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Abstract
Magnetoplasmons are the coupling of an external magnetic field and a plasmon or a localized plasmon, in the case of nanoparticles. We present a theoretical study, in the quasi-static limit, of the plasmonic response of nanoparticles when a constant magnetic field is applied. The plasmonic modes split into two satellite peaks with a frequency shift proportional to the magnetic field. The constant of proportionality is the effective Bohr magneton. This splitting of the fundamental plasmonic mode is akin to the splitting of energy levels in the Zeeman effect. The results are valid for any material that has a plasmonic response. For higher magnetic fields, the frequency shift of the splitting becomes non-linear with the magnetic field as what happens with the non-linear Zeeman effect.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The coupling of a constant magnetic field with a surface plasmon-polariton changes its dispersion relation and gives rise to magnetoplasmons (MPs). Magnetoplasmon studies go back to the work of Keyes [1] and Palik [2–5]. InSb was of particular interest because of the low effective mass of the charge carriers. It was found that MPs could be excited with magnetic fields of a few Teslas [2]. The change in the plasmonic response with an external magnetic field happens when an optically isotropic material becomes strongly anisotropic depending on the magnetic field’s magnitude and direction. Kushwaha [6] did an extensive review of the fundamentals of MPs in semiconductors.
The renewed interest in magnetoplasmonics is their potential use in THz plasmonics. Experimentally, the excitation of MPs of InSb and InAs on a dielectric interface for THz sensor applications with applied magnetic fields of a few tenths of a Tesla has been demonstrated [7]. The magneto-optical properties of InSb for different carriers and carrier concentration were also measured as well as the polar Kerr rotation [8]. The possibility of having a strong MP at room temperature and low magnetic fields makes these systems ideal for THz applications, such as polarization converters [9,10]. Graphene also exhibits an excellent magneto-plasmonic coupling [11–13], however, two distinct modes are possible: edge modes [14] and bulk modes that depend on the doping of graphene [15]. Black-phosphorus is also an excellent candidate for magneto-plasmonic material [16] as well as 2D electron gases [17]. Composite nanoporous layer systems made of Ag/CoFeB/Ag can be used to excite MPs [18]. Geometry can also be used to control the plasmonic response in the THz. Arrays of disks are another approach that has been using metasurfaces with a non-reciprocal magneto-optical response with a magnetic field parallel to the surfaces (Voigt configuration) [10,19,20].
Applying an external magnetic field to an optically isotropic material, induces an anisotropy in the dielectric function. This anisotropy is controlled by the intensity of the magnetic field and its orientation with respect to the semiconductor’s surface [6,21–23]. Changing the dielectric function with an external field can induce Van der Waals torques in nano structures [24], change the Casimir force in micro and nano-electromechanical devices [25], and alter the radiative heat flow in the near field between semiconductors [26].
Nanoparticles can also exhibit localized magnetoplasmons. A strong magneto-optical response of oleylamine-protected Ag nanoparticles was measured. For magnetic fields of 1.6 T, surface MP’s were detected [27]. A theoretical study of the lifetime of surface magnetoplasmons in metallic nanoparticles was done by Weick et al. [28] that showed that in the presence of an external magnetic field two collective electronic modes appear corresponding to two magnetoplasmon modes. The magnetic fields needed are at least 10 T which limits the experimental accessibility.
However, as with many materials of interest in plasmonics, it is now possible to synthesize InSb nanoparticles [29–31]. Magnetoplasmonic effects have been calculated for InSb nanoparticle dimers with each particle of the dimer at different temperatures, to achieve active control of the near-field radiative heat transfer [32].
Size [33], shape [34,35], and dielectric function determine the plasmonic response of nanoparticles. The absorption efficiency can be calculated in the quasi-static limit for small particles [33], using Mie scattering or the discrete dipole approximation [34,36]. In this work, we calculate, within the quasi-static approximation, the absorption efficiency of spherical nanoparticles of InSb as a function of an applied constant magnetic field. The otherwise isotropic nanoparticle becomes anisotropic, giving rise to two plasmonic peaks. The magnitude and separation of the peaks depend on the applied field. We show that this is a classical analog of the Zeeman effect.
2. Theory
For a material with a local isotropic dielectric function $\epsilon (\omega )$, application of a constant magnetic field $\vec {B}$ leads to an anisotropic dielectric function [6,23]. This anisotropy arises from the magnetic part of the Lorentz force. The dielectric tensor is
If the magnetic field is fixed along the $z$ axis and the dielectric tensor Eq. (1) becomes [32]
Where $\epsilon _{yy}=\epsilon _{xx}$. The components are
In this work we consider a nanosphere of radius $R$ made of InSb. An external magnetic field is applied along the $z$ axis. The particle is illuminated with linearly polarized light in the $xz$ plane forming an angle of $\phi$ with the $x$ axis, as shown in Fig. 1.
