In two dimensions, the plasmon energy dispersion is gapless, thus is capable of being excited (and absorbing energy) at arbitrary low energies. For q denoting the wave vector, for q --> 0 (under the condition that ω/(q v_F) >>1, where v_F is the Fermi velocity), to leading order one obtains ω_p ~ a q^{1/2}, where "a" is a constant (note that for q --> 0, indeed q^{1/2}/(q v_F) >> 1). This is in contrast to three dimensions (in both cases assuming electrons to interact through the long-range Coulomb potential) where the plasmon energy dispersion is gapped (the gap is typically relatively large - in three-dimensional electron gas, in Hartree atomic units it is equal to 3^{1/2)/r_s^{3/2}, where r_s is the dimensionless Wigner-Seitz radius; for metals r_s is typically between 2 and 6 and 1 Hartree ~ 27 eV). For details see:

For two-dimensional systems:

Frank Stern, Polarizability of a Two-Dimensional Electron Gas

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.18.546

For three-dimensional systems:

David Pines and Philippe Nozières, The Theory of Quantum Liquids, Vol. I (W.A. Benjamin, New York, 1966)

Good answer, i'm not sure though, whether it matches the question. In finite size films there is also the surface plasmon mode (at constant charge density) and the corresponding surface plasmon (or plasmon polariton) in particles -- both at finite yet successively smaller energies than the bulk 3D plasmon.

The latter two carry a considerable dipole moment and may therefore serve as to enhance absorption.

[again, I am not sure whether what I wrote hits the original interest of the questioner]

Dear Kai, if you consider the text following the headline of the question ("In organic solar cells, surface plasmons are used to improve light absorption ..."), I believe that I gave the answer exactly matching the question.

Dear Behman, Would you please give the basics of plasmons in organic solar cells. Actually I'm expecting answers for the following questions

1). What are surface plasmons. Why it is orginated in metallic nanoparticle or meatallic layers and it is possible in insulator or semiconductor.

2.) How these plasmons can enhance organic photovoltaics characteristics.

3.) I knew these plasmons enhance light absorption without increasing active layer thickness, but how is it ? what mechanism or physics behind it.

4.) Why the lower band gap material absorb more light.

I'd have a few questions then:

1) in the limit q->0, what is the description (in words) of the 2D plasmon mode? (What motion is obtained?)

2) What is the associated dipole moment or oscillator strength? Are these plasmons effective in absorbing light?

3) What are the objects in real solar cell devices that are used for the effect the questioner describes?

4) is the relevant process in these devices given by the 2D plasmon odes you describe or is it more related to the surface plasmon (In my mind the surface plasmon is a different beast than what I imagine to be your 2D plasmons. "My" surface plasmons derive their properties from being embedded in 3D. For small particles this would be the "Mie resonance").

I'll be happy to learn on the subject.

Dear Shamjid, your questions are numerous and detailed. Given my commitments elsewhere, I shall be brief.

Regarding (1), Plasmons are collective charge-density excitations in any dimensions (they belong to the category of the so-called neutral excitations of the system). You can find a classical description of these in the large wavelength limit in Chapter 1 of the book Solid State Physics, by Ashcroft & Mermin (the physics being that on negative charge in a positive rigid background piling up somewhere, an electric field (a "dipole" field) builds up, forcing the charge to move in the opposite direction, and this -- the existence of a restoring force -- gives rise to a self-sustained charge-density oscillation, plasma oscillation, in the long wave-length limit).

Plasmons also exist in insulators, however their excitation spectrum is gapped. This can be understood by realising that plasmon excitations correspond to coherent electron-hole excitations in a system (this can be best understood by considering the Feynman diagrams for the polarization function). It follows that when the single-particle excitations are gapped, plasmon excitations must also be gapped (the converse is not true, as exemplified by the Coulomb-interacting electrons in three dimensions, where at long wavelengths plasmon excitations are gapped).

Regarding (2). The more channels for absorption of light a system has to absorb light energy over a given range of energies, the more efficient it can be as a photovoltaic device. I am not a photovoltaic expert, however can imagine that plasmons that absorb light's energy will in turn decay by giving off their energy through exciting electrons from the valence band to the conduction band; these secondary excitations complement the primary excitations.

Considering light in the visible part of the spectrum, they correspond to very small momenta q (the wavelengths of visible light are between 400 to 700 nm, to be contrasted with atomic distances that are of the order of 1 nm or less) . This is why the plasmon dispersion ω_p ~ a q^{1/2} I referred to earlier is relevant.

Regarding (3). Answer is included in my answer to (2).

