Boston University
College of Engineering
Term Paper
Optical Nonlinear Properties of Noble Metal Nanoparticles and Composites
By
Abdulkadir Yurt
Submitted in partial fulfillment of
the requirements for CAS PY 543
April , 2009
Contents:
1. Introduction:
pg: 3
2. Theory and Discussions:
pg: 3
a. Linear Optical Properties:
pg: 3
b. Nonlinear Optical Properties:
pg: 6
c. Effect of the surrounding medium on the
nonlinear response:
pg: 10
3. Selected Nonlinear Phenomena and Applications:
pg: 11
4. References:
pg: 14
5. Figures:
pg: 15
1. Introduction:
Metallic particles have been used in optical and electrical devices more than decades.
However recent developments in fabrication techniques have made it possible to have
a great control on the design of nanometer sized systems involving metallic particles
in particular noble metals. The unique property of noble metals is the fact that they
have strong resonant optical characteristics of surface plasma (SP) oscillations in the
visible spectrum range. In particular the strong field enhancement and large extinction
properties around SP resonance has found interesting applications during last decade
[1]. One other unique property of metal nanoparticles which has drawn less attention
is that they intrinsically posses considerably large optical nonlinearity comparing
dielectric and semiconducting materials [2]. Unlike the organic molecules these
nonlinearities are also extremely fast. Therefore these material systems can be used in
applications which require fast modulation speed and compact structures for large
scale integration such as all-optical photonic device technology and active
plasmonics.
In this paper, the following questions are investigated: How does the electronic band
structure affect the nonlinearity of metal nanoparticles and how does the dielectric
host medium play a role defining the nonlinear response of metal nanoparticles? The
paper organized as follows: In section 2a, the linear optical properties of the metal
nanoparticles are briefly explained. Then in 2b the nonlinear properties are described
in a more detailed manner. After that, in 2c an effective medium theory is presented
to understand how nonlinear properties are modified by surrounding medium. Finally
in section 3, a few selected device applications is discussed regarding the theory
elaborated in section 2.
2. Theory and Discussion:
a. Linear Optical Properties:
The permittivity of noble metals has been extensively investigated in literature, a
review can be found in [3]. Unlike the alkali metals, electronic response of the noble
metals cannot be described merely contribution from quasi-free electrons (sp band
electrons). The effect of the bound electrons (d band) electrons should also be
considered. Therefore the permittivity can be described as a sum of electronic
transitions within sp bands (intraband) and transitions from d band to sp bands
(interband).
(1)
The second term εintra can be defined by classical Drude model:
(2)
Γ is the damping constant, ω is the applied angular frequency and ω p is the plasma
frequency which is defined as:
(3)
where e, N, m* are the charge, density and effective mass of the sp band electrons
respectively. The first term in equation (1) can be found using classical approach of
Drude-Sommerfeld model [4] or solving Fermi’s golden rule for the band structure of
the noble metals in order to find the imaginary part and then applying KramersKronig relation to find the real part of the interband permittivity. For the sake of
brevity, only the expression for the Drude-Sommerfeld is given here:
(4)
where ω’p is the corresponding plasma frequency in analogy to intraband expression
with a different physical meaning. γ is the damping ratio for bound electrons and
w0   / m where
is the spring constant of the potential that keeps electrons in
place.
In the quasi-static approximation, SP resonance can be analytically derived solving
Laplace’s Equation for a single spherical particle surrounded by a homogeneous
medium. Then the local field inside the sphere is written as:
(5)
where E0 is the applied field and εd and εm are the permittivity of the surrounding
medium and metal sphere respectively. The local field factor f will be defined the
ratio of the field inside the sphere and field applied. The complex quantity f will also
be used defining the nonlinear properties. The equation (5) diverges at εm(ωSPR) = 2εd
as the denominator goes to zero where the SP resonant wavelength is defined as ωSPR.
Therefore, local field factor f exhibits a strong enhancement at wavelengths close to
ωSPR. Figure 1 show the magnitude of the f for silver and gold nanoparticles [5].
