Experimental Surface Plasmon Resonance in a Thin Gold Film

Sebastian Duque1

1Grupo de Fsica Atomica y Molecular, Instituto de Fsica, Facultad de Ciencias Exactas y Naturales,Universidad de Antioquia UdeA; Calle 70 No. 52-21, Medelln, Colombia.

(Dated: October 2, 2014)

By shining monochromatic light into a thin gold metal film through a prism the surface plasmonresonance curve is pretended to be observed. In this write-up the experimental progress up to thedate is described.

INTRODUCTION

Surface plasmons are a collective oscillations of elec-trons in a solid (or liquid) excited by photons incident onthe metal-glass interface. Surface plasmons are a quan-tum phenomena, however, they are well described byclassical electromagnetic theory as waves traveling alongthe interface of two different media [1]. Surface plasmonresonance (SPR) is an experimental technique for mea-suring changes in the index of refraction and is widelyuse for studying binding interactions of biomolecules in-cluding antigen/antibody, complementary DNA probes,enzyme/substrate and receptor/ligand interactions.

Although SPR is theoretically observable in any con-ducting metal, silver and gold are the metals of choicebecause its plasma resonance is observed in the visiblepart of the electromagnetic spectrum [2].

The dispersion relation for surface waves propagatingalong the interface semi-infinite dielectric bounded byvacuum is

kx =

c

(()

() + 1

)1/2(1)

with the complex dielectric function of the medium [1].This surface plasmons cannot be excited directly by lightbeams since light propagating in vacuum wont matchthe frequency and wave vector of them. An increase on

FIG. 1.

the magnitude of the free propagating light wave vectorcan be achieved by passing the incident light through amedium, such as glass, allowing some of the incident lightto excite the surface plasmon (see figure 1). This couplingscheme is known as attenuated total internal reflectionand involves tunneling of the fields of the excitation beamto the interface where surface plasmon excitation takesplace [3]. Since this kind of surface plasma waves consistonly of evanescent waves, they dont emit light, hence arecalled non-radiative surface plasma waves [4].

This report is organized as follows: first the underlyingtheory of surface plasmon resonance effect is reviewed.After the possible schemes of optical excitation of thesurface plasmons are commented and using the S-matrixformulation the reflectivity is calculated. Last, experimen-tal observation of Surface Plasmon is presented discussingthe sample preparation, the experimental set-up and ex-perimental results.

THEORY

A Surface Plasmon (SP) is a surface-bound electromag-netic wave that propagates along the interface of a metaland a dielectric. SP are charge density fluctuations (seefig. 2) in the free electron gas (plasma), meaning thatSP excitation is only achieved in metals well describedby the free electron gas model. The resonance width ofthin silver films is smaller than for gold leading to highersensitivity but the inertness and ease of functionalizationof gold makes it the most aplicable metal.

Electromagnetic waves propagating in media are de-scribed by Maxwells equations. These equations canbe combined to yield wave equations for B and E withplane waves as solution. The wave equation imposes aconstraint (known as the dispersion relation) between themagnitude of the wave vector k and the frequency . Forexample, for waves propagating in vacuum

k2 =2

c2 . (2)

In nonmagnetic materials the permeability = 1; ina dielectric material, such as glass, the permittivity usually varies with the frequency .

In a more complicated situation, such as a plane bound-

ResaltadoResumir algo sobre los resultados

Resaltado"Todos" los fenmenos son cunticos, as que si son bien descritos por la teora electromagntica de Maxwell, entonces son fenmenos "clsicos" por definicin.

Nota adhesivaNota: 4.7Excelente marco terico, simulacin y presentacin de la regin donde se puede observar el fenmeno.Sin embargo, el anlisis de los resultados experimentales es pobre y falta homogeneidad en la presentacin de las grficas para una interpretacin ms expedita. Tambin falta mayor claridad en el desarrollo experimental, cuando se describen los equipos usados debe hacerse referencia al modelo y a la precisin con realizan la medida.lo resultados y las perspectivas les falta un mayor desarrollo.

