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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.
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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.
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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!
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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