Plasmonic resonances in metallic nanoarrays Bachelor Thesis
Jana Huber
Spring/Summer 2015
Faculty of Science Department of Physics First Evaluator: Prof. Dr. Elke Scheer Second Evaluator: Ass. Prof. Dr. Vassilios Kapaklis
Department of Physics and Astronomy Materials Physics Supervisor: Ass. Prof. Dr. Vassilios Kapaklis
Abstract
The optical and magneto-optical response of plasmonic resonances in metallic nanoarrays out of square structures, either in holes or islands, were investigated. The excitation of the Bragg Plasmons (BP) takes place within a grating. Signi cant dierences in the excited plasmon modes were seen by using p- or s-polarized light as well between the holes and islands sample. In order to investigate magneto-optical response from the magnetic nanostrucures, transverse magneto-optical Kerr eect (TMOKE) measurements were done with the result that there is a dierence in holes and islands sample. Contrary to what is generally expected for the polarization dependence of TMOKE, a TMOKE signal for s-polarized light on the holes sample was measured.
Deutsche Zusammenfassung
Plasmonen sind quantisierte Schwingungen des freien Elektronengases in einem Metall. Diese können an der Grenz äche zu einem Dielektrikum Ober ächenplasmonen ausbilden. Dabei sind die Dichteschwingungen im Metall an das elektromagnetische Feld im Dielektrikum gekoppelt. In der folgenden Arbeit wurden plasmonische Resonanzen in verschiedenen Nanostrukturen untersucht. Diese unterscheiden sich in der Struktur (Löcher oder Inseln) wie auch in der Gröÿe. Da die Anregung über ein periodisches Gitter passiert, wurde die Periodizität der Nanostrukturen in allen Fällen gleich gehalten. Die Resonanzen wurden mittels optischen Messungen in Re ektivität und Transmission untersucht. Dabei wurde jeweils p- sowie s-polarisiertes Licht verwendet. Auallend hierbei ist, dass es möglich ist mit den zwei verschieden Polarisierungsmöglichkeiten unterschiedliche Resonanzen anzure- gen, die jeweils in die Polarisierungsrichtung des elektrischen Feldvektors laufen. Ebenfalls wurden markante Unterschiede zwischen Löcher- und Inselstrukturen gefunden. Allerdings sind diese Strukturen nicht komplementär in ihren Gröÿen, weshalb die unterschiedliche Reaktion nicht nur eine Ursache der unterschiedlichen Struktur, sondern auch der unterschiedlichen Gröÿenverhältnissen sein könnte. Um darüber hinaus Aussagen über die magneto-optische Reaktion machen zu können wurden transversale magneto-optische Kerr-Eekt Messungen durchgeführt. Diese zeigen ebenfalls deutliche Unterschiede zwischen Löcher- und Inselstruktur. Für den transversalen magneto-optischen Kerr-Eekt wird das Magnetfeld parallel zur Probe und senkrecht zur Einfallsebene des Lichtes ausgerichtet, weshalb nur für p-polarisiertes Licht ein Signal er- wartet wird. Dementgegen wurde für die Löcherstruktur ein deutliches Signal mit s-polarisiertem Licht gemessen. Die Ursache dafür ist noch nicht erklärt.
Contents
1 Introduction 1
2 Theoretical Background 2 2.1 Optics ...... 2 2.1.1 States of Light ...... 2 2.1.2 Light in Matter ...... 3 2.1.3 Optical Properties ...... 4 2.1.4 Re ection and Transmission ...... 5 2.1.5 Babinet's Principle ...... 6 2.2 Plasmons ...... 7 2.2.1 Volume Plasmons ...... 7 2.2.2 Localized Plasmons (LSPs) ...... 8 2.2.3 Surface Plasmons Polaritons (SPPs) ...... 9 2.2.4 Excitation of Surface Plasmon Polaritons ...... 12 2.3 Magnetism ...... 13 2.3.1 Ferromagnetism ...... 14 2.3.2 Magneto-Optical Kerr Eect ...... 16 2.4 Sample ...... 20 2.4.1 Deposition ...... 20 2.4.2 Patterning ...... 20 2.4.3 Structure ...... 21 2.4.4 Grating Coupling for Square Structures ...... 22 2.5 Measurement Techniques and Equipment ...... 24 2.5.1 AFM ...... 24 2.5.2 TMOKE ...... 25 2.5.3 Experimental Set-Up ...... 26
3 Measurements, Results and Discussion 27 3.1 AFM-Measurements ...... 27 3.1.1 Calculation of the Amount of Metal ...... 28 3.2 Optical-Measurements ...... 28 3.2.1 Strength of Resonances γ ...... 29 3.2.2 Re ectivity ...... 29 3.2.3 Transmission ...... 36 3.2.4 Absorption ...... 38 3.2.5 Babinet's Principle ...... 39 3.3 Magneto-Optical Measurements ...... 40
4 Conclusion and Outlook 44
5 Acknowledgment 45
6 List of Figures 46
7 List of Tables 48
8 References 49 1 Introduction
Optics, the science of light, has amounted to plenty of useful tools for the humanity. Ev- ery morning we take a look in the mirror, switch on the light and get the news from the internet over a glass ber directly to the breakfast table. Unfortunately the size of optical devices, like glass bers, are constrained by the diraction limit [1]. In order to avoid interference eects between two neighboring light waves, the width of the glass ber must be at least half of the light's wavelength [1]. For devices used in computer technology the sizes of optical components are a lot bigger than that of electronic devices. However they can carry a lot more data and are also faster than the electronic devices. If it would be possible to squeeze down the size of optical devices it would be possible to transfer a huge amount of data quickly across a computer chip. Indeed researchers have found a technology to squeeze light down to a sub-wavelength scale. This is possible if one shines light at the interface between a dielectric and a metal. Under certain circumstances the oscillation of the free electrons in the metal can couple to the electromagnetic eld in the dielectric and a coupled state between these two oc- curs, called a Surface Plasmon Polariton (SPP). This charge density wave travels along the metal/dielectric interface, like ripples on a water pond. The SPP waves can be used in future applications to transfer data very fast on computer chips. [1] In order to use SPPs in such applications it is very important to know how to in uence and control these resonances. Material properties as well as the structure on which the excitation takes place is a big playground to gure out how the resonances could be con- trolled optimally. This work focuses on the in uence of dierent structures to the plasmon resonances. But one can not only tune the light with optical devices. As it was shown by M. Faraday and J. Kerr in the 19th century also magnetic elds can in uence the properties of light. Magneto-optics was born and is used nowadays in plenty of applications for example in data storage devices. To use magneto-optical materials in plasmonics opens the possibility to use external magnetic elds in order to control the plasmon resonances. An interesting application might be the improvement of magneto-optic data storage. [2] To understand the physics of such a wave one has to take a closer look at the optical and magneto-optical interplay between a metal and a dielectric.
1 2 Theoretical Background
2.1 Optics
Light and optics is omnipresent in our everyday life. We wear glasses to see better, use light to send data around the world and even our life rhythm is adjusted to the sun circulation. It is no wonder that optics is one of the oldest parts of physics. Since the antiquity people have been curious about the nature of light. Already in this time simple optical devices like mirrors and magni ers were used. In the 17th century the development of the telescope lead to a revolution of the concept of the world. The desire to explain the nature of light became huge and two competitive theories were discussed. Is light a particle or a wave? This question was rst answered with the development of the quantum mechanics in the 20thcentury. New optical devices like the LASER in uences our life deeply and it would look completely dierent without this invention. Nowadays optics is hidden and gets smaller and smaller. In the bar-code scanner in the supermarket, in the CD player or remote controls. But there are a lot more scenarios thinkable how we could use light to improve our life. They reach from the improvement of computer chips to cancer treatment or even invisibility cloaks. To understand the idea behind these potential future applications one has to look closer at the interplay between light and matter for dierent materials. J. C. Maxwell made it with his development of the Maxwell equations evident that light can be described as an electromagnetic wave. The curl-Maxwell equations state that a time varying electric eld E~ causes a magnetic eld B~ which is perpendicular to the changing direction of the electric eld. The same is true for a changing magnetic eld which generates an electric eld. In this way they can recreate each other in an endless cycle. [3] The electromagnetic wave can be considered as a plane wave, traveling in the direction of the wave vector ~k with the angular frequency ω and an amplitude E0. The connection between the absolute value of the wave vector ~k and the angular frequency ω is named as the dispersion relation and is given for an electromagnetic wave by: [3] ω k = c where c is the speed of light in vacuum.
2.1.1 States of Light
Electromagnetic waves are transversal meaning electric and magnetic elds are oscillating perpendicular to the propagation direction ~k. One can de ne dierent polarization states depending on the orientation of the electric eld vector to the wave vector, sketched in gure 1. When the electric eld vector has a xed orientation in the space one speaks about linear polarized light. Whereas one speaks about circularly polarized light when the electric eld vector describes a circular path. The most general case is elliptically polarized light. Here the direction and the strength of the electric eld is changing with time. The electric eld vector describes an ellipse.
2 Figure 1: Assuming the electromagnetic wave is travelling in z direction and the electric eld vector E is oscillating in the xy-plane, the three dierent kinds of polariza- tion states are sketched.
2.1.2 Light in Matter
When an electromagnetic wave propagates through matter its electric eld will interact with the elds of the atoms in the material. Electron oscillations are induced and depending on the material's properties dierent optic and magnetic eects occur. Depending on if one is looking at bound or free electrons, there will be a dierent response to a penetrating electromagnetic wave. At an interface between such two materials the excitation of surface plasmons will take place. The dierent response can be reduced to one physical quantity called the dielectric function (ω), where ω is the angular frequency. The dielectric function will be calculated for two dierent models, the Lorentz Model which is associated with bound electrons and the Drude Model which refers to the free electrons. According to the Lorentz Model the case of bound electrons can be described as a positive core surrounded by an electron cloud. By applying an external oscillating electric eld the positive and negative charges are induced to harmonic oscillations. As a result they get separated and produce their own electric eld which leads to the polarization P~ :
P~ = 0 · [χ] · E~
Where 0 is the vacuum permittivity, χ the electric susceptibility, which states how easy it is to move the charges and E~ is the electric eld. The susceptibility is a symmetric tensor but can be replaced by a scalar for cubic crystals and amorphous materials. However the in uence of the tensor can be mostly neglected and in these cases the tensor will be replaced by a scalar. The same is valid for the dielectric tensor [] without the in uence of an external magnetic eld. The case with an external magnetic eld will be discussed later. The dielectric and susceptibility tensor are connected by:
[] = [1] + [χ] (1) The described polarization is a result of the separated charges induced by the harmonic oscillations which can be described by a damped and driven harmonic oscillator:
∂2~r ∂~r m + mγ + mω ~r = −qE~ (2) ∂t2 ∂t 0 With m as the mass of the electrons, ~r as the space coordinate, t as the time γ as a dissipation factor, ω0 as the eigenfrequency and q as the particle's charge.