2FIG. 2.

ary between a region (z > 0) with dielectric constant 1real and positive and a region (z < 0) with dielectricconstant 2 that can be complex (as depicted in figure 2),there are a set of solutions called surface waves whichexist at the boundary between medium 1 and 2 propagat-ing along the surface z = 0. In this case the constraintfrom the dispersion relation is

k2z = 2

c2 k2x . (3)

Notice that kz must be real, meaning that kz itself iseither real, implying wave propagation in the z direction,or is imaginary, implying an exponential decay of the fieldin the z direction. From the boundary conditions the nextcondition must be fulfilled

1k2x 22/c2 = 2

k2x 12/c2 . (4)

Only if 2 < 0 the condition above can be fulfilled. Solvingfor , the dispersion relation for surface waves:

2 = (ckx)2

(1

1+

1

2

). (5)

The above equation implies that 2 < 0 and |2| > |1|.If the interface is formed by air 1 1 and a metal well

described by the free electron model

2() = 1(p

)2, (6)

where p is the plasma frequency, the dispersion relationfor surface waves between a metal and air is

2 = (ckx)2

(1 +

1

1 (p/)2). (7)

With all of the above one can notice that for all valuesof kx > 0 the dispersion curve for surface plasmons prop-agating the metal-air boundary lies to the right of thedispersion curve for electromagnetic waves in air, = ckx(see fig. 1). Because the dispersion curve for propagationin vacuum does not intersect the disperson curve for sur-face plasmons, it is not possible to match the frequencyand wave vector of the surface plasmons to the frequencyand wave vector of incident electromagnetic radiation

(light). Since this plasma waves dont propagate in vac-uum they are called non-radiative. However, if light passby a medium, such as glass, the magnitude of the wavevector is increased and the dispersion curves (light andplasmon curves) will intersect. Experimentally a prismis used to couple light and plasmons. The wave-vectorcomponent of the incident light parallel to the boundaryis matched to the surface plasmon wave vector by chang-ing the incidence angle i relative to the normal of theglass/metal-film interface (see figure 3) by means of

kglass sini = kplasmon . (8)

When the critical angle for plasmon excitation is achievedthen energy of the beam is taken by the plasmon and adeep in the reflectivity is seen.

Fresnels theory relates the amplitude of the surfaceplasmon with the amplitude of the incident radiationusing the electromagnetic boundary conditions at theinterfaces [? ]. However, the S-matrix method uses thefact that there are only boundary conditions at thoseinterfaces and the wave propagation can be derived as asimple matrix operation if the incident field is known [?]. In fact, on biosensing applications there are more thanthree media involved modifying the plasmon resonanceand solving analytically Maxwells equations becomes atedious task; on the other hand, S-matrix formulation canbe employed to solve for an arbitrary number of layers [?].

Consider a stack of N + 1 smooth and perfectly parallellayers with varying thicknesses dj , and complex refrac-tive index nj . The incident layer and the last layer areconsidered as non-absorbing. The relation between the

FIG. 3. Prism coupling and surface plasmon dispersionrelation. Figure taken from [3].

GermanResaltado

GermanResaltado

GermanResaltado

3electric field vectors at two points z1 and z2 is given by[E+(z1)E(z1)

]= M

[E+(z2)E(z2)

](9)

where M denotes the scattering matrix. When the pointsare located within the same layer the relation can bewritten as [

E+(z1)E(z1)

]= Lj

[E+(z2)E(z2)

](10)

wherein Lj is the layer matrix of j-th layer, given by

Lj =[eij 0

0 eij

], (11)

describing the phase shift undergone upon propagation.The phase shift is given by

j =2pidj

nj cosj (12)

where nj is the complex refractive index of the layer, djis the layer thickness and j is the incidence angle. If thetwo points are located within two adjacent layers then[

