The Near-Field of Gold
Plasmonic Nanoparticles
June 27, 2017
AbstractResearch on metal nanoparticles is rising due to their applications in medicine and sustainable energy. Local heating and electric near-field effects are seen due to plasmon resonance but it is often very difficult to distinguish both effects. Von Hauff’s research group examines gold nanoparticles because of their high stability and facile synthesis and tries to untangle both effects using Raman spectroscopy. This is important because the separate effects are useful for different applications. Local heating effects could be used for destroying cancer cells while near-field effects can be used to enhance solar cell efficiencies. This thesis is part of this project and it presents a Matlab model that simulates the electric near-field around spherical gold nanoparticles. Parameters like the wave-length of light and the dielectric constant of the medium can be varied in order to investigate their effect on the near-field. Consecutively, an ex-periment with gold nanorods is executed to measure the near-field using Raman spectroscopy. Raman intensity enhancements are seen that could be caused by the near-field, but this experimental setup only measured Stokes shifts and therefore local heating can not be excluded as a cause. Further research at a different facility is necessary to measure the anti-Stokes spectrum in order to make a distinction between local heating and near-field effects.
Bachelor’s Thesis
Name Thijs van Eeden
Student nr. 10627499
Supervisor dr. Elizabeth von Hauff
2nd Examinator dhr. prof. dr. Tom Gregorkiewicz Specialisation Physics of Energy
University Universiteit van Amsterdam & Vrije Universiteit Faculty Physics and Astronomy
Credits 15 EC
Samenvatting
Wetenschappelijk onderzoek naar gouden nanodeeltjes is in de afgelopen 30 jaar sterk gegroeid vanwege toepassingen in geneeskunde en zonne-energie. Wanneer gouden deeltjes met een grootte tussen de 1 en 100 nm met een laser wor-den beschenen, kunnen deze deeltjes als kleine antennes gaan werken. Als de golflengte van de laser en de eigenschappen van het omringende materiaal goed zijn afgesteld, dan gaan de vrije elektronen van het goud meetrillen met het elek-tromagnetische veld van de laser. Dit resulteert in een lokale opwarming en een versterking van het elektromagnetische veld rondom de nanodeeltjes. De lokale opwarming kan worden gebruikt voor het uitschakelen van vijandige kanker-cellen en de elektromagnetische versterking heeft toepassingen in het effici¨enter maken van zonnecellen. Er is veel onderzoek gedaan naar deze eigenschappen van gouden nanodeeltjes, maar men heeft tot nu toe grote moeite met het on-derscheiden van deze twee verschijnselen. Temperatuurmetingen zijn erg lastig op de nanometer schaal en hierdoor is het lastig om de verschijnselen van elkaar te onderscheiden. Deze scriptie is deel van een groter onderzoek dat de lokale opwarming van de elektromagnetische versterking wil onderscheiden met behulp van Raman spectroscopie. Raman spectroscopie is een techniek die wordt ge-bruikt om de verschillen in energieniveaus van de vibraties van moleculen te meten. De moleculen worden beschenen met een laser en er wordt bijgehouden in welke golflengtes de moleculen het licht verstrooien. Als het verstrooide licht dezelfde golflengte heeft als de golflengte van de laser, dan heet het Rayleigh ver-strooiing. Als de golflengte groter wordt, dan treedt er roodverschuiving op en heet het een Stokes verschuiving. Als de golflengte kleiner wordt, dan treedt er blauwverschuiving op en heet het een anti-Stokes verschuiving. Bij Raman spec-troscopie worden al deze verstrooiingen bijgehouden en de verhouding van Stokes en anti-Stokes verschuivingen kan worden gebruikt om de lokale temperatuur te bepalen. Ook kan Raman spectroscopie gebruikt worden om de versterking van het elektromagnetische veld in kaart te brengen. Deze scriptie focust op de ver-sterking van het elektromagnetische veld door een computermodel te schrijven voor het elektromagnetische veld rondom bolvormige gouden nanodeeltjes. Met dit model kunnen de specifieke voorwaardes voor het meetrillen van de vrije elek-tronen worden onderzocht en wordt de eerste stap gezet naar het onderscheiden van de lokale opwarming van de versterking van het elektromagnetische veld. Vervolgens zijn er in het laboratorium gouden nanodeeltjes gemaakt en zijn deze behandeld met een stabilisator. De gouden nanodeeltjes zijn onderzocht met behulp van Raman spectroscopie en er is gekeken of er een versterking van het signaal van de stabilisator zichtbaar was. Deze resultaten zijn vergeleken met de resultaten uit het computermodel om zo de elektromagnetische versterking uit de metingen te halen. Dit is uiteindelijk niet gelukt omdat de beschikbare ex-perimentele opstelling niet in staat was om het anti-Stokes spectrum te meten. Hierdoor kon de temperatuur niet worden bepaald en was het niet mogelijk om de lokale opwarming te onderscheiden van de elektromagnetische versterking. Vervolgprojecten zouden de anti-Stokes verschuivingen van gouden nanodeelt-jes moeten meten en het model verder uitbreiden zodat ook anders gevormde nanodeeltjes kunnen worden gesimuleerd.
Contents
1 Introduction 4
2 Plasmonics 5
2.1 Nanoparticles . . . 5
2.2 Localized Surface Plasmon Resonance . . . 6
2.3 Near-Field Enhancement . . . 6
2.4 Cross Sections and Efficiencies . . . 9
2.5 Local Heating . . . 9
3 Raman Spectroscopy 10 3.1 Fundamentals . . . 10
3.2 Near-Field Effects . . . 11
3.3 Temperature . . . 12
3.4 Untangling Temperature and Near-Field Effects . . . 13
4 Method 13 4.1 Matlab Model . . . 13 4.1.1 Code Implementation . . . 14 4.2 Experimental Setup . . . 15 4.2.1 Synthesis of Nanoparticles . . . 15 4.2.2 Preparing Substrates . . . 16 4.2.3 SERS Measurement . . . 16 5 Results 17 5.1 Model Results . . . 17 5.2 Experimental Results . . . 20 6 Discussion 23 References 25
Appendix A Raman Spectra 28
Appendix B Optical Constants Gold 30
1
Introduction
Research on various types of metal nanoparticles (NPs) has exponentially grown in the past 30 years [1]. The growth in the number of publications is due to a series of discoveries of altering properties of metals on the nanoscale. These dis-coveries now form the base for the field of plasmonics, which is a part of the field of nanophotonics [2]. Nanophotonics investigates how electromagnetic fields can be confined and manipulated in the nanoscale range. The field of plasmonics is based on the interaction between electromagnetic waves and conduction elec-trons leading to local heating and electric near-field effects. Applications for these effects are found in a wide range of industries and research groups. Plas-monic particles have applications in increasing the efficiency of solar cells [3], targeting cancer cells in medicine [4], storing data using spectral coding [5], plasmon-resonance microscopy in biology [6] and metal-enhanced fluorescence [7, 8].
