3 Distinction Between Localized Surface Plasmon Resonance (LSPR) and Surface Plasmon Resonance (SPR)
Tải bản đầy đủ - 0trang
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
O. Pluchery
this creates a dynamic polarization P(r, ω). This quantity expresses how far
the electric field succeeds in displacing the electrons relative to the core
atoms. The polarized atoms are the sources of a depolarizing field and all
these effects are combined into the electric displacement D(r, ω) linked
to the excitation electric field by the relationshipii : (see ref. 1 for a good
introduction to the optical properties of condensed matter).
D r, ω = ε0 ε (ω) E r, ω
(1)
ε(ω) is the dielectric function of the metal and it captures the entire
response of a metal to the electromagnetic excitation wave for the whole
frequency spectrum, starting from radio frequencies up to X-ray including
of course optical frequencies. If the medium is vacuum, ε(ω) = 1 and the
displacement D is simply proportional to E, but ε accounts for any kind
of materials (metals, insulators, transparent or opaque media) and in the
general case, it is a complex function expressed as:
ε(ω) = ε1 (ω) + iε2 (ω)
(2)
For metals, ε is dominated by its real part which is a negative function.
ε is linked to the optical complex index n˜ by
ε = n˜ 2 = [n + ik]2 = n2 − k 2 + 2nki
(3)
ε, n˜ are functions of the angular frequency ω (linked to the wavelength
λ) and this dependence will not be expressly indicated in the followings to
simplify notations, but it should be kept in mind since the plasmon resonance
is directly linked to it.
The oscillator model is the simplest analytical way to describe the
polarization and gives rise to the Drude dielectric function1−3 expressed
as follows:
εDrude = 1 −
ωp is the plasma frequency and
ωp2
ω2 + i ω
≈1−
ωp2
ω2
the damping constant.
ii The SI system of units is used for all the equations of this chapter.
46
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
Optical Properties of Gold Nanoparticles
3.3.2 The dielectric function of gold
As explained in the previous section, the Drude model is a good start for
describing the dielectric functions of metals. The formula given above is
slightly modified by replacing the factor 1 by a constant ε∞ that accounts for
transitions that do not need to be explicitly expressed in the visible range.
εDrude = ε∞ −
ωp2
(4)
ω2 + i ω
In the case of gold, the plasma frequency and the damping constant are
given by4
−1
16
h¯ ωp = 8.95 eV i.e. ωp = 1.36 × 10 rad.s
(5)
h¯ = 65.8 meV i.e. = 1.0 × 1014 rad.s−1
ε∞ = 9.5
These values are obtained by fitting experimental values with equation (4)
in an energy range where the free electrons are the major contributions
to the dielectric function5,6 and these values fluctuate from one author to
another.
However when a precise model is needed, the Drude model is far too
simplified because it only takes into account the free electrons (intraband
transitions) and completely dismisses the contribution from the bound electrons (interband transitions). The latter plays an important role for gold. This
is illustrated in Fig. 3.1 where the complex dielectric function is plotted.
The accurate measurement of this function is crucial for reasonably modelling the optical response of AuNP. One of the preferred measurements
was performed by Johnson and Christy in 19725 and an approximated analytical model has been published by Etchegoin.7 The measures provided
by the Handbook of Optical Constants have some problems near the interband threshold for gold and must be used with great care.8 In the present
chapter we are using values measured by H. Arwin from the University of
Linköping (Sweden)9,10 because they cover a wide range of wavelength.
3.3.3 Plasmon resonance at surfaces, SPR
An electromagnetic wave in the visible range is rapidly screened by a metal.
It does not penetrate into metal much farther than the so-called skin depth
47
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
O. Pluchery
Fig. 3.1. The complex dielectric function of gold ε = ε1 + iε2 obtained from an ellipsometric
measurement9,10 (thin line) and from an analytical model7 (large dots) based on Johnson and Christy’s
data.5 The dotted line shows the plot of the Drude model discussed in the text. This model accounts
well for the free electron contribution but not for the bound electrons (intraband transitions). This
discrepancy is clearly visible in the range of 300 nm to 530 nm.
√
given by δ = 2ρ/µ0 µr ω.1,3 In the case of light beam impinging on gold
with a wavelength of 500 nm the skin depth is δ = 20 nm and strongly
depends on the wavelength. Therefore, the conduction electrons are excited
by the electromagnetic field within an ultrathin sheet of metal close to the
surface. This occurs in two cases: in the case of a flat and infinite interface
and in the case of structures whose size is of the order of magnitude of
the skin depth. We will quickly consider the first case and then focus our
attention on the second one.
