Spring 009 EE 710: Nanoscience and Engineering Part 10: Surface Plasmons in Metals Images and figures supplied from Hornyak, Dutta, Tibbals, and Rao, Introduction to Nanoscience, CRC Press Boca Raton, 007 Prasad, Nanophotonics, John Wiley and Sons, New Jersey 004 Instructor: John D. Williams, Ph.D. Assistant Professor of Electrical and Computer Engineering Associate Director of the Nano and Micro Devices Center University of Alabama in Huntsville 406 Optics Building Huntsville, AL 35899 Phone: (56) 84-898898 Fax: (56) 84-898 email: williams@eng.uah.edu 1
Introduction Plasmon: quasi-particle used to describe a collection on interacting electrons Occur on the surface of a metal Quantized Consistent of collective longitudinal (or transverse) oscillations of conduction electrons Induced by the excitation of metal surface electrons by light Wavelength and strength th depend d on: incident radiation dielectric constants Size Shape Orientation surface roughness JDW, UAHuntsville ECE, Spring 009
Introduction Plasmon energy taken from e ω = p Extinction Coefficient (measure of absorption and scattering of plasmons on a surface) is: 1/ 18 π NV ε h ε kex = λ ε + ε + ε N e ε Where λ is the wavelength of light, ε h is the dielectric constant of the medium, ε 1 and ε are the real and imaginary dielectric constants of the metal and are frequency independent. If ε is small or weakly dependent on frequency then the absorption maximum to the resonance is : o m [ ] ε1 = ε h Size dependence of the surface plasmon comes from the size dependence of the dielectric constant ε of the metal Two types of dielectric constants in the metal One is from the inner d orbital electrons in the conduction band Free electrons using the Drude Model γ Is the damping constant relating to ωp ε D( ω) = 1 the width of the plasmon and ω iγω provides size dependence on the plasmon in nanoparticles JDW, UAHuntsville ECE, Spring 009 h 3
Introduction Higher electron density of states blue shifting of the plasmon wavelength Noble metals demonstrate shorter plasmon wavelengths than transition metals Metallic nanoparticles and clusters demonstrate shorter wavelengths due to enhanced surface oscillations Noble (Au, Ag, Cu) and alkali metals demonstrate strong plasmon resonances Gold red shifts below 5 nm and blue shifts above 5 nm in size However plasmonic scattering in materials occurs over approximately -5 times the wavelength 4 JDW, UAHuntsville ECE, Spring 009
Plasmonic Resonance Consider an electric field incident on a nanoparticle The geometry of the nanoparticle and its orientation to the electric field determines the polarization vectors strength and direction Thus scattering of the field off of nanoparticles of different sizes and orientations is significantly different For spheres less than 0 nm in diameter, the dipolar resonance is: ε o = -ε m ε m is the real component of the complex dielectric function ε o is the real component of the refractive index Size of the particle causes a change in the physical properties For larger particles multipole expansions must be considered and the effect of size on electronic and optical properties is significantly reduced JDW, UAHuntsville ECE, Spring 009 5
Nanoparticles and Plasmonic Resonance Two types of cluster size effects that affect plasmonic resonance Eti Extrinsic: i Classical l E&M using bulk optical constants t can be applied safely to large clusters, because the atomic structure of large clusters is similar to that of bulk materials However optical properties are not the same Restrictive boundary conditions inherent in metallic clusters allow collective electron oscillations known as Mie resonances Example: purple color of colloidal gold Blue/grey color of large diameter colloidal silver Intrinsic: Changes in internal structure of small clusters due to surface free energy/ bulk free energy requirements Occur at the threshold between bulk and quantum domains Structural free energy affects many properties Optical and electronic Ionization potential Binding energy and chemical reactivity Melting temperaturet Plasmonic resonance 6 JDW, UAHuntsville ECE, Spring 009