In the discussion of permittivity of the metal nanoparticles above, the finite size
effects are not considered explicitly. However it is experimentally and theoretically
shown that the bulk values of the permittivity fails when the particle sizes are on the
order of a few nanometers. In particular the SP resonance location and its spectral
features are altered dramatically. A review can be found in [6]. Briefly the origin of
the deviation can be explained classically as the mean free path approaches to radius
of the nanoparticle an extra scattering term must be included to compensate the
scattering from particle surface. The most widely used approach is the modification
of the damping factor Γ in Drude expression (Equation (2)) as [5]:
(6)
where Γinf , vf and r are the bulk collision constant, Fermi velocity and R the particle
radius respectively. Although this simple expression qualitatively succeeds to
incorporate the surface effect into conduction band electrons it does not account the
modification of d band structure. Therefore for really small particle sizes one has to
do quantum calculations to fully appreciate the finite size effect.
b. Nonlinear Optical Properties:
When the light field intensity is large enough, the induced polarization P in the
materials is no more linear to electric field component of the incoming field.
(7)
χ(n) is called the nth order nonlinear susceptibility which is a tensor rank n+1. In
centrosymmetric media with inversion symmetry the even order susceptibilities
vanish. Then the first nonlinear susceptibility is then the third order. However it
should be noted that this assumption does not hold for the cases such as aspherical
nanoparticles and aggregated clusters since the inversion symmetry is broken. These
types of materials can also show second order nonlinearities such as SHG etc. In this
section however the second order effects are neglected. A more appropriate
expression of the third order nonlinear polarization can be written as:
(8)
term is a fourth-rank tensor having 81 elements. However assuming isotropic
material, there remain 21 elements where only 3 of them are independent. Consider
the incoming field is a linearly polarized plane wave. Then induced polarization
vector will be sum of two propagating terms oscillating at ω and 3ω. The amplitudes
of these polarizations are:
(9)
(10)
where the susceptibilities are simplified as:
(11)
(12)
Equation (9) indicates the induced polarization which affects the wave propagation at
frequency ω which is called optical Kerr Effect and Equation (10) indicates the
contribution to third harmonic generation (THG).
The optical Kerr effect causes an intensity dependant change in the permittivity of
nonlinear materials. The modified refractive indexes can be written as follows:
(13)
(14)
Where the nonlinear refractive indexes n2 and α2 can be related to third order
nonlinear susceptibility as:
(15)
(16)
where
. There are various experimental methods to measure the
third order nonlinearities of the materials. Most well-known methods are DFWM, Zscan, Transient Transmission (pump-probe). Each of these measures nonlinear
susceptibilities under specific conditions therefore, one has to be careful defining the
nonlinear susceptibility values.
The nonlinear response of the materials observed in experiments comes from two
main contributions: electronic and thermal nonlinearities. In short pulse length and
low repetition regime, electronic contribution is the only origin whereas in longer
pulse regime and high repetition rate regime thermal nonlinearities dominate over
electronic contribution. In general electronic nonlinearities have small magnitudes but
fast response (~1 ps-10ps) on the other hand thermal nonlinearities are larger in
magnitude but also very slow (~1ms-1s). In the following discussion, I will focus on
the electronic contribution which stems from non-equilibrium electron distribution
originated from fs pulses [7-10]. In their pioneering work [10] Hache et al observed
three contributions to the nonlinear susceptibility of the metal nanoparticles:
(17)
The first and the second term on the right hand side result from the same coherent
transitions as those in the linear susceptibility. The third term is an incoherent
contribution that result from the modification of the populations of the electron states
caused by the elevation of their temperature subsequent to the absorption of photons
in the resonant process but before the heat is transferred to the lattice. This latter
incoherent process takes a few picoseconds to occur. Since the heat capacity of the
conduction electrons is weak due to Pauli exclusion the electrons may attain very
high temperatures. (Figure 2)
The intraband contribution
that is electric dipole nonlinear susceptibility
associated with transitions involving the only conduction electrons (sp band
electrons). It is approximately given by:
(18)
(19)
Where T1, T2, g numbers are the energy lifetime, dephasing time and some real
numbers on the order of 1 respectively. This response is expected to depend on size of
the nanoparticles as 1/a3 where a0>>a. Note
The term
term vanishes for bulk materials.
