðθ; ωÞ cos θ −ξm ðωÞb3 ðθÞ j2 þ F >< ð0Þnz ðθ; ωÞ sin θe

þ jðI < ðkjj ; ωÞ yy ðkjj ; ωÞ

;>
0) can be written via the corre↔>
ð0Þ ðk; ω; zÞ is the filed in the upper semispace of the pseudovacuum (when the magnetic layer is absent) induced by the 0 currents J< ext ðk; ω; z Þ. Assuming that the thickness of the magnetic film is small compared to the characteristic length (i.e., wavelength of the incident radiation, thickness of the plasmonic film), Eq. (A2) can be presented in the form > E> ðmÞ ðk; ω; zÞ ≅ Eð0Þ ðk; ω; zÞ ↔>>

− iμ0 ωhm Gð0Þ ðk; ω; z; lm ÞJ> ðmÞ ðk; ω; lm Þ; ðA3Þ with lm ¼ hm =2. Taking into account the constitutive equation, which connects local current with the local field, ↔

> J> ðmÞ ðk; ω; lm Þ ¼ −iωε0 χ ðmÞ ðωÞEðmÞ ðk; ω; lm Þ:

Equation (A3) can be written as > E> ðmÞ ðk; ω; zÞ ≅ Eð0Þ ðk; ω; zÞ ↔>>

↔

− k20 hm Gð0Þ ðk; ω; z; lm Þ χ ðmÞ ðωÞE> ðmÞ ðk; ω; lm Þ; ðA5Þ where k0 ¼ ω=c. The linear response to the local field can be expressed via dielectric function of the magnetic layer, ↔

↔

↔ðMÞ

namely, χ ðmÞ ðωÞ ¼ ε ðω; BÞ − U , which is dependent on an external magnetic field. Combining Eqs. (A1) and (A5), the unknown Green’s ↔>


>

↔

− k20 hm Gð0Þ ðk; ω; z; lm Þ χ ðmÞ ðωÞ ↔>


ðmÞ ðk; ω; zÞ ¼ −iμ0 ω

> E> ðmÞ ðk; ω; zÞ ¼ Eð0Þ ðk; ω; zÞ Z ↔>> 00 − iμ0 ω dz00 Gð0Þ ðk; ω; z; z00 ÞJ> ðmÞ ðk; ω; z Þ;

ðA1Þ where integration is over the field source. The field E> ðmÞ ðk; ω; zÞ induces currents inside the magnetic layer, 00 J> ðmÞ ðk; ω; z Þ, which are the source of the reradiated field. Then, the total electric field in the upper semispace (z > 0) can be written as

↔>
>

↔

Ω ðmÞ ðk; ω; lm Þ ¼ ½U þ k20 hm Gð0Þ ðk; ω; lm ; lm Þ χ ðmÞ ðωÞ−1 : ðA8Þ

When substituting Eq. (A7) into Eq. (A6), the explicit form of the Green’s function of the system takes the form

Demidenko et al. ↔>

>

↔

− k20 hm Gð0Þ ðk; ω; z; lm ÞX ðmÞ ðk; ω; lm ; lm Þ ↔>