In the case of a flat interface between a metal and an insulating medium
whose dielectric functions are respectively ε and εdiel it is possible to launch
a surface wave that stays confined very close to the interface. This wave is
a charge density wave with longitudinal structure (unlike light waves that
propagate in vacuum with a transverse structure). The charge wave is called
a polariton wave and is coupled to an electromagnetic wave, which is the
surface plasmon wave. This is the reason why this SPR is sometimes called
surface plasmon polariton (SPP).4 This propagation is characterized by the
following dispersion relation:
kx =
ω
·
c
ε(ω) · εdiel (ω)
ε(ω) + εdiel (ω)
48
(6)
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
Optical Properties of Gold Nanoparticles
The dispersion relation plays a key role in electromagnetism because it
controls the propagation of the wave. It is an expression of the link between
fundamental quantities in physics: energy (hω)
¯ and momentum (h¯ k). For a
monochromatic light beam of photon energy ω, only waves whose wavevector has a component parallel to the interface kx given by relation (6) can be
excited. In this relationship the number in the square root is greater than
one for metals, meaning that the wave vector of the plasmon wave is greater
than that of any wave travelling in free space. As a consequence, surface
plasmons can only be launched with special setups. The most usual way is
to excite the surface plasmon with an evanescent wave resulting from a total
internal reflection from a prism in the so-called Kretschmann configuration
depicted in Fig. 3.2.11
When the geometrical conditions are fulfilled to excite the surface plasmon, this is the surface plasmon resonance (SPR). It shows up as a dip in
the reflection spectrum when the incident beam impinges on the gold film
with a given angle (see Fig. 3.2-b).
Since SPR only occurs on flat surfaces approximated to planes of infinite extension, we will not discuss this kind of plasmon resonance further
and restrict ourselves to charge oscillations within particles of nanometer
dimensions in three-dimensional space.
Fig. 3.2. Surface Plasmon Resonance: (a) sketch of the prism for coupling an excitation wave of a
laser to the surface plasmon (SP) wave. The coupling is controlled by the incidence angle. (b) when the
coupling is achieved the reflected beam undergoes a strong intensity drop measured at θint = 41.8◦ in
the present case.
49
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
O. Pluchery
3.3.4 Localized surface plasmon resonance
in nanoparticles, LSPR
In nanoparticles, the charge oscillation is different from the SPR described
above. Qualitatively, if particles have sizes much smaller than the wavelength of light and smaller than the penetration depth of the field (i.e. particle size around 20 nm), the electron cloud of the particle is entirely probed
by the electric field. The whole assembly of electrons is polarized, and
this creates surface charges that accumulate alternately on opposite ends of
the particle. This oscillating polarization of the particles creates an electric
field opposed to the excitation field and results in a restoring force. This
oscillation is partially damped. The damping occurs through two channels:
creation of heat and light scattering. All this can be described as a dipolar oscillator characterized by a resonance frequency ωplasmon that will be
discussed and used throughout this book.
In the following chapter, this resonance will be called Localized Surface Plasmon Resonance (LSPR). This denomination is generally accepted
although this localized plasmon oscillation is not primarily a surface effect,
but a bulk effect taking place in the very small and confined volume of metallic nano-objects. Thus, some authors prefer using other denominations such
as nanoparticle plasmon (NPP12 ), surface plasmon on metal nanoparticles
(SPN13 ).
The main properties of LSPR can be understood within the dipolar model
and can be summarized as follows:
• Spectrally, the plasmon resonance appears in the visible or near-infrared
range for gold or silver nanoparticles. A light beam going through an
assembly of homogeneous nanoparticles is partially absorbed at the plasmon resonance frequency so that the emerging beam displays a spectrum
with a sharp absorption at ωplasmon . At the same time the nanoparticles
exhibit light scattering with a cross section much larger than conventional
dye.14,15
• The LSPR strongly depends on the environment close to the particle surface. For example, if the layer of molecules adsorbed on the NP changes,
the SPR shifts. The AuNP typical shift for a protein interaction is of the
order of magnitude of 10 nm (see Chapter 11). This effect is the basis of
plasmonic biosensing.
50
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
Optical Properties of Gold Nanoparticles
• When excited at the resonance, the dipole radiates a near-field electromagnetic wave, whose amplitude can be enhanced by a factor up to 10.
This plasmon amplification is widely used for enhancing the sensitivity
of biosensors (Chapter 10).