Noble metals are excellent materials for plasmonic applications because their resonance frequencies, at which the near-field enhancement and absorption ef-ficiency are at a maximum, lie in the visible spectrum of light [4]. Gold (Au), Silver (Ag) and Copper (Cu) NPs all show these effects, but Au is used most often because of it’s relative high stability. Furthermore, Au NPs can easily be synthesized with basic chemistry knowledge using the reduction of Au ions in a solution [9].
This thesis is part of a bigger project that focuses on local heating and near-field effects due to excitation of Au NPs around their resonance frequency. Dr. Von Hauff’s research group is attempting to untangle these phenomena using Raman spectroscopy to gain a better understanding in the separate effects of both phenomena. This thesis contributes to this project by looking into the near-field enhancement. Research shows that the near-field enhancement can reach up to 5 orders of magnitude compared to the applied electromagnetic field in specific experimental setups [10, 11]. A model will be proposed to simulate the enhancement of the electromagnetic field to investigate the dependence on the wavelength of the incident field and the dielectric properties of the materi-als. The focus will be on spherical Au NPs because of their visible resonance frequency, high stability and facile synthesis. Au NPs will be synthesized in the VU chemistry laboratory and stabilized using Cetrimonium bromide (CTAB). CTAB prevents the Au NPs from aggregating and the electromagnetic field around the NPs will be measured using Raman Spectroscopy and compared to the model. The Raman setup at the VU is equiped with lasers of 532 and 785 nm. In order to measure the near-field using Raman spectroscopy, we need NPs that absorb at both the excitation wavelength and the red-shifted wavelength due to the molecular vibrations of CTAB. This will result in an enhancement of the Raman scattered light that is scattered by the stabilizer CTAB around the Au NPs. Unfortunately, it is very difficult to synthesize spherical Au NPs that resonate at these wavelengths and therefore Au nanorods were synthesized instead. These nanorods resonate at approximately 600 nm and they can en-hance the Raman shifts of CTAB and can be measured using the 532 nm laser. The results will then be compared to the model. The research question of this thesis is: How can we model the near-field enhancement around spherical Au NPs and how can we measure this enhancement using Raman spectroscopy? First, the general physics behind metal NPs will be discussed in the
plasmon-ics section followed by a review of theory on plasmon resonance, the near-field enhancement, cross sections and local heating. The second section will be on Raman measurements for obtaining the near-field enhancement. The next sec-tion will be on the implementasec-tion of this theory into a Matlab model that simulates the near-field around spherical Au NPs. Thereafter, an experiment will be described to synthesize and measure the near-field around Au nanorods using a Raman spectrometer. The results of both the model and the experi-ment will then be shown, compared and discussed in the results and discussion section.
We expect that we can model the near-field enhancement in order to gain a deeper understanding in the physics and characteristics of this phenomenon. There are currently advanced models using Mie theory and the Finite Difference Time Domain (FDTD) approach that give exact and numerical solutions for the electromagnetic field around NPs of different sizes and shapes [12, 13, 14]. This research group is interested in how the near-field enhancement can be measured using Raman spectroscopy and how we can distinguish this phenomenon from local heating effects.
2
Plasmonics
In this section the theory of optics and electromagnetism of NPs will be sum-marized. Firstly, the characteristics of NPs will be discussed followed by the localized surface plasmon resonance (LSPR) that occurs with them. This will be used in the next paragraph of this section to understand the near-field en-hancement. The final paragraph will briefly summarize local heating effects of excited NPs.
2.1
Nanoparticles
NPs are with respect to the size a bridge between single molecules and bulk solids [13]. Where atoms and molecules are in the order of an ˚Angstr¨om (10−10m or 0.1nm), typical spherical NPs range between 1 and 100 nm. The collective response of free electrons in these NPs strongly depends on the shape, size and material of the particle and environment. If a small particle is excited with light, the energy can both be absorbed or scattered depending on the characteristics of the particle and the wavelength of light. The extinction efficiency of a particle is the sum of the absorption and scattering efficiencies.
Furthermore, there are two fundamental excitations of plasmonic particles, namely; surface plasmon polaritons (SPPs) and localized surface plasmon res-onances (LSPRs) [2]. SPPs are propagating and dispersive electromagnetic waves coupled to the electron plasma at a dielectric interface of a conductor. They only exist for magnetic (P) polarization and not for transverse-electric (S) polarization of light and are confined to the perpendicular direction. LSPRs are non-propagating excitations of free electrons coupled to the applied electromagnetic field. They occur when an electromagnetic field drives the col-lective oscillations of the NP’s free electrons into resonance [15]. This classical effect was theoretically described by Mie in 1908 by solving Maxwell’s equations for a plane wave incident on a metal sphere surrounded by a dielectric medium.
LSPRs account for the near-field enhancement around NPs and they will be discussed in the next paragraph.
2.2
Localized Surface Plasmon Resonance
An electromagnetic field with a certain frequency can oscillate the free elec-trons in spherical NPs with their size much smaller than the wavelength of light (d λ) [11, 4]. This oscillation is often referred to as the LSPR. For noble metals like Au and Ag, the resonance frequencies lie in the visible range of the electromagnetic spectrum. The LSPR of the free electrons results in a strong enhancement of the particle’s absorption and scattering of electromagnetic ra-diation and a strong local enhancement of the applied electric field. These characteristics of the LSPR are dependent on the the size and shape of the NP, the dielectric function of the metal and the dielectric constant of the surround-ing medium. Figure 1 shows the LSPR of free electrons in a metal sphere in response to an electric field.
Figure 1: An illustration of the localized surface plasmon resonance effect [10]. The spectral width, amplitude and temporal decay of the SPR are influenced by the damping of the surface plasmons. This damping of the SPR is due to scattering and absorption of electromagnetic waves and is further discussed in the next paragraph.
2.3
Near-Field Enhancement
The interaction of NPs with diameter d with the applied electromagnetic field can be understood within the quasi-static approximation [2, 3]. This approx-imation only holds in the limit d λ and for particles smaller than 100 nm because then the phase of the harmonically oscillating field is practically con-stant over the volume of the particle. This limit is called the Rayleigh limit and then the harmonic time dependence can simply be added to the applied electrostatic field [13]. In the electrostatic approach, the Maxwell equations may be replaced by the Laplace equations of electrostatics [16]. We start with a perfect dipole in an static and uniform electromagnetic field where mis the
dielectric function of the metal, d the dielectric constant of the medium, E0the
the particle and Einis the electromagnetic field inside the NP. See image 2 for
a sketch of a NP in a dielectric medium.
Figure 2: Sketch of a metal NP in static electromagnetic field E0with potential
φ0and dielectric constant d outside the particle. Inside we have Ein, φinand
m.
