Plasmon energies measured in metals.

## Abstract

Recent experimental data shown a promising direction in employing nano-plasmonics for increasing efficiencies of the solar cells. The effect is due to metallic nanoparticles’ plasmons mediating energy transfer from the incoming e-m wave to the semiconductor in a regime violating limits in energy transitions imposed by the momentum conservation, due to translational invariance departure in surface nano-modified system. The chapter presents analysis of material dependence of near-field coupling to band electrons of surface plazmons in metallic nanoparticles deposited on the top of semiconductor substrate in nano-modified solar cells. Various materials for metal and substrate are comparatively studied upon the quantum Fermi Golden Rule approach in theoretical quantitative modeling of the plasmon-electron coupling that enhances ordinary PV effect. The material dependence of the plasmon-mediated efficiency growth in two types of solar cells, multi-crystalline Si and CIGS (copper-indium-gallium-diselenide), modified by various surface-deposited metallic nanoparticles is additionally illustrated by the experimental data.

### Keywords

- plasmons
- metallic nanoparticles
- photo effect
- solar cells

## 1. Introduction

The plasmon-mediated sunlight energy harvesting in metal-nano-modified solar cells is caused by three effects: the strong concentration of electric field of plasmon oscillations close to metallic components with local large curvature, the large amplitude of plasmon oscillations in metallic nanoparticles and the enhancement of the probability of interband excitations in semiconductor substrate caused by breaking of the translational symmetry for a nanoparticle and the dipole near-field coupling of surface plasmons with semiconductor band electrons [1, 2, 3, 4, 5, 6, 7]. The transition probability for transfer of electrons from the valence band to the conduction band in a semiconductor, essential for efficiency of the photovoltaic effect, grows due to the electric field amplitude enhancement and due to admission of all oblique transitions not here prohibited by the momentum conservation [4]. In the ordinary photo effect Kiriejew [8], the interband transitions are confined to only vertical ones between states with almost the same momentum due to the momentum conservation and the fact that the sunlight photons have very small momentum (owing to large light velocity,

However, for interaction of band electrons with surface plasmon from the metallic nanoparticle deposited on the semiconductor surface, the situation changes significantly. In the near-field regime [9], the potential of the plasmon dipole on the nanosphere is proportional to

The resulted effect of oblique interband transitions can be accounted for via the Fermi Golden Rule (FGR). According the FGR scheme [10], the probability of interband transitions is proportional to matrix element of the perturbation potential between initial and final states and summed up over all initial states in the valence band and over all final states in the conduction band assuming only the energy conservation,

The chapter is organized as follows. In Section 2, we present the quantum calculation of the efficiency of photo effect mediated by plasmons in metallic nanoparticles deposited on the top of a semiconductor photodiode. This efficiency has been accounted by application of the Fermi golden rule to the near-field coupling of dipole-plasmons with band electrons in the semiconductor substrate. The resulted transition probability is next utilized to the derivation of the plasmon damping rate due to coupling with band electrons which we present in Section 3. Section 4 addressed to analysis of the by-plasmon enhanced photo effect efficiency in various materials, including various metals for nanoparticles with plasmons and various semiconductor substrates. Section 5 contains also comparison with experiment both for laboratory Si photodiode covered with metallic nanoparticles as well as for standard solar cells, Si-multi-crystal and CIGS.

## 2. Plasmon-mediated photo effect: Fermi Golden Rule calculus of probability of electron interband excitation due to plasmons

The perturbation of electron band system in the substrate semiconductor due to the presence of dipole surface plasmon oscillations in metallic nanosphere (with a radius

and

(

The term

According to the FGR [10], the interband transition probability is proportional to

where the Bloch states in the conduction and valence bands are assumed as planar waves (for simplicity),

The matrix element,

can be found analytically by a direct integration, which gives the formula (

Next, we must sum up overall initial and final states in both bands. Thus, for the total interband transition probability we have,

where

After some also analytical integration in the above formula, we arrive at the expression,

according to assumed band dispersions,

In the latter case in Eq. (10), the following approximation was applied,

whereas in the former one,

With regard to two limiting cases,

One can notice that the above formula, Eq. (9) and its explicit form in limiting situations given by Eq. (10), is the generalization of to the ordinary photo effect, for which the transition probability is different [8],

The number of photons of the

(factor (4) corresponds here to spin degeneration of band electrons).

