Lec 19: Localized surface plasmon resonance (LSPR)

Main ideas / concepts covered (Lecture 19: Localized Surface Plasmon Resonance, LSPR)

What LSPR is

• Surface plasmon polaritons (SPPs) (from previous lecture) are propagating, dispersive electromagnetic waves coupled to electron plasma at a metal–dielectric interface.
• Localized surface plasmon resonances (LSPRs) are non-propagating excitations that:
  • Occur in metallic nanostructures (e.g., nanoparticles)
  • Are driven by incident electromagnetic radiation
  • Produce strong near-field localization/amplification

Induced oscillating dipole picture

LSPR behavior can be explained via an induced oscillating dipole:

1. An incident oscillating electric field displaces the electron cloud, creating charge separation (a dipole).
2. The dipole oscillates with the field and radiates.
3. The nanoparticle’s curved surface provides a restoring force (likened to a “jelly” deformation analogy).
4. Field amplification occurs inside and near the particle, producing resonance when the electron natural frequency matches the incident frequency.

Key advantage over SPPs

• No phase-matching condition is required for LSPR.
• Nanostructures can be excited by direct illumination.

Why damping matters

Resonances in any resonator have damping, which controls the:

• Q factor
• Resonance linewidth

Damping depends on:

• Nanoparticle size and shape
• Ensemble interactions
• Material loss channels, including radiative and non-radiative mechanisms

Examples of plasmonic color effects

• Gold and silver nanoparticles support LSPR in the visible range, leading to strong absorption/scattering and vivid colors.
• Illustrative/historical examples:
  • Stained glass colors (e.g., Notre-Dame style) from metal nanoparticles embedded in glass.
  • A leica/cup-like example: color changes depending on whether the light source is behind or in front—resonant absorption/scattering alters which colors reach the observer.

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Methodology / theory presented

1) Exact electromagnetic solution: Mie theory (for spheres)

Purpose

• Provide exact scattering and absorption for spherical particles.
• Explain optical responses (including color) as a function of particle size.

High-level approach

• Start from linear Maxwell equations: total field = incident + scattered + internal (standing-wave) contributions.
• Use spherical coordinate harmonics and enforce boundary conditions.

Conceptual procedure

1. Use the time-harmonic scalar wave equation (electric potential form).
2. Expand fields in vector spherical harmonics:
   • ( \mathbf{M} ) and ( \mathbf{N} ) functions
3. Separate variables in spherical coordinates.
4. Determine coefficients by applying boundary conditions at the sphere surface:
   • Tangential components of ( \mathbf{E} ) and ( \mathbf{H} ) are continuous
5. Express external, internal, and scattered fields via Mie coefficients.
6. Compute:
   • Scattering
   • Absorption
   • Extinction: ( C_{\text{ext}} = C_{\text{sca}} + C_{\text{abs}} )

2) Simplified limit: Quasi-static approximation (for small particles)

When valid

• Particle radius (a \ll \lambda), so the electromagnetic phase variation across the particle is nearly constant.

Replacement

• Use electrostatics instead of full electrodynamics:
  • Solve Laplace’s equation for the potential.

Setup

• Homogeneous isotropic sphere of radius (a)
  • Metal permittivity: ( \varepsilon_m )
  • Surrounding dielectric permittivity: ( \varepsilon_D )

Key results

• The particle behaves like an induced dipole with polarizability ( \alpha ).
• Resonance occurs when the denominator of polarizability is near zero:
  • Resonance condition (Frohlich condition):
    • ( \varepsilon_m(\Omega) + 2\varepsilon_D = 0 )
    • equivalently ( \varepsilon_m(\Omega) = -2\varepsilon_D )

Resonance frequency shift (Drude-like, ignoring damping initially)

• LSPR frequency for a sphere shifts from bulk plasma frequency to:

  • [ \Omega_{LSPR} = \frac{\Omega_p}{\sqrt{1 + 2\varepsilon_D}} ]

• Comparison of related plasmon conditions:

  • Bulk plasmon: ( \varepsilon_m(\Omega)=0 ) at ( \Omega=\Omega_p )
  • Surface plasmon (flat interface): ( \varepsilon_m(\Omega)+\varepsilon_D=0 )
  • Localized (sphere): ( \varepsilon_m(\Omega)+2\varepsilon_D=0 )

Polarizability and dipole picture

As presented conceptually:

• [ \alpha \propto a^3 \frac{\varepsilon_m - \varepsilon_D}{\varepsilon_m + 2\varepsilon_D} ]

Damping effects (Q factor and linewidth)

Including damping (via an imaginary part in the metal permittivity through parameter ( \gamma )):

• Resonance position is roughly similar
• The peak broadens (lower Q)