Fig. 1. Nanoparticle of radius $R$. A constant magnetic field is applied along the $z$ axis. Light impinges on the particle along the $y$ direction as linearly polarized light within the $xz$ plane forming an angle of $\phi$ with the $x$ axis. The sphere is embedded in a medium of dielectric constant $\epsilon _m$. The figure is not to scale, we will work only with wavelengths much larger than the diameter of the particle.
In the isotropic case, the polarizability of the particle of radius $R$ embedded in a medium with a dielectric function $\epsilon _m$ is
and the absorption efficiency for particles much smaller than the wavelength of the incident light is [33] where $k=2\pi \sqrt {\epsilon _m}/\lambda$. This relation (dipolar contribution) holds for small spheres compared with the wavelength, which is the case for nanoparticles ($\sim 10^{-9}m)$ in the THz region ($\sim 10^{-3}m-10^{-4}m)$.When the external magnetic field is applied, the nanoparticle becomes optically anisotropic (Eq. (2)) and in order to calculate the extinction coefficient we have to work along the principal axis of the system [37], where the diagonalized dielectric tensor is
This can be written as $\tilde {\epsilon }^{\prime }=diag(\epsilon _x^{\prime },\epsilon _y^{\prime },\epsilon _z^{\prime })$ where the corresponding eigenvectors are $e_x'=(i,1,0)$, $e_y'=(-i,1,0)$ and $e_z'=(0,0,1)$. If the electric field of the incident light is ${\vec E}=(a,0,c)E_0$, then the polarization can be written as $tan(\phi )=c/a$.
In this frame, the polarizability tensor takes the form $\tilde {\alpha }^{\prime }=diag(\alpha _{xx}^{\prime },\alpha _{yy}^{\prime },\alpha _{zz}^{\prime })$ where the components of the polarizability are the same as in Eq. (6), for example $\alpha ^{\prime }_{xx}= 4 \pi R^3(\epsilon _{xx}^{\prime }-\epsilon _m)/(\epsilon _{xx}^{\prime }+2\epsilon _m)$, and similarly for the other two components. The total absorption efficiency in the original coordinate system can be written in terms of the polarizability components along the principal axes as, [37]
The expressions for different polarization angles are
In Appendix (A) we show the derivation of these equations.
3. Results
We consider nanoparticles of InSb of radius $R=20$ nm in air ($\epsilon _m=1$). No finite-size corrections are needed [38,39] for this particle sizes. The parameters of the dielectric function [2–5] are $\epsilon _{\infty }=15.7$, $\omega _p=3.41\times 10^{13}$ rad/s, and $\gamma =3.71\times 10^{12}$ rad/s. The electronic effective mass is $m^*= 0.022m_e$. These values are for n-doped InSb, since we are using the dielectric function in the Drude-limit.
As shown in the previous section, the two parameters that can be changed are the external magnetic field and the polarization of the incident light. In Fig. 2 we present the total absorption efficiency for different polarization angles (a) $\phi =0^{\circ }$, (b) $\phi =30^{\circ }$, (c) $\phi =45^{\circ }$ and (d) $\phi =60^{\circ }$. In each panel, the different curves indicate the values of the external magnetic fields indicated. For $B=0$ the total absorption shows the characteristic plasmon resonance for a sphere centered at $\omega _0=29$ THz. Throughout the paper we define $Q_{abs}(\omega _0)=Q_0$. As the magnetic field increases, two satellite peaks around the original resonance begin to form. The separation of the peaks increases as the magnetic field increases. The resonance that is blue-shifted is defined as $\omega _+$ and the red-shifted resonance corresponds to a mode $\omega _{-}$. Depending on the polarization, the amplitude of the central plasmonic resonance at $\omega _0$ changes. We observe that for light polarized at $\phi =0^{\circ }$, the central resonance is not present after a certain value of the magnetic field. However, for other polarizations the central resonance decreases in amplitude.
Fig. 2. Absorption efficiency (in arbitrary units) of an InSb nanoparticle of radius $R=20$ nm, for different polarization angles (a) $\phi =0^{\circ }$, (b) $\phi =30^{\circ }$, (c) $\phi =45^{\circ }$, (d) $\phi =60^{\circ }$.
To understand the splitting, we fix the magnetic field at $B=1$T. In Fig. 3, we show the absorption efficiency as a function of the polarization angle. The position of the satellite resonances $\omega _+/\omega _0$ and $\omega _-/\omega _0$ are indicated by the dashed lines. We observe that the central PR at this particular value of the magnetic field is evident for polarization angles larger that $25^{\circ }$ and the satellite peaks decrease as we approach the maximum polarization angle, until they vanish at a polarization angle of $\phi =90^{\circ }$.