3 q One has the acceleration force, the frictional force, the restoring force with K ω0 = m where K is the spring constant and the electric force. Solving this equations leads to [4]:
q E~ (ω) ~r(ω) = − 2 2 me ω0 − ω − iγω with ω as the frequency of the external electric eld. The result shows the displacement of the electron cloud referring to the former position. For the susceptibility follows: [4]
2 ωp χ = 2 2 ω0 − ω − iωγ Where Nq2 is the plasma frequency with as the electron density. Further for the ωp = m N dielectric function0 follows with equation (1):
ω2 p (3) (ω) = 1 + χ(ω) = 1 + 2 2 ω0 − ω − iγω This equation can be split in real and imaginary parts which is useful for later discussions:
2 0 00 2 ω0 − ω 2 ωγ (ω) = (ω) + i (ω) = 1 + ωp 2 2 2 2 + iωp 2 2 2 (ω0 − ω ) + ω γ (ω0 − ω) + ω A metal can be described by a gas of free moving electrons with the density N, oscillating against the positive core. Due to the free electrons there is no restoring force as in the formulation (2) which was used for the bound electrons. Thus one can set ω0 = 0 in equation (3) and for the dielectric function follows:
ω2 (ω) = 1 − p ω + iγω In a metal one has plenty of free moving electrons. As a result the dielectric constant becomes negative for frequencies below the plasma frequency and an incoming electro- magnetic wave can not penetrate inside the material because the electrons will always compensate the incoming eld. As a result the incoming electromagnetic wave gets re- ected. Only for frequencies higher than the plasma frequencies the electrons can not longer follow the oscillation of the incoming eld and the electromagnetic eld can travel through the metal. Metals become transparent for frequencies higher than the plasma frequency. [4]
2.1.3 Optical Properties
As mentioned before a dielectric and a metal have dierent optical properties. Metals are shiny in the visible spectrum whereas for instance air is transparent. The dierences in the refractive index and the absorption coe cient are coming from the dierent complex dielectric functions out of the Loretz or Drude Model. The optical quantities are connected to the complex dielectric function by:
(ω) = 0(ω) + i00(ω) = (n(ω) + iκ(ω))2 =n ˜2
4 where n is the refractive index, κ is the extinction coe cient and n˜ is the complex refractive index. The refractive index is connected to the phase velocity vph of light in a medium by: c n = vph with c as the speed of light in vacuum. An electromagnetic wave will travel slower in a medium than in vacuum. The absorption coe cient κ gives the amount of light which gets absorbed from the ma- terial.
2.1.4 Re ection and Transmission
When light is hitting the interface between two dielectrics with dierent refractive indices, the beam will either be re ected or refracted as it is sketched in gure 2 [3]. For the re ec- tion the angle-of-incidence (θi) equals the angle-of-re ection (θr). This is the statement of the re ection law:
θi = θr The light can also be refracted and the appearing refracted beam is depending on the refractive indices of the two media, given by Snell's Law:
ni sin(θi) = nt sin(θt)
Where n is the refractive index and θ the angle for either incident (i) or transmitted (t) beam (compare gure 2).
Figure 2: Re ection and refraction of an incident light beam. In the sketch is ni The amount of re ected or transmitted light can be calculated with the Fresnel equations (which can be found e.g. in [3]). These equations take the wavelength λ and the angle of incidence θ and the complex refractive index n˜ into account. Further the interaction at the interface is depending on the plane in which the light is oscillating. 5 Figure 3: P-polarized light is oscillating in the plane of incidence whereas s-polarized light is oscillating perpendicular to the plane of incidence. P-polarized light is de ned in the plane of incident whereas s-polarized light is perpendic- ular to this, sketched in gure (3). For the p-polarized light one nds an angle of incident by which one gets no re ection back. This angle is called Brewster angle θB and is given by: n2 θB = arctan , n2 > n1 (4) n1 Here n1 and n2 are material speci c refraction indices. The transmitted light is linearly polarized and the eect can be used as a polarizer. However the Fresnel equations neglect absorption eects, which occur by the excitation of surface plasmons and are therefore not valid in these cases. 2.1.5 Babinet's Principle The used samples are patterned with a complementary holes and islands grating. The contribution of light diracted by such complementary screens (opaque at places where the other is transparent and vice versa) is given by the Babinet's principle [5]. The complex displacement of the light due to the presence of the diraction object in a point p is U1(p) for the rst and U2(p) for the second screen when they are placed alone between the light source and the observation screen. If there is no diraction object one gets a unobstructed beam which is considered as U. U1 and U2 can be calculated by the integral over the openings of the two screens [5]. The superposition of both complementary openings is the same as a completely transparent screen and therefore one can write: [5] U1 + U2 = U This is known as Babinet's principle. At points where U = 0 it follows that U1 = −U2. Therefore U1 and U2 are opposite in phase and, because the intensity is the absolute square of the amplitude, equal in intensity! There is no dierence in diraction if one is using a holes or a islands patterned nanostructures with the same dimensions. If one assumes a perfect metal which re ects 100% of the 6 incoming light, re ection R and transmission T for such complementary holes (H) and islands (I) structures are given by: RH = TI RI = TH This is sketched in gure 4. Figure 4: Babinet's principle for complementary holes and islands structures. 2.2 Plasmons Plasmons can be observed by looking at the impressing colorful church windows. It turns out that tiny metal particles in the glass produce the colors [6]. From our everyday expe- rience it is known that metals interact strongly with light. For example a mirror, which forces the light to change direction after hitting the surface. The strong interaction comes from the free moving electrons in the metal. They will start oscillating due to the incom- ing electric eld and trying to compensate the incoming eld. As a result the light cannot enter the metal and gets totally re ected. The introduced oscillations to the free moving electrons in the metal can only have certain frequencies, which refer to the allowed standing waves. A plasmon is now the oscillation of free moving electrons at a certain frequency. However there are dierent kinds of plasmons with dierent physical properties. 2.2.1 Volume Plasmons One can look at a thin lm of metal with free moving conducting electrons described by the free electron gas ( gure 5). The electron density N can oscillate longitudinally (parallel to the propagation direction) against the positive ions. A collective displacement of the electron clouds results in a surface charge density: σ = −N · e · u. Where u is considered as the displacement from the electron cloud in respect to the ion core and e is the elementary charge. 7 Figure 5: In a thin lm of metal electron density oscillations can occur described by the displacement of the electron cloud u. As a result a homogeneous electric eld Neu occurs, which acts like a restoring force. E = The following equation of motion describes this0 case: [4] 2 2 2 2 ∂ u N e u ∂ u 2 Nm 2 = − ⇒ 2 + ωp · u = 0 ∂t 0 ∂t with m as the mass of the electrons and t as the time. Here ωp is the eigenfrequency of the electron gas. The quanta of these plasma oscillations is called a plasmon, or more precisely bulk plasmon. Due to the longitudinal character of the bulk plasmons they can not couple to an electromagnetic wave which is a transverse wave. [6] Surface Plasmons These charge density oscillations can also take place at the interface between a metal and a dielectric, with the result of the occurrence of a bound surface wave, called a surface plasmon. Because of the missing neighboring particles the longitudinally charge oscillations gain a transverse moment and they oscillate just like water ripples on a pond along a dielec- tric/metal interface. Surface plasmons can be divided into two groups. The Surface Plasmon Polaritons (SPPs) and the Localized Surface Plasmons (LSPs). 2.2.2 Localized Plasmons (LSPs) A very famous example for the localized surface or particle plasmons is the Lycurgus Cup, shown in gure 6. Transmitted light through the glass is red, whereas light which is re ected appears green [7]. The occurring colors are due to resonant excitation of collective oscillations of electrons in the embedded particle. Localized surface plasmons are non propagating surface plasmons, which can be excited by an incoming electromagnetic wave. 8 Figure 6: Re ected light from the Lycurgus Cup is green whereas transmitted light appears red. [8] The arising resonances can be explained by a semi-classical model. If one looks at small particles (diameter a lot smaller than the incident wavelength) the excitation light is able to penetrate the particle. As a result the conduction electrons are shifted in respected to the positive lattice ions (shown in gure 7). At the particle's surface a surface charge occurs which leads to a repulsive force to the next electron. Due to the restoring force the complete electron cloud (all conductive electrons) behaves as an oscillating system. Figure 7: The incoming light leads to a separation of the electrons in respect to the ion core. The oscillating surface charges behave like dipoles and radiate electromagnetic waves. If one matches exactly the eigenfrequency of this system, even a small exciting light eld leads to a strong oscillation. The eigenfrequency depends mainly on the strength of the restoring force and this force depends on the surface charges for incident the particle size, shape and environment of the particles. Depending on the size, shape and material of the metal particle dierent conditions for the plasmons excitation appear. In the stained glass windows the dierent metal particles absorb dierent wavelengths and consequently the window is transmitting dierent colors. [9] 2.2.3 Surface Plasmons Polaritons (SPPs) SPPs occur at the interface between a dielectric and a metal. They are a coupled state between the charge oscillation of the free electron gas in the metal and an electromagnetic wave on the dielectric side. This is sketched in gure 8. 9 Figure 8: The coupled state between the density charge oscillation in the metal and the electric eld in the dielectric is called a Surface Plasmon Polariton. When a photon hits the interface it can give rise to this surface wave under certain cir- cumstances. The wave is propagating like the ripples on a water pond along the surface, constrained by the boundary condition of the two materials. The SPP makes it possible to con ne light on a sub-wavelength scale which opens the possibility to plenty of interesting applications. [10] The physical quantities of a SPP can be discussed based on the dispersion relation. To calculate the dispersion relation for the SPPs one can look at a at interface between a dielectric half space (assuming z>0) and a conducting half space (assuming z<0), sketched in gure 9. The interface should lie in the xy-plane, and the wave is propagating in x- direction. Figure 9: Interface between a dielectric (z>0) and metal (z<0) in the xy-plane where x is assumed as the propagation direction. 1 will be used as the dielectric constant for the dielectric and 2 as the dielectric tensor for the metal. In order to describe the SPPs one can use the wave equations for p- and s-polarized light. Where the p-polarized light only contains the Ex, Ez, and Hy component and the s- polarized light the Hx, Hy and Ey component, where H is the magnetic eld. They are given by the Helmholtz equation with harmonic time dependence and by using the Maxwell equations. Referring to [6] one gets for p-polarized light: ∂2H y + (k2 − k2)H = 0 (5) ∂2z 0 x y And for s-polarized light: ∂2E y + (k2 − k2)E = 0 (6) ∂z2 0 x y 10 Where is the wave vector and w is the wave vector of a propagating wave in vacuum, k k0 = c where c is the light velocity. As one can see here, there is no possibility to excite SPPs modes by s-polarized light. There is only an E component parallel to the surface, which can not provide the needed perpendicular wave vector. Whereas by the p-polarized light one can get a component parallel and perpendicular to the surface and therefore it is possible to excite the SPPs. For the p-polarized light the magnetic eld is given by: Hy = A2 exp(ikxx) exp(−k2z) z > 0 Hy = A2 exp(ikxx) exp(k1z) z < 0 With the continuity conditions Ez,1 = Ez,2, Hy,1 = Hy,2 and A1 = A2 one gets [6] k 2 = − 1 (7) k1 2 Furthermore Hy has to ful ll the wave equation (5) which leads to two more conditions: 2 2 2 2 2 2 (8) k1 = β − k02 k1 = β − k01 Combing (7) and (8) leads to the dispersion relation: r 1 · 2 kSPP = k · (9) 1 + 2 In gure 10 the dispersion relations from a photon and the SPP are sketched. The disper- sion relation of a photon is: w k = (10) p c With ω as the frequency and c as the speed of light. Figure 10: The dispersion relations of a photon (eq. 10) and a SPP (eq. 9) do not match in respect of the conservation of momentum and energy. The two dispersion relations do not match in respect of the conservation of energy and momentum, the SPP's momentum is much larger than the momentum of the incident light by the same energy. As a result it is impossible to excite plasmons only by shining light onto a surface. Anyhow there are dierent ways to add an extra momentum to the incident light in order to get the two dispersion relations to match. 