Plasmons, Surface Plasmons and Plasmonics Plasmons govern the high frequency optical properties of materials since they determine resonances in the dielectric function ε(ω) and hence in the refraction index n( ω ) ε ( ω) µ ( ω) n( ω ) ε ( ω) Often written as for non ferromagnetic materials. Above the plasma frequency, electric fields penetrate into the matter which becomes therefore transparent to electromagnetic radiation (ultraviolet transparency of alkali metals). Surface plasmons determine the high frequency surface response function, governing the screening of external fields and electron transmission at the interface, determining e.g. photoemission intensities. Plasmons exist also for nearly -dimensional systems, formed e.g. in charge inversion layers and artificially layered materials [1]. Such D plasmons may have low energy and govern many dynamical processes involving electrons and phonons and mediate the formation of Cooper pairs in superconductors [] [1] M.H. March and M.P. Tosi, Adv. in Physics 44, 99 (1995) [] J. Ruvalds, Nature 8, 99 (1987)
Plasmons and Surface plasmons: collective excited states of the D electron gas Equation of motion for a free electron of mass m and charge e in presence of a time varying electric field EE(t) : d x iωt i t m ee E E o e dt x x e ω o ee r o ne r xo and the polarization reads P E mωω mωω r ne r r D ( 1 4π ) E ε ( ω) E mω while the electric displacement field is: For ε0 E 0 also when D0, i.e. when there are no external charges. ε0 determines therefore the condition for which self sustaining polarization waves can exist at the frequency ω For the free electron gas it follows: p 4πe m n ω ε 1 ω p
What determines the magnitude of plasmon frequency? h d ao me a d a o 6 4πne 4πe m e ω p 6 m m 8h k F distance between atoms in a solid (approx. twice the Bohr radius) fcc lattice spacing containing 4 atoms per unit cell electron density πm e h 8 6 n 6 π m e 1/ me π n k ( ) 6 F π 8h h E F 4 a d 8a E o 6 m e hω 8h 6 p p me h 4 4 h m e me (π ) (π ) 4 m 4h 8h in terms of the energy of the fundamental state of the H atom (Rydberg0.5 Hartree) E E E H p F me h 4 4π E 4π E p EH, 98E 1 4 π F h k m F H 0.989 E H EF ( π ), 01 4 The ratio is slightly smaller than unity because the plasmon is a collective rather than a single particle excitation E 4 H π
Demonstration that k F π Let s start with the free electron gas model of the solid: n E p m h k m For a linear chain of length L with N atoms separated by a lattice spacing a, LNa we have standing waves whenever the electron wavelength λ satisfies the requirement : nλ L i.e. k π λ nπ L nπ Na n1,,...n π The largest value of k is when nn, i.e. kmax, independent of N a (having more atoms implies a higher density of k points in the dispersion, not a larger k max since there are also more states) The largest energy E max is thus: E max h kmax h π m ma
with κ continuous variable for very large n κ π ma h ) ( 1 n n n ma E + + π h For a particle in a square box quantization implies Since n 1, n and n are positive, the total number of states N is then given by the volume of the octant of a sphere 8 6 4 8 1 ) ( E m h V me V E N π π π πκ h
8 6 4 8 1 ) ( E m h V me V E N π π π πκ h de E g E de m h V E dn ) ( 4 ) ( π Differentiating the above eq. we get the density of states per energy interval the number of electrons dn is twice as large, due to the spin degeneration the spin degeneration and the total number of electrons is obtained by integrating dn from 0 up to E max E E E m h V E de m h V E dn n 0 0 16 8 ) ( π π 16 m k m h V n F h π k F n π and hence
Volume plasmon dispersion depends on electronic polarizability α and is quadratic in transferred momentum Example Ag
Surface plasmons Surface plasmons are the normal modes of charge fluctuation at a metallic surface and govern the long range interaction between the metal and the rest of the world. At the surface, if σ is the charge density, the Maxwell equations read: E z D z z< 0 : πσ z> 0 : πσ z< 0 : πσ z > 0 : πεσ E z vacuum ε1 metal ε -1 πσ + + E z πσ + z D z is continuous at the interface so that -πσ πεσ ε(ω) -1 ω sp ω p nearly free electrons The frequency of a proper surface plasmon is thus determined by its volume dielectric function