is on the other hand originates from electric-dipole transitions
between the d band to sp bands. This term has importance only about excitations
about bandgap between d band and Fermi level of the sp bands due to Pauli
exclusion. It is size independent and exists in bulk as well. The approximate
expression is given as:
(20)
Where A4 is angular form factor (~0.2) and
and
are the energy lifetime and
dephasing time respectively for the interband transitions respectively. The terms J(w)
and P denotes joint density of states between sp and d bands and an average matrix
element of the momentum operator respectively. Although the hot electron
contribution is not a purely electronic effect it can be included into the third order
nonlinear susceptibility with certain approximations [10] as follows:
(21)
Where τ0 is the electron cooling time, γT is the specific heat of the conduction
electrons and ε”D and ε”L are the imaginary parts of the Drude and interband
contributions to ε. Note that hot electron contribution is also not size dependant.
Having the values of the physical parameters from experimental results, these
contributions can be compared to each other. For gold nanoparticles:
-
Im(
) ≈ -10-10 esu with parameters a0 = 136 Ǻ, EF = 5.5 eV, T2= 2.10-14 s,
T1= 10-13 s, w = 3.55.1015 Hz and a = 5 nm
-
Im(
) ≈ -1.7.10-8 esu with A4 = 0.2, T’2 = 2.10-14 s, T‘1 = 2.10-13s and |P|2 =
3.4 10-39 esu
-
Im(
) ≈ -1.1.10-7 esu with ε”Dτ0 = 5.1.10-18s and γ = 66 Jm-3K-2
c. Effect of the surrounding medium on the nonlinear response:
So far the environment of the nanoparticles has been ignored. However in real
applications nanoparticles have to be incorporated in some sort of a medium. The new
system has thus different optical properties of its constituents. In order to study the
new optical properties, various effective medium (EM) theories have been developed
[11]. In these theories, fictitious homogenous medium which have the same
macroscopic response of heterogeneous
material
is defined with certain
approximations. Following the method presented in [5], a general expression for
effective χ(3) in the quasi-static approximation can be derived from Maxwell’s
Equations.
(22)
Where V denotes the volume of the heterogeneous material, E0 is the spatially
averaged applied field. Note that the frequency dependency is omitted for clarity.
Most of the time, the nanoparticles inclusions have larger nonlinear susceptibility
than the surrounding matrix. Assuming the value of the nonlinear susceptibility is the
same and constant for all nanoparticles. Equation (17) can be simplified as:
(23)
If the electric field inside the nanoparticles is homogenous as it is noted in the
previous section, then this expression can be written as:
(24)
Where p is the volume fraction of nanoparticles and f(r) is the local field factor. In
low density and quasi-static limit f(r) is defined as in equation (5). Therefore χeff(3)
can be written:
(25)
From the expression above following conclusions can be drawn: Response of a
composite medium can be very different in terms of sign and magnitude from its
constituents because both χm(3) and f terms are complex quantities. For instance,
neglecting the dispersion of χm(3) and assuming it has a constant value of (-1+5i).10-8
esu, the resulting χeff(3) is indicated on Figure 3. The figure clearly shows how χ eff(3) is
altered in the vicinity of SP resonance region [5].
In addition to dispersion relation of the nonlinearity in dielectric materials, one should
also keep the following fact in mind: The nonlinear properties depend strongly on
some other factors such as shape and concentration of the nanoparticles, nonlinear
properties of the surrounding medium and excitation characteristics etc.
3. Selected Nonlinear Phenomena and Applications:
In this section, I will briefly summarize various nonlinear phenomena and
experimental results regarding the third order nonlinear susceptibility.