11 2.2.4 Excitation of Surface Plasmon Polaritons There are two methods to excite the SPPs using the fact that the dispersion relation can be shifted if the light is traveling into an optical denser medium like for instance glass. The total internal re ection of the light gives rise to an evanescent wave which results in the SPP's excitation. Figure 11: For exciting the SPPs either the Otto con guration (a) or the Kretscham con- gurtaion (b) can be used. In the Otto con guration seen in gure 11a) one has a prism which is separated by a dielectric layer (e.g. air) to the metal. The SPPs occur at the dielectric/metal interface. [10] By the Kretschman con guration seen in gure 11b) one has a prism with a thin metal lm. The light hitting the prism gets totally re ected and gives rise to the evanescent wave which travels through the metal and gives rise to the SPPs at the metal/air interface. [10] Grating Coupling In this case the sample is a periodic structure, which can be used to add the missing momentum G by the grating's scattering properties. If light is hitting a grating it will get diracted. Constructive interference takes place when the Bragg condition is satis ed: [4] 2a sin(θ) = mλ Where a is seen as the distances in the diractive grating, θ is the incident angle, λ the wavelength and m gives the diraction order. The missing momentum which is necessary to excite the SPPs can be added by a reciprocal lattice vector, given by the Bragg diraction and sketched in gure 12 : kSPP = k · sin(θ) ± n · G (11) with as the incident angle and 2π a reciprocal lattice vector which depends on the θ G = a lattice distance a and n is an integer. 12 Figure 12: To overcome the miss match between the photon and the plasmon dispersion a grating can be used. The incoming photon, sketched is the parallel component to the grating kx, gains the missing momentum from a reciprocal lattice vector G and then gives rise to the plasmon excitation. The incoming photon gains the missing momentum from the reciprocal lattice vector and gives rise to the plasmon excitation. The in a grating excited plasmons are named Bragg Plasmons (BP) to specify how they are excited. All excited plasmons in this thesis are excited within a grating, therefore the further discussion will be always about Bragg Plas- mons. The grating will diract light with dierent wavelengths into dierent directions and con- sequently plasmons can be excited by dierent combinations of wavelengths and angles of incident, given by the dispersion relation which will be discussed in chapter 2.4.4. 2.3 Magnetism It was shown that the SPPs depend on the dielectric function of the used metal. In a mag- netic material a magnetization leads to a change of the material's properties. Consequently the dielectric function changes and hence also the properties of the SPPs get in uenced! To investigate the interplay between SPPs and a magnetic eld a ferromagnetic sample is investigated. For that reason it is useful to look at the theory of a ferromagnet. An important physical quantity to describe magnetism is the magnetic eld strength H~ with [H] = A/m, where A is ampere the unit of the electric current. Former the magnetic ux density is de ned for a homogeneous magnetic eld in the vacuum as: [11] B~ = µ0H~ − with [B] = 1 T (Tesla) = 1 N/Am and µ0 = 4 π · 10 7 Vs /Am the permeability of the free space. Atomic dipole moments give rise to magnetism in matter. Out of the angular moment J~ of an electron one can derive the magnetic moment ~µ: e ~µ = − J~ 2m where e is the elementary charge and m the electron mass. Due to the quantization of the angular moment the magnetic moment is also quantized. The quantization is given by the Bohr magneton: e µ = − ~ = 9.274 · 10−24 J/T B 2m 13 With the Planck's constant 6.62606957 −34m2 kg . ~ = · 10 s The magnetic moment ~m is the sum over all atomic magnetic moments ~µi. If one divides the magnetic moment by the volume V one gets the magnetization M~ : ~m M~ = V Applying a magnetic eld to a solid gives rise to the magnetization M~ and the magnetic ux density becomes: [11] B~ = µ0(H~ + M~ ) In so called linear materials the magnetization is linearly related to the magnetic strength H~ by M~ = χ · H~ where χ is the dimensionless magnetic susceptibility. For this case one can construct again a linear condition between B~ and H~ : B~ = µ0(1 + χ)H~ = µ0µrH~ with the relative permeability µr which is given by µr = 1 + χ. For χ < 0 one speaks about Diamagnetism and for χ> 0 about Paramagnetism [11]. These are two other types of magnetism which are for this case not relevant and consequently not discussed any deeper. For more details see [11]. 2.3.1 Ferromagnetism Ferromagnets have a long range ordering under a speci c temperature. Depending on this ordering one can dier ferromagnets, antiferromagntes and ferrimagnets. To get a spontaneous ordering of the atomic magnetic moments one needs an interaction between the neighboring moments. This interaction is called exchange interaction and can only be explained by quantum mechanics. The origin of the interaction comes out of the overlapping of the wave function of the magnetic ions and can be described by the Heisenberg-Hamilton-Operator: [12] X H = − Jij · S~iS~j i,j J is called the exchange integral and gives the strength of the coupling between the two spin moments S~i and S~j by looking at the energy dierence between the parallel and anti-parallel state of the spin alignment. [12] Jij = E↑↓ − E↑↑ One speaks about ferromagnetitism if all neighboring spins align parallel, this is the case for a positive exchange integral J>0. For a negative exchange integral the spins will align anti parallel and one gets an antiferromagnetic or for the case of two dierent magnetic ions the ferrimagnetic state. 14 To explain the most characteristic of a ferromagnet - the spontaneous magnetization, one can look at the mean- eld approximation founded by P. Weiss. He postulated the so called molecular eld, which has the same properties as the exchange eld and is proportional to the magnetization: [12] B~ E = λM~ with λ as the temperature independent constant. Besides the external magnetic eld, in every magnetic moment a molecular eld occurs due to the neighboring moments. The eective magnetic eld is given through: [12] B~ eff = B~ + λM~ Against the exchange interaction one also has thermal uctuation which destroy the long range ordering above the so called Curie-Temperature TC . As a result the material becomes paramagnetic and the former high susceptibility of the ferromagnet decreases according to the Curie-Weiÿ law: [13] C χ = T − TC with the temperature T , C as the Curie constant and TC the Curie-temperature. [12] Hysteresis A further characteristic for a ferromagnet is the hysteresis loop which arise if one applies rst an increasing and then a decreasing eld to a ferromagnetic material. The hysteresis loop is a useful tool to make sure that a sample is completely magnetized. In this case it was used to gure out the strength of the magnetic eld to saturate the sample for the TMOKE measurements (compare chapter 2.3.2 and 2.5.2). A typical hysteresis loop is sketched in gure 13. Figure 13: Hysteresis loop. If one starts with an unmagnetized state the magnetization follows the so called virgin curve until the material is saturated. At this point all spins are aligned in the eld direction. 15 If one now decreases the eld the magnetization decreases again. As one can see there is a magnetization left if one decreases the eld back to zero. This magnetization is the remanent magnetization which will rst vanish by applying a magnetic eld in the opposite direction. The coercive eld gives the eld which is needed to remove the remanent magnetization. If one increases the eld again one gets to the saturation point in the other magnetization direction - now the spins are aligned in the opposite direction. By applying the eld in the other direction again the material follows a shifted curve in the respect to the virgin curve. 2.3.2 Magneto-Optical Kerr Eect As it was discussed before, BPs can be in uenced by optical and magnetic properties which opens the possibility to investigate the nature of BPs with magneto-optical measurements. The magneto-optics started in the 19th century with Michael Faraday and John Kerr, who discovered the interaction of an electromagnetic wave with a magnetic material. The Faraday eect is associated with the transmission and the Kerr eect with the re ection of the electromagnetic wave. In both cases a rotation of the polarization plane of the incident light is observed. In this case only the magneto-optical Kerr eect (MOKE) is relevant, however the theory stays the same. To explain the origin of the MOKE one has to take a closer look at the interaction between light and matter in the presence of an external magnetic eld. As mentioned before the dielectric tensor [(ω)] is a good quantity to describe the material's properties (compare chapter 2.1). The dielectric tensor is given by: D~ = 0[(ω)] · E~ with D~ as the dielectric displacement. To keep things simple one can consider a cubic material with the magnetization M~ in z-direction. In this case the dielectric tensor can no more be seen as a scalar as in chapter (2.1). However it can be written as: [14] xx xy 0 ~ [(M, ω)] = −xy xx 0 (12) 0 0 zz Where all the components 0 00 might be complex (compare chapter 2.1) and ij = ij + iij depend on the magnetization. Further they follow the so called Onsager relation which states that "the diagonal components of the dielectric tensor are even functions of M~ , while the non-diagonal ones are odd functions of M~ " [14]: ij(−M,~ ω) = ji(M,~ ω) Out of the dielectric tensor in (12) it is obvious that an external magnetic eld which induces a magnetization M~ to a material leads to an optical anisotropy, because all the optical properties as shown in chapter 2.1 are depending on this dielectric tensor which is no longer symmetric. In order to get a deeper understanding how the dielectric tensor in uences the optical prop- erties one has to look at the Maxwell equations and its solution for plane electromagnetic 16 waves. ∂H~ ∂E~ ∇ × E~ = −µ ∇ × H~ = (13) 0 ∂t 0 ∂t E~ = E~0 exp[−i(ωt − ~k · ~r)] H~ = H~ 0 exp[−i(ωt − ~k · ~r)] (14) with ω as the frequency and ~k as the wave vector of the electromagnetic wave. Out of these four equations one can get: [14] 2 2 ~k(~k · E~ ) − k E~ + 0µ0ω · E~ = 0 (15) By de ning the complex refractive index ~k and considering only the case that the light n˜ = k is propagating along the z-direction one can nd the two following conditions for nonzero solutions of Ex and Ey. [14] 2 (16) n˜± = xx ± ixy ±iEx = Ey (17) Linearly polarized light can always be described by the superposition of two circularly polarized waves as sketched in gure 14. These are the two helicities of light σ+ and σ−. Figure 14: The two circularly polarized waves σ+ and σ− add up to a linearly polarized wave (red arrow) . If one considers that the ± sign refers to two circularly polarized waves, one can see that these waves travel due to e.g. 17 with dierent refractive indices (n˜±). As it was shown earlier the complex refractive index can be split in the real part, which refers to the phase velocity by n = c and the imaginary part iκ which states the absorption in the material. vph Because of the connection between refractive index and phase velocity, a dierence in the refractive index leads to two dierent propagation velocities in the material. Consequently the two arrows of the circular waves will not have completed the same path of the circle and the resulting amplitudes appear rotated by the angles ϑK from its original position ( gure 15). This phenomenon is called optical rotation. 17 Figure 15: Because of the dierent propagation velocities of σ+ and σ− the two arrows do not nish the circle at the same time and the amplitude is rotated by the angle ϑk. Figure 16: The Kerr Ellipticity ηk is induced by dierent absorption coe cens for the two circularly polarized waves. The arrows add up to an ellipse (red) and the amplitude of the linearly polarized wave changes from a maximum by σ+ + σ− to a minimum by σ+ - σ−. The left- and right-handed polarized light is also aected dierently by the extinction coe cient κ. By traveling through the material they experience not the same amount of absorption and hence the amplitude of the arrows changes. The arrows are following dierently sized circles which add up to an ellipse, shown in gure 16. This is called the ellipticity ηK . The Kerr-rotation ϑK and the Kerr-ellipticity ηK can be expressed as: [14] 0 ϑ = − xy K n(n2 − 1) 00 η = − xy K n(n2 − 1) The Kerr-rotation therefore depends on the real part of the dielectric function, which was connected to the phase velocity and the Kerr-ellipticity refers to the imaginary part of the dielectric function which was identi ed by the absorption coe cient. In the general case both features will occur and the re ected light follows a rotated ellipse shown in gure (17). 