The electromagnetic fields are given by
Ein= ∇φin (1)
E0= ∇φ0 (2)
and the potentials by
∇2φin= 0 (3)
∇2φ
0= 0. (4)
At the boundary of the particle, the condition Vin= V0 results in
m
δφin
δr = d δφ0
δr (5)
where r is measured from the center of the particle. The potential outside the NP is given by the potential of a dipole with the poles separated by a distance of 2R as shown in equation 6. φ0= m− d m+ 2d E0R3 cosθ r2 (6)
We add the induced dipole potential to the potential of the applied electromag-netic field to obtain the potential φout outside the NP:
φout= −E0rcosθ +
m− d
m+ 2d
E0R3
cosθ
r2 , (7)
The first term represents the applied electrostatic field and the second term represents the harmonic oscillation in the form of a dipole at the center of the particle. We can rewrite this equation using the dipole moment p as
φout= −E0rcosθ +
p · r 4π0dr3 (8) p = 4π0dR3 m− d m+ 2d E0. (9)
The polarizability α is defined as
α = 4πR3 m− d m+ 2d
Be aware that the dielectric function of the metal is a complex function and angular frequency dependent [17]. It is defined as
m(ω) = r(ω) + ii(ω) (11)
where the real and imaginary part are angular are given by
r(ω) = n2− κ2 (12)
i(ω) = 2nκ. (13)
κ is the extinction coefficient and n the refractive index. When we look at the polarizability, we find a resonant enhancement when |m+ 2d| is at a
mini-mum. In the case of a small or slowly-varying Im[m] around the resonance, the
condition becomes
Re[m(ω)] = −2d. (14)
The near-field enhancement around metal NPs can be obtained from the poten-tial using
Eout = −∇φout = E0+
3n(n · p) − p 4π0dr3
(15) where n is the vector from the position of the dipole to the position where the field is being measured and p is defined as
p = 0dαE0. (16)
In equation 15 we can see that the near-field enhancement has a r−3dependence. The analytical solution for the magnitude of the electromagnetic field outside the particle can alternatively be expressed in the polarizability, radius of the particle and applied electric field [10]. Eout can be converted to Cartesian
coordinates and is given by
Eout= E0ˆx − gR3E0 ˆx r3 − 3x r5(xˆx + y ˆy + zˆz) (17) g = m− d m+ 2d . (18)
where x, y and z are the Cartesian coordinates, ˆx, ˆy and ˆz are unit vectors and Eout is the amplitude of the applied electromagnetic field. In this equation, we
can see that the applied electromagnetic field is polarized in the x-direction. When a NP is excited at resonance, the induced resonance can enhance the applied electromagnetic field by a factor 105 [11].
Wang et al. carried out an experiment to map the electromagnetic near field enhancement [18]. Au NPs of 50 nm diameter in size were placed in arrays, excited and measured using microspectrophotometry. The light was polarized in x- or y-direction and they showed that the enhancement is mainly in the direction of the polarization of the electromagnetic field as shown in figure 3.
Figure 3: Heat map of the distribution of the field amplitude enhancement for x- and y-polarization. [18].
2.4
Cross Sections and Efficiencies
The scattering and absorption efficiencies can be written down in the small-particle limit as Qabs= 4kIm m− d m+ 2d (19) Qsca = 8 3k 4 m− d m+ 2d 2 . (20)
where k = 2πλ [12]. The cross sections for absorption and scattering are Cabs= kIm(α) = 4πkR3Im m− d m+ 2d (21) Csca= k4 6π|α| 2= 8 3πk 4R6 m− d m+ 2d 2 . (22)
For very small particles, scattering becomes very small compared to absorption and therefore we can write
Cext≈ Cabs= kIm(α). (23)
We can see that the absorption cross section dominates for small particles be-cause of the R3 dependence instead of R6 dependence for scattering. The cross
sections have the same resonant condition as the near-field enhancement that comes with the polarizability.
2.5
Local Heating
Heat is a form of energy that is associated with the kinetic energy of atoms and molecules in objects, fluids and gasses [19]. This energy can be transmitted
by radiation, convection or conduction and flows from more energetic to less energetic locations. We call this the diffusion of heat and this effect occurs when plasmonic NPs absorb light and heat up. Newton first discussed conductive heat transfer in 1701 with the law of cooling [20, 21]. Equation 24 shows Newton’s law of cooling where q is the heatflux, k is the heat conductivity and ∇T is the difference in temperature between spot a and b.
˙
q = k∇T = k(Ta− Tb) (24)
Next to conductive heat transfer, heat dissipation can occur via convection or thermal radiation. Also, heat conduction can be reduced at interfaces of two dissimilar materials. At an interface, there is an impedance that slows the heat transfer. Knowledge of the latter is crucial to describe the heat dissipation from Au NPs to a dielectric surrounding.
3
Raman Spectroscopy
This section will summarize Raman spectroscopy and how it can be used to measure the near-field enhancement and local heating of plasmonic NPs. Firstly, the fundamentals of Raman spectroscopy will be discussed followed by the near-field effects in Raman measurements. The next paragraph will be on measuring local heating and temperature using Raman spectroscopy. The section will conclude with a discussion on distinguishing local heating from near-field effects.
3.1
Fundamentals
Raman spectroscopy is a technique that can be used to examine the vibrational and rotational modes of molecules [20]. A laser interacts with the modes in a molecule and that results in different types of scattering. This is called Rayleigh scattering when the incident and scattered photon have the same wavelength. Alternatively this is called elastic scattering. The molecule gets excited by the incident photon and relaxes to the same energy state as before. When the scattering is inelastic, the molecule relaxes to a different state than before. The scattered wavelength and energy of the photon is different than that of the incident photon. This inelastic scattering is called Raman scattering. The energy lost or gained in Raman scattering is defined as
∆ER= Ei− Es (25)
where ERis the Raman shift, Ei is the energy of the incident photon and Esis
the energy of the scattered photon. This shift can either be positive or negative and is often converted to wave numbers (cm−1) by
Ramanshif t(cm−1) = 1 λ0(nm) − 1 λR(nm) 107nm cm (26)
where Ramanshif t(cm−1) is the shift in wave numbers, λ0 the wavelength of
the incident photon and λR the wavelength of the Raman scattered photon in
nm. These shifts are called Stokes when they are positive, and anti-Stokes when they are negative. Anti-Stokes scattering only occurs when the molecule is in a vibrational state above the ground state in order to relax to a lower energy
state after Raman scattering. See figure 4 for an illustration on Stokes and anti-Stokes scattering. When the temperature of a system increases, more excited states will be occupied and therefore the chance for anti-Stokes scattering gets enhanced. Simultaneously the chance for Stokes scattering decreases and the ratio between Stokes and anti-Stokes scattering can be used to calculate the temperature [22]. This will be discussed in 3.3.
Figure 4: An illustration of Rayleigh and Raman scattering with Stokes and anti-Stokes shifts [23].