Fig. 3. Absorption efficiency for a fixed value of the external magnetic field ($B=1$T) as a function of the polarization angle. The position of the central resonance and the position of the satellite resonances $\omega _+/\omega _0$ and $\omega _-/\omega _0$ are indicated by the white dashed lines. For small polarization angles, the satellite peaks are more intense. For a polarization of $\phi =90^{\circ }$ only the central peak is present, indicated by the white solid line.
The splitting of the plasmonic resonances has been reported in planar structures. Indeed this is expected when exciting magnetoplasmons due to the anisotropic dielectric function [6]. The difference is that in planar structures, the direction of the magnetic field induces different optical responses [6]. When the magnetic field is parallel to the surface and perpendicular to the direction of propagation of surface plasmons, the behavior is similar to what we report here. In the case of a metamaterial made of a photonic crystal filled with a ferrofluid, the transmission coefficient shows the plasmonic resonances and their splitting as a function of the magnetic field intensity [40]. The same effect is observed in magnetized InSb under circularly polarized light [10].
4. Similarities with the Zeeman effect
In the presence of an external magnetic field, the atomic spectral lines split into several components. The normal Zeeman effect does not take into account the spin of the electrons, and depends on the magnitude of the applied field. The transverse observation of the atomic spectra changes with the presence of a polarizer. No splitting is observed if the polarizer is parallel to the magnetic field. Nevertheless, when the field is perpendicular to the polarization filter, the central line vanishes and only the satellite lines can be observed. Another characteristic of the quantum Zeeman effect is that the separation of the split-lines is constant and linearly proportional to Bohr magnetron for small magnetic fields. As shown in Figs. 2 and 3, the same dependence with polarization is observed for the plasmonic case.
The linear dependence of the frequency shift of the resonances $\omega _{+(-)}$ with the magnetic field is clearly seen in the contour plot of the absorption efficiency for the different angles shown in Fig. 4. The white dotted line shows the best fit for small magnetic fields to the data and both $\omega _+$ and $\omega _-$ vary linearly with the magnetic field as
The shift in frequency can be written in terms of the effective Bohr magneton $\mu _B^*=q \hbar B/2m^*$. Multiplying Eq. (14) by Planck’s constant $\hbar$ these equations can be written as
and the energy difference is $\Delta E=2 \mu ^*_B B$. In the atomic Zeeman effect the energy difference is the same, except that the Bohr magneton depends on the bare mass of the electron.Fig. 4. Shift in the frequency position of the satellite resonances $\omega _+$ and $\omega _-$ as a function of the magnetic field. The different panels correspond to different polarization angles as indicated. The best fit for small magnetic fields is indicated by the white-dotted line. As the magnetic field increases, the shift in frequency of the plasmonic resonances is no longer linear and quadratic dependence is observed, as indicated by the yellow-dashed line.
As the magnetic field becomes larger, the splitting is no longer linear with the applied magnetic field. In addition to the linear fit for small fields (dotted line) shown in Fig. 4, the non-linear behavior is also shown (dashed line), and corresponds to Eq. (20) derived in the Discussion section.
This non-linear behavior with the magnetic field has also been observed in the Zeeman effect of Rydberg atoms, where the application of an external magnetic field is used to obtain electromagnetic induced transparency [41].
It is important to emphasize that the observed effect is an analog of the quantum Zeeman effect (QZE). In the QZE we have the ground state of an atomic level, and excited states, while in the plasmonic Zeeman effect the plasmonic resonances do not have the equivalent excited state. There are many instances where there is a classical phenomena similar to a quantum phenomena, being a typical example the acoustic and photonic crystals, Anderson localization, Bloch oscillations among others [42]. In Table 1 we make a comparison between the quantum and plasmonic Zeeman effect to understand the similarities and differences.
Table 1. Comparison between the quantum Zeeman effect and the plasmonic Zeeman effect.
Finally, it is important to notice that it is possible to have changes in the plasmonic response induced by the quantum Zeeman effect. In the case of faceted nanocrystals, an external magnetic field induces an excitonic Zeeman effect that changes the plasmonic response of the crystal [43]. We emphasize that this is different from the plasmonic Zeeman effect.
5. General validity
The linear dependence of the frequency shift given by Eq. (14) as well as the quadratic behavior (Eq. (16)) is observed for other materials. We present a contour plot of the absorption efficiency of the absorption efficiency as a function of the normalized frequency $\omega /\omega _0$ and the normalized cyclotron frequency $\Omega _c=\omega _c/\omega _0$ for GaAs and Na. In each case, the frequency of the characteristic plasmonic resonance ($\omega _0$) corresponds to the material, this being $\omega _0($GaAs$) = 307$ THz and $\omega _0($Na$) = 797$ THz. As a reference, the magnetic field is also indicated on the top axis. Figure 5 (a) shows the splitting of the magnetoplasmons for GaAs. As the effective mass increases, larger magnetic fields are needed since $\omega _c$ decreases. Finally, we show the results for a Na nanoparticle in panel (b). The magnetic fields needed to observe the splitting are quite large and unattainable experimentally. The results are consistent with those obtained by Weick et al. [28] using a full quantum approach for the linear behavior. However, for higher magnetic fields our model was able to predict a quadratic behavior. In all cases, the linear dependence with the magnetic field is shown when the condition $\omega _0>\omega _c$ is fulfilled, as indicated by the dashed lines in the figures and the slope in each case is $q/2m^*$ or multiplying by $\hbar$ the slope is $\mu _B^*$.