18 Figure 17: The general case. Left- and right-handed circularly polarized waves are aected dierently in their absorption and refractive coe cient. The result is an ellip- tical polarization. One can look at three dierent con gurations of the MOKE, shown in gure 18. They are dierent in the magnetization direction referring to the incident plane and the sample surface [13]. Figure 18: The polar MOKE set-up (a) contains a magnetic eld which is perpendicular to the sample surface and parallel to the plane of incidence. Whereas the longitu- dinal MOKE (b) has a parallel magnetic eld to both. In the transverse MOKE (c) the magnetic eld is parallel to the surface of the sample but perpendicular to the plane of incidence. The occurring changes in re ectivity and polarization can be nicely explained by the Lorentz force. Of course one could calculate all these cases by applying the right boundary conditions to the Maxwell equations and the dielectric tensor as shown in [15]. If one rst looks at the polar con guration with the magnetization perpendicular to the surface, the linearly polarized light will induce oscillations to the electrons parallel to the plane of polarization. Without a magnetization the polarization of the re ected beam would lie in the same plane as the incident light, named as R~ N . Applying a magnetization to the surface gives rise to the Lorentz force: F~L = e · (~v×B~ ) with e the elementary charge, v the velocity and B as the magnetic ux density. The Lorentz force induces a secondary amplitude, called the Kerr amplitude R~ K in the direction of ~vLor = −M~ × E~ where M~ is the magnetization and E~ the electric eld. The superposition of R~ N and R~ K leads to the rotated polarization plane as sketched in gure 19. [13] As far as we are going to use the TMOKE set-up we will have a closer look at this situation. In the TMOKE set-up no change in the polarization direction is observed because the induced Kerr-amplitude lies in the same plane as the incident polarization ( gure 20). 19 Figure 19: PMOKE: The induced Figure 20: TMOKE: The plane in which the Kerr-amplitude Rk (blue induced Kerr-amplitude lies is arrow) lies out of plane in parallel to the incident polariza- respect to the incident tion. Consequently no polariza- polarization. Consequently tion rotation is observed. How- a rotation of the polarization ever the re ected beam has a dif- is observed, sketched by ferent direction and a change in the green arrow. the re ectivity is observable. However the re ected beam RK has a dierent direction and the superposition gives a change in the amplitude referring to the incident beam. This is only possible for p- polarized light because for the s-polarized case the cross product in the Lorentz force will be always zero. 2.4 Sample 2.4.1 Deposition The samples are made out of F e20P d80 which was fabricated by co-sputtering from two elemental targets of Fe and Pd. The process takes place in an ultra-high vacuum magnetron sputtering system with a base pressure of lower than 10-7 Pa. To ensure that the ITO substrate is clean it was baked for 30 min at 300 degree Celsius. Afterwards the deposition takes place. An Argon sputtering gas was used with a pressure of 0.8 Pa while the sample stage was rotating in order to get a uniform thickness of the metallic lm. The sample was kept at room temperature during growth. [16] 2.4.2 Patterning The samples were fabricated with electron beam lithography (EBL) which is a special technique to fabricate extremely small structures in the nm-range. In this process one uses an electron sensitive resist, which covers the substrate and gets exposed to the electron beam. The electron beam scans the sample and draws the desired structure on it. The resist gets chemically altered by the electron beam and in a following etching process either 20 the exposed or resist (depending on the resists) is taken away and only the desired structure is left. Since this technique uses raster scanning it is inherently slow and therefore large areas are expensive and time-consuming to manufacture. [17] For the used samples in this thesis, a thin lm of F e20P d80 was sputtered and thereafter covered by a resist which is exposed by the electron beam. The patterns are de ned in a CAD-program associated with the JEOL-5000 e-beam printer and is fully automated. The samples were patterned in the Center for Functional Nanomaterials, Brookhaven National Laboratory. 2.4.3 Structure The samples are all periodic nanostructures with a highly periodic array of squares- either as holes or islands. The material of the samples is a combination of a noble and a magnetic metal in order to excite plasmons, which is provided by the noble metal, and also to obtain an active plasmonic nanostructure by the magnetic material. To be able to look at the dierences in plasmonic excitation due to sizes of holes and islands, four dierent sample with dierently sized holes and islands, but same periodicity, were measured. The four samples have been lithographically patterned whereof two with islands of various sizes as well as two with holes with various sizes. The patterned area on each sample is 1.5 mm × 3 mm. All samples with sizes are sketched in gure 21. They will be named by their size (big/small) and structure (holes/islands) as SH, BH, SI and BI. Figure 21: Samples de nition The material of the samples is F e20P d80 with a thickness of 30 nm. The sample is placed on a 100 nm thick ITO substrate on top of glass. 21 2.4.4 Grating Coupling for Square Structures For these speci c square structures a conditions for the excitation of the Bragg plasmons can be calculated (compare chapter 2.2.4). Therefore a perfect square structure with xed periodicity is assumed. For the derivation one has to take a closer look at the diraction geometry ( gure 22). Figure 22: Diraction geometry: The xy-plane is de ned as the sample's surface with the incident plane in the zx-plane. The incident wave vector k (green) lies in the zx-plane, the scattered wave vector k0 (red) points in an arbitrary direction and ∆ k (blue) is the scattering vector. Further the angle of incidence ϑ and the scattered angle ϑ2 is sketched, as well as the angle ϕ which is the angle between the xy-component of the scattered vector k0 and the x-axis. Incident and scattered wave vectors can be seen in gure 22 and are given as: 2π ~k = sin(ϑ)~xˆ − cos(ϑ)~zˆ λ 2π k~0 = sin(ϑ ) cos(ϕ)~xˆ + sin(ϑ ) sin(ϕ)~yˆ + cos(ϑ )~zˆ λ 2 2 2 Here λ is the wavelength of the incident light and ϑ and ϕ are angles de ned as in gure 22. The diraction condition (18) states that the scattering vector ∆~k must be equal to a reciprocal lattice vector G~ . ∆~k = k~0 − ~k = G~ (18) As it was explained in chapter 2.2.4 due to this equation the incident photon gains the missing momentum in order to add up to the plasmon dispersion and consequently the excitation of the BP is possible. In this case we are looking at a 2D square lattice. If one considers the nearest neighbor spacing is in the real space, the spacing in the reciprocal space is 2π . Therefore ~ in a a G 22 equation (18) must have a form like 2π and 2π , where and . Inserting these p · a q · a p q Z two condition in (18) leads to: 2π 2π (sin(ϑ ) cos(ϕ) − sin(ϑ)) = p λ 2 a 2π 2π sin(ϑ ) sin(ϕ) = q λ 2 a As far as one is not interested in the direction of propagating of the BPs, one can solve this two equations for sin(ϕ) and cos(ϕ) in order to eliminate ϕ. Using cos2(α) + sin2(α) = 1 ◦ and setting ϑ2 to 90 because one is only interested in parallel to the surface scattered light, one can nd a quadratic equation in . Replacing by 2π leads to the searched λ λ E solution: hc (p2 + q2) E = (19) a pp2 + q2 cos2(ϑ) − q sin(ϑ) In this case our samples are on a transparent glass substrate. Consequently light can be transmitted through the sample into the substrate, get re ected on the backside and then becomes diracted. As a result the incident angle and energy must be adjusted with the refractive index of the glass substrate to sin(ϑ) and . Applying this to equation (19) n n nE leads to: hc (p2 + q2) E10 = (20) a −k sin(ϑ) + pn2(p2 + q2) − q2 sin2(ϑ) One can also calculate the dispersion relation for illumination from the [11] direction instead from the [10] direction.Therefore the incident wave vector has to be changed: 2π sin(ϑ) sin(ϑ) ~k = √ ~xˆ + √ ~yˆ − cos(ϑ)~zˆ λ 2 2 With the same arguments as before one gets the following contribution: √ hc 2(p + q) E11 = √ (21) a −(p2 + q2) sin(ϑ) + 2n2(p2 + q2) − (p − q)2 sin2(ϑ) The now calculated dispersion relations give the conditions for which angles and energies the excitation of the BP is possible. Under the condition of grazing incidence the incoming light can couple to the charge density wave in the sample which results in the BP. This surface wave is observed by a drop in re ectivity along the dispersion relations, meaning the sample is absorbing the energy in order to give rise to the surface wave. When light hits the patterned sample it will get diracted in dierent directions, shown in gure 23a). Each direction refers to one dispersion relation. Labeled in round brackets. If the diracted paths look the same out of the view from the incoming light they are described by the same dispersion relation. As for the (01) and (0-1) in gure 23a.) which become one dispersion line in 23b.) (green line). If one rotate the sample 45 ◦ the light sees dierent paths and other dispersion relations can be observed as shown in gure 23c.). Now the paths for the (01) and (10) as well as for the (0-1) and (-10) direction look the same and they end up in one dispersion line (turquoise for the rst and purple for the second case). The (-1-1) direction becomes a separated dispersion line which makes it possible to observe this mode alone. 23 Figure 23: For dierent propagation direction of the light, sketched as dierent colored arrows in a.) and c.), dierent dispersion relations are found, sketched in b.) and d.). Where a.) and b.) corresponds to the [10] and c.) and d.) to the [11] crystal alignment. 2.5 Measurement Techniques and Equipment 2.5.1 AFM The atomic force microscope (AFM) is a microscope which has a resolution in the nanome- ter scale. It was invented by Gerd Binning in 1986. With the AFM it is possible to scan the topography of a material until atomic resolution. Figure 24: Sketch of AFM set-up A very thin and in the best case atomic tip is rastered above the material's surface con- 24 trolled by piezoelectric motors. The tip is xed on a cantilever which acts like a spring. Depending on the surface features the force on the tip is dierent. To detect the changing force, a laser is focused on the cantilever and re ected to a sensitive photo diode. A chang- ing force leads to a movement of the cantilever and of the re ection of the laser which is detected by the photo diode. The interaction between tip and surface can be described by the Lennard-Jones-Potential: 1 1 ϕ(r) = −c − (22) r6 r12 with c as a constant and r as the space coordinate. The rst term corresponds to the attractive van der Waals interaction and the second part to the repulsive force due to the Pauli-Principle. There are dierent types of operating modes, depending on the sample and what one would like to measure. In this case the contact mode was used because the sample was stable enough to resist the small forces from the tip. In the contact mode the tip is in mechanical contact with the surface of the sample and the repulsive force is measured by holding the force constant while using a feedback loop. The re ected laser beam from the cantilever gives information about its de ection and torsion. The de ection is linearly connected with the acting normal force where the spring constant is the proportional factor. These values are compared with the set point. A deviation caused by the de ection will give a signal to the z-piezo to restore the set point. The feedback control signal is connected to the x-y position of the tip and is the foundation of the topography picture. The other measuring mode is the non-contact or tapping mode. In this case the tip is not in contact with the surface. In this mode the cantilever gets excited close to its eigenfrequency by using the piezo crystal. If the cantilever is close to the surface the interaction will cause changes in the spring constant of the cantilever and as a result in uences changes in the oscillation amplitude. These changes are detected and the acting force can be calculated. Out of this the topography map can be created. 2.5.2 TMOKE The transversal magneto-optical Kerr eect signal is de ned as: RM+ − RM− TMOKEasymmetrie = · 100% (23) RM+ + RM− It is the normalized dierence between the re ectivity (R) measured with a magnetic eld pointing in the up direction (M+) and the re ectivity with a magnetic eld pointing in down (M-), compare gure 18c.), where (M+) is orientated in the arrow direction and (M-) consequently in the opposite direction. As it was explained in chapter 2.3.2 the TMOKE in uences only the re ectivity of the beam and not the polarization plane. The eect of exciting plasmons is visible as a decrease of the re ectivity. Consequently this should aect the TMOKE signal as well and due to the change in re ectivity by the TMOKE, the TMOKE asymmetry gets enhanced in the region where plasmons are excited. As far as the TMOKE signal depends on the overall re ectivity (denominator) it is also in uenced by eects like the Brewster angle. 