Optical Nonlinear Properties Ofnoble Metal Nanoparticles and Composites

BostonUniversity

College of Engineering

Term Paper

Optical Nonlinear Properties ofNoble 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

1. Linear Optical Properties:pg: 3
2. Nonlinear Optical Properties:pg: 6
3. Effect of the surrounding medium on the

nonlinear response:pg: 10

1. Selected Nonlinear Phenomena and Applications:pg: 11
2. References:pg: 14
3. 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 havestrong resonant opticalcharacteristics 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 isdiscussed regarding the theory elaborated in section 2.

1. Theory and Discussion:

1. 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 ωp 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 Kramers-Kronig 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 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)= 2εdas 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 rare 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.

1. 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 anintensity 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, Z-scan, 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 alobserved 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 term vanishes for bulk materials.The term 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 susceptibilitywith 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 = 136Ǻ, EF = 5.5eV, T2= 2.10-14s, T1= 10-13s, w = 3.55.1015Hz and a = 5nm

-Im() ≈ -1.7.10-8esu with A4 = 0.2, T’2= 2.10-14s, 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

1. 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 valueof (-1+5i).10-8 esu, the resulting χeff(3) is indicated on Figure 3. The figure clearly shows how χeff(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.

1. 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. El-Sayed 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:SiO2composite 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 thismodification 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.

1. 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 all-optical photonic applications in near future.