In the case of mediation by plasmons, all oblique interband transitions contribute, not only vertical ones (as it was for the interaction with the planar wave in the ordinary photo effect). This results in an enhancement of the transition probability for the near-field coupling in comparison to the photon (planar wave) absorption rate in a semiconductor in the ordinary photo effect. The enhancement of the probability of transition due to hopping not conserving momentum, is, however, gradually quenched with the radius

The probability of energy absorption in the semiconductor via mediation of surface plasmons per single photon incident on the metallic nanospheres,

## 3. Damping rate for plasmons in a metallic nanoparticle deposited on a top of a semicoductor

Assuming that the energy acquired by the semiconductor band system,

where

By

For example, for nanospheres of Au deposited on the Si layer, we obtain for Mie self-frequency

for light(heavy) carriers in Si,

In another scenario when the output of the plasmon energy is recovered by continuous income from the sunlight, one can consider the energy-balanced state. In an idealized case, whole incoming energy of the monochromatic

The incident sunlight dispersion covers the visible spectrum and also some UV and infra-red tails. The total efficiency of the plasmon channel corresponds to a sum (integration) overall Fourier components

where

The ratio,

This ratio turns out to be of order of

## 4. Efficiency of the light absorption channel via plasmon for various materials

Nanoparticles of gold and silver (sometimes also of copper) are mostly used in plasmon photovoltaics because their surface plasmon resonances are located within the visible light spectrum. These nanoparticles can be deposited on various semiconductor substrates with different material parameters. We list here the appropriate parameters usable for comparison with experiment for various configurations of the plasmon solar cell systems. In order to compare with the experiment, we can estimate the photocurrent in the case of a semiconductor photodiode with the metallically modified photoactive surface. This photocurrent is given by

where the ratio

In Tables 1, 2, 3, we list parameters for several semiconductor substrates and for a metallic nanoparticle few materials, which allow for comparison of the ratio

metal | Bulk pl. (eV) | Surface pl. (eV) |
---|---|---|

Li | 6.6 | 3.4 |

Na | 5.4 | 3.3 |

K | 3.8 | 2.4 |

Mg | 10.7 | 6.7 |

Al | 15.1 | 8.8 |

Fe | 10.3 | 5.0 |

Cu | 6 | 3.5 |

Ag | 3.8 | 3.5 |

Au | 4.67 | 2.7 |

metal | Au | Ag | Cu |
---|---|---|---|

Mie frequency |

Semiconductor | |||
---|---|---|---|

Si | 1.12 eV | ||

GaAs | 1.35 eV | ||

CIGS |

Formula (18) is exemplified in Figure 1 for Au nanoparticles deposited on Si semiconductor (continuous line)–this reproduces well the experimental behavior (red dashed/dotted) [14]. Both channels of photon absorption resulting in photocurrent in the semiconductor sample are included, the direct ordinary photo effect absorption with probability of transitions given by

In Figure 2, we present the spectral dependence of the plasmonic efficiency enhancement with respect to substrate change (Si, CIGS and GaAs) for the same Au nanoparticles with radius

For nanoparticles of gold (Au) and silver (Ag) of size,

Worth noting is an agreement of experimentally observed difference in the increase of the efficiency due to the plasmon effect in both cases, of mc-Si and CIGS cells, if one compares the results of application of Au and Ag particles (at the same size of metallic nanoparticles and the same their surface concentration). This behavior agrees with the theoretical study of the material dependence of the plasmon effect, as shown above. From Figures 2, 3, 4, 5, we see that for Si substrate Au nanoparticles with radii 50 nm better utilize the solar light spectrum than Ag or Cu particles (cf. Figure 5), and indeed in the experiment (cf. Figure 8) for Au nanoparticles the efficiency growth is ca. 10% larger than for Ag nanoparticles of the same size and concentration on the substrate m-Si solar cell. Interestingly, for the substrate CIGS cell, the effect is weaker and inverted, cf. Figure 9. This also is noticeable from the theoretical modeling—due to different

## 5. Conclusion

We have demonstrated by application of the Fermi Golden Rule scheme, that the efficiency of the energy transfer channel between the surface plasmon oscillations in a metallic nanoparticles and a substrate semiconductor depends on parameters of both deposited metallic particles (its radius and material) as well as on semiconductor parameters (energy gap, and effective masses of electron and holes). Found by us formula which generalizes the ordinary photo effect onto the plasmon-mediated one, agrees well with the experimental measurements in laboratory photodiode configuration. The measured ratio of photocurrent in the setup with and without metallic nano-components is compared with the theoretically predicted scenario. The quantitative consistence is obtained both in the shape of the spectral characteristics and in the particle size dependence (as illustrated for Si diode with deposited Au nanoparticles with radii 25, 40 and 50 nm). The qualitative agreement has been achieved also for complete solar cells where the plasmon effect is obscured by other elements of the long series of effects resulting in overall solar cell efficiency beyond only efficiency of the absorption of photons. We have compared the experimental data for multi-crystalline Si solar cell and CIGS (copper-indium-gallium-diselenide) solar cell covered or not with gold and silver nanoparticles with radii of order of 50 nm. The increase of the overall photovoltaic efficiency for metallically modified cells varies between 1.5 (CIGS) and 6% (Si), depending on nanoparticle concentration (for too dense concentration the efficiency drops down). A bit better increase (ca. 10% difference) causes Au nanoparticles for Si cell in comparison to Ag nanoparticles, whereas for CIGS cell, the difference between effect of Ag nanoparticles and Au ones is inverted and strongly reduced. This also agrees qualitatively with theory predictions taken into account differences in Mie frequency in Au and Ag and also different semiconductor parameters for Si and CIGS.