Photoemission spectroscopy Plasmon and surface plasmon are observed in photoemission spectra. Their relative intensity depends on the kinetic energy of the electrons. The bulk plasmon can be excited while the electron is inside the solid, the surface plasmon when the electron leaves the surface on the trajectory to the analyser
The photoemission probability may be strongly affected by the surface response function as shown for the case of the surface Shockley state of Al and Be. No photoemission is observed at photon energies coinciding with the plasma resonance since then the crystal becomes transparent to the electric fields Right: In phase (solid) and out of phase (dashed) contributions to the normal component of the electric field near a jellium surface
Surface plasmon dispersion Contrary to its frequency, the dispersion of the surface plasmon is determined by surface properties and in particular by the position of the centroid of the screening charge with respect to the geometric surface, defined by the d parameters which correspond to the centroid of the screening charge for electric fields vertical and parallel to the surface. ω 1 ( q ) ωsp (0)(1 ( d ( ω) d ) q o( q sp + )) The field associated to the surface plasmon oscillates along the surface and decreases exponentially towards the bulk. r r iq q Φ( r ) Φ e e The charge density felt by the surface plasmon depends thus on q o z
The position of the centroid of induced charge is located outside of the surface in the low density electron spill out region since there the electron gas is more dilute and thus more compressible d(ω sp )>0 dispersion slope is negative for free electron metals The position of the centroid of the screening charge vs frequency diverges towards the interior of the metal at ω p. At the surface palsmon frequency it is still positive.
Surface plasmon dispersion for free electron metals The dispersion is initially linear, the quadratic terms dominates at large q
Measurement of surface plasmon dispersion by HREELS (high resolution electron energy loss spectroscopy) and ELS-LEED Energy conservation: Momentum conservation: Dipolar energy loss cross section:
HREELS High Resolution Electron Energy Loss Spectroscopy measurement of surface plasmon dispersion Energy conservation: Momentum conservation: M..Rocca, Surface Science Reports,1 (1995)
Multipole plasmon mode +++ - - - +++ - - - - - - +++ - - - +++ only the multipole mode is observed in photoyield experiments
Surface plasmon dispersion in presence of d-electrons: Ag and Au In presence of d-electrons, the plasmon frequency is displaced to lower frequencies due to the contribution of interband transitions to the dielectric function ε ω) ε1( ω) + iε ( ω) ( For Ag, plasmon damping is small at ω p but ε0 is shifted from 9 ev to.8 ev H. Raether, Springer tracts in Mod. Phys. Vol 88, 1980 J. Daniels, Z.Phys. 0, 5 (1967)
Surface plasmon dispersion in presence of d-electrons: Ag and Au Liebsch model: the d-electrons are schematized by a dielectric medium which extends up to a distance z d from the geometric surface The interaction of the electric potential, associated with the plasmon, with the dielectric medium causes the shift of the plasmon energy ω s 6.5 ev.7 ev at large q the induced fields penetrate less and the shift is smaller and hence ω s higher if this effect overcompensates the negative slope due to the position of the centroid of induced charge POSITIVE DISPERSION M. Rocca, Low energy EELS investigation of electronic excitations at metal surfaces, Surf. Sci. Rep., 1 (1995) A. Liebsch, Electronic excitations at metal surfaces, Plenum Press (1997)
Surface plasmon dispersion for Ag M. Rocca et al. PRL 64, 98 (1990)
Multipole Surface plasmon at Ag surfaces F. Moresco et al. PRB 54, 14 (1996) confirmed by photoyield measurements in 001
Mie resonance hω hω 0 + m Re( d ( ω )) R 0 ε d ε Plasmons in thin films of thickness t t ω ω(q ) on surfaces ω ω(1/t) on thin films ω ω(1/r) on clusters
The same holds true in presence of d-electrons Mie Resonance thin films of thickness t ε d t ε d The influence of the polarizable medium scales with the surface to volume ratio!!
