Drude model parameters to fit the dielectric function of free electron metals including plasma frequencies and damping constants for silver (Ag), aluminum (Al), gold (Au), copper (Cu), potassium (K), sodium (Na), platinum (Pt)


Drude model parameters to fit the dielectric function of free electron metals including plasma frequencies and damping constants for Ag, Al, Au, Cu, K, Na, Pt


Metal plasma [eV/cm-1/PHz] damping [meV/cm-1/THz] source
Ag 9.6*/77430/2.32122.8*/183.9/5.513 Blaber
9.013/72700*/2.18 18/145.2*/4.353 Ordal
9.04*/72920/2.18621.25*/171.4/5.139 Zeman
8.6*/69370/2.0845*/363/10.88 Hooper
Al 15.3*/123000/3.7598.4*/48.27/144.7 Blaber
14.75/119000*/3.57 81.8/660*/19.79 Ordal
12.04*/97110/2.911128.7*/1038/31.12 Zeman
Au 8.55*/69000/2.068 18.4*/148.4/4.449 Blaber
9.026/72800*/2.183 26.7/215*/6.46 Ordal
8.89*/71710/2.1570.88*/571.7/17.14 Zeman
9*/72590/2.17670*/564.6/17.71 Berciaud
8.951/72200/2.165 69.1/557.4/17.94 Grady
7.9*/63720/1.91 / / Kreiter
Cu 7.389/59600*/1.914 9.075/73.2*/8.34 Ordal
8.76*/70660/2.118 95.5*/770.3/23.09 Zeman
K 3.72*/30000/0.8896 18.4*/148.4/4.449 Blaber
Na 5.71*/46100/1.381 27.6*/222.6/6.674 Blaber
5.93*/47830/1.434380*/3065/91.89 Zeman
Pt 5.145/41500*/1.244 69.2/558*/16.73 Ordal

The respective source values are indicated by a star. Conversion between different units performed by using
Spectroscopic Unit Converter

In order to avoid any confusion, the above values in cm-1 correspond to the so-called wavenumber=1/lambda, i.e. a direct inverse of the wavelength without 2\pi prefactor. The above values in Hz correspond to the ordinary frequency, i.e. not to the radial frequency given in rad/s.

Do you also wonder why the experimental data are so much scattered? You are not alone! Some differences in the data may be accounted for by differences in samples preparation. Yet the differences in the data appear to be too big to be explained merely by the differences in samples preparation. Let me know, please, if you find a satisfactorily explanation.

Recipes so that films with optimal optical properties can be routinely obtained have recently been provided by Norris et al. [11]. They also quote that: "due to inexperience with deposition methods, many plasmonics researchers deposit metals under the wrong conditions, severely limiting performance unnecessarily. This is then compounded as others follow their published procedures."

Some data on graphene layers you can find in Ref. [12].


Note in passing that unless you are cautious, your calculations based on using a Drude fit in the region where the real part of experimental dielectric constant is between approximately -10 and zero may not even be qualitatively correct!

The region (where the real part of experimental dielectric constant is between approximately -10 and zero) is rather narrow region with a relatively little weight in an overall Drude fit, which stretches down to the values of the real part of dielectric constant below of -10**4. A Drude fit is, by its very definition, optimized for frequencies (wavelengths) lower (larger) than a plasma frequency (wavelength) and thus not so much concerned in the region where the real part of dielectric constant is in the proximity of zero. Consequently the latter can be significantly shifted compared to the real material data. By that I mean more than hundreds of nanometers!

Do you want an example? Whereas a Drude fit to the silver data employs a plasma frequency of 9.6 eV (wavelength 129 nm), the experimental material data exhibits zero of the real part of silver dielectric function at approximately 328 nm, what makes a difference by a factor higher than 2.5!

Unfortunately, the region where the real part of dielectric constant is between approximately -10 and zero is the most interesting region from the view of plasmonics and various plasmonic applications. You may try to replace the plasma frequency (wavelength) in that region by the frequency (wavelength) which corresponds to zero of the experimental real part of dielectric function, as I did in in Ref. [9]. But then you face the problem that a Drude fit yields typically much smaller slope of the real part of the dielectric function compared to experimental data.

