Chapter 2 Study and application of metallic nano-particles system
2.4 Future Development of LSPR
The development of innovative concepts, unprecedented applications, and fabrication of sensors with previously unimagined selectivity and sensitivity are seeing daylight at an exponentially increasing rate. The above reviews of fundamental property and development give me a strike on a large number creation of new idea generated from researcher in this field. Every idea provides high potential for the realization of all kinds of nanoscale optical devices and sensors.
Nevertheless, there is much more to be done, much more to be hoped for. Full realization of the potential of surface plasmons requires a much better understanding off the underlying fundamentals and a greater cooperation between physicists, chemists, and material scientists.
References
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2. Rongchao Jin,1 YunWei Cao,1 Chad A. Mirkin,1 K. L. Kelly,George C.
Schatz,1 J. G. Zheng, “Photoinduced Conversion of Silver Nanospheres to Nanoprisms”, SCIENCE 294, 1901, (2001)
3. J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanopaticles”, Journal of Chemical Physics 116,6755 (2002)
4. Sylvia Underwood and Paul Mulvaney, Langmuir, 10, 3427 (1994)
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8. Stephan Link and Mostafa A. El-Sayed, J. Phys. Chem. B 103, 8410 (1999) 9. W. L. Barnes, T. W. Preist, S. J. Kitson, J. R. Sambles, Phys. Rev. B 54, 6227
(1996)
10. V. Vlasko-Vlasov, A. Rydh, J. Pearson and U. Welp, “Spectroscopy of surface plasmons in metal films with nanostructures”, Appl. Phys. Lett., 88, 173112 (2006)
11. William L. Barnes1, Alain Dereux and Thomas W. Ebbesen, “Surface plasmon subwavelength optics”, Nature 424, 826 (2003)
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“Single-target molecule detection with nonbleaching multicolor optical immunolabels”, PNAS 97, 996 (2000)
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15. Robert M. Dickson and L. Andrew Lyon, “Unidirectional Plasmon Propagation in Metallic Nanowires,” J. Phys, Chem. B 104, 6095-6098 (2000)
16. L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles”, Physical Review B 71, 235408 (2005)
17. Stefan A. Maier, Mark L. Brongersma, Pieter G. Kik, and Harry A. Atwater,
“Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy” PHYSICAL REVIEW B 65, 193408 (2002)
18. Wataru Nomura, Motoichi Ohtsu, and Takashi Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion”, Appl.
Phys. Lett 86, 181108(2005)
19. H. Ditlbacher, J. R. Krenn, G. Schider, A. Leitner, and F. R. Aussenegg,
“Two-dimensional optics with surface plasmon polaritons”, Appl. Phys. Lett. 81, 1762 (2002)
20. Ignacy Gryczynski, Joanna Malicka, Wen Jiang, Hans Fischer, Warren C. W.
Chan, Zygmunt Gryczynski, Wojciech Grudzinski, and Joseph R. Lakowicz,
“Surface-Plasmon-Coupled Emission of Quantum Dots”, J. Phys. Chem. B 109, 1088 (2005)
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Chapter 3
Theoretical background
The excitation of plasmon polaritons in small metal particles gives rise to complex optical extinction spectra consisting both of true absorption and scattering. In the present contribution we show the experimental separation of such scattering and absorption losses. The optical extinction of spherical particals was calculated already at the beginning of this century by Mie and Debye within the frame of Maxwell’s theory [1,2], and a few extensions due to longitudinal excitation modes and due to the size dependent dielectric response of the particle material have been added since then.
Based on the experimental observation, it is clear that the dielectric constant of nano metal particles can be quite different from the dielectric constant of bulk metals.
However, the dielectric property of metal material is determined by the collective oscillation of electron. Figure 3.1 (a) shows a collective electron oscillation is induced in the metal particle due to the probe light. In Figure 3.2, various scattering light band are observed due to the Ag nanoparticles in various size and shape. The phenomenon of wavelength dependent bright colors attracts so much interest.
In this chapter, the physical phase damping of electron in metal nanoparticle will be discussed. And then I will introduce a geometry factor into the analysis of dielectric property of metal material. The follow is Mie theoty, the theoretical calculation of extinction cross section will be shown also. The separation of plasmon-polariton modes of small metal particles will be reviewed in the final section.
Figure 3.1: (a) Metal nanoparticle is probed by a incident light and then irradiates scattering light. (b) the scattering light from Ag nanopartilces in dark-field microscopy.
3.1 Plasmon damping in metal [3, 4]
The damping of a surface plasmon resonance of a metal nanoparticle is due to the dephasing of the coherent conduction electron motion with time. Figure 3.2 shows a schematic of the different processes involved in the damping of a surface plasmon excitation excited by a photon of energy hv.