Regarding (4). Band gap determines the threshold energy beyond which an insulating system can absorb energy (higher-order processes are also possible, but their so-called oscillator strengths are so small that can be neglected); band gap is directly relevant to first-order absorption processes. Here I am considering pure insulators; insulators that contain defects have localized energy levels inside the gap of their pure state. Restricting ourselves to light in the visible spectrum, their momenta are very small and therefore direct band gaps becomes relevant (see my answer to your question 2). It follows that the smaller the direct band gap of an insulator, the larger its ability to absorb the photon energy over a wider spectrum of the visible light. There is of course a limit to the band gap of a semiconductor that is to operate under the sun light; the gap must not be so small that at room temperature the system becomes practically a metal.

Dear Kai, here are my brief answers:

Regarding (1): Plasmons are just charge-density oscillations, best understood in the q-->0 limit by the classical argument that charge imbalance creates electric field in a direction that forces the charge imbalance to be restored. This restoring force gives rise to self-sustaining charge-density oscillations. There is a Landau-damping region, where plasmons become unstable (acquiring finite lifetimes), but even in d=2 there is a region where within the RPA the plasmons have infinite lifetimes.

Regarding (2): Yes they can absorb light. This can be understood in the q->0 limit by expanding the exponential exp(i q.r) in the expression for the dynamical dielectric response function in powers of q, which immediately gives rise to dipole-moment matrix-elements to appear in the latter expression (elements involving the combination q.r). One can immediately see that the imaginary part of this function (or of its inverse) is in the long wavelength limit q--> 0 determined by the values of the dipole-moment matrix elements.

Regarding (3): As I indicated in my response to Shamjid (my previous post on this page), I am not an expert in solar cells. However, I can mention that they consist of systems with gaps in their optical spectrum (neglecting electron-hole interaction, i.e. excitonic effects, optical gap is the same as the quasi-particle gap). The transformation of light's energy into electrical energy/current is through electron-hole excitations. Plasmons are noting but coherent electron-hole excitations.

Regarding (4): It is important to realise that here, as elsewhere in applied physics, 2D systems are always embedded in a 3D background. The relevant electrodynamics is therefore the electrodynamics of 3D, not that of 2D (i.e. that of Schwinger and Dyson, or Polyakov). The way one works is as follows: One describes the single-particle wavefunctions of the 3D system as products of functions that depend on two variables, corresponding to the plane of the "2D" system, and functions (often a single function) that depend on one variable, corresponding to the direction perpendicular to the plane of the "2D" system; one effects a separation of variables, but often because of the strong confinement along the z direction, approximates the behaviour of the system in the z direction by considering a single localized function (often a Gaussian -- one talks of "the lowest subband"). Integrating out the relevant equations with respect to z, one arrives at an effective two-dimensional problem.

I hope I have answered your questions.

To Shamjid Palappra: what sources (of literature) have you consulted so far, concerning the topic? Any specific questions that arose from that study? (it will be easier to help when knowing your specific context)

Dear Behnam Farid, thanks a lot. Sounds good at first reading. But I shall need some time to really swallow all of that with comprehension. Good opportunity to revisit good old Pines...

You are welcome Kai! The following publication is of some relevance to the discussion at hand:

http://scitation.aip.org/content/aip/journal/apl/60/15/10.1063/1.107142

Dear Behnam Farid, I have taken a quick look into F. Stern's PRL you quoted in your first post. (Actually the embedding in 3D is explicitely explained - nice) The response he calculates is to a longitudinal field, which is not the situation corresponding to irradiation by light, I guess. Also, in the discussion following eqn (7), it is mentioned that in the low q limit the plasmons cannot radiate. That would lead me to imply that they cannot be excited by radiation neither.

I'd appreciate your opinion and correction.

Dear Kai, you are right about the non-radiative plasmons in 2D; ω_p(q) starts off scaling like q at very small q, going over to the q^{1/2} at somewhat larger q. In principle, radiative coupling with the incident light would exist if light impinged on the 2D system at zero grazing angle. The issue was originally raised by Ferrell here (which I had forgotten about until now -- Stern refers to it):

http://journals.aps.org/pr/abstract/10.1103/PhysRev.111.1214

In practice, there is however radiative coupling due to surface roughness, but also due to Umklapp processes that are neglected in the uniform-electron-gas approach (in so-called jellium models), whereby wave vector is conserved only up to reciprocal-lattice vectors. This is in a way very similar to how phonon spectra are measured over the entire Brillouin zone with the aid of the Raman scattering. Further, in slab geometries, which I believe to be better models for photovoltaic devices, the surface charges on both sides of the slap interact and give rise to radiative surface plasmon excitations (one has the usual hybridization process that gives rise to splitting of energy levels; one has a similar situation in coupled quantum wells). The following review is relevant to this discussion:

http://iopscience.iop.org/0034-4885/70/1/R01/

Here (p. 10) one reads also about a third mechanism, referred to as 'attenuated total reflection' (ATR).