Particle Depolarization and Assume a spherical particle Screening Polarization is therefore independent of orientation thus only single resonance value exist (one absorption peak) Prolated particles Polarization is dependent on direction leaving two strong resonance values (two absorption peaks) Three terms aid in the understanding the role of particle orientation on optical resonance Eccentricity, ζ or f of the ellipsoid: how much the ellipsoid deviates in shape from a sphere Depolarization factor, q: magnitude of the depolarization based on shape, q = 0 1 Screening parameter, κ: screening that takes place in a composite composition. I.e. the transparency of the 7 JDW, UAHuntsville ECE, Spring 009
Eccentricity Eccentricity, ζ or f of the ellipsoid How much the ellipsoid deviates in shape from a sphere f b a a = a b Oblate ξ = Prolate a 8 JDW, UAHuntsville ECE, Spring 009
Depolarization Factor Depolarization factor, q Magnitude of the depolarization based on shape, q = 0 1 1 = q i + q j + qk i,j,k are the principle axes for the system qi 1 ξ 1 1+ ξ = ln 1 ξ ξ 1 ξ Scattering of Prolated particle facing the field For an ellipsoid with geometrical elongation along only 1 principle axis, the depolarization along the other two axis can be equated q 1/ i = i 1/ i + 1/ j + 1/ k j = k q i 1/ i = 1/ i + / q = q = j k j 1 q i 9 JDW, UAHuntsville ECE, Spring 009
Screening Parameter Screening parameter, κ Screening that takes place in a composite composition. Effective transparency of the object = 1 q i κ K = 1 for spheres, for cylindrical rods lined up parallel to impinging fields K= 0 for structures perpendicular to impinging radiation (oblate particles facing the field) K = infinity for structures parallel to impinging radiation (particles oblated along the field ) q i 10 JDW, UAHuntsville ECE, Spring 009
Blue Shifting Due to Elongation JDW, UAHuntsville ECE, Spring 009 11
Effects of Orientation 1 is perpendicular to E 5 is parallel to E 1 JDW, UAHuntsville ECE, Spring 009
Calculating the Dielectric Function Composition of complex dielectrics and distribution in a medium is accounted for by the fraction of metal inclusion such that: f m ε m εo ε c εo = ε m + κεo ε c + κεo Where fm is the volume fraction of metal inclusion, c m is the metal dielectric o is the oxide dielectric c is the composite dielectric measured JDW, UAHuntsville ECE, Spring 009 13
Very Tiny Nanoparticles What happens when a nanoparticle is 100 times smaller than the wavelength of incident id light? Oscillations of the field are then relatively show compared to motions of free electrons As a result, only dipolar resonance of the plasmon contributes t to the optical response Problem essentially becomes an electrostatic in nature Larger particles with a radius greater than 10 nm or so require Mie models to describe their optical response. JDW, UAHuntsville ECE, Spring 009 14
SERS Surface Enhanced Raman Spectroscopy Uses plasmonic resonance to increase Raman spectroscopic signal by a factor of 100 Normally requires noble metals due to their highly efficient plasmonic field generation Resonance is achieved for very high real refractive index on very small spherical particles. The electromagnetic enhancement is: where: 1 ε m ε o ε m εo G m o m = χ ε + ε ε + ε λ o S r χ = r + d r is the radius of the spherical surface, d is the distance to the analyze molecule lambda is the laser excitation field S is the stokes field 15 JDW, UAHuntsville ECE, Spring 009
Subwavelength Aperture Photonic Applications i Light emerging from an aperture of sizes less than its wavelength is diffracted in all directions Transmission of light through a subwavelength hole is normally extremely low However if the aperture is fabricated in a periodic array is plasmonically coupled then extraordinary optical emission occurs, enhancing the transmission of light by orders of magnitude. Light cannot be coupled directly to the surface plasmon mode on a flat metal surface. This is because the wavevector of the plasmon is significantly larger than that of light propagating in vacuum or air. Various geometries are therefore used to match the wavevector of surface plasmons and light to simultaneously conserve energy and momentum allowing wave propagation JDW, UAHuntsville ECE, Spring 009 16
Plasmonic Wave Guiding Dielectric medium whose dimensions are controlled by refractive index contrasts provide guiding of plasmons Limited by diffraction limit of light Minimum confinement size is on the order of λ/n where n is the effective refractive index Propagation is extremely lossy but may provide adequate signal transduction for nano-optical and small micro-optical photonic structures JDW, UAHuntsville ECE, Spring 009 17