Figure 3.2: Energy relaxation of a surface plasmon. The relaxation takes place either via radiative or non-radiative processes. The nonradiative relaxation channel leads to the creation of electron-hole pairs and the subsequent relaxation into hot electrons and phonons [3].
The damping of the plasmon resonance is in general described via a total dephasing time T2 and an energy relaxation time T1. Both are related to the homogeneous linewidth Γ of the surface plasmon resonance via
(1)
T2* is called the “pure dephasing time” which describes quasi-elastic electron scattering events that change the electron wavevector but not its energy. Its contribution to the total dephasing time T2 is often put into the energy relaxation time T1, yielding T2 = 2T1. The damping of the plasmon resonance is thus determined by the energy relaxation time T1, defined as
(2)
The energy relaxation of a plasmon oscillation is composed of a non-radiative decay channel with time constant τnr and a radiative decay channel with time constant τr.
Figure 3.3: (a) A damped oscillation in time corresponds via the Fourier transformation to (b) a broad peak in the frequency spectrum.
The damping oscillation of electron leads to broad Fourier spectrum. Figure 3.3, a damped oscillation in time (a) corresponds via the Fourier transformation to a broad peak in the frequency spectrum (b). Here, an example for an oscillation with frequency f = 1015 and damping time T=10-13s is shown. The broad of spectrum occurred due to the damping time of electron in the metal.
The nonradiative damping is thus due to a dephasing of the oscillation of individual electrons. In terms of the Drude-Sommerfeld model this is described by
scattering events with phonons, lattice ions, other conduction or core electrons, the metal surface, impurities, etc.. Because of the Pauli-exclusion principle, the electrons can only be excited into empty states in the conduction band. These excitations can be divided into inter- and intraband excitations by the origin of the electron either in the d-band or the conduction band.
3.2 Optical Dielectric Property of metal material
In what follows, we investigate the dielectric response of a metallic nano-particle based on the classical motion of the electrons. We assume that the sizes of the nano-particles are much smaller than the wavelength of light (a <<λ), and much larger than the De Broglie wavelength of the electrons so that quantum-well effect can be ignored.
3.2.1 Dielectric response of metal material in quasi electric field
Consider a free electron gas in a rectangular box (e.g., a solid cube of silver, gold or copper). The solid is electrically neutral. The negative charges of the free electrons (in the conduction band) are completely neutralized by the positive ionic lattice. The electrons are free to move inside the box while maintaining the charge neutrality.
Now, to consider the displacement of the electron gas by a small distance Δx along the x-axis, as shown in Figure 3.4, the displacement creates a space charge separation and a polarization P along the x-direction given by
x e N
P= (− )Δ (3)
where N is the number of free electrons per unit volume, (-e) is the electron charge.
+
Figure 3.4: Space charge separation in a slab due to the displacement of the electron gas relative to the ionic crystal lattice.
In addition, an electric field inside the box is created by the space charge separation. For a parallel slab (ignoring the fringing field), the electric field in the box is given by where ε0 is the dielectric constant of vacuum. The electric field acting on the electron gas leads to a restoring force that tends to pull the free electrons back to their original position. This force is given by
Ne x
where V is the volume of the box. Following the physical motion of the electrons induced by electromagnetic field, the equation of motion of the free electron gas can be written known as the plasma frequency,
0
Eq. (5) is an equation of motion of a simple harmonic oscillator with a natural oscillation frequency of ωp. So, if the free electrons are displaced from their equilibrium position and then released, the free electrons will undergo a simple harmonic oscillation with a frequency of ωp. If collisions occur, the equation of motion can be written
x
where τ is the mean collision time. The oscillation will decay exponentially with a relaxation time of τ.
Figure 3.5: Space charge separation in an ellipsoid of revolution.
For a sphere or an ellipsoid (shown in Figure3.5), the electric field inside the ellipsoid due to the space charge separation is given by
Ne x
where Q is a depolarization factor which depends on the shape of the ellipsoid. For a sphere, Q is equal to 1/3. For an ellipsoid of revolution, this factor varies between 0 and 1. For a nano-slab, Q is equal to1 (with E-field perpendicular to the slab). For an ellipsoid of revolution with semi-axes (a, a, c), the depolarization factor is Qc for electric field parallel to the axis of revolution, Qa for electric field perpendicular to the
axis of revolution. The depolarization factors Qa and Qc of an ellipsoid of revolution are plotted in Figure 3.6.