3.2
Near-Field Effects
The enhancement of the electromagnetic field can be measured using Raman spectroscopy [10]. The intensity of the applied field is enhanced at the surface of the NP so if we evaluate equation 17 at r = R and using I ∝ |E|2, we get
Iraman∝ |Eout|2= |E0|2|1 − g|2+ 3cos2θ(2Re(g) + |g|2)
(27) where θ is the angle between the field vector of the applied electromagnetic field and the vector pointing towards positions on the sphere’s surface. The Raman intensity peaks in the polarization direction at θ = 0 and θ = π. When g is large, the maximum intensity becomes
|Eout|2= |E0|2|g|2(1 + 3cos2θ). (28)
If cosθ is at a maximum we get
|Eout|2→ 4|E0|2|g|2 (29)
and at a minimum
|Eout|2→ |E0|2|g|2. (30)
The radially averaged of equation 27 is
|Eout|2= 2|E0|2|g|2 (31)
and this equation is generally used in Raman measurements of the near-field. The applied field induces a oscillating and radiating dipole on the surface and
there is a small probability that this radiation is Stokes or anti-Stokes shifted. It is possible to evaluate equation 27 at the shifted frequency. Kerker et al. fully evaluated this approach [24], but we give the first-order approximation:
EF = |Eout|
2|E0 out|2
|E0|2
= 4|g|2|g0|2, (32)
where the prime indicates that it has to be measured at the Raman scattered frequency. This expression is called the Surface-Enhanced Raman Spectroscopy (SERS) electromagnetic enhancement factor (EF). When the Stokes shift is fairly small, g and g0 are almost the same and then the EF scales as g4. In an
experimental setup equation 32 is often given as EF = ISERS/Nsurf
IN RS/Nvol
(33) that describes the enhancement for both the incident electromagnetic field and the Stokes-shifted fields [25]. In this equation ISERS is the surface-enhanced
Raman intensity, Nsurf the number of molecules bound to the enhanced metallic
substrate, IN RS the normal Raman intensity and Nvolthe number of molecules
in the excited volume.
The field enhancement decays as we showed before with r−3 so the EF should scale as r−12. We have to take into account that increased surface scales as r2 because we consider shells of molecules at an increased distance from the NP, we get ISERS = a + r a −10 , (34)
where ISERSis the intensity of the Raman mode, r the distance from the surface
of the NP to the absorbing molecule and a the average size of the field enhancing features on the surface.
3.3
Temperature
Previous research has showed that temperature can be measured using Raman spectroscopy [26, 27, 22]. The ratio ρ between the Stokes intensity (IS) and
anti-Stokes intensity (IaS) is given by
ρ = IS IaS = A τ σ 0 SIL hνL + e−hνmkbT (35) where νm is the shift frequency, τ the vibrational excited state lifetime, σ0S the
Raman cross section, T the temperature and subscript L denotes the laser. The first term represents vibrational pumping while the second term represents the thermal population. The asymmetry factor A is given by
A = ηaS ηS σaS0 σ0 S ELEaS ELES 2 (36) where η is the detection efficiency of unpolarized light and E the local electro-magnetic field strength. The Raman cross section is given by
σ0m= σ0,mνL(νLνm)3 (37)
for both Stokes and anti-Stokes shifts where σ0 is the Raman cross section
3.4
Untangling Temperature and Near-Field Effects
Local heating and near-field EFs both influence the Stokes/anti-Stokes ratio [22]. Within a hotspot caused by local heating, the Stokes and anti-Stokes EFs are wavelength dependent and this modulates the ratio ρ. Simultaneously local heating increases diffusion from these hotspots which could modify ρ. Deviations of ρ have been found in more than 1.5 order of magnitude and this is accounted for by EF variations and local heating. Simulations can be used to calculate the EF variations in order to distinguish effects from local heating and the near-field. Once the electromagnetic field is known, it is possible to calculate the local temperature using ρ. Molecules with Raman modes at terahertz shift frequencies (50 − 500 cm−1) have a symmetry factor A that approaches unity. This results in a higher accuracy for temperature measurements.
4
Method
In this chapter the implementation of the theory in chapter 2 into a mathemat-ical framework is discussed that can be solved by computer. The model was designed in Matlab with the purpose to compute the near-field enhancement around spherical Au NPs with several adjustable parameters. After the imple-mentation of the model, the setup for an experiment is described to measure the near-field enhancement of Au nanorods using Raman spectroscopy.
4.1
Matlab Model
A N xN matrix was constructed with a circle in the middle with diameter d = 0.2N to represent the top view from above a NP. In this implementation the particle takes a considerable amount of space in the matrix and is therefore clearly visible, but there is enough space left in the matrix for the electromag-netic enhancement. The cells in the matrix that are outside the particle were given the integer 0 and cells within the NP the integer 1.
The first two constants in the model are the number of cells N and the diameter d in m. With these parameters, the N xN matrix’ cells get physical dimensions because the N and d are linked by d = 0.2N . N = 1000 was set because it gave a well defined circular NP while keeping the run-time of the model under 1 second. See figure 5 for a sketch of the model.
The following input parameter is the wavelength λ (nm) and intensity I0 in
photons · s−1A−1 of the incident electromagnetic field. With this wavelength, equations 11, 12 and 13 and experimental measurements by Johnson et al., we can calculate the complex dielectric function mof Au [17]. The table of Johnson
et al. with the optical constants of gold is added to the appendix. The positive dielectric constant d of the medium is added so that the model calculates the
polarizability α and the scattering and absorption efficiencies and cross sections with the theory from chapter 2.
Figure 5: A sketch of the Matlab model. A top view from above a spherical NP placed within a N xN array with size d relative to the dimensions of the array.
4.1.1 Code Implementation
To calculate the near-field enhancement it is necessary to calculate the distance from each cell in the array to the NP. Previous research showed that the en-hancement is in the direction of the polarization of the exciting electromagnetic field [18]. To calculate the electromagnetic field outside a NP we use
Eout= E0ˆx − gR3E0 ˆx r3 − 3x r5(xˆx + y ˆy + zˆz)
from chapter 2. A loop runs through the whole N xN matrix to calculate the distance r from cells that are outside of the particle to the center of the NP. Cartesian coordinates x and y are used to calculate r using Pythagoras and z = 0 is set because we are looking in the x − y plane. See figure 6 for a sketch.
Figure 6: The distance for each cell with indices i and j are calculated to the center of the NP.
The model stores an electromagnetic field vector in every cell and calculates the amplitude for every cell using
|E| =qE2 x+ Ey2.
There is no z-component of the electromagnetic field because the coordinate z is always zero in the x − y plane. The amplitudes are normalized to the applied electromagnetic field and they are displayed in a 2-D heatmap. The model also creates a plot of the normalized enhanced amplitude versus the distance from the center of the particle in the polarization axis.
4.2
Experimental Setup
In this paragraph the experimental setup is discussed. Firstly, the preparation of the Au nanorods is briefly discussed followed by the preparation of substrates with and without the nanorods. Finally the method of the measurements with the Raman spectrometer will be discussed.
4.2.1 Synthesis of Nanoparticles
The Au nanorods were synthesised by Farooq Kyeyune using the recipe by Vidgerman et al. [28]. Cetrimonium bromide (CTAB) was used to stabilize the nanorods and they were dispersed in Milli-Q H2O. See figure 7a for the chemical
structure of CTAB and figure 7b for the bonding of CTAB to Au nanorods. The absorption peak of the dispersion is located at 526 and 598 nm, see figure 17b in the next chapter.
(a)
(b)
Figure 7: (a) shows the chemical structure of CTAB [29]. (b) shows the bonding of CTAB to the Au nanorods [30].
4.2.2 Preparing Substrates
Glass samples of 25x25 mm were cut and cleaned in an ultrasonic bath by the following recipe:
• 5 minutes in Milli-Q H2O.
• 5 minutes in 2-propanol. • 5 minutes in ethanol. • 5 minutes in acetone.