One of the most common methods to study third order nonlinear effect is Third
Harmonic Generation (THG). As seen in equation (10) the third order susceptibility is
directly related to the frequency conversion efficiency via nonlinear polarization.
Lippitz et al. reported [12] an efficient THG using gold nanoparticles in colloids at
1500nm excitation wavelength and 1ps pulse length. They claim the efficiency is
large due to the fact that the third harmonic is close to the surface Plasmon resonance
thus enhancing the generation. These small nanoparticles can be used efficiently for
labeling the biomolecules since THG provides great contrast from the background
scattering of the excitation light.
One other application of harmonic generation is photo-thermal cancer therapy. ElSayed et al. has mastered the method of photo-thermal cancer therapy using gold
nanoparticles which are conjugated to specific antibody in last decade. In their
original study [13] they showed the effectiveness of the method can be further
increased using Second Harmonic Generation (SHG) on gold nanoparticles. Since the
human tissue is maximally transparent at NIR, high power ultrafast laser pulses can
be used at 800nm to generate second harmonic light on the gold nanoparticles. The
generated light will be absorbed efficiently and turned into heat by the nanoparticles
because it is close to interband transition and SP resonance region (Figure 4). They
report that the threshold to destroy the cancer cell is reduced 20 times using
nanoparticles. However note that SHG is not a third order nonlinear effect. SHG can
only be realized in case of destroyed centrosymmetric structure.
Another promising application of the nonlinearity of the metallic structures is the
emerging field called active plasmonics. Like the historical development of the
electronics and photonics, plasmonics also is also undergoing a transition from
passive to active structures. Active plasmonic devices enable optical signals in the
form of SP polaritons to be switched by an electrical or optical stimulus [14]. The
previous reports of these kinds of devices have been based on thermo-optic nonlinear
media, quantum dots which have a large nonlinear susceptibility in the expense of
modulation bandwidth [15]. Alternatively, metal nanoparticles systems offer both
ultra-fast speed and relatively large nonlinearity which is required for plasmonic
based circuitry. In this regard I would like to give two recently reported examples.
The first one [16] demonstrates all-optical switching in subwavelength metallic
grating structure containing Au:SiO2 composite material. Figure 5 shows the
geometry of the structure and near-field time averaged intensity distribution during
on/off states of the switch. It basically works in pump and probe method. Existence of
the strong pump signal will tune the refractive index of the composite material so that
SP resonance of the grating shifts enough to transmit or block the probe signal. Figure
6 indicates the time and spectral responses. As they claim, this basic device has 200fs
switching time ignoring the material response of the nonlinear material. As it is
demonstrated in the previous sections the nonlinear material (Au:SiO2) can have very
fast response time if pulse length of the control signal is less than 1ps. Therefore
overall switching time can be on the order of picoseconds.
Zheludev et al. recently reported a study on modulating the surface plasmon polariton
(SPP) propagation on the femtosecond timescale by direct ultrafast optical excitation
of the metal [17]. The geometry of the experiment is shown on Figure 7. The
excitation beam is focused on coupling grating and it generates SPP wave on the
aluminum/silica interface. After traveling 5um on the interface, the decoupling
gratings scatter the SP waves into propagating wave so that it is sensed at the output.
The optical control pulses were incident on the waveguide region between the
coupling and decoupling region such that the transient effect of the control signal is
thus monitored at the output. They discuss that control pulse generates locally
refractive index variation which leads to a change in reflectivity and plasmon decay
length and this modification generates modulation on the SPP signal in two different
timescale. Figure 8 indicates these two responses. The fast component has almost
instantaneous response whereas the slow component decays on the order of tens of
picoseconds. The physical explanation behind this observation is the following: the
fast response is due to combination of a coherent nonlinearity and coupling of the
wavelength degenerate pump into the probe through a transient grating created by
pump beam and probe SPP wave. The slow response is on the other hand due to
Fermi smearing and followed by thermal and elastic relaxation of the energy into
environment. The initial relaxation is dominated by Fermi Smearing and after a
couple of picoseconds electron-electron scattering minimizes this effect and then
thermal effects govern the response of the material nonlinear response. The main
advantages of this switching technology include simplicity of both geometry and
material composition with existing CMOS fabrication techniques.