3.4 Theoretical Description of the Localized
Plasmon Resonance
3.4.1 About Mie theory
An exact theory in case of spherical particles of any size is provided by Mie
theory16 (Gustav Mie, 1908). It gives the exact solution of a plane wave interacting with a metallic sphere. The electromagnetic fields are expanded in
multipole contributions and the expansion coefficients are found by applying
the correct boundary conditions for electromagnetic fields at the interface
between the metallic nanoparticle and its surroundings. For small particles
(<60 nm), it is sufficient to restrain the multipole expansion to its first term,
which is dipolar. This approximation is the dipolar approximation, also
called the quasistatic or Rayleigh limit. We will develop this approximation
which is sufficient to grasp the working principles of plasmonics; readers
interested in the full Mie theory should refer to textbooks such as Born and
Wolf 3 or Bohren and Huffmann.17
3.4.2 The quasistatic approximation for describing
the localized plasmon resonance
Let us consider a metallic spherical particle of radius R which is submitted
to an excitation electric field aligned along the x-axis: E = E(r, t)ex (see
Fig. 3.3). In the quasistatic limit, the electronic polarization is exactly in
phase with the excitation field (no retardation effect) and the electrons are
displaced as a whole. Therefore the charge distribution in the particle can
be treated as if it were a static distribution.4,17 The electric field should obey
the Laplace equation:
V =0
(7)
Where V is the electric potential linked to the electric field by E = −∇V .
51
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
O. Pluchery
Fig. 3.3. (a) Sketch of the metallic sphere and the coordinates used in the electrostatic model to
calculate the external electric field. (b) Under the influence of an excitation wave with a wavelength
greater than the dimension of the sphere, the electrons oscillate as a whole and the system can be
handled as an oscillating dipole.
Due to the symmetry of the problem, spherical coordinates are used.
Moreover the x-axis is an axis of symmetry so that the third coordinate usually required for spherical coordinates becomes useless. Therefore equation (7) written in spherical coordinates becomes:
1 ∂
1
∂
∂V
(rV ) + 2
sin θ
2
r ∂r
r sin θ ∂θ
∂θ
=0
(8)
The solutions of this differential equation are the spherical harmonics with
the following form:
∞
An r n +
V (r, θ) =
n=0
Bn
Pn (cos θ)
r n+1
(9)
where An and Bn are coefficients to be determined. Pn are the Legendre
polynomials which often appear in physics problems expressed in spherical
coordinates.iii
This electric potential has two different forms Vint (r, θ) and Vext (r, θ)
inside and outside the metallic particle. Moreover it should obey the
iii The Legendre Polynomials are obtained as solutions of the Legendre equations or the nth element
of the recurrence relation of Bonnet. The first four terms are: P0 (X) = 1; P1 (X) = X; P2 (X) =
3/2X 2 − 1/2; P3 (X) = 5/2X 3 − 3/2X.
52
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
Optical Properties of Gold Nanoparticles
following boundary conditions:
• The electric field should be properly defined at r = 0: ∂V∂rint r=0 exists.
• The electric field far away from the particle should match the excitation field, which is constant in the electrostatic approximation, so that
lim Vext = 0.
r→∞
• At the interface, the electric fields obey the continuity equation at the
particle surface: ε(ω)Eint (r = R) = εdiel (ω)Eext (r = R).
• Finally, the potential should be continuous at the particle surface:
Vint (r = R) = Vext (r = R).
Applying these conditions allows us to determine the coefficients for the
electric potentials inside and outside the particle. For example, for the external potential one determines that all the An coefficients are null except one:
A1 = −E0 and similarly for the Bn coefficients, the only non-zero coefficient
is given by:
ε − εdiel
B1 = E0 4πε0 R3
(10)
ε + 2εdiel
Once the electric potential is obtained, the electric field is deduced by using
the gradient operator: E = −∇V .
After some calculations one obtains the following expression for the
electric field outside of the nanoparticle, which is a sum of the incident field
and a second field produced by the particle:
sin θ
cos θ
Eext = E0 − αE0 −2 3 ur − 3 uθ
(11)
r
r
Where α is the sphere polarizability given by
ε − εdiel
(12)
α = 4πε0 R3
ε + 2εdiel
The electromagnetic response of the particle is captured in the polarizability.
It is clear that the external field will go through a maximum when the
polarizability is maximized. The parameter that depends on frequency is
ε = ε(ω). Therefore |α| is maximized when the following relationship is
fulfilled:
|ε + 2εdiel | is a minimum
(13)
As a good approximation the dielectric permittivity of the surrounding
medium, εdiel is a constant and real parameter. Therefore equation (13)
53
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
O. Pluchery
leads to the following condition applied to the real part of ε: ε1 + 2εdiel = 0.