E+(z1)E(z1)

]= Iij

[E+(z2)E(z2)

](13)

where the interface matrix Iij is given by

Iij =1

ij

[1 rijrij 1

](14)

with ij and rij the transmission and reflection Fresnelcoefficients

rij =nj cosi ni cosjnj cosi + ni cosj

, (15)

ij =2ni cosi

nj cosi + ni cosj. (16)

In the general case where z1 and z2 are separate withinnon-adjacent layers[

E+(z1)E(z1)

]= M

[E+(z2)E(z2)

](17)

where

M =

N1j=1

I(j1)jLj

I(N1)N . (18)With all of the above, the reflection r = E(0)/E+(0)and transmission = E+(N)/E+(0) coefficients of thelayer stack can be expressed in terms of the scattering

2.2 Stratified medium matrix model 15

Incident light Reflected light

xy

z

0th layer, (; n0)

1st layer, (d1; n1+ik1)

2nd layer, (d2; n2+ik2)

Nth layer, (; nN)

E||

E0

jth layer, (dj; nj+ikj)

1

2

j

NFigure 2.3: Layer stack in the stratified medium model. The 0th and Nth layers are

semi-infinite and have real refractive indices. The incident wave is p- (TM, k) or s- (TE,?) polarized. The planes show the interfaces, Iij .

p, TM or k) or perpendicular- (denoted s, TE or ?) polarized. The Cartesiancoordinate system in Figure 2.3 defines the z-direction as parallel to the plane of

incidence with the positive direction into the layer stack. Electric fields will be su-

perscripted + for positive and - for negative z-direction corresponding to refracted

and reflected waves respectively. For our purposes it is convenient (although not

necessary) to consider all layers isotropic, hence all fields are independent of x or

y. Just as before, it is assumed that the permeability = 0 for all layers.

Derivation

Since there is no dependence on x or y, we have for the total electric field amplitude

at a certain distance along the z-axis:

Etotz = E+z +E

z (2.25)

Where the subscript indicates the z-dependence. Eqn. 2.25 holds for both TM-

polarized and TE-polarized light respectively. For the relation between the electric

field vectors at two points, z1 and z2 we have:E+z1Ez1

=

M11 M12M21 M22

E+z2Ez2

=M

E+z2Ez2

(2.26)

From the German word senkrecht, meaning orthogonal.

FIG. 4. Prism coupling and surface plasmon dispersionrelation. Figure taken from [3].

matrix elements

r =M21M11

, =1

M11. (19)

EXPERIMENTAL SURFACE PLASMONRESONANCE

Sample preparationTwo gold samples (10 nm and100 nm) were prepared using the sputtering technique.The main purpose of using two samples is to observe theinfluence of the thickness and oxidation in the surfaceplasmon resonance. Figure 5 shows the 10 nm gold samplespectra taken using a Scanning Electron Microscope. Itis easy to see the Au peaks, all other peaks come fromthe substrate (glass).SimulationUsing the S-matrix formalism described in

Sec. the Surface Plasmon Resonance curve is calculatedwithin the model presented in fig. 6. The reflectance is

Au 6/16/2014 2:43:39 PM Project: 2014-06-16 Owner: INCA Site: Site of Interest 1

Sample: Au Type: Default ID:

Comment:

Spectrum processing : No peaks omitted Processing option : All elements analyzed (Normalised) Number of iterations = 3 Standard : C CaCO3 1-Jun-1999 12:00 AM O SiO2 1-Jun-1999 12:00 AM Na Albite 1-Jun-1999 12:00 AM Mg MgO 1-Jun-1999 12:00 AM Al Al2O3 1-Jun-1999 12:00 AM Si SiO2 1-Jun-1999 12:00 AM K MAD-10 Feldspar 1-Jun-1999 12:00 AM Ca Wollastonite 1-Jun-1999 12:00 AM Au Au 1-Jun-1999 12:00 AM