A concentrated dispersion of CTAB-stabilized Au nanorods in H2O was diluted
25, 50 and 100 times with Milli-Q H2O. These new solutions were dropcasted
on several substrates and dried within the fume hood. Furthermore solutions of CTAB were made in ethanol with concentrations ranging from 0.0625 to 0.25 M and spin coated on the substrates. These samples were used as a control group for the Au NPs. The rpm was set on 600 or 1000 for a time duration of 30 seconds.
4.2.3 SERS Measurement
For this experiment the Renishaw Raman spectrometer in the VU laserlab was used. See figure 8 for a schematic view of the Raman spectrometer. The Raman spectrometer is equiped with lasers of 532 and 785 nm. The 532 nm laser was used to excite the samples with an exposure time of 5.00 s and 50 accumulations. The laser power with the CTAB control group was varied from 1.0 % to 10 %. The gold samples were excited with with a maximum of 1.0 % laser power. The laser power for Au was lower than for CTAB to protect the detector and the sample from high intensities due to the the near-field enhancement. This ex-perimental setup was only able to do measurements on Stokes shift frequencies.
Figure 8: Schematic view of a Raman spectrometer [31]. The spectrometer at the VU laserlab received the scattered light via the objective lens.
5
Results
This chapter will discuss the results and conclusions of this Bachelor’s thesis. Firstly, the results from the simulations of near-field enhancements of spherical Au NPs will be shown. This will be followed by the Raman spectra of Au nanorods and the NP stabilizer CTAB. Finally, we will draw conclusions on determining the near-field enhancement and local heating.
5.1
Model Results
The simulations where run with m= 3.0, d = 30 nm, an intensity of 5 photons ·
s−1A−1 and the light polarized in the x-direction. In figures 9a and 9b the heatmaps are displayed with the exciting wavelength at λ = 530 and 570 nm. According to the theory, the resonant wavelength should be in the middle of these wavelengths.
(a) (b)
Figure 9: Heatmaps of the near-field enhancement |Eout|
|E0| around a Au NP with
d = 30 nm and intensity of 5 photons · s−1A−1. The wavelength λ is 530 nm in (a) and 570 nm in (b).
In these heatmaps there is almost no enhancement visible and according to the theory the resonance condition should be met at a wavelength of approximately 550 nm. When we run the model with this wavelength, we get the result in figure 10.
Figure 10: Heatmaps of the near-field enhancement|Eout|
|E0| around a Au NP with
In this heatmap we can clearly see an enhancement of the electromagnetic field in the direction of polarization. The enhancement of the first and last simulation is plotted against the distance from the surface of the NP in the direction of the polarization in figure 11a and 11b.
(a) (b)
Figure 11: Plots of the enhancement |Eout|
|E0| and distance to the boundary of the
particle in the direction of polarization. Both plots are with d = 30 nm and intensity 5 photons · s−1A−1 where (a) has λ 530 nm and (b) 550 nm.
The enhancement increases to 7.5 and 100, respectively, and decreases with the r−3 dependence as showed in the theory. When we look at the plot with λ = 550 nm, we can see a much higher enhancement then with a wavelength of 530 and 570 nm.
We see that the hotspots near the boundary of the particle have an enhancement that can reach up too 100 at a wavelength of 550 nm. We see the same distance dependence in all simulations but the decay distance scales linearly with the polarizability.
The next series of simulations were with a fixed excitation λ = 550 nm, d = 30 nm, intensity 5 photons · s−1A−1 and a varying dielectric constant m. This
was done to get a better understanding of the resonant condition presented in the previous chapters. mwas increased from 2.7 to 3.2 and figure 12 shows the
(a) (b)
(c) (d)
(e) (f)
Figure 12: Simulations of |Eout|
|E0| of a spherical Au NP with d = 30 nm, λ = 530
nm and intensity 5 photons · s−1A−1. The dielectric constant of the medium d was for (a) 2.7, (b) 2.8, (c) 2.9, (d) 3.0, (e) 3.1, (f) 3.2. In figure 12c the
resonance condition is met and therefore a strong enhancement is visible. We can see that in 12c the relative near-field enhancement |Eout|
|E0| is at a
maxi-mum. The dielectric function of Au at λ = 550 nm is m= −5.84 + 2.11i.
The resonant condition |m+ 2d| evaluates to
|m+ 2d| = −0.04 + 2.11i.
The small value of the real part results in a strong increase of the polarizability α and gives this hotspot in the simulation. When the dielectric constant is either increased of decreased, the hotspots disappear according to the resonant condition.
5.2
Experimental Results
The Au nanorods were stabilized using CTAB and figure 13 shows the Raman spectra of substrates prepared with solely CTAB. The exciting wavelength of the laser was 532 nm. We did a correction for the background noise by subtracting the bass line of each measurement.
(a) (b)
Figure 13: Raman measurements on CTAB substrates that were spin coated with rpm 600 using 180 µl of a 0.25 M CTAB solution in ethanol. (a) shows an extended measurement with a clearly visible peak around 2800 cm−1. (b) shows the peaks with smaller wavenumbers. We see peaks at 451, 751, 1062 and 1464 cm−1.
Figure 13a shows the extended measurement from 100-3500 cm−1and figure 13b shows measurements focused on the 400-2000 cm−1region. In 13a we see peaks at 2800 cm−1 that are associated with the CH2 symmetrical and asymmetrical
stretching of CTAB molecules [32]. When we look at 13b we can see peaks at 1464 cm−1 that are associated with various CH2 twists, at 1062 cm−1 with the
C-C stretch, at 751 cm−1 with the CN+ stretch and at 451 cm−1 with C 4N+
deformation. Only 2 measurements are shown here for a better overview, but the rest of the Raman spectra are disclosed in the appendix.
In figure 14 measurements are shown on clean glass substrates for reference.
(a) (b)
Figure 14: Raman measurements on Clean glass substrates for reference. (a) shows an extended measurement with peaks at 2400, 1100, 800 and 600 cm−1.
We can see the same peaks in (b) when we look in a smaller range of wavenum-bers.
The following measurements were done on Au nanorods stabilized with CTAB. Figure 15 shows a screenshot of the Au + CTAB samples viewed through a 50x ocular.
(a) (b)
Figure 15: Two screenshots of the Au + CTAB substrates viewed through a 50x ocular. The green dot in (b) represents the approximate size of the laser spot. The laser spot is in the µm range while the NPs are in the nm range. Figure 16 shows the Raman spectrum of substrates with Au + CTAB. Multiple measurements were done and there were strong deviations in the spectrum. This was due to the size of the laser spot compared to the particles. The laser spot size was in the µm range while the NPs’ size were in the nm range. The laser spot therefore covered a variable amount of nanorods and this resulted in deviating Raman spectra. Figure 16 shows one of the spectra with clear peaks. The other measurements are disclosed in the appendix.