4. Conclusion:
In summary, the nonlinear effects of noble metal nanoparticles have been discussed
including basic theory and applications. Although the fundamentals of the optical
theory of metals was developed almost two decades ago, the first device applications
in many different field has only recently been realized due to the improved control on
the fabrication of nano-size systems. In that regard, impressive nonlinear properties of
noble metals seem to open new horizons in the field of active plasmonics and alloptical photonic applications in near future.
5. References:
1. S. Lal, S. Link and N.J. Halas, Nature Photonics 1 641-648 (2007)
2. R. W. Boyd, Nonlinear Optics, Academic Press, Boston, (1992)
3. Palik, E.D. (ed.): Handbook of Optical Constants of Solids, vols. I and II,
Academic, New York (1985/1991)
4. L. Novotny and B. Hecht, Principles of Nano-Optics Cambridge University Press,
New York, NY, 2006
5. M. G. Papadopoulos, A. J. Sadlej and J. Leszczynski, NonlinearOptical Properties
of Matter, Springer, Dordrecht, 2006
6. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, Springer-Verlag,
Berlin, (1995)
7. Y. Guillet, M. Rashidi-Huyeh, and B. Palpant, Phys. Rev. B 79, 045410 (2009)
8. C. Voisin, D. Christofilos, P. A. Loukakos, N. D. Fatti, F. Vallée, J. Lermé,
M. Gaudry, E. Cottancin, M. Pellarin, M. Broyer, Phys. Rev. B 69,195416 (2004)
9. J. Y. Bigot, V. Halté, J. -C. Merle, A. Daunois, Chem. Phys. 251,181-203 (2000)
10. F. Hache, D. Ricard, and C. Flytzanis, J. Opt. Soc. Am. B 3, 1647 (1986)
11. J.P. Huang and K.W. Yu, Phys. Rep. 431 (2006)
12. Lippitz, M., van Dijk, M.A., Orrit, M. ,Nano Lett. 5, 799–802 (2005)
13. X. Huang, W. Qian, I. H. El-Sayed, M. A. El-Sayed, Lasers in Surgery and
Medicine, 39(9), 747 – 753, (2007).
14. L. Cao and M. Brongersma, Nature Photonics, January (2009)
15. Pala, R. A.; Shimizu, K. T.; Melosh, N. A.; Brongersma, M. L. Nano Lett. 8(5);
1506-1510 (2008)
16. C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G.
Yang, Opt. Lett. 33, 869-871 (2008)
17. MacDonald, K. F., Sámson, Z. L., Stockman, M. I. and Zheludev, N. I. Nature
Photon. 3, 55–58 (2009)
6. Figures:
Figure 1: Modulus of the local field factor f. [5]
Figure 2: a. Schematic of the band structure of noble metals. The arrows show the interband and intraband
transitions. b. Change of electronic band occupation in the vicinity of Fermi level. Dash line show initial
distribution whereas the solid line shows the thermalized case. [8]
Figure 3: Real, imaginary and modulus of the effective nonlinear susceptibility within visible range. Please see
the text for the details. [8]
Figure 4: Absorption spectra of gold nanoparticles in different solutions. [13]
(c)
Figure 5: a&b. Near-field time-averaged intensity distribution of the probe signal. c. Schematic of the optical
switch under study. [16]
Figure 6: a. Normalized transmission of signal light as a function of time. b. Far field transmission of the signal
light obtained with pump on and off. [16]
Figure 7: Schematic of the device under study. [17]
Figure 8: Ultrafast SPP modulation dynamics and pump pulse fluence scaling. Transient pump induced change
in fast component (a) and slow component (b). c. Corresponding peak amplitudes of the fast and slow
components as a function pump fluence. [17]