25 2.5.3 Experimental Set-Up Figure 25: Experimental set-up A sketch of the experimental set-up is shown in gure 25. In order to examine the sample with certain energies and polarizations one needs monochromatic and linearly polarized light which is provided by a super continuum laser with a polarizer in front of it. Here energy scans from 1.55 eV to 3.15 eV were done with a resolution of 0.1 eV. The sample is placed on a goniometer which makes it possible to rotate the sample in precise angular steps in θ. This is used to investigate the dependence of the angle of the incident beam. In this case scans from 4◦ to 45◦ in θ were done (optical convention). For the transmission measurements the detector is placed behind the sample whereas for the re ectivity measurements it has to be placed by 2θ in order to detect the re ected light. To get the relative intensity Ir in percent independent of the incident intensity, the intensity of the light without hitting the sample (I0) was measured. The relative intensity is de ned as the measured intensity with the sample Im divided by the intensity without the sample I0. Further, to investigate also magneto-optical properties of the sample, one has four magnetic coils to get a homogeneous eld in every direction. These were used to measure the TMOKE asymmetry where the magnetization is parallel to the sample but perpendicular to the incident plane. 26 3 Measurements, Results and Discussion 3.1 AFM-Measurements To characterize the samples respectively size and shape, AFM pictures were made, shown in gure 26. All AFM-pictures were taken in the contact mode. The resolution is in the range of 10-40 nm depending on the size of the scan area. Figure 26: Results from the AFM measurements: a.) SH with a diameter of (130±10) nm and a periodicity of (510±10) nm b.) BH with a diameter of (190±5) nm and a periodicity of (508±5) nm c.) SI with a diameter of (400±10) nm and a periodicity of (512±10) nm d.) BI with a diameter of (420±5) nm and a periodicity of (509±5) nm. All four samples show a high periodicity and it does not seem that there are missing a lot of holes or islands. However not all the samples have a nicely de ned square shape. The small islands show more a circular shape and also the big islands do not have well de ned edges. Whereas both hole structures show sharp edges and a square shape. To excite the plasmons it is important to have a grating with a periodic spacing between holes or islands. The periodicity can be read out of the AFM-pictures and is for all four samples around 510 nm (compare table (1)). All further calculations will be done with a periodicity value of 510 nm. 27 Table 1: Periodicity of the used samples. The uncertainty is due to the resolution of the AFM Sample Periodicity in [nm] SH 510±10 BH 508±5 SI 512±10 BI 509±5 3.1.1 Calculation of the Amount of Metal In order to make a prediction about the re ection and transmission of the samples, the metal coverage of every sample was calculated. Therefore it was assumed that the structure is perfectly square. With the help of the Software Gwyddion a height pro le was extracted out of which the diameter of the holes and islands can be read out. With the knowledge of this and the number of holes or islands on the sample the overall coverage of metal can be calculated. The results are shown in table (2): Table 2: Calculated metal coverage on each sample. The uncertainty is given by the max- imal resolution of the AFM-picture, for the percentage the biggest uncertainty was calculated. Sample Diameter in [nm] Amount of metal in [%] SH 130±10 93.3±2.1 BH 190±5 85.6±1.5 SI 400±10 64.1±6.4 BI 420±5 70.6±3.4 3.2 Optical-Measurements For all the four samples re ectivity and transmission measurements with p- and s-polarized light were made in the high symmetry direction [10] and as well for two of them in the [11] direction. The measurements were done for certain wavelengths between 393.7 nm (3.15 eV) up to 751.5 nm (1.65 eV). For these wavelengths the angular dependent response of the sample in re ection or transmission was measured form 4◦ up to 45◦ in the optical convention which means 0◦ refers to normal incidence! To visualize the measured values re ectivity or transmission maps were created with the help of the MATLAB software. In these maps one can see the incident angle θ in degrees on the x-axis. On the y-axis the energy is plotted in eV which refers to the laser emission in nm by the equation: 1239.84 E = λ The colors from blue to red refer to the normalized re ectivity/transmittivity. Further dierent dispersion relations are plotted in the maps, which have been explained 28 in chapter (2.4.4) and are calculated with equation (19) for the front side modes and (20) for the backside modes. For illumination from the [11] crystal alignment equation (21) is used. The white dispersion lines refer to the plasmons witch get excited from the front side and the black dispersion line is the (10) mode which is excited from the backside. This mode will be labeled by (10)b and is excited by light which rst propagates through the sample, gets re ected from the glass and then gives rise to the plasmon resonances. 3.2.1 Strength of Resonances γ To compare dierent resonances the strength of the resonances were calculated with the following formula: R− γ = 100 − · 100 (24) R+ Where R+ refers to the maximum re ectivity before the re ectivity drops due to the plasmon excitation. Consequently the re ectivity drop refers to R−. If these two features are observed with the contrast given by optical measurements one can see that the plasmon excitation takes place. However one has to be careful to which dispersion the excitement refers because they overlap at a lot of points so it also diers a lot at which energy the values are calculated. Furthermore the values are depending on the background. Therefore it is not possible to compare the values between dierent samples and within a sample one only should look at the dierence between the values and not at the absolute value. 3.2.2 Re ectivity Comparison small and big holes Two samples with dierent holes sizes were analyzed. The small holes have a diameter of 130±10 nm and the metal coverage of the sample is 93.5%. For the big holes the diameter is 190±5 nm and the metal coverage is 85.2%. Out of the AFM-pictures one can assume well de ned square holes. The easiest way to get an overview is to compare the re ectivity maps for p-polarized light. The maps are shown in gure 27 and 28. The highest re ectivity value in percent is given by the red color and is for both, holes and islands sample, by 60%. The lowest value for the re ectivity is given by the blue color. For the holes sample this is about 30%, whereas it is by the islands about 10%. The re ectivity for the SH is higher than for the BH because the higher metal coverage for the SH. The observed features match well with the calculated dispersion relations. Here features along the (±10) and (-1±1) direction are visible. For the small holes the resonances look a lot weaker than for the big holes. To prove this observation the values for all possible resonances were calculated and presented in table 3 and 5 in the appendix. The strongest response is for the (10) and (-10) mode for both samples. The next strongest resonance is the (-1±1) mode. For energies over 2.85 eV this mode starts to overlap with the (10) mode. Consequently the values are a bit intermixed which sometimes makes it di cult to assess the strength of a resonance. However both modes are visible. 29 Figure 27: Re ectivity map SH Figure 28: Re ectivity map BH with p-polarized light. with p-polarized light. The overlapping process is nicely observable by plotting the re ectivity curves in this energy range, done in gure 29 and 30 for both the SH and BH. In the gure for the BH it is visible that there is another resonance excited. This could be the (0±1) mode. But this mode would indicate a re ectivity maximum followed by a minimum and the minimum is not clearly visible. There is also no sharp resonance peak visible as it is observed for the other modes. Figure 29: SH: In the plot four dierent en- Figure 30: BH: Same energy scans for the big ergy scans are shown. For the holes. As one can see the re ectiv- 2.75 eV (red) one can see two sep- ity maxima and minima are a lot arated re ectivity maxima which sharper and stronger. The same shift together for higher energies overlapping process is visible. and become for 3.05 eV (green) There is also a hump occurring one re ectivity maximum. The and moving in the direction which rst peak for the 2.75 eV refers to would be expected for the (0±1) the (10) mode, the second to the mode. (-1±1) mode. One can also see However for this mode one would the (-20) mode for 2.95 eV by 41◦ need a re ection maximum fol- which shifts for 3.05 eV to smaller lowed by a minimum which is not angles (38◦). really clearly visible. 30 How does the size of the holes in uence the plasmon resonances? The interesting dierence between the SH and BH is the size of the holes (the big holes are 60 nm bigger in diameter). To compare the resonances from the SH and BH the energy scans for 2.85 eV and 1.85 eV were plotted together in one graph shown in gure 31 and 32. Figure 31: SH (black) vs. BH (blue) Figure 32: SH (black) vs. BH (blue) at 2.85 eV. at 1.85 eV. The rst dispersion which gets crossed (red horizontal line) in the plot in gure 31 is the (10) mode. This mode is excited by light witch is diracted in the forward (10) direction. The dispersion is up going, i. e. the energy rises with the angle of incidence. Therefore one expects rst a maximum in re ectivity followed by a drop in re ectivity. The plasmon excitation does not take place at the intersection point with the dispersion line but slightly shifted to higher energies. For up going dispersion lines this is a shift to higher angles, for down going ones (like the (-10)) to lower angles. This is what is observed in the energy scans. For the (10) mode one can rst see a peak by 9◦, followed by a re ectivity drop. This drop refers as well to the (-1±1) mode, which rst has this drop followed by a peak by 20◦. For the (-10) mode ( gure 32), which is down going (decrease in energy for higher angles of incidence) one can rst see a drop in re ectivity followed by a peak. One can see that the re ectivity maxima and minima are found for both samples at the same θ values. The red horizontal lines are marking the points at which the dispersion lines get crossed. These matches well with the observed features. The re ectivity values in % are higher for the SH which makes sense because the amount of metal is nearly 10% higher. In the plot for 2.85 eV one can see that for the BH the (10) and (-1±1) mode are excited with the same strength whereas for the SH the (10) is stronger than the (-1±1). In general the intensity maxima and minima are stronger for the BH than for the SH. This is also visible in the plot for 1.85 eV. As a conclusion one could say that the resonances get excited more strongly for the BH than for the SH. This might be a result out of the dierent sizes. Maybe the SH are a bit too small to excite the modes in the same manner as the big ones. Also the amount of metal could play a role in terms that the biggest part of the incident light gets re ected back without interacting with the sample. To really make a prediction about the in uence of the size to the resonances one should look at a third sample. 31 Comparison p- and s-polarized light excited modes The samples are patterned with a 2D grating. This makes it possible to get plasmon modes with p- and s-polarized light. In order to investigate a dierence in the optical response one can compare the re ectivity maps for p- and s-polarized light from the same sample. This is shown in gure 33 and 34 for the BI which have a diameter of 420 nm and a metal coverage of 69.8 %. By looking at these re ectivity maps for p- and s-polarized light a clear trend shows up. For the p-polarization the (10) and the (-10) mode and for the s-polarization the (0±1) mode gets excited. Figure 33: Re ectivity map BI Figure 34: Re ectivity map BI with p-polarized light. with s-polarized light. To get a better feeling how big the dierence between p- and s-polarized excitement is, the re ectivity scans were plotted for 2.95 eV and 1.85 eV for p- and s-polarized light, shown in gure 35 and 36. Figure 35: Re ectivity scan at 2.95 eV Figure 36: Re ectivity scan at 1.85 eV for p- and s-polarized light for p- and s-polarized light for the BI. Black curve is p- for the BI. Black curve is p- and blue curve is s-polarized. and blue curve is s-polarized. In these two plots it gets quite obvious that dierent modes get excited. For the 2.95 eV one has the (10) mode visible by a sharp peak by 10◦ for the p-polarized light (black curve) where for the s-polarized light (blue curve) there is no peak noticeable. In comparison the (0±1) mode is pronounced by the s-polarized light with a re ectivity maximum at 34◦ where for the p-polarized light no resonance is observable. For the 1.85 eV the same is happening for the (-10) mode which is only excited by the p-polarized light. 