Mie resonance for clusters experiment theory K Ag Tiggesbaeumker et al. 199 Similar result for thin Ag films Y. Borensztein Eur. Phys. Lett.1, 4 (1995)
Notre Dame Paris The bright colors of stained glasses of gothic cathedrals were obtained by nanosized gold particles which resonate at the Mie resonance. This phenomenon corresponds to light confinement
Plasmon confinement in nanostructured Ag films Deposition at 00 K yields isolated flat clusters at 90 K a percolated layer Plasmons are however localized in both cases as demonstrated by the absence of dispersion at small q t r STM image F. Moresco et al. Phys. Rev. Lett. 8, 8 (1999)
Plasmons in nanostructured ultrathin Ag/Si(111)7x7 slope depends on surface to volume ratio Confinement in percolating layer due to frequency mismatch at the touching points of the clusters t r F. Moresco et al. Phys. Rev. Lett. 8, 8 (1999)
The interaction with light: surface plasmon polaritons Unlike the case of volume plasmons light can interact with surface plasmons due to the lower symmetry giving rize to an avoided crossing of the dispersion curve. The mixed mode is called surface plasmon polariton. The light cone does not cross the surface plasmon dispersion curve unless it has an imaginary q //z (evanescent waves) or if the missing momentum is provided by surface roughness or by a nanometric superlattice.
Measurement of Surface Plasmon Polaritons: Attenuated total reflection
D surface plasmons: Ag monolayer on Si x structure ω q N D areal density of electrons ε(si) dielectric constant of Si 10.5-11.5 m* effective electron mass in the film Similar plasmons observed in charge inversion layers at semiconductor surfaces March and Tosi Adv. Phys. (1995)
D surface plasmons: Ag monolayer on Si x structure
D surface plasmons at bare metal surfaces Two dimensional electron gases exist an all metal surfaces supporting Shockley surface states in band gaps. Such states may generate plasmons with acoustic linear dispersion The linear rather than squareroot dispersion implies possible applications to devices since no distortion occurs when converting light into the plasmon and backwards (nanooptical devices and metamaterials with adjustable dielectric properties)
Plasmons, Surface Plasmons and Plasmonics Plasmons govern the high frequency optical properties of materials since they determine resonances in the dielectric function ε(ω) and hence in the refraction index n( ω ) ε ( ω) µ ( ω) The plasmon resonances causes a strong variation of the dielectric function which may become very small or even negative. ε 1 ε 1 When both ε(ω) and µ(ω) become negative then also n(ω) may be negative since the dephasing introduced both by ε(ω) and µ(ω) corresponds to positive angles (delays)
More recent concepts: Plasmonics and plasmonic materials Plasmon resonances are used to build metamaterials, i.e. artificial materials with very small (ENZ - Epsilon Near Zero) or even negative index of refraction (NIR). ENZ were proposed for the construction of invisibility cloaks by which even massive objects could be made perfectly transparent to light.
Metamaterials with negative refraction (NIR) NIR act as a sort of optical antimatter and were recently used for the construction of perfect superlenses with no limit to the resolving power. R. A. Shelby et al., Science 9, 77-79 (001)
To keep in mind Surface plasmons control the high frequency optical response of materials. For usual D surface plasmons, the frequency is a bulk property, but the dispersion is determined by the surface electronic structure. The dispersion is then negative for simple metals and positive for d-metals. At very small wavevectors the dispersion cuts the light cone giving rise to surface plasmon polaritons. For clusters surface plasmons correspond to the Mie resonance showing the same properties vs inverse cluster size. For thin film the surface plasmon frequency scales with inverse film thickness. Surface plasmons exist also for D electron gases and are then acoustic with squareroot like dispersion. For intrinsic two dimensional surface states the interaction with the underlying electron gas implies a surface plasmon with acoustic linear dispersion which can couple with light via surface roughness or surface nanostructures and looks promising for applications Further reading: M. Rocca, Low energy EELS investigation of surface electronic excitations, Surf. Sci. Rep., 1 (1995) A. Liebsch, Excitations at metal surfaces (Plenum Press London (1977)) J. Pendry, Playing tricks with light, Science 85, 1687 (1999) Subsequent Science and Nature articles 1999-007