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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and the dipole near-field coupling of surface plasmons with semiconductor band electrons [1–7]. 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, c) which almost does not change electron momentum at scattering: for excitation energy ħω beyond the forbidden gap, Eg, in the substrate semiconductor, <sup>ħ</sup><sup>ω</sup> <sup>¼</sup> cq gives <sup>q</sup> <sup>≪</sup> <sup>p</sup>, where <sup>p</sup> � <sup>π</sup><sup>ħ</sup> <sup>l</sup> is the semiconductor band quasi-momentum scale in the Brillouin zone (l denotes here the elementary cell linear size). Thus the change of the band electron momentum p<sup>1</sup> ¼ p<sup>2</sup> þ q is negligible on the scale of the Brillouin zone and p<sup>1</sup> ≃p<sup>2</sup> (because <sup>c</sup> <sup>¼</sup> 108 m/s) and only the vertical, conserving momentum, interband transitions contribute to the ordinary photo effect, i.e., when the transition is caused by free photons with momentum q and energy ħω ¼ cq.

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-

Plasmonic Enhancement of Solar Cells Efficiency: Material Dependence in Semiconductor Metallic Surface Nano…

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 a) deposited on the semiconductor surface, has the form of the potential of the e-m field of an oscillating dipole. The Fourier components of the electric E<sup>ω</sup> and magnetic B<sup>ω</sup> fields produced in the distance R from the center of considered nanosphere with the dipole of surface plasmon with the fre-

<sup>þ</sup> <sup>n</sup>bð Þ� <sup>n</sup><sup>b</sup> � <sup>D</sup><sup>0</sup>

<sup>R</sup> � <sup>1</sup> R2 � �<sup>e</sup>

. In Eqs. (1) and (2), we used the notation for the retarded argument, <sup>i</sup><sup>ω</sup> <sup>t</sup> � <sup>R</sup>

� � � � <sup>e</sup>

<sup>ε</sup> <sup>p</sup> ½ � <sup>D</sup><sup>0</sup> � <sup>n</sup><sup>b</sup> ik

(ε is the dielectric permittivity). In the case of the spherical symmetry, the dipole of plasmon is considered as pinned to the center of the nanosphere (the origin of the reference frame system),

<sup>R</sup>, <sup>ω</sup> <sup>¼</sup> ck, momentum <sup>p</sup> <sup>¼</sup> <sup>ħ</sup>k. The terms with denominators <sup>R</sup><sup>3</sup>

referred to near-, medium- and far-field zones of the dipole radiation, correspondingly. Because we consider the interaction with a closely adjacent layer of the substrate semiconductor, all terms with denominators R<sup>2</sup> and R we neglect as small in comparison to the term with R<sup>3</sup> denominator—this is the near-field zone approximation (the magnetic field disappears and the electric field is of the form of a static dipole field [9]). Therefore the related perturbation

<sup>ε</sup>R<sup>2</sup> <sup>n</sup><sup>b</sup> � <sup>D</sup><sup>0</sup> sin ð Þ¼ <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>α</sup> <sup>w</sup>þ<sup>e</sup>

2

<sup>2</sup><sup>i</sup> <sup>n</sup><sup>b</sup> � <sup>D</sup><sup>0</sup> describes emission, i.e., the case of our interest.

k 2 <sup>R</sup> � <sup>3</sup>ik <sup>R</sup><sup>2</sup> <sup>þ</sup>

3 R3

<sup>i</sup>ω<sup>t</sup> <sup>þ</sup> <sup>w</sup>�<sup>e</sup>

�iωt

<sup>δ</sup> Epð Þ� <sup>k</sup><sup>1</sup> Enð Þþ <sup>k</sup><sup>2</sup> <sup>ħ</sup><sup>ω</sup> � �, (4)

ikR, (2)

http://dx.doi.org/10.5772/intechopen.79113

119

ikR (1)

c � � <sup>¼</sup>

, R<sup>2</sup> and R are

: (3)

2. Plasmon-mediated photo effect: Fermi Golden Rule calculus of

probability of electron interband excitation due to plasmons

� �

<sup>B</sup><sup>ω</sup> <sup>¼</sup> ik ffiffiffi

potential added to the system Hamiltonian attains the form,

2π

e

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

<sup>ħ</sup> <sup>&</sup>lt; <sup>k</sup>1jw<sup>þ</sup> j j <sup>j</sup>k<sup>2</sup> <sup>&</sup>gt;

w ¼ eψð Þ¼ R; t

εR<sup>2</sup> eiα

wð Þ¼ k1; k<sup>2</sup>

crystal and CIGS.

quency ω, have the form [9],

and

<sup>D</sup> <sup>¼</sup> <sup>D</sup>0e�iω<sup>t</sup>

<sup>i</sup>ω<sup>t</sup> � ikR, <sup>n</sup><sup>b</sup> <sup>¼</sup> <sup>R</sup>

The term <sup>w</sup><sup>þ</sup> <sup>¼</sup> <sup>w</sup>� ð Þ<sup>∗</sup> <sup>¼</sup> <sup>e</sup>

<sup>E</sup><sup>ω</sup> <sup>¼</sup> <sup>1</sup> ε D<sup>0</sup> k 2 R þ ik <sup>R</sup><sup>2</sup> � <sup>1</sup> R3

However, for interaction of band electrons with surface plasmon from the metallic nanoparticle deposited on the semiconductor surface, the situation changes significantly. In the nearfield regime [9], the potential of the plasmon dipole on the nanosphere is proportional to <sup>1</sup> <sup>R</sup><sup>2</sup> (R is a distance from the sphere center), which has the infinite decomposition in Fourier picture and thus overlaps with all quasi-momenta in the substrate semiconductor Brillouin zone. This is in contrary to the potential of the free photon which contributes via only single e<sup>i</sup>ð Þ <sup>q</sup>�r�ħω<sup>t</sup> <sup>=</sup><sup>ħ</sup> plane-wave Fourier component.

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, Ep p<sup>1</sup> � � <sup>þ</sup> <sup>ħ</sup><sup>ω</sup> <sup>¼</sup> En <sup>p</sup><sup>2</sup> � �, where Ep nð Þð Þ <sup>p</sup> is the valence-p (conduction-n) band dispersion and ħω is the excitation energy related to damped and forced by sunlight surface plasmon oscillations with the bare self-energy value <sup>ħ</sup>ω<sup>1</sup> <sup>¼</sup> <sup>ħ</sup>ω<sup>p</sup> ffiffi 3 p (i.e., the Mie energy [11, 12], ħω<sup>p</sup> ¼ ħ ffiffiffiffiffiffiffi nee<sup>2</sup> m∗ε<sup>0</sup> q is the bulk-plasmon energy in metal [13], ne is the density of collective electrons in metal, m<sup>∗</sup> is the effective mass of electron in metal, e is the

electron charge and ε<sup>0</sup> is the dielectric constant) with not-defined momentum, however. The initial momentum, p1, and the final one, p2, can be arbitrary because the momentum conservation is rule out by the matrix element of the local dipole interaction.

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-multicrystal and CIGS.