32 As a consequence one can say that the p-polarized light excite mainly the (10) and (-10) direction which is the forward and backward propagation direction of the electric eld vector. The same is happening for the s-polarization. Here the (01) and (0-1) directions get excited which is also the forward and backward propagation direction of the electric eld vector. A sketch is shown in gure 37. Figure 37: With dierent polarization stats dierent modes can be pronounced depending on which direction the electric eld vector is pointing. The green arrow reers to the s-polarization and the red one to the p- polarization. The same comparison can be made for the BH. The re ectivity maps are shown in gure 38 and 39. The same trend is observed as for the BI. The p-polarized light excites the (10) and the (-10) mode and the s-polarized consequently the (0±1) one. Figure 38: Re ectivity map BH Figure 39: Re ectivity map BH with p-polarized light. with s-polarized light. As before the re ectivity scans for 2.85 eV and 1.85 eV were plotted. Shown in gure 40 and 41. In the gure 40 it becomes quite clear that the (10) mode gets only excited with p-polarized light (black curve). There is absolutely no hint for a resonance with the s-polarized light (blue curve). For the (-1±1) mode there is some re ectivity maximum visible for the s- polarized light but a lot weaker than for the p-polarized light. Whereas the (0±1) mode is clearly visible for the s-polarized case and look completely dierent to what was observed for the p-polarized case. In gure 41 it becomes also quite clear that the (-10) is only excited by the p-polarized light. However there is a small re ectivity increase at the same θ value for the s-polarized light. 33 Figure 40: Re ectivity scan at 2.85 eV Figure 41: Re ectivity scan at 1.85 eV for p- and s-polarized light for p- and s-polarized light for the BH. Black curve is p- for the BH. Black curve is p- and blue curve is s-polarized. and blue curve is s-polarized. The samples do not have a perfectly square shape which could result in a superposition of p- and s-polarized light, the electric eld component will gain some component of the other polarization by following the smeared out shape. This explains the observed small hint of the mode which gets mainly excited by the other polarization. Comparison holes and islands For the comparison from holes and islands the samples with the BH and BI were chosen because here the strongest features were observed and the square shape is better than for the small ones. In gure 42 and 43 the re ectivity maps for the BH and BI for p-polarized light and in 44 and 45 with s-polarized are shown. Figure 42: Re ectivity map BH Figure 43: Re ectivity map BI with p-polarized light. with p-polarized light. Due to Babinet's Principle holes and islands produce the same diraction pattern (except from the intensity) and therefore the same plasmon excitations are expected. For the calculation of the dispersion relation only the periodicity plays a role and not the shape or size of the grating. Consequently the equation is valid for all the samples. In the maps one can see that the same (main) resonances get excited, which are the (10) and (-10) for the p-polarized case and the (0 ±1) for the s-polarized case. However there are modes which are only excited in the holes or islands sample. 34 Figure 44: Re ectivity map BH Figure 45: Re ectivity map BI with s-polarized light. with s-polarized light. So one can see the (0±1) mode for the BH and the (10)b for the BI. This gets obvious in the two following plots ( gure 46 and 47), where the re ectivity scans for the BH and BI were plotted for the same energy. Figure 46: Re ectivity scans BH (black) Figure 47: Re ectivity scans BH (black) and BI (blue) at 2.85 eV and BI (blue) at 2.05 eV and p-polarized light. and p-polarized light. In gure 46 the (10) mode lies almost perfectly at each other (but notice the dierent scaling!) with the dierence that the resonance for the BI is stronger than for the BH. One can also nicely see that the peak for (-1± 1) for the BI (blue curve) is missing whereas it is very strongly pronounced for the BH (black curve). The (0±1) mode is barley or not at all excited. For the BI one can also observe the (10)b and the (-20) mode. The (10)b mode is clearly visible in gure 47 whereas for the BH nothing at all is visible. An explanation for the occurrence of the backside mode could be that for the BI a lot more light can go through the sample because of the lower amount of metal (about 15% less!). Consequently a lot more light is able to hit the glass backside and can get back re ected to give rise to the resonances. This should not be possible for the BH sample because the holes are just too small to let enough light through. The (-1±1) mode is not in one of the electric eld vector directions. But it still can get excited by the fact that the propagation vector of the BPs is a superposition of the incoming wave vector, which must be parallel to the surface, and two lattice vectors which have either an x or a y component (compare grating coupling): ~ ~ kBP = k|| + p · Gx + q · Gy (25) 35 Out of this equation one can see that the BPs can gain some extra momentum in another direction than that of the incident beam. The occurrence of the (-1±1) mode for only the holes sample could be a result out of the better de ned structure. The holes sample has a lot sharper edges than the islands sample, where the edges are more round than spiky. A more de ned structure may lead to more pronounced resonances. There where some hints observed for the (-1 ±1) mode for the BI but a lot weaker than for the BH. Therefore the structure might be not well enough structured to excited the (-1±1) mode as strong as for the better de ned holes structure. With the measurements in the [11] crystal direction the (-1-1) direction can be observed separately (compare gure 23). These measurements were made for the BH and BI with the result that the (-1-1) direction only gets excited in the BH sample. The maps are shown in gure 48 and 49. Figure 48: Re ectivity map for BH Figure 49: Re ectivity map for BI in the [11] crystal direction in the [11] crystal direction with p-polarized light. with p-polarized light. The same dierences as discussed before are distinguishable. For the BH the (-1-1) mode is visible whereas for the BH the (10)b mode is excited. In this way of measuring the (-1-1) resonances look a bit broader because there is not only one distance between corner to corner but a lot dierent paths with dierent lengths. As a result the (-1-1) mode looks broader than the other two. This is also true for the measurements in the [10] direction. However the comparison from the BH with the BI is a bit di cult because the samples dier a lot in sizes and amount of metal, which, as is was shown before, in uences the resonances. If the observed dierences between holes and islands excited resonances is coming from the dierent structure has to be checked again with better comparable samples. Anyway it might be reasonable that the response from the holes and the islands is dierent, as far as the islands can oscillate by themselves and couple to each other. Whereas the resonances in the holes grating have to behave dierently. 3.2.3 Transmission The same measurements as for re ection were made for transmission. Comparison p-and s-polarized light excited modes The transmission maps for the BH and BI with p- and s-polarized light are shown in gure 50 - 53. 36 Figure 50: Transmission map BH Figure 51: Transmission map BH with p-polarized light. with s-polarized light. Figure 52: Transmission map BI Figure 53: Transmission map BI with p-polarized light. with s-polarized light. The transmission measurements for the holes sample were a bit di cult because the amount of transmitted light was very low. To get a reasonable result the measurement for the s- polarized light for the BH was done with a circular aperture before the detector in order to only detect the transmitted light from the sample and avoid other in uences. Therefore the transmission for the BH with s-polarized light is lower than for the p-polarized light where the measurements were done without the circular aperture. This is the origin of the red (about 2% transmission) looking background in the transmis- sion map for the BH with p-polarized light in gure 50. The transmission through the islands sample is much higher than for the holes sample. This is due to the the lower metal coverage of the islands sample. As for the re ectivity the possibility of switching excited modes with the change of the polarization is observed for both BH and BI sample. In the transmission measurements the backside mode (black dispersion line) for the BI is observed for both p- and s-polarized light. This might be a result of the not precise de ned structure. I. e. the s-polarized light can gain some p-polarized component to excite the (10)b mode. Because the measurements were done in transmission the backside mode is detected much more pronounced and as a result the eect shows up much stronger in transmission than in re ection. Comparison holes and islands As before the comparison from holes and islands the BH and BI were chosen because they have the best de ned structure, shown in gure 54 and 55. 37 Figure 54: Transmission map BH Figure 55: Transmission map BI with p-polarized light. with p-polarized light. For the BH sample the transmission is around a few percent which is a lot smaller than for the BI and might be a result out of the very small holes in combination with a high metal coverage. The possibility to excite dierent modes with holes and island sample, as it was seen in re ection, is also clearly visible in transmission. However the samples are dierent in holes and islands size, the metal coverage and the shape of the holes and islands. Therefore it is di cult to get a conclusion out of these comparisons. 3.2.4 Absorption To prove the excitation of plasmons one can consider to look at the absorption. If a plasmon gets excited an absorption maximum is expected because the energy of the incident beam gets transferred to the plasmon and consequently the amount of re ected and transmitted light goes down which end up in a maximum for the absorption. One more thing has to be considered. Light impinging on a non- at structure can not only be re ected, transmitted or absorbed, it also gets diracted. The amount of diracted light we could not measure. As a result the plot of the absorption contains also the amount of the diracted light. We will see a wavelength dependent (but not angle dependent) background because the amount of diracted light scales with the wavelength. For incidence, blue light is diracted more than red light, therefore the value of the background at the energy of blue light (around 2.85 eV) will be higher than for red light (1.9 eV). The absorption maps were calculated with the following equation: 100% − Re ectivity[%] − Transmission[%] = Absorption[%] + Diraction[%] (26) The results are shown in gure (57, 59 and 61) Figure 56: Re ectivity map BH Figure 57: Absorption+Diraction map BH with p-polarized light. with p-polarized light. 38 Figure 58: Re ectivity map SI Figure 59: Absorption+Diraction map SI with p-polarized light. with p-polarized light. Figure 60: Re ectivity map BI Figure 61: Absorption+Diraction map BI with p-polarized light. with p-polarized light. In the calculated absorption (plus diraction) maps one can observe the change in the absorption. If a BP is excited, energy is transferred to the sample and an increase in the absorption appears. These increase in the absorption value is nicely visible in the absorption maps in gure 57, 59 and 61. Therefore the absorption maps are a nice tool to observe the occurrence of BP. For comparison the re ectivity maps are plotted as well. The features in the absorption maps are a bit more clearly visible than in the pure re ection maps. 3.2.5 Babinet's Principle The structures we were measuring on are complementary structures. For such complemen- tary structures Babinet's principle states that the transmission from one sample should equal the re ection from the complementary structure, as discussed in chapter (2.1.5). This is valid for a perfect metal, which re ects 100% and perfect structures. Perfect struc- tures implies the holes and islands have exactly the same size. For the samples which were investigated here are complementary in the sense that there are holes and islands but the sizes of holes and islands as well as the amount of metal diers. This might change a lot in the optical response. If one applies this principle for our measurements big deviations are observed, as it is visible in gure 62. The compared maps do not really look the same. 39 Figure 62: Re ection BH vs. transmission BI with p-polarized light. Figure 63: Re ection BI vs. transmission BH with p-polarized light. Babinet's principle does not seem to work for our samples. At least not in this simple and ideal case formulation. The structures from the samples dier a lot from the assumed ideal case. The metal in this case is 30 nm thick and consequently a lot of the light gets transmitted! Also the sizes of holes and islands are not the same. The holes have a diameter of 420 nm and the islands of 190 nm which is quite a huge dierence! Further the amount of metal diers signi cantly. The islands sample has about 15% less metal coverage. All these parameters might in uence the excitation of BPs and therefore changes the optical response for holes and islands samples. All these might lead to the big deviation from the Babinet's principle for the here measured samples. In order to make a prediction about the Babinet's principle with respect to the optical response complementary magneto-plasmonic nanostructures with the same hole and island sizes as well as some with the same amount of metal have to be investigated. There also should not be any transmission through the metal. That means the metal needs to be a lot thicker! 