For the influence of surface roughness, consult in particular the following paper:

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.511

and for the ATR, the following:

http://www.fkphy.uni-duesseldorf.de/oftencitedarticles/Z-Physik-ATR.pdf

To kai, I have visited the following journals that cause my previous question

1. http://pubs.acs.org/doi/abs/10.1021/nn102295p

2. http://scitation.aip.org/content/aip/journal/apl/93/7/10.1063/1.2967471

I think from these you will give me the more specific answer about my questions..

So both papers report on results by incorporating metal nanoparticles in their devices, Au in one case, Ag in the other.

These particles exhibit a strong resonant absorption, the surface plasmon (-polariton) mode. It has been mentioned in posts above. This is also known as the Mie resonance (G. Mie, Ann. Phys. (1908)). The physics of that is treated in books like M. Born, optics (pretty old, but good), and see also book titles by Stratton, Bohren & Huffman, Kreibig & Vollmer. Also more recent books on nano-optics covering metallic structures (e.g. B. Hecht) will cover this. [Actually if you look for the citations of G. Mie's original 1908 work, you will find out that it's been cited in the thousands in recent years]. There is no shortcut to actively studying the Mie resonance at least to some extent, I believe [an electrodynamics topic, quite some math, if you really want to get at it]. Very briefly, Think of a collective motion of the conduction electrons with respect to the atoms as Behnam wrote above. There is a restoring force and therefore this is an oscillator. The resonance frequency and peak width depends on material, DF of the surrounding, coupling to the surrounding, weakly on size at small sizes. For Gold it's usually in the green range, for Ag in the violet->UV.

As I said before, this has a pretty large absorption cross section for particles with diameters well below the wavelength of the radiation (for larger particles, (Mie-) scattering becomes dominant). Your solar cell can only profit, if that absorption helps the separation of charges. The title figure of your first paper suggests that there is coupling between the plasmon mode in the particle and excitons in the organic material (the usual excitation channel in organic solar cell materials). So, the intention is to have a transfer of energy from the metallic particle to the organic material.

This is probably most efficient if the Mie absorption peak overlaps the excitonic excitation range [I have not looked into details of this in the papers - you should be able to do this on your own].

Hope this helps.

Edit: forgot to mention that the resonance splits up into more than one mode of different energy in case the shape is anisotropic (i.e. prolate ellipsoids [=cigar shape, rod shape]). The differences in peak positions may be considerable at high aspect ratio. this may or may not be usefol for a particular device.

To kai and Behnam,

Thank you so much about the detailed explanations...

You are welcome Shamjid. Two remarks. First, the excitons mentioned in the first paper, are just bound electron-hole excitations, which are generated in all photovoltaic cells. Second, the publications cited by you reveal additional information that were not included in your main, original, question.

Blending the relevant systems with (metallic) nanoparticles (whether gold and silver) breaks translational symmetry, which in a way acts as the surface roughness I referred to earlier. Breaking translational symmetry in the layers implies that momentum in these layers is not a conserved quantity, leading to 1. change of the excitation spectrum of plasmons in the active layer of the device, and 2. the fact that while the incoming light of a given energy has a certain momentum (or equivalently wavelength), on entering the active layer of the device it generates light at all momenta/wavelengths. These two aspects complicate (at least mathematically) an otherwise simply picture that one has of an idealized two-dimensional system. The above second aspect is related to the Mie scattering Kai has referred to. On this just one remark. Mie scattering is a relatively simply phenomenon in the so-called far-field region (i.e. at distances that are large in comparison with the relevant length scales of the problem), but not so in the near-field region, where in addition to propagating modes one has to do with evanescent modes, modes that decay exponentially with distance from the scatterer and do not survive to reach the far-field region. I believe that for these photovoltaic devices, near-field modes are important.

I strongly support the latter aspect Behnam mentions. I have forgotten to mention it. the excitation of the plasmon modes may locally create strong fields that will most probably play a role in the energy transfer.

This field enhancement was used e.g. in Raman scattering and is known as surface enhanced R.S. (SERS). if you look for literature on this, you will find the field enhancement discussed.