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Cross sections and size scaling (quasi-static regime)

Using dipole polarizability ( \alpha ), the lecture states:

• Scattering cross section:
  • ( C_{sca} \propto k^4 |\alpha|^2 )
• Absorption cross section:
  • ( C_{abs} \propto k\,\mathrm{Im}(\alpha) )
• Extinction cross section:
  • ( C_{ext} = C_{sca} + C_{abs} )

Scaling trends

• Scattering scales like (a^6)
• Absorption scales like (a^3)

Therefore:

• For small particles, absorption dominates
• For larger particles, scattering becomes dominant

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Beyond quasi-static: limitations and corrections

Limits of quasi-static approximation

Not valid for:

• Very large particles (> ~100 nm): retardation and radiative effects become significant → need Mie theory or corrections
• Very small particles (< ~10 nm): quantum effects become important

Intermediate regime corrections (tens to ~100 nm)

Add to the quasi-static picture:

• Redshift due to retardation (phase lag across the particle)
• Broadening from radiative decay
• Higher-order resonances

Damping pathways for LSPR (particle-plus-one)

• Radiative decay: photon emission → contributes to linewidth broadening
• Non-radiative decay: electronic transitions, including:
  • Interband (e.g., d-band → sp-band)
  • Intraband (within band, lower → higher energy states)

Very small particles (< ~10 nm)

Additional damping mechanisms:

• Finite electron mean free path
  • Example given: gold mean free path ~42 nm
  • If particle size < mean free path, electrons collide with the particle boundary, increasing damping
• Quantum regime
  • Classical electromagnetic theory fails near ~1 nm scale
  • Condition described:
    • Energy gained per electron excitation ( \Delta E ) should be comparable to or exceed ( k_B T )

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Resonances for non-spherical structures (rods/ellipsoids)

Ellipsoid quasi-static polarizability

• An ellipsoid has three semi-axes (a_1, a_2, a_3).
• Polarizability depends on direction and includes geometrical factors (L_i).
• Spherical symmetry: (L_1=L_2=L_3=1/3)

Nanorods and aspect ratio effects

As aspect ratio increases:

• Two main plasmon modes emerge:
  • Longitudinal dipolar resonance (along the long axis)
    • Shifts strongly (large redshift with increasing aspect ratio)
  • Transverse dipolar resonance (across the short axis)
    • Shifts much less
• Longitudinal peak moves to longer wavelengths (redshift), decreasing energy.

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Additional LSPR type: void plasmon and nanoshells

Void plasmon (holes in metal)

• When a void (subwavelength hole) exists, the resonance condition swaps the roles of metal and dielectric permittivities.

• Resonance condition:

  • [ \varepsilon_D + 2\varepsilon_m = 0 ]

  • equivalently ( \varepsilon_m(\Omega) = -\varepsilon_D/2 )

  • Stated resonance frequency:
  • [ \Omega_{void} = \sqrt{\frac{2}{3}}\,\Omega_p ]

Nanoshell plasmon hybridization

A nanoshell supports coupled inner + outer dipole oscillations, producing:

• Hybridized modes: bonding and antibonding
  • Antibonding: inside and outside dipoles oscillate in opposite directions
  • Bonding: inside and outside dipoles oscillate in the same direction → overall energy lowered
• Two tunable resonant frequencies (expressed as ( \Omega_+ ) and ( \Omega_- ) via a hybridization formula)

Practical takeaway

• By changing shell thickness and metal properties, nanoshell LSPR is highly tunable, enabling design for target operating wavelengths.

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Overall lessons / takeaways

• LSPR is the localized, non-propagating plasmon resonance in metal nanostructures, driven by direct illumination and explained through an induced dipole.
• In the quasi-static (small particle) sphere limit, resonance satisfies the Frohlich condition:
  • ( \varepsilon_m(\Omega) + 2\varepsilon_D = 0 )
• LSPR frequency shifts with:
  • the environment/dielectric permittivity ( \varepsilon_D )
  • geometry (shape/aspect ratio)
• Scattering vs absorption strongly depends on size:
  • scattering grows faster with radius
• Quasi-static is approximate; for larger/smaller particles you must include:
  • retardation and radiative decay
  • non-radiative damping mechanisms
  • quantum effects
• More complex shapes (ellipsoids, voids, nanoshells) yield multiple resonant modes with strong tunability.

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Speakers / sources featured

• Speaker: Unnamed lecturer (course instructor for the “online course on Nanophotonics plasmonics and metamaterials”)
• Sources / referenced authors/books/scientists:
  • Gustav Mie — Mie theory / Mie scattering (1908)
  • Bohren and Huffman — Absorption and Scattering of Light by Small Particles