Consequently in the visible and near infrared you might arrive at contradicting conclusions when using a Drude fit to the metal data instead of true experimental data!

A spectacular example has been provided in Ref. [10], where photonic band structure of two-dimensional metallic and metallo-dielectric photonic crystals was calculated. It has been demonstrated there that on using a Drude fit to silver data, an optimal lattice structure with the largest complete photonic band gap (CPBG) was a square lattice, whereas on using experimental silver data, an optimal lattice structure with the largest CPBG turned out to be a triangular lattice. (An additional information can be found here.) The reason behind the above behavior was that CPBG width turned out to be sensitive to the slope of the dielectric constant in the proximity of zero. One can verify that a Drude fit yields typically much smaller slope of the real part of the dielectric function compared to the true experimental data in the proximity of zero of the real part.


In the case of small nanoparticles, the so-called surface scattering correction to a free-electron metal dielectric function, also known as a free path effect, has to be taken into account, resulting in the so-called size-corrected dielectric function. Importance of the size correction to a metal dielectric function in describing the non-radiative rates near the dipolar surface plasmon resonance has been recently discused in Read more about the surface scattering corrections here.


REFERENCES

  1. M. G. Blaber, M. D. Arnold, and M. J. Ford, Search for the Ideal Plasmonic Nanoshell: The Effects of Surface Scattering and Alternatives to Gold and Silver, J. Phys. Chem. C 113(8), 3041-3045 (2009) (see compilation from different sources in Table 1 therein).

  2. N. K. Grady, N. J. Halas, and P. Nordlander, Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles, Chem. Phys. Lett. 399(1-3), 167-171 (2004)

  3. M. H. Hider and P. T. Leung, Nonlocal electrodynamic modeling of fluorescence characteristics for molecules in a spherical cavity, Phys. Rev. B 66, 195106 (2002).

  4. I. R. Hooper and J. R. Sambles, Dispersion of surface plasmon polaritons on short-pitch metal gratings, Phys. Rev. B 65, 165432 (2002).

  5. M. Kreiter, S. Mittler, W. Knoll, and J.R. Sambles, Surface plasmon-related resonances on deep and asymmetric gold gratings, Phys. Rev. B 65, 125415 (2002).

  6. M. A. Ordal, R. J. Bell, R. W. Alexander, Jr., L. L. Long, and M. R. Querry, Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W, Appl. Opt. 24, 4493-4499 (1985) (see compilation from different sources in Table 1 therein).

  7. E. J. Zeman and G. C. Schatz, An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, gallium, indium, zinc, and cadmium, J. Phys. Chem. 91(3), 634-643 (1987) (see compilation from different sources in Table 1 therein).

  8. A. D. Rakic, A. B. Djuric, J. M. Elazar, and M. L. Majewski, Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic Devices, Appl. Opt. 37(22), 5271-5283 (1998).
    (They presented models for the optical functions of 11 metals used as mirrors and contacts in optoelectronic and optical devices: noble metals (Ag, Au, Cu), aluminum, beryllium, and transition metals (Cr, Ni, Pd, Pt, Ti, W). The values for the respective plasma frequencies are summarized in Table 1 therein; the remaining Lorentz-Drude model parameters are summarized in Table 2 therein.)

  9. A. Moroz, Three-dimensional complete photonic bandgap structures in the visible, Phys. Rev. Lett. 83, 5274-5277 (1999).
    [Demonstrated for the first time the possiblity of a complete photonic bandgap (i.e. common for both polarizations and for all propagation directions) in a 3D fcc lattice of metallic spheres embedded in a dielectric matrix.]
  10. H. van der Lem and A. Moroz, Towards two-dimensional complete photonic-bandgap structures below infrared wavelengths, J. Opt. A: Pure Appl. Opt. 2, 395-399 (2000) [pdf].
  11. K. M. McPeak, S. V. Jayanti, S. J. P. Kress, S. Meyer, S. Iotti, A. Rossinelli, and D. J. Norris, Plasmonic Films Can Easily Be Better: Rules and Recipes, ACS Photonics 2(3), 326-333 (2015).
  12. M. Bruna and S. Borini, Optical constants of graphene layers in the visible range Appl. Phys. Lett. 94(3), 031901 (2009).


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Alexander Moroz August 16, 2009 (last updated on July 21, 2015)

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