Figure 16: Raman spectrum of substrates with Au nanorods stabilized with CTAB. The Au + CTAB solution was dropcasted on clean glass substrates and dried. Measurements were done on various spots and this illustration shows a measurements with clear peaks around 1480-1520 cm−1 and 1130-1140 cm−1. Figure 16 shows two peaks around 1480-1520 cm−1 and 1130-1140 cm−1. In the other measurements, peaks were barely visible and they came with a strong irregular background noise. Next we compare this spectrum with the spectra obtained from glass and CTAB and to the absoprtion spectrum of Au + CTAB. See figure 17a for all spectra with the wavenumbers converted to wavelengths and figure 17b for the absorption spectrum of Au + CTAB.
(a) (b)
Figure 17: (a) Raman measurements on CTAB, Au nanorods + CTAB and glass placed with an vertical offset to enhance clarity. The peak of Au + CTAB around 566 nm (1130-1140 cm−1) coincides with the peak of the glass with a
red-shift. The peak around 580 nm (1480-1520 cm−1) coincides with CTAB’s peaks with a red-shift as well. (b) shows the normalized absorption spectrum of the Au nanorods + CTAB. The absorption peaks are located at 526 and 598 nm.
The 566 nm (1130-1140 cm−1) peak of Au + CTAB coincides with the peak of the glass with a small red-shift. This peak of Au + CTAB coincides with some peaks of CTAB as well, but the peak in Au + CTAB peak probably rises from Raman scattering from the glass. The other peak in Au + CTAB at 580 nm (1480-1520 cm−1) coincides with peaks from the CH2 symmetrical
and asymmetrical stretching of CTAB molecules. The peak is shifted to the red, wider and less sharp. This is probably due to the lower concentration of CTAB on the Au + CTAB substrates compared to the CTAB substrates. This makes the peaks less high and sharp and gives more background noise. It is not possible to compare the peak heights of CTAB and Au + CTAB because of the difference in concentration of CTAB on both substrates. CTAB substrates were spin coated and it was possible to control the thickness of the layer while the Au + CTAB substrates were dropcasted. The CTAB measurements were done on a relative thick layer compared to the dropcasted layer of Au + CTAB. The wavelength of the Au + CTAB peak at 579 nm coincides which the ab-sorption peak of Au + CTAB. Figure 17b shows two peaks in the abab-sorption spectrum at 526 and 598 nm. The first peak is associated with the spherical part of the Au NPs, where the second peak is associated with the elongated part of the nanorod. If you further elongate the nanorods, the rightsided peak will shift more towards the red. The Raman peak of wavelength 579 nm of Au + CTAB is approximately at the absorption peak and the Au + CTAB absorb very efficient at this wavelength. Effects of local heating and plasmon resonance are enhancing the Raman intensity that is coming from the CTAB molecules. These phenomena can explain the peak in the Au + CTAB Raman spectra, but unfortunately it is not possible to untangle local heating from near-field effects because the anti-Stokes region is missing. The experimental setup only supports Stokes measurements and without anti-Stokes it is not possible to measure temperature differences due to local heating.
6
Discussion
The first point of discussion is that it is impossible to ensure that the enhanced Raman intensities are from the near-field enhancement. In this experimental setup it is not possible to distinguish the local heating effects from near-field effects and measured Raman peaks could be enhanced by both. In order to untangle these effects, we need to measure anti-Stokes shifts. Prior to this project, the research group reckoned that these measurements could be done with the femtosecond Raman setup at the VU laserlab, but unfortunately this turned out to be impossible within the framework of this project. Anti-Stokes measurements should be done in a different facility in order to continue this research.
Secondly, the model presented in this thesis is designed to simulate the electro-magnetic field of spherical particles. The symmetry of spherical NPs makes the programming of the near-field elegant with respect to the parameters λ, d and d. Unfortunately it was not possible to synthesize spherical NPs that can be
measured using the Raman spectrometer in the VU laserlab. The laser wave-lengths of 532 and 785 nm are able to resonate spherical NPs, but we wanted NPs that resonated at both the exciting wavelength of the laser and the red-shifted wavelength due to molecular vibration of CTAB. Therefore we chose to synthesize comparable nanorods with resonance wavelengths that would give visible enhancements of the CTAB peaks. Within this approach it was possi-ble to measure near-field enhancements of Au nanorods, but it is very difficult to compare these results to the model. Follow-up projects should expand the model to simulate different shapes of NPs. Another direction could be synthe-sizing spherical Au NPs and measure them elsewhere in a experimental setup with lasers with different wavelengths.
Thirdly, it was not possible to spin coat NPs dissolved in water because it evaporates slowly. For this reason the NPs were dropcasted on the substrate and this makes it difficult to control thickness of the layer. The CTAB control samples were dissolved in ethanol and were be spin coated and dropcasted. The next step would be to either change the environment of the NPs from water to ethanol or to functionalize the glass substrates. The glass can be functionalized using piranha solution to create a hydroxyl surface [33]. The coupling agent (3-Mercaptopropyl)trimethoxysilane will bind the Au NPs to the glass and this will enhance spin coating results. When Au + CTAB solutions are spin coated as well as the CTAB, it is possible to compare substrates of similar thickness. This enables a quantitative evaluation of the near-field effect induced by the presence of Au NPs.
Fourthly, the optical constants of Au are taken from bulk measurements. Small particles need corrections to these constants and this will result in an altering dielectric function [16]. The optical constants used in the model were taken from Johnson et al. who did measurements in a wide range of wavelengths. The spacing between the wavelengths they used is approximately 30 nm and this is relatively large. The polarizability of NPs blows up at the resonant condition and a smaller spacing between the optical constants data would increase the accuracy of the model.
Fifthly, the Raman spectra of Au + CTAB deviated at separate spots and the peaks that were shown in the results section appeared occasionally. This has to
do with the size of the laser spot compared to the size of the NPs, as discussed in the previous section. Another reason for this altering spectra could be that the near-field and local heating effects could lead to the loss of the CTAB ligands. This could be prevented by lowering the laser intensity, but a lower intensity also results in less sharp peaks. The loss of ligands can be monitored by measuring the same spot several times to see if there is a change in the Raman spectrum. Furthermore, the Matlab model simulates NPs surrounded by a single dielectric medium, but this is not the case in our experimental setup. When NPs are placed on a substrate, the NPs are surrounded by CTAB and they will be at a glass-air interface. This makes the situation more complex and should be covered in follow-up projects.
Acknowledgements
I would like to thank my thesis supervisor dr. Elizabeth von Hauff for her support during my project. She was enthusiastic and always approachable and she gave me the opportunity to gain experience in theoretical, experimental and programming work. I would also like to thank dr. Simon Boehme for our endless discussions on Raman spectrometers, spin coating and working holidays in Bhutan. Furthermore I would like to thank Farooq Kyeyune for synthesizing gold nanoparticles and prof. dr. Tom Gregorkiewicz for examining this thesis. Last but not least I would like to thank Zazo Meijs for assisting me with the simulations and my fellow students for keeping me company at the VU.
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A
Raman Spectra
Figure 18: Raman measurements on CTAB substrates that were spin coated with rpm 600 using 180 µl of a 0.25 M CTAB solution in ethanol. The laser intensity was 10.0% with 50 accumulations and an exposure time of 5.0 s. 4 different spots were measured and plotted in this figure. We see peaks at 451, 751, 1062 and 1464 cm−1. The background noise was removed and the baselines of the spectra were set at 0.