As a result of the shape of the ellipsoid, the restoring force is decreased by a factor of Q. The smaller restoring force leads to a smaller natural oscillation frequency
) 1 0
( ellipsoid
2 0
2 2
0 = ω ≤ ≤
= ε
ω Q Q
m
Q Ne p (11)
Figure 3.6: Depolarization factor Qa and Qc vs c/a, where c and a are semi-axes of the ellipsoid of revolution. c is the semi-axis of the axis of revolution. Qc is the depolarization factor for electric field in the direction of the axis of revolution.
Now, the shape factor (shown in Table 3.1) is considered into the dielectric function. It is still necessary to define a complex factor for particle with un-regular shape.
Table 3.1: Depolarization factor due to various shape of particle.
a c
Shape thin slab pancake sphere cigar
c/a 0 1/3 1 3
Qa= Qb 0 0.181 0.33333 0.4456
Qc 1 0.638 0.33333 0.1087
3.2.2 Forced oscillation via an incident electromagnetic radiation
At present, we consider the displacement of the electron gas via the incidence of an electromagnetic radiation. If the dimension of the box is much smaller than the wavelength of the radiation, then we can use the dipole approximation by assuming a time-harmonic electric field. The forced oscillation of the whole free electron gas in a
"nano-box" can thus be written
t
The steady state solution is
τ
The electric field induced polarization is thus given by
t
The dielectric constant of the "nano-box" can thus be written
⎟⎟ half-maximum of the resonance is approximately
τ
= ω
Δ 1/ (16)
For a nano-ellipsoid, the dielectric constant can be written
⎟⎟
where Q is the dimensionless depolarization factor. We note that the resonance frequency is ω0 = Qωp for the case of a nano-ellipsoid.
The width of the resonance is also approximately Δω=1/τ for ellipsoids with a well-defined shape and orientation. If ellipsoids consist of different shapes and orientations, then a broadening of the resonance occurs. This broadening can be much bigger than the intrinsic width of Δω=1/τ.
This dielectric constant is quite different from that of a bulk free electron gas (Q=0),
which exhibits no sharp resonance. The lack of resonance in bulk free electron gas is a result of the periodic variation of the space charge separation, as shown in Figure 3.7, leading to a near zero restoring force for the whole electron gas.
+
Figure 3.7: When the box is larger than the wavelength, the periodic variation of the space charge separation leads to no net restoring force for the whole electron gas.
In conclusion, resonant oscillation of the electron gas occurs in nano-particles when the particle dimension is much smaller than the wavelength of light. The resonant oscillation of the electrons inside a nano-particle leads to a dielectric constant which can be quite different from that of the bulk metals.
3.2.3 Theoretical and Experimental optical constant [5-9]
The experimental optical constant of metals, such as silver and gold, adapted from Johnson’s measurement is compared with the theoretical prediction. Based on a precious optical constant of metal material, the extinction and scattering cross sections of metal nano-sphere can be exactly calculated out via Mie theory.
The dielectric function of metals is contributed not only from the free electron in conduction band but also from the bound electron in the deeper band. Considering the contribution both free- and bound- electron in metals, the dielectric function can be wrote as follows, distribution function of conduction electron of energy hx at the temperature T with Fermi energy EF; γb represents the damping constant in the band to band transition and Qbulk is a proportionally factor.
At first, we focus on the contribution of free electron in the metal material.
Figure 3.8 shows the real part and imaginary part of the dielectric constant from Eq.
19. They are the function of wavelength of incident light. The real part of dielectric decreases with the increase of wavelength. But the imaginary part increases when increasing the wavelength.
)
Figure 3.8: (a) Real and (b) imaginary part of the dielectric function of silver
In Figure 3.9, we can find that not only Real part but also imaginary part of the theoretical dielectric function are consistent with the Johnson’s measurement. The consideration of contribution from bound electron in material is necessary for the theoretical analysis.
Figure 3.9: (a) Real and (b) imaginary part of the theoretical dielectric function and Johnson’s measurement.
3.3 Mie theory [10-12]
Since the excitation of localized surface plasmons on metal nanoparticle leads to a strong light scattering and absorption at the resonance wavelength, the total extinction cross section is defined as Cext =Cabs +Csca, where Cabs and Csca are absorption and scattering cross sections, respectively. The scattering light can be given by
Io is the intensity of the incident light, Isca is the intensity of the scattered light in a point at a large distance r from the particle. And k is the wave number defined by
λ π
=2
k , where λ is the wavelength in the surrounding medium. Since Isca must be proportional to I0 and r-2, we may write I ( ) F(2 ,2 )
function of the direction. It also depends on the orientation of the particle with respect to the incident wave and on the state of polarization of the incident wave.