3.3 Magneto-Optical Measurements In order to investigate the magneto-optical response of the BPs by an applied magnetic eld, TMOKE measurements were done for the BH and BI. The data is presented in the same way as it was done before for the re ectivity and transmission measurements. Since the TMOKE asymmetry is based on the re ectivity dierences the signal is a lot smaller than for the re ectivity measurements. Therefore multiple measurements have been averaged to reduce the measurements errors. Other than that the same scans were done as for the re ecticity/transmission measurements. The two TMOKE maps are presented in gure 64 and 65. 40 Figure 64: TMOKE map for BH Figure 65: TMOKE map for BI with p-polarized light. with p-polarized light. The colors refer now to the TMOKE asymmetry which was de ned as: RM+ − RM− TMOKEasymmetry = · 100% RM+ + RM− Red refers to a positive TMOKE asymmetry which is arising when the re ectivity for the up pointing magnetization RM+ is higher than for the down pointing one RM−. And vice versa for the blue color which refers to a negative TMOKE asymmetry. An applied magnetization leads to non-negligible o diagonal elements in the dielectric tensor [(ω)] which is directly connected to the optical constants. Consequently the mag- netization leads to a change in the optical quantities. These changes are very small and for the used samples out of FePd in the ± 0.05% range. However it is interesting to look at these small changes because, as one can see out of these two maps the applying of a mag- netic eld leads to completely dierent magneto-optical responses in the holes and island sample. The holes sample shows mainly for low angles of incidence a TMOKE response, whereas the islands sample shows features over the complete scan range. The combination of optics and magnetism gives a new tool which can be used to control plasmon resonances. The dispersion relation of the BPs is directly connected to the dielectric tensor: s d · [m] kBP = k0 · (27) d + [m] As a result the intersection from the light dispersion ( ω ) with the help of a kphoton = c reciprocal lattice vector takes place at another point and the plasmon resonance appears slightly shifted in the TMOKE scan. Dierent propagation directions of the BPs seem to have a dierent in uence to the RM+ and RM−. The up going (10) mode is followed by a TMOKE asymmetry maximum (red) whereas the (-10) mode is followed by a TMOKE asymmetry minimum (blue). This could be explained by the Lorentz force FL = e(~v × B~ ). The Lorentz force will point in dierent directions for dierent propagation directions (sign inversion of the velocity vector ~v). Consequently the (10) mode (forward propagation) will have the opposite response to that of the (-10) (backward propagation) mode. This sign inversion for the forward or backward propagation is observable for all calculated dispersion relations and is nicely visible in the TMOKE maps. Excitement-wise the same trend as discussed before is discernible. The holes excite besides the (±10) mode the (-1±1) one and the islands the (10)b and also the quite strongly pronounced the (-20) mode. 41 Another point to mention is the completely dierent looking background in the two maps. If one would look at a at thin lm the TMOKE signal will be quite at for low angles of incidence and then will rise quickly near the Brewster angle. This is the trend which is seen for the BH sample. The positive signal (red in the map) is probably mainly due to the Brewster angle. For the BI sample the map looks completely dierent. The modes seem to overlap the in uence of the Brewster angle completely. The in uence of the (10)b mode is visible over the whole measured angular range whereas the modes are vanishing in the background by the holes sample. This big dierence in the magneto-optical response was not visible in the optical maps. Therefore it is useful to look at the TMOKE signal. TMOKE scans with s-polarized light TMOKE measurements with s-polarized light should show no TMOKE signal because the electric eld vector is pointing in the same direction as the magnetic eld. Consequently the cross product in the Lorentz force is zero and no Kerr-amplitude can be induced. For the holes and islands samples s-polarized TMOKE scans were measured for 2.75 eV and 1.95 eV in order to cover all available modes. The results of these scans are shown in gure 66, 67, 68 and 69. To avoid measurements errors due to the small signals, a lot of scans were averaged and the standard deviation was calculated and as well plotted in the gure 68 - 69. Figure 66: TMOKE scan BH at 1.95 eV Figure 67: TMOKE scan BH at 2.75 eV with s-polarized light. with s-polarized light. In the gures one can see that one gets no TMOKE signal for the islands sample. But for the holes the scan at 2.75 eV shows quite clearly two features. These two features could refer to the (-1±1) and the (0±1) mode. Where the rst has a negative impact and the second a positive. For 1.95 eV nothing is observed, but for this energy there was also no plasmonic mode excited before with s-polarized light. 42 Figure 68: TMOKE scan BI at 1.95 eV Figure 69: TMOKE scan BI at 2.95 eV with s-polarized light. with s-polarized light. In order to make a statement about the TMOKE response from s-polarized light one should look at more well de ned samples. As it was seen before the BH sample shows more clearly pronounced features than the BI sample which could be due to the better de ned shape of the BH sample. This might be also the explanation for the occurrence of the TMOKE signal only for the BH but not for the BI. To gure out where the signal is coming from more measurements with also dierent sized holes and islands should be done in order to investigate if this eect is due to the structure (island/holes) or maybe properties like diameter or the amount of metal. 43 4 Conclusion and Outlook In order to investigate optical- and magneto-optical properties of patterned nanostructures measurements in re ection, transmission as well as TMOKE were done. The patterning of the samples was either a hole or a complementary island grating. It turned out that the size of the holes (130nm for the SH and 190nm for the BH) in uence the strength of the resonances. For the big holes sample the resonances were a lot more pronounced however the same modes were excited. Whereas by the measurements with light in either p- or s-polarization it was observed that dierent modes can be excited, which are always in the direction of the electric eld vector. Due to the a bit smeared out shapes there were as well often traces for the other polarization modes visible, but these signals were a lot weaker than the main resonances. Holes and islands samples seem to be able to give rise to dierent modes. For the holes samples the (-1±1) mode was observed whereas for the island sample the (10)b mode showed up. However one has to be a bit careful by the comparison of these two samples. The sizes of the holes and islands dier a lot as well as the amount of metal. Further the holes sample is much better de ned, which could explain the appearances of the (-1±1) mode. The backside mode might be a result from the lower amount of surface coverage. To get sure which modes get excited by holes and which by islands one should repeat these measurements with better de ned samples with the same sizes as well as with the same amount of metal. As it was shown by the comparison of the two dierent sized hole samples, these properties do in uence the occurring resonances. The transmission measurements showed the same trends as the ones in re ection. The backside modes are a bit more pronounced which makes sense as far as the measurements were done from the backside. The absorption maps were calculated to prove that energy gets absorbed along the dis- persion relations which is a ngerprint for the excitation of BPs. However one has to be aware of the amount of diracted light. If it would be possible to measure the amount of diracted light the pure absorption could be calculated. However it gets already quite clear which resonances get excited. Babinet's Principle in respect to the here measured samples is a bit di cult. The principle assume a perfect metal and perfect complementary structures. Unfortunately the sizes of holes and islands are quite dierent also the amount of metal and the thickness of the metal is only 30 nm, which means that there will be a lot of transmitted light. All these factors are deviations from the ideal case and will in uence the occurring BPs which will then in uence the optical response. Babinet's Principle in this simple model does not t with our measurements. To get as well some information about the magneto-optical response of the samples, TMOKE measurements were done. The measured maps show a lot of dierent features but as mentioned before one has to take the dierent design of the samples into account. However it gets quite clear that holes and islands samples have a dierent response to magneto-optical measurement. If this is a result of the dierent patterning (holes or is- lands) or if it comes out of the sizes, shape or amount of metal one has to check with more comparable samples. The same is true for the TMOKE measurements with s-polarized light. Out of this one measurement it is not clear where the signal is coming from. If it is an eect due to the patterning or the sample de nitions must be checked as well with better de ned and com- 44 parable samples. In a nutshell it was shown that properties like size, sharpness, amount of metal as well as the polarization state of the incoming light have a huge in uence in the excitation of BPs. 5 Acknowledgment Thank you to my supervisors in Uppsala, Ass. Prof. Dr. Vassilios Kapaklis and Emil Melander for the project idea and the great support during my measurements and the writing of this thesis. Thank you as well to the materials physics group & friends for the nice working environment and the amazing time I had in Uppsala. Thank you to my supervisor in Konstanz, Prof. Dr. Elke Scheer, for giving me the possibility and support to do my bachelor thesis abroad. 45 6 List of Figures 1 Assuming the electromagnetic wave is travelling in z direction and the elec- tric eld vector E is oscillating in the xy-plane, the three dierent kinds of polarization states are sketched...... 3 2 Re ection and refraction of an incident light beam. In the sketch is ni component to the grating kx, gains the missing momentum from a reciprocal lattice vector G and then gives rise to the plasmon excitation...... 13 13 Hysteresis loop...... 15 14 The two circularly polarized waves σ+ and σ− add up to a linearly polarized wave (red arrow) ...... 17 15 Because of the dierent propagation velocities of σ+ and σ− the two arrows do not nish the circle at the same time and the amplitude is rotated by the angle ϑk...... 18 16 The Kerr Ellipticity ηk is induced by dierent absorption coe cens for the two circularly polarized waves. The arrows add up to an ellipse (red) and the amplitude of the linearly polarized wave changes from a maximum by σ+ + σ− to a minimum by σ+ - σ−...... 18 17 The general case. Left- and right-handed circularly polarized waves are aected dierently in their absorption and refractive coe cient. The result is an elliptical polarization...... 19 18 The polar MOKE set-up (a) contains a magnetic eld which is perpendic- ular to the sample surface and parallel to the plane of incidence. Whereas the longitudinal MOKE (b) has a parallel magnetic eld to both. In the transverse MOKE (c) the magnetic eld is parallel to the surface of the sample but perpendicular to the plane of incidence...... 19 46 19 PMOKE: The induced Kerr-amplitude Rk (blue arrow) lies out of plane in respect to the incident polarization. Consequently a rotation of the polar- ization is observed, sketched by the green arrow...... 20 20 TMOKE: The plane in which the induced Kerr-amplitude lies is parallel to the incident polarization. Consequently no polarization rotation is observed. However the re ected beam has a dierent direction and a change in the re ectivity is observable...... 20 21 Samples de nition ...... 21 22 Diraction geometry: The xy-plane is de ned as the sample's surface with the incident plane in the zx-plane. The incident wave vector k (green) lies in the zx-plane, the scattered wave vector k0 (red) points in an arbitrary direction and ∆ k (blue) is the scattering vector. Further the angle of incidence ϑ and the scattered angle ϑ2 is sketched, as well as the angle ϕ which is the angle between the xy-component of the scattered vector k0 and the x-axis...... 22 23 For dierent propagation direction of the light, sketched as dierent colored arrows in a.) and c.), dierent dispersion relations are found, sketched in b.) and d.). Where a.) and b.) corresponds to the [10] and c.) and d.) to the [11] crystal alignment...... 