Element App Intensity Weight% Weight% Atomic% Conc. Corrn. Sigma C K 0.62 0.2795 9.15 3.97 17.03 O K 4.89 0.6382 31.81 1.89 44.44 Na K 1.62 0.9650 6.94 0.53 6.74 Mg K 0.33 0.7749 1.76 0.26 1.62 Al K 0.16 0.8651 0.78 0.20 0.64 Si K 7.04 0.9435 30.80 1.58 24.51 K K 0.23 0.9494 0.99 0.20 0.56 Ca K 1.22 0.9208 5.47 0.42 3.05 Au M 1.91 0.6430 12.29 1.32 1.39 Totals 100.00

FIG. 5.

4given by

R =

r01(1 + r12r23e2i2) + e2i1(r12 + r23e2i2)1 + r01e2i1(r12 + r23e2i2) + r12r23e2i22(20)

where j is given by equation 12 and rij , ij are thereflection and transmission coefficients.

The results are plotted in figure 7 showing that goldfilm thickness modify strongly the plasmon resonanceangle and deep. In fact, we supposed a 300 nm thickair layer but due to experimental limitations it cannotbe controlled. Figure 8 shows how plasmon resonanceangle is modified and suppressed as a function of the airlayer thickness. An optimal gold and air layer thicknessshould be found in order to improve the experimentalmeasurement of the plasmon resonance.

Set-upFor the experimental realization a = 633laser were used. A pupil and filters clean-up the laserspot for measurement and a polarizer ensures the correctpolarization to excite the plasmon. A lock-in with a lightchopper clean up the signal of any undesired noise. Theprism and the sensor were mounted in a rotary stage tovary the light incidence angle. The experimental set-upis shown in figure 9.

ResultsThe results obtained for a 10 nm gold filmalong with the simulation (supposing a 800 nm thick airlayer) are shown in figure 10. It may seems that thesimulation and the experimental measurement coincidesbut this is not a complete proof of the experimentalexcitation of the plasmon.

For a 100 nm film a detailed measurement were notperformed, but the angle of total internal reflection werefound and a interesting behavior appears around the totalinternal reflection angle (figure 11). Results are shownalong the simulation supposing again a 800 nm thick airlayer.

DiscussionAlthough non-trivial effects due to thepresence of the gold film are evident it is not yet stablishedif they really correspond to surface plasmon resonance.One of the main problems of our experiment is the distancebetween the prism and the gold film (the thickness of theair layer), since we were not able to control it we arenot sure if the measure was perform within the regime ofexistence of plasmon resonances (see fig. 8).

PerspectivesIf the experimental technique is improvedthen a characterization device can be built so, using

18 Surface plasmon resonance

Glass prismFree electron metal filmAdsorbate layerAmbient

00123

Figure 2.4: Four layer model of an SPR biosensor. In the most simple case an SPR

sensor can be modeled as a four layer stratified medium where the constituents are a glass

prism, coated with a thin metal film, to which the organic sensing layer is adsorbed. The

semi-infinite ambient layer is typically aqueous buer.

sion coecients, Eqn. 2.41 are, however, dierent for dierent states of polarization.

rkij =

nj cosinicosjnj cosi+nicosj

kij =

2ni cosinj cosi+nicosj

r?ij =ni cosinjcosjni cosi+njcosj

?ij =2ni cosi

ni cosi+njcosj

(2.41)

The intensity reflectance and transmittances of the layer stack are found as the

square modulus of the coecients in Eqn. 2.33.