Let the total energy scattered in all directions be equal to the energy of the incident wave falling on the area Csca. By this definition and by the preceding equation we have Csca = k12
∫
F(θ,ϕ)dΩ, where dΩ=sinθdθdϕ is the element of solid angle and the integral is taken over all directions. The energy absorbed inside the particle may be definition be put equal the energy incident on the areaCabs, and the energy removed from the original beam may be definition be put equal to the energy incident on the area Cext. The conservation of energy requires that Cext =Cabs +Csca. The efficiencies Qsca, Qabs, and Qext can be obtained by normalizing the cross section to the two times particle’s geometrical cross-section 2πr2,The response of a metal sphere to a probe electromagnetic field can be estimated by solving Maxwell’s equations. The exact analytical electrodynamical treatment is attributed to Mie (1908). The spherical symmetry suggests the use of a multipole
extension of the fields, here numbered by n. The Rayleigh-type plasmon resonance, discussed in the previous sections, corresponds to the dipole mode n=1. In Mie’s theory, Maxwell’s equation is solved in a sphere boundary condition and the scattering and extinction efficiencies are calculated by:
∑
∞We use the following parameter into the calculation of cross section
τ
p : Dielectric constant of gold nm
c : Plasmon resonant wavelength
sec
a=50 nm: sphere radius
We normalized the cross section by 2 2 aπ
≡ σ
Σ (dimensionless). Figure 3.10 shows the extinction cross section of Au nano-particles with R=50 nm without high order mode (n=1). The optical constants used here are only contributed from the free electron in metal (the same in Fig. 3.11). It is noted that the cross section is a function of wavelength and the maximum cross section of spectrum means the occurrence of strong excitation of surface plasmons. The spectra range from 400 to 800 nm.
Figure 3.10: Extinction cross section of Au nanoparticle with a radius of 50 nm, three high orders plasmon modes can be found in the spectra (n=1).
Figure 3.11 shows the extinction cross section of Au nano-particles with R=50 nm without high order mode (n>1). In the previous works, the high order plasmon mode is neglected due to the small particle size. Only dipole is induced by the electromagnetic field. High order modes are needed to be considered when the particle size increased.
Figure 3.11: Extinction cross section of Au nanoparticle with a radius of 50 nm, three high orders plasmon modes can be found in the spectra (n>1).
In Figure 3.12, the experimental absorbance is compared with the theoretical extinction cross section. The theoretical optical constants used here are from the contribution of free electrons and bound electrons. We find that the theoretical prediction is in a good agreement with the experimental data. The transition from extinction cross section to absorbance property can be done by
1 ) log( 1 A NQ
≡ − (27) where A is the absorbance, N is the particle density, and Q is the extinction cross section. After the transition, the same absorbance curve can be found as the experimental data.
Figure 3.12: Comparison of experimental absorbance and theoretical extinction cross section of Ag spherical nanoparticle.
Figure 3.13 shows the extinction cross section varies with the particle radius from 40 to 70 nm. The spectrum shifts to red band as the particle radius is increased.
Mie’s theory mode provides a simply approach to calculate the scattering and extinction cross section of spherical metal nanoparticle.
Figure 3.13: Extinction cross section of Au spherical nanoparticle in various radius from 40 to 70 nm.
3.3.1 Dielectric dependent cross section
Figure 3.14 shows that the cross section varies with the refractive index of surrounding environment. A sensitive red shift is observed as increasing the value of refraction index. In Figure 3.13, a red shift is also observed due to the increase of particle size. The dependence of size and dielectric property on the extinction cross section leads to a wild application to bio-sensor.
\
Figure 3.14: Extinction cross section is a function and varies with the dielectric property
3.3.2 Scattering, absorption and extinction cross section
Figure 3.15 and 3.16 shows the extinction, absorption and scattering cross section of silver nanoparticle in size from 2-9nm. The ratio of scattering cross section to extinction cross section increases with the particle size. But to absorption cross section, it was decreased when the particle size is increased. The particle with a diameter of 10 nm shows the 50 % scattering and 50 % absorption to extinction cross section. The distance of the particle diameter is approached to the skin depth of gold material is about 11 nm. The electromagnetic energy seems absorbed and loss 50 %
Figure 3.15 and 3.16 shows the extinction, absorption and scattering cross section of silver nanoparticle in size from 2-9nm. The ratio of scattering cross section to extinction cross section increases with the particle size. But to absorption cross section, it was decreased when the particle size is increased. The particle with a diameter of 10 nm shows the 50 % scattering and 50 % absorption to extinction cross section. The distance of the particle diameter is approached to the skin depth of gold material is about 11 nm. The electromagnetic energy seems absorbed and loss 50 %