24 24 Sketch of AFM set-up ...... 24 25 Experimental set-up ...... 26 26 Results from the AFM measurements: a.) SH with a diameter of (130±10) nm and a periodicity of (510±10) nm b.) BH with a diameter of (190±5) nm and a periodicity of (508±5) nm c.) SI with a diameter of (400±10) nm and a periodicity of (512±10) nm d.) BI with a diameter of (420±5) nm and a periodicity of (509±5) nm...... 27 27 Re ectivity map SH with p-polarized light...... 30 28 Re ectivity map BH with p-polarized light...... 30 29 SH: In the plot four dierent energy scans are shown. For the 2.75 eV (red) one can see two separated re ectivity maxima which shift together for higher energies and become for 3.05 eV (green) one re ectivity maximum. The rst peak for the 2.75 eV refers to the (10) mode, the second to the (-1±1) mode. One can also see the (-20) mode for 2.95 eV by 41◦ which shifts for 3.05 eV to smaller angles (38◦)...... 30 30 BH: Same energy scans for the big holes. As one can see the re ectivity maxima and minima are a lot sharper and stronger. The same overlapping process is visible. There is also a hump occurring and moving in the direction which would be expected for the (0±1) mode. However for this mode one would need a re ection maximum followed by a minimum which is not really clearly visible...... 30 31 SH (black) vs. BH (blue) at 2.85 eV...... 31 32 SH (black) vs. BH (blue) at 1.85 eV...... 31 33 Re ectivity map BI with p-polarized light...... 32 34 Re ectivity map BI with s-polarized light...... 32 35 Re ectivity scan at 2.95 eV for p- and s-polarized light for the BI. Black curve is p- and blue curve is s-polarized...... 32 47 36 Re ectivity scan at 1.85 eV for p- and s-polarized light for the BI. Black curve is p- and blue curve is s-polarized...... 32 37 With dierent polarization stats dierent modes can be pronounced depend- ing on which direction the electric eld vector is pointing. The green arrow reers to the s-polarization and the red one to the p-polarization...... 33 38 Re ectivity map BH with p-polarized light...... 33 39 Re ectivity map BH with s-polarized light...... 33 40 Re ectivity scan at 2.85 eV for p- and s-polarized light for the BH. Black curve is p- and blue curve is s-polarized...... 34 41 Re ectivity scan at 1.85 eV for p- and s-polarized light for the BH. Black curve is p- and blue curve is s-polarized...... 34 42 Re ectivity map BH with p-polarized light...... 34 43 Re ectivity map BI with p-polarized light...... 34 44 Re ectivity map BH with s-polarized light...... 35 45 Re ectivity map BI with s-polarized light...... 35 46 Re ectivity scans BH (black) and BI (blue) at 2.85 eV and p-polarized light. 35 47 Re ectivity scans BH (black) and BI (blue) at 2.05 eV and p-polarized light. 35 48 Re ectivity map for BH in the [11] crystal direction with p-polarized light. . 36 49 Re ectivity map for BI in the [11] crystal direction with p-polarized light. . 36 50 Transmission map BH with p-polarized light...... 37 51 Transmission map BH with s-polarized light...... 37 52 Transmission map BI with p-polarized light...... 37 53 Transmission map BI with s-polarized light...... 37 54 Transmission map BH with p-polarized light...... 38 55 Transmission map BI with p-polarized light...... 38 56 Re ectivity map BH with p-polarized light...... 38 57 Absorption+Diraction map BH with p-polarized light...... 38 58 Re ectivity map SI with p-polarized light...... 39 59 Absorption+Diraction map SI with p-polarized light...... 39 60 Re ectivity map BI with p-polarized light...... 39 61 Absorption+Diraction map BI with p-polarized light...... 39 62 Re ection BH vs. transmission BI with p-polarized light...... 40 63 Re ection BI vs. transmission BH with p-polarized light...... 40 64 TMOKE map for BH with p-polarized light...... 41 65 TMOKE map for BI with p-polarized light...... 41 66 TMOKE scan BH at 1.95 eV with s-polarized light...... 42 67 TMOKE scan BH at 2.75 eV with s-polarized light...... 42 68 TMOKE scan BI at 1.95 eV with s-polarized light...... 43 69 TMOKE scan BI at 2.95 eV with s-polarized light...... 43 7 List of Tables 1 Periodicity of the used samples. The uncertainty is due to the resolution of the AFM ...... 28 48 2 Calculated metal coverage on each sample. The uncertainty is given by the maximal resolution of the AFM-picture, for the percentage the biggest uncertainty was calculated...... 28 3 Calculated values for the strength of the resonances on SH with p-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by minium (or vice versa) ...... 51 4 Calculated values for the strength of the resonances on SH with s-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by minium (or vice versa) ...... 51 5 Calculated values for the strength of the resonances on BH with p-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by a minimum or vice versa ? strange hump showing up- moving like (0±1) but not really obvious max and min x* overlap (10)and (0±1) . 52 6 Calculated values for the strength of the resonances on BH with s-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by a minimum or vice versa ...... 53 7 Calculated values for the strength of the resonances for SI with p-pol light -: the condition for plasmon excitation is not ful lled x: condition for γ not ful lled * intersection (-10) with (10)b ** intersection (-1 ±1) with (10b) *** interaction (-1 ± 1) with (0 ± 1) **** interaction (10) with (-1 ± 1) . . 54 8 Calculated values for the strength of the resonances for SI with s-pol light -: the condition for plasmon excitation is not ful lled x: condition for γ not ful lled (11)b is visible! ...... 55 9 Calculated values for the strength of the resonances on BI with p-pol light -: contribution for plasmon excitation not ful lled x: no re ection maximum followed by a minimum or vice versa ...... 55 10 Calculated values for the strength of the resonances for BI with s-pol light -: the condition for plasmon excitation is not ful lled x: condition for γ not ful lled ...... 56 8 References [1] Atwater, H. A. : The Promise of Plasmonics, Scienti c American, vol 296, pp. 56-62, 2007 [2] Nature Photonics, Editorial: Commercializing Plasmonics, July 2015 [3] Hecht, Eugene: Optik, DE GRUYTER, 6. Au age, München, 2014 [4] Kittel, Charles: Einführung in die Festkörperphysik, Oldenbourg Wissenschaftsverlag GmbH, 14. Au age, 2006 [5] Born and Wolf: Principles of optics, Cambridge University Press, 7th edition, 1999 [6] Maier, Stefan A. : Plasmonics: Fundamentals and Application, Springer Science +Business Media LLC, New York, 2007 [7] Stockman, Mark I. : Nanoplasmonics: The physics behind the applications, Physics Today, February 2011 49 [8] Figure Lycurgus Cup: http://www.theblaze.com/stories/2013/08/28/ ancient-nanotechnology-exhibited-in-this-1600-year-old-roman-goblet/, 9.9.2015 [9] Sönnichsen. Carsten: Dissertation der Fakultät für Physik der Ludwigs- Maximilians- Universität München: Plasmons in metal nanostructure, München, June 2001 [10] Pluchery, Vayron, Van: Laboratory experiments for exploring the surface plasmon resonance, European Journal of Physics, 2011 [11] Blundell, Stephan: Magnetism in Condensed Matter, Oxford University Press, 2001 [12] Korenivski V. and Slonczewski J. C. :Introduction to Spintronics, Stockholm 2007 [13] Hubert A. and Schäfer R. : Magnetic Domains- The Analysis of Magnetic Microstruc- tures, Springer, 1998 [14] Sugano S. and Kojima N. : Magneto-Optics, Springer, 2000 [15] Visnovski, S. : Optics is Magnetic Multilayers and Nanostructures, Taylor & Francis Group, LLC, Florida, 2006 [16] E. Östman: Hysteresis free switching between vortex and collinear magnetic states, New Journal of Physics, 2014 [17] J.P. Morgan: Thermal ground-state ordering and elementary excitation in arti cial magnetic square ice, Nature Physics, 2010 [18] Aussenegg, F.; Ditlbacher H. : Plasmonen als Lichttransproter, Physik unserer Zeit, May 2006 Appendix Tables with calculated values for the strength of resonances To compare dierent resonances the strength of the resonances were calculated with the following formula: R− γ = 100 − · 100 (28) R+ Where R+ refers to the maximum re ectivity before the re ectivity drops due to the plasmon excitation. Consequently the re ectivity drop refers to R−. All calculated values for the strength of resonances are shown in the following tables. 50 Table 3: Calculated values for the strength of the resonances on SH with p-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by minium (or vice versa) SH- p-pol [10] [-10] [0 ± 1] [-1 ± 1] [-20] 1.55 eV - 0.8 - - - 1.65 eV - 1.8 - - - 1.75 eV - 1.4 - - - 1.85 eV - 1.9 - - - 1.95 eV - 2.1 - - - 2.05 eV - 2.4 - - - 2.15 eV - 3.5 - - - 2.25 eV - 1.9 - - - 2.35 eV - - - - - 2.45 eV - - x x - 2.55 eV - - x x - 2.65 eV 4.3 - x x - 2.75 eV 6.5 - x 0.5 - 2.85 eV 4.7 - x 1.2 x 2.95 eV 1.2 - x 0.3 x 3.05 eV 7.3 - x 3.6 0.9 3.15 eV 7 - x 3.5 0.3 Table 4: Calculated values for the strength of the resonances on SH with s-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by minium (or vice versa) SH s-pol [10] [-10] [0 ±1] [-1 ± 1] 1.55 eV - x - - 1.65 eV - x - - 1.75 eV - x - - 1.85 eV - x - - 1.95 eV - x - - 2.05 eV - x - - 2.15 eV - x - - 2.25 eV - x - - 2.35 eV - - - - 2.45 eV - - - - 2.55 eV - - 0.9 - 2.65 eV x - 0.2 x 2.75 eV x - x x 2.85 eV x - x x 2.95 eV x - x x 3.05 eV x - x x 3.15 eV 0.2 - x x 51 Table 5: Calculated values for the strength of the resonances on BH with p-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by a minimum or vice versa ? strange hump showing up- moving like (0±1) but not really obvious max and min x* overlap (10)and (0±1) BH- p-pol (10) (-10) (0±1) (-1±1) (10)b 1.55 eV - 18.5 - - - 1.65 eV - 18.8 - - - 1.75 eV - 17.3 - - x 1.85 eV - 18.8 - - x 1.95 eV - 18.8 - - x 2.05 eV - 22.3 - - x 2.15 eV - 25.7 - - x 2.25 eV - - - - 4 2.35 eV - - - - x 2.45 eV - - - - x 2.55 eV - - 0.4 11.1 x 2.65 eV 30.3 - ? 28.4 x 2.75 eV 23.2 - ? 20.5 x 2.85 eV 10.3 - ? 9.9 x 2.95 eV x* - x ? x 3.05 eV x* - x ? x 3.15 eV x - ? 2.2 - 52 Table 6: Calculated values for the strength of the resonances on BH with s-pol light. -: condition for plasmon excitation not ful lled x: no re ection maximum followed by a minimum or vice versa BH -s-pol (10) (-10) (0 ± 1) (-1±1) (10)b 1.55 eV - x - - - 1.65 eV - x - - - 1.75 eV - x - - x 1.85 eV - x - - x 1.95 eV - x - - 1.7 2.05 eV - 3.8 - - 0.2 2.15 eV - x - - 0.2 2.25 eV - x - - x 2.35 eV - - - - x 2.45 eV - - 24.2 - x 2.55 eV - - 11.6 x x 2.65 eV x - 5.8 x x 2.75 eV x - 10.1 4.5 ?? x 2.85 eV x - 7 x x 2.95 eV x - 4 x x 3.05 eV x - 1.1 x x 3.15 eV 2.2 - 0.3 x - 53 Table 7: Calculated values for the strength of the resonances for SI with p-pol light -: the condition for plasmon excitation is not ful lled x: condition for γ not ful lled * intersection (-10) with (10)b ** intersection (-1 ±1) with (10b) *** interaction (-1 ± 1) with (0 ± 1) **** interaction (10) with (-1 ± 1) SI -p-pol (10) (-1 ± 1) (0 ± 1) (-10) (10)b 1.55 eV - - - 48.3 - 1.65 eV - - - 40 - 1.75eV - - - 27.8 25.4 1.85 eV - - - 28.6 15.7 1.95 eV - - - 35.7 * 10 * 2.05 eV - - - 36.9 20.8 2.15 eV - - - 43 18.4 2.25 eV - - - - 25.7 2.35 eV - - - - 45 2.45 eV - - 10.8 - 45.9 2.55 eV - 44.6** 16.2 - 44.6** 2.65 eV - 16.4** 14.8 - 47.7** 2.75 eV 19.6 16.6*** 5*** - 52.6 2.85 eV 20 25 6.3 - 51.1 2.95 eV 17.5**** 5.3**** 13.2 - x 3.05 eV 2.8**** 5.7**** 3.3 - - 3.15 eV 18.4 21 x - - 54 Table 8: Calculated values for the strength of the resonances for SI with s-pol light -: the condition for plasmon excitation is not ful lled x: condition for γ not ful lled (11)b is visible! SI- s-pol (-10) (10)b (0 ± 1) (-1 ± 1) (10) 1.55 eV x - - - - 1.65 eV x - - - - 1.75 eV x x - - - 1.85 eV x x - - - 1.95 eV x x - - - 2.05 eV x 1.1 - - - 2.15 eV x 4.4 - - - 2.25 eV x 8.5 - - - 2.35 eV - 2.7 - - - 2.45 eV - x 44.3 - - 2.55 eV - x 31.5 x - 2.65 eV - x 20.6 x x 2.75 eV - x 20 31.6 x 2.85 eV - x 14.6 35.4 x 2.95 eV - x 3.6 17.6 x 3.05 eV - x 1.2 30 x 3.15 eV - x 4.8 29.8 x Table 9: Calculated values for the strength of the resonances on BI with p-pol light -: contribution for plasmon excitation not ful lled x: no re ection maximum followed by a minimum or vice versa BI- p-pol (10) (-10) (0 ± 1) (-1 ± 1) (-20) (10)b 1.55 eV - 32.8 - - - - 1.65 eV - 32.6 - - - - 1.75 eV - 48.1 - - - 31.5 1.85 eV - 64.1 - - - 22.2 1.95 eV - 54.3 - - - 31.1 2.05 eV - 47.1 - - - 16.9 2.15 eV - 27.4 - - - 21.6 2.25 eV - - - - - 35.2 2.35 eV - - - - - 25.4 2.45 eV - - x x - x ? 2.55 eV - - x x - x ? 2.65 eV 40.4 - 2.9 x? - x? 2.75 eV 37.8 - x x - 16 2.85 eV 37 - 2.6 x - 1.1 2.95 eV 50.5 - x 1.4 47.1 10.8 3.05 eV 38.5 - x x 37.9 20 3.15 eV 36 - x x 39.6 - 55 Table 10: Calculated values for the strength of the resonances for BI with s-pol light -: the condition for plasmon excitation is not ful lled x: condition for γ not ful lled BI -s-pol (-10) (10) (0 ± 1) (-1 ±1) (-20) (10)b 1.55 eV x - - - - - 1.65 eV x - - - - - 1.75 eV x - - - - x 1.85 eV x - - - - x 1.95 eV x - - - - 1.27 2.05 eV x - - - - 2.1 2.15 eV x - - - - 4.1 2.25 eV x - - - - 4.1 2.35 eV - - - - - 5.2 2.45 eV - - 28.8 - - 6 2.55 eV - - 19.3 5.3 - x 2.65 eV - x 11.8 x - x 2.75 eV - x 10.7 x - x 2.85 eV - x 6.6 x 7.1 x 2.95 eV - x 6.1 x 2.8 x 3.05 eV - x 5.2 x 4.6 x 3.15 eV - x 2.3 x x x 56