Figure 19: Raman measurements on glass substrates cleaned following the recipe in section 4. Two spots were measured using a laser intensity of 10.0% with 50 accumulations and an exposure time of 5.0 s. We see peaks at 1100, 800 and 600 cm−1. The background noise was removed and the baselines of the spectra were set at 0.
Figure 20: Raman measurements on substrates with Au nanorods stabilized with CTAB. The Au + CTAB dispersion was dropcasted on clean glass sub-strates and dried. Measurements were done on various spots and this illus-tration shows the measurements with clear peaks around 1480-1520 cm−1 and 1130-1140 cm−1. The laser intensity was 1.0% with 50 accumulations and an exposure time of 5.0 s. The background noise was removed and the baselines of the spectra were set at 0.
Figure 21: Raman measurements on Au + CTAB substrates. Measurements were done on various spots and this illustration shows the measurements without clear peaks. The laser intensity was 1.0% with 50 accumulations and an exposure time of 5.0 s. The background noise was removed and the baselines of the spectra were set at 0.
B
Optical Constants Gold
Table 1: Optical Constants Gold
E (eV) λ (nm) n κ 1.39 891.7 0.17 5.663 1.51 820.8 0.16 5.083 1.64 755.8 0.14 4.542 1.76 704.2 0.13 4.103 1.88 659.3 0.14 3.697 2.01 616.6 0.21 3.272 2.13 581.9 0.29 2.863 2.26 548.3 0.43 2.455 2.38 520.8 0.62 2.081 2.50 495.8 1.04 1.833 2.63 471.3 1.31 1.849 2.75 450.7 1.38 1.914 2.88 430.4 1.45 1.948
Note: Table 1 Optical constants measured from Au by Johnson et al. [17].
C
Matlab Code
1 % p a r a m e t e r s 2 N _ s i z e = 1 0 0 0 ; % s i z e g r i d 3 r e a l _ s i z e = 3 0 * 1 0 ^ ( - 9 ) ; % d i a m e t e r of n a n o p a r t i c l e in ( m ) 4 i n t e n s i t y = 5; % i n t e n s i t y of the l i g h t ( p h o t o n s / s * A - > p h o t o n s ) and I ~ | E |^2 5 l a m b d a = 5 5 0 ; % w a v e l e n g t h of l i g h t ( nm ) 6 e p s i l o n _ d = 3; % p e r m i t t i v i t y of d i e l e c t r i c 7 8 % c o n s t a n t s 9 c = 2 . 9 9 7 * 1 0 ^ 8 ; % ( m / s ) s p e e d of l i g h t 10 h = 4 . 1 3 5 6 6 7 * 1 0 ^ ( - 1 5 ) ; % ( eV * s ) p l a n c k c o n s t a n t 11 nm = 1 * 1 0 ^ - 9 ; % ( m ) n a n o m e t e r e x p r e s s e d in m e t e r 12 k = (2* pi ) /( l a m b d a * nm ) ; % w a v e n u m b e r k 13 14 % v a r i a b l e s 15 r a d i u s = N _ s i z e / 1 0 ; % r a d i u s p a r t i c l e r e l a t i v e to s i z e g r i d 16 c e n t e r = N _ s i z e /2; % c e n t e r of the g r i d 17 d i a m e t e r = 2* r a d i u s ; % p a r t i c l e d i a m e t e r in p i x e l s 18 r e s o l u t i o n = r e a l _ s i z e / d i a m e t e r ; % g i v e s r e a l s i z e of e v e r y p i x e l in m o d e l 19 E_0 = s q r t ( i n t e n s i t y ) ; % i n c i d e n t em f i e l d I ~ | E |^2 20 21 c o l o r m a p ( hot ) ; % plot - s e t t i n g s 22 23 % c a l c u l a t e w a v e d e p e n d e n t c o m p l e x d i e l e c t r i c f u n c t i o n g o l d ( j o h n s o n ) 24 j o h n s o n _ e v = [0.64 , 0.77 , 0.89 , 1.02 , 1.14 , 1.26 , 1.39 , 1.51 , 1.64 , 1.76 , 1.88 , ... 25 2.01 , 2.13 , 2.26 , 2.38 , 2.50 , 2.63 , 2.75 , 2.88 , 3.00 , 3.12 , 3.25 , ... 26 3.37 , 3.50 , 3.62 , 3.74 , 3.87 , 3.99 , 4.12 , 4.24 , 4.36 , 4.49 , 4.61 , ... 27 4.74 , 4.86 , 4.98 , 5.11 , 5.23 , 5.36 , 5.48 , 5.60 , 5.73 , 5.85 , 5.98 , ... 28 6.10 , 6.22 , 6.35 , 6.47 , 6 . 6 0 ] ; 29 30 % c o n v e r t e l e c t r o n v o l t to w a v e l e n g t h in nm 31 j o h n s o n _ l a m b d a = z e r o s (49 ,1) ; 32 for j = 1 : 1 : 4 9 33 j o h n s o n _ l a m b d a ( j ) = ((( c ) *( h ) ) / j o h n s o n _ e v ( j ) ) /( nm ) ; 34 end 35 36 % o p t i c a l c o n s t a n t s f r o m j o h n s o n w i t h r o w 1 = n r o w 2 = k 37 j o h n s o n _ o p t i c a l = [0.92 , 0.56 , 0.43 , 0.35 , 0.27 , 0.22 , 0.17 , 0.16 , 0.14 , 0.13 , 0.14 , ...38 0.21 , 0.29 , 0.43 , 0.62 , 1.04 , 1.31 , 1.38 , 1.45 , 1.46 , 1.47 , 1.46 , ... 39 1.48 , 1.50 , 1.48 , 1.48 , 1.54 , 1.53 , 1.53 , 1.49 , 1.47 , 1.43 , 1.38 , ... 40 1.35 , 1.33 , 1.33 , 1.32 , 1.32 , 1.30 , 1.31 , 1.30 , 1.30 , 1.30 , 1.30 , ... 41 1.33 , 1.33 , 1.34 , 1.32 , 1 . 2 8 ; 13.78 , 11.21 , 9.519 , 8.145 , 7.150 , ... 42 6.350 , 5.663 , 5.083 , 4.542 , 4.103 , 3.697 , 3.272 , 2.863 , 2.455 , 2.081 , ... 43 1.833 , 1.849 , 1.914 , 1.948 , 1.958 , 1.952 , 1.933 , 1.895 , 1.866 , 1.871 , ... 44 1.883 , 1.898 , 1.893 , 1.889 , 1.878 , 1.869 , 1.847 , 1.803 , 1.749 , 1.688 , ... 45 1.631 , 1.577 , 1.536 , 1.497 , 1.460 , 1.427 , 1.387 , 1.350 , 1.304 , 1.277 , ... 46 1.251 , 1.226 , 1.203 , 1 . 