Implementation

As the number of layers increase, the matrix algebra required to find the reflection

coecient of the stack quickly becomes tedious. However, the stratified medium

matrix model as described v.s. can easily be implemented in a computer program,

for instance MATLAB. A special case which is somewhat useful in SPR biosensing

concerns a stack consisting of two thin film layers on top of a glass prism and

in an aqueous ambient (Figure 2.4). Some algebra will give an expression for the

intensity reflection from such a layer stack:

R =

r01(1 + r12r23e2i'2) + e2i'1(r12 + r23e2i'2)1 + r01e2i'1(r12 + r23e2i'2) + r12r23e2i'22 (2.42)

In the above expression, which is valid for nj = nj + ikj , the response upon intro-

duction of an analyte can be modeled as a thickness increase or as an increase in

refractive index of the adsorbate layer. The former is most suitable for immobi-

lization of biomacromolecules to a two-dimensional organic linker layer, whereas

the latter can be employed when a sensing matrix (for instance a hydrogel) is used.

By eliminating all terms with r23 in Eqn. 2.42, an expression for the more simple

Prism

Air

Gold

Glass

nprism = 1.52

nair = 1ngold = 0.467 + 2.415i

nglass = 1.5

dair = 300 nm

= 633 nm

FIG. 6.

Critical Angle

Total internal reflection

dgold = 100 nm

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 aHradL0.00.2

0.4

0.6

0.8

1.0RR0

Plasmon Resonance

dgold = 10nm

FIG. 7.

optimization programs and along with simulations, thedielectric constant of materials and layer thickness canbe measured.

AcknowledgementsI specially acknowledge JuanSerna from Grupo de Optica y Fotonica for advisingthe present project, for his patience and interesting dis-cussion. I also thanks Grupo de Optica y Fotonica forallowing me to use their working space and equipment.

[1] H. J. Simon, D. E. Mitchell, and J. G. Watson, AmericanJournal of Physics 43 (1975).

[2] C. Rhodes, S. Franzen, J.-P. Maria, M. Losego, D. N.Leonard, B. Laughlin, G. Duscher, and S. Weibel, Journalof Applied Physics 100, 054905 (2006).

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.6

0.7

0.7

0.7

0.7

0.7

0.7

0.8

0.80.8

0.8

0.9

0.0 0.5 1.0 1.50

200

400

600

800

1000

a

d air

(rad)

(nm)

FIG. 8.

ResaltadoWhat does it mean "correct" in this case?

ResaltadoExcelente anlisis!

5Prism and rotary stage

Film

Sensor and rotary stage

PolarizerLight chopper

Filters

Pupil

Aligning mirrors

FIG. 9.

2"

2.05"

2.1"

2.15"

2.2"

2.25"

2.3"

2.35"

2.4"

2.45"

2.5"

320" 322" 324" 326" 328" 330" 332" 334" 336" 338" 340"

Serie1" Serie2" Serie3" Serie4"Data 1 Data 2 Data 3 Average

Degrees

Intensity(a.u.)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 aHradL0.00.2

0.4

0.6

0.8

1.0RR0

d = 800 nm

10 nm film

FIG. 10.

[3] S. A. Maier, Plasmonics: Fundamentals and Applications(SpringerScience+BusinessMedia, New York, 2007).

[4] A. Otto, Zeitschrift fur Physik 216, 398 (1968).

0"

0.2"

0.4"

0.6"

0.8"

1"

1.2"

1.4"

1.6"

45" 47" 49" 51" 53" 55" 57" 59" 61" 63" 65"

Serie1" Serie2" Serie3"

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 aHradL0.00.2

0.4

0.6

0.8

1.0RR0

100 nm film

Data 1 Data 2 Average

Intensity(a.u.)

Degrees

FIG. 11.

Nota adhesivaAbsorcin de reflexin total interna------------------------------------------Grfca terica en radianes (Absorcin en 37 grados)y grfica experimental en grados (absorcin en 360-323=37grados).

Nota adhesivaLos picos de absorcin no coinciden. ??La curva experimental empieza a caer en 58 grados.La curva terica en 0.85rds=48.7grados

Experimental Surface Plasmon Resonance in a Thin Gold Film AbstractIntroductionTheoryExperimental Surface Plasmon ResonanceReferences