In case of a nanoparticle in water, εdiel = 1.77, and the condition becomes
ε1 = −2 × 1.77 = −3.54. From the plot of the dielectric function of gold
in Fig. 3.1, it is easy to check that this condition leads to a plasmon resonance at 520 nm. If the particle was in air, the plasmon resonance would be at
504 nm. These calculated values for the plasmon resonance in water or in air
correspond closely to experimental values of spherical gold nanoparticles.
3.4.3 Extinction and scattering cross sections
However, in order to proceed one step further and calculate the intensity
of light being scattered or absorbed, the electrostatic model has to be completed to take into account that light is a wave and not a static electric field.
Upon interaction with the particle, light is absorbed (absorption cross section σabs ) and scattered (scattering cross section σscatt ). As a result, the beam
going through a particle undergoes an extinction characterized by the extinction cross section (σext ). The three cross sections are linked by the simple
relation
σabs = σext − σscatt
(14)
These phenomena are strongly frequency dependent and, therefore, time
dependent. The model of the radiating dipole allows for calculating the
extinction and scattering cross sections and links them to the polarizability
found in equation (12):
2π √
σext = 3
εdiel Im(α)
λ
(15-a & 15-b)
2π 3/2
Im(ε)
= 9 εdiel V
λ
|ε + 2εdiel |2
σscatt = 3
(2π)3 2
ε |α|2
λ4 diel
(2π)3
ε − εdiel
= 3 4 ε2diel V 2
λ
ε + 2εdiel
2
(16-a & 16-b)
This description is acceptable for homogeneous, metallic spherical
nanoparticles whose diameter is roughly between 10 and 60 nm (see the
following section).
54
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
Optical Properties of Gold Nanoparticles
Fig. 3.4. Comparison of the extinction (solid line), absorption (dashed line) and scattering (black
dots) efficiencies as a function of particle diameter for gold nanoparticles. In the case of particles of
20 nm diameter (graph a), absorption is largely dominating as well as for 40 nm (graph b). For 80 nm
NP, scattering and absorption are of the same order of magnitude (graph c). Calculations are made with
the Mie theory. Reprinted with permission from Ref. 15. Copyright 2006, American Chemical Society.
For small particles, only absorption is in play with negligible scattering. A proper calculation within the Discrete Dipole Approximation (DDA)
formalism has been conducted by El-Sayed and coworkers15 and shows
that for 20 nm AuNP, no scattering occurs and the particles only absorb
radiation. For diameters of 80 nm, both cross sections are equivalent, and
for larger particles scattering dominates (see Fig. 3.4). In other words, if
one wants to have brilliant particles, they must choose particles with a large
diameter, for example larger than 40 nm. For biomedical applications where
AuNPs can be used as markers, this will be important. AuNP nanospheres
with a diameter of 40 nm have a calculated absorption cross-section of
2.93 × 10−15 m2 (thus corresponding to a molar absorption coefficient ε
of 7.66 × 109 M−1 .cm−1 ) at a plasmon resonance wavelength maximum
λmax of 528 nm. This value is five orders larger than the molar extinction
coefficient for indocyanine green ε = 1.08 × 104 M−1 .cm−1 at 778 nm),
a NIR dye commonly used in laser photothermal tumor therapy (see
Chapter 10).
3.4.4 Experimental illustrations
The electrostatic approximation and the analytical model provided by equation (15) accurately describe simple experimental situations such as spherical nanoparticles in suspensions with a concentration such that the particles
stay far from each other (a few particle diameters). In this case one usually
55
b1370-ch03
November 15, 2012
11:28
9in x 6in
Gold Nanoparticles for Physics, Biology and Chemistry
O. Pluchery
Fig. 3.5. Comparison of measured and calculated extinction spectra of 14.2 nm gold nanoparticles
prepared by the Turkevich method. The experimental spectrum is taken in a b = 1 mm thick cuvette
and the density of AuNP is n = 1.76 × 1018 m−3 . A beginning of particle aggregation is visible at
700 nm on the experimental spectrum and indicated on the graph.
measures the absorbance of a solution placed in the cuvette of spectrophotometer. The absorbance is linked to the extinction cross section by:
A=−
1
nbσext
ln 10
(17)
where n is the number of nanoparticles per unit volume, b the length probed
by the optical beam (thickness of the cuvette) and σext is the cross section
given by relation (15). As an example, Fig. 3.5 shows a comparison of the
measured absorbance of 14.2 ± 1.3 nm gold nanoparticles in suspension in
water with the calculated absorbance.
Systematic checks have been conducted and show that the plasmon resonance is measured at 519 nm in aqueous solutions and does not shift for
diameters from 4 to 35 nm. The extinction coefficients are precisely measured and can be used in a Beer-Lambert law.18 According to such a rule,
the measured absorbance writes:
A = εbC
56
(18)
b1370-ch03