1 1 8 ] ; 47 48 % f i n d c l o s e s t w a v e l e n g t h in a r r a y 49 tmp = abs ( j o h n s o n _ l a m b d a - l a m b d a ) ; 50 [~ , idx ] = min ( tmp ) ; % i n d e x of c l o s e s t v a l u e 51 52 % o p t i c a l c o n s t a n t s of i n c i d i d e n t e l e c t r o m a g n e t i c f i e l d 53 ev = j o h n s o n _ e v ( idx ) ; % e n e r g y of l i g h t in ( eV ) 54 n = j o h n s o n _ o p t i c a l (1 , idx ) ; % r e f r a c t i v e i n d e x 55 k a p p a = j o h n s o n _ o p t i c a l (2 , idx ) ; % e x t i n c t i o n c o e f f i c i e n t 56 57 % c o m p l e x d i e l e c t r i c f u n c t i o n 58 e p s i l o n _ 1 = n ^2 - k a p p a ^2; 59 e p s i l o n _ 2 = 2* n * k a p p a ; 60 e p s i l o n _ m = e p s i l o n _ 1 + e p s i l o n _ 2 *1 i ; 61 62 % c a l c u l a t e p o l a r i z a b i l i t y 63 n o m i n a t o r = e p s i l o n _ m - e p s i l o n _ d ; 64 d e n o m i n a t o r = e p s i l o n _ m + 2* e p s i l o n _ d ; 65 66 % t a k e the r e a l p a r t for e n h a n c e m e n t f a c t o r 67 g = ( n o m i n a t o r * c o n j ( d e n o m i n a t o r ) ) /( d e n o m i n a t o r * c o n j ( d e n o m i n a t o r ) ) ; 68 g _ r e a l = abs ( r e a l ( n o m i n a t o r ) / r e a l ( d e n o m i n a t o r ) ) ; 69 g _ i m = abs ( i m a g ( n o m i n a t o r ) / i m a g ( d e n o m i n a t o r ) ) ; 70 71 a l p h a = 4* g _ r e a l * pi *( r e a l _ s i z e /2) ^3; % p o l a r i z a b i l i t y 72 73 % c a l c u l a t e c r o s s s e c t i o n s 74 c _ s c a t = (( k ^4) *( abs ( a l p h a ) ^2) ) / ( 6 * pi ) ; 75 c _ a b s = k * i m a g ( a l p h a ) ; 76 77 % c r e a t e c i r c l e w i t h radius , c e n t e r e d in an N _ s i z e X N _ s i z e i m a g e
78 % n u m e r i c a l v a l u e 1 is i n s i d e the p a r t i c l e , 0 o u t s i d e 79 [ x , y ] = m e s h g r i d (1: N _ s i z e ) ; 80 C = s q r t (( x - c e n t e r ) . ^ 2 + ( y - c e n t e r ) . ^ 2 ) <= r a d i u s ; 81 82 % d i s t a n c e to p a r t i c l e m a t r i x 83 D = z e r o s ( N _ s i z e ) ; 84 85 % e n h a n c e m e n t of e l e c t r i c f i e l d m a t r i x 86 E = z e r o s ( N _ s i z e ) ; 87 88 % l o o p t h r o u g h the w h o l e 2 D a r r a y 89 for i = 1 : 1 : N _ s i z e 90 for j = 1 : 1 : N _ s i z e 91 % c h e c k if p o s i t i o n is o u t s i d e the p a r t i c l e 92 % e l e c t r i c f i e l d i n s i d e the p a r t i c l e is 0 93 if ( C ( j , i ) == 0) 94 95 % d i s t a n c e s to the c e n t e r of the p a r t i c l e 96 x = abs ( i - c e n t e r ) * r e s o l u t i o n ; 97 y = abs ( j - c e n t e r ) * r e s o l u t i o n ; 98 99 % d i s t a n c e to p a r t i c l e 100 D ( j , i ) = s q r t (( i - c e n t e r ) ^ 2 + ( j - c e n t e r ) ^2) * r e s o l u t i o n ; 101 102 % c o m p o n e n t s of the e - f i e l d
103 E_x = E_0 - E_0 *( g _ r e a l ) *(( r e a l _ s i z e /2) ^3) *(( D ( j , i ) ^( -3) ) - (3* x * x ) *( D ( j , i ) ^( -5) ) ) ; 104 E_y = E_0 *( g _ r e a l ) *(( r e a l _ s i z e /2) ^3) * ( 3 * x * y ) *( D ( j , i ) ^( -5) ) ; 105 106 % f i e l d a m p l i t u d e 107 E _ a m p = s q r t ( E_x ^2 + E_y ^2) ; 108 109 % s t o r e the a m p l i t u d e r e l a t i v e to i n c i d e n t em f i e l d 110 E ( j , i ) = E _ a m p / E_0 ; 111 112 end 113 end 114 end 115 116 % e n h a n c e m e n t d i s t a n c e p l o t 117 X = z e r o s (1 , r o u n d (( center - r a d i u s ) /( nm / r e s o l u t i o n ) ) ) ; 118 % s t o r e E w i t h 1 nm s t e p s f r o m s u r f a c e 119 I = 1; 120 for i = ( c e n t e r + r a d i u s + 1) : r o u n d ( nm / r e s o l u t i o n ) : N _ s i z e 121 % s c a l i n g to the a p p l i e d e l e c t r i c a l f i e l d 122 X ( I ) = ( E ( center , i ) ) ; 123 I = I + 1;
124 end 125 126 % get a x i s r e s o l u t i o n for h e a t m a p 127 a x i s r e s o l u t i o n = ( r e s o l u t i o n / nm ) * N _ s i z e ; 128 129 % h e a t m a p s e t t i n g s 130 i m a g e ('XData ',[ - a x i s r e s o l u t i o n /2 , a x i s r e s o l u t i o n /2] ,' Y D a t a',[ - a x i s r e s o l u t i o n /2 , a x i s r e s o l u t i o n /2] ,'CData ' , E ) ; 131 a x i s t i g h t ; 132 x l a b e l ('x (nm)') ; 133 y l a b e l ('y (nm)') ; 134 c = c o l o r b a r ; 135 c . L a b e l . S t r i n g = 'Eoutside / E_0 '; 136 s a v e a s ( gcf ,'heatmap . png ') 137 138 % e n h a n c e m e n t vs d i s t a n c e p l o t s e t t i n g s 139 p l o t ( X ) ; 140 t i t l e ('Relative intentensity ')
141 x l a b e l ('distance from surface (nm)')
142 y l a b e l ('E outside / E_0 ') 143 s a v e a s ( gcf ,'plot . png ') 144 145 % p r i n t d a t a 146 f p r i n t f ('Diameter nanoparticle is %s m or %s nm .\n', n u m 2 s t r ( r e a l _ s i z e ) , n u m 2 s t r ( r e a l _ s i z e / nm ) ) ;
147 f p r i n t f ('The incident light has wavelength %s nm with
i n t e n s i t y % s p h o t o n s / A * s .\ n', n u m 2 s t r ( l a m b d a ) , n u m 2 s t r ( i n t e n s i t y ) ) ;
148 f p r i n t f ('The polarizability alpha is %s and
e n h a n c e m e n t f a c t o r g is % s .\ n', n u m 2 s t r ( a l p h a ) , n u m 2 s t r ( g _ r e a l ) ) ;