Fig. 5. Contour plot of the absorption efficiencies ($Q/Q_{0}$) as a function of the normalized frequency $\omega /\omega _0$ and the normalized cyclotron frequency $\Omega _c=\omega _c/\omega _0$. The top axis in each panel shows the value of the magnetic field. The panels are (a) GaAs and (b) Na. For panel (a) $\omega _0($GaAs$) = 307$ THz while for panel (b) $\omega _0($Na$) = 797$ THz
The results so far presented, are valid for any semiconductor. However, InSb, due to its low effective mass, is a suitable material to excite magneto-plasmons, because the magnetic fields needed are small. Furthermore, the plasmon resonance of InSb is in the THz, which makes it a convenient material for many applications. Nonetheless, the PZE is not unique to InSb.
6. Discussion
To understand the results from a classical point of view, we calculate the frequency of the two modes $\omega _+$ and $\omega _-$. The magnetic part of the Lorentz force in a charge carrier of effective mass $m^*$ and charge $q$ is $\vec {F}=q\vec {v}\times \vec {B}$. The velocity vector is $\vec {v}=(v_x,v_y,v_z)$. With the magnetic vector along the $z$ axis, the components of the Lorentz force are $\vec {F}=(v_y B, -v_x B,0)$. The electrons are bounded to the metal and there is a restitution force with a constant $k$. This leads to the equations of motion
From the third equation we obtain the normal frequency $\omega _0^2=k/m^*$, and from the other two equations the frequencies for the system are
and when $\omega _c < \omega _0$ the two modes of Eq. (14) are recovered. This is $\omega _+=-(\omega _0+\omega _c/2)$ and $\omega _+=+(\omega _0-\omega _c/2)$. The shift is linear with magnetic field for small values of the field in accordance to Fig. 3. The motion of the carriers in the two modes is on the $x-y$ plane where one corresponds to clockwise motion while the other one to counter clockwise, as indicated by the sign. This is depicted in Fig. 6Fig. 6. Schematics of the two possible satellite modes of the total extinction efficiency for a polarization angle of $\pi /4$.
7. Conclusions
We presented a theoretical calculation of the plasmonic response of spherical nanoparticles in the presence of an external magnetic field. Without a magnetic field, the nanoparticles exhibit the single plasmon resonance typical of spherical particles. The application of an external magnetic field splits the primary plasmonic response into two satellite peaks. The frequency shift of the peaks is proportional to the effective Bohr magneton as in the normal Zeeman effect and is linear in the magnetic field for small fields. A simple model describes the splitting in the dipolar excitation approximation, which predicts the linear behavior. As the external magnetic field increases, the frequency shift in the plasmonic peaks becomes nonlinear, showing a quadratic dependence in the magnetic field, that is akin to the quadratic Zeeman effect observed in Rydberg atoms. In the atomic Zeeman effect (normal), the energy level splits into two satellite levels and the original level remains. In our case, the plasmonic resonances behave in like manner, obeying an analogous equation for the shifting, thus our claim of a plasmonic Zeeman behavior. The work we presented can be extended to other shapes of nanoparticles. For example ellipsoids or nanocubes that will have a richest plasmonic structure. However, for nanocubes, analytical solutions are not possible and numerical approaches such as the discrete dipole approximation [44,45] has to be considered.
Appendix A
We briefly present the derivation of Eqs. (10–13) as shown in Bohren Ref. ([37]).
Let’s asume an incident electromagnetic wave with a wavevector $\vec {k} = (0,k,0)$, in such a way that the electric field is polarized within the xz-plane, this can be written as
where $(a,0,c)$ is a unit vector, thereby satisfying thatThe polarization angle $\phi$ is given by
from equations (2) and (3) it follows thatOn another note, since the principal axes system satisfies
Therefore, the absorption efficiency takes the form
As an example, in the case where $\phi = 45^{\circ }$, then $a = c = \frac {1}{\sqrt {2}}$ and consequently
Similarly, for
Funding
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN110819).
Acknowledgements
The authors thank G. Pirruccio and J. A. Reyes-Coronado for their valuable comments to the manuscript.
Disclosures
The authors declare no conflict of interest.
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