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 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 frequency ω, have the form [9],

$$\mathbf{E}\_{\omega} = \frac{1}{\varepsilon} \left\{ \mathbf{D}\_{0} \left( \frac{k^{2}}{R} + \frac{i\mathbf{k}}{R^{2}} - \frac{1}{R^{3}} \right) + \widehat{\mathbf{n}} (\widehat{\mathbf{n}} \cdot \mathbf{D}\_{0}) \left( -\frac{k^{2}}{R} - \frac{3i\mathbf{k}}{R^{2}} + \frac{3}{R^{3}} \right) \right\} e^{i\mathbf{k}\mathbf{R}} \tag{1}$$

and

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 semicon-

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

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>

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>

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

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>

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 conser-

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.

ffiffiffiffiffiffiffi nee<sup>2</sup> m∗ε<sup>0</sup> q

vation is rule out by the matrix element of the local dipole interaction.

� � <sup>þ</sup> <sup>ħ</sup><sup>ω</sup> <sup>¼</sup> En <sup>p</sup><sup>2</sup>

is the bulk-plasmon energy in metal [13], ne is the

<sup>l</sup> is the semiconductor band quasi-momentum scale in

<sup>R</sup><sup>2</sup> (R

ffiffi 3 p

� �, where Ep nð Þð Þ <sup>p</sup> is the

ductor, <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>

momentum q and energy ħω ¼ cq.

118 Plasmonics

plane-wave Fourier component.

(i.e., the Mie energy [11, 12], ħω<sup>p</sup> ¼ ħ

band assuming only the energy conservation, Ep p<sup>1</sup>

$$\mathbf{B}\_{\omega} = \frac{i\mathbf{k}}{\sqrt{\varepsilon}} [\mathbf{D}\_0 \times \hat{\mathbf{n}}] \left( \frac{i\mathbf{k}}{R} - \frac{1}{R^2} \right) e^{i\mathbf{k}\cdot\mathbf{R}},\tag{2}$$

(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>D</sup> <sup>¼</sup> <sup>D</sup>0e�iω<sup>t</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> c � � <sup>¼</sup> <sup>i</sup>ω<sup>t</sup> � ikR, <sup>n</sup><sup>b</sup> <sup>¼</sup> <sup>R</sup> <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> , R<sup>2</sup> and R are 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 potential added to the system Hamiltonian attains the form,

$$w = e\psi(\mathbf{R}, t) = \frac{e}{\varepsilon \mathbf{R}^2} \hat{\mathbf{n}} \cdot \mathbf{D}\_0 \sin \left(\omega t + \alpha \right) = w^+ e^{i\omega t} + \pi^- e^{-i\omega t}.\tag{3}$$

The term <sup>w</sup><sup>þ</sup> <sup>¼</sup> <sup>w</sup>� ð Þ<sup>∗</sup> <sup>¼</sup> <sup>e</sup> εR<sup>2</sup> eiα <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. According to the FGR [10], the interband transition probability is proportional to

$$w(\mathbf{k}\_1, \mathbf{k}\_2) = \frac{2\pi}{\hbar} | < \mathbf{k}\_1 | w^+ | \mathbf{k}\_2 > |^2 \delta \left( E\_p(\mathbf{k}\_1) - E\_n(\mathbf{k}\_2) + \hbar \omega \right) \tag{4}$$

where the Bloch states in the conduction and valence bands are assumed as planar waves (for simplicity), <sup>Ψ</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>=<sup>2</sup> <sup>e</sup><sup>i</sup>k�R�iEn pð Þð Þ <sup>k</sup> <sup>t</sup>=<sup>ħ</sup>, Epð Þ¼� <sup>k</sup> <sup>ħ</sup><sup>2</sup> k2 2m<sup>∗</sup> p � Eg, Enð Þ¼ <sup>k</sup> <sup>ħ</sup>2k<sup>2</sup> 2m<sup>∗</sup> n (indices n, p refer to electrons from the conduction and valence bands, respectively, Eg is the forbidden gap).

The matrix element,

$$<\mathbf{k}\_1|w^+|\mathbf{k}\_2> = \frac{1}{\left(2\pi\right)^3} \int d^3 \mathbf{R} \frac{e}{\varepsilon \mathcal{D}} e^{i\mathbf{\hat{n}}} \widehat{\mathbf{n}} \cdot \mathbf{D}\_0 \frac{1}{R^2} e^{-i(\mathbf{k}\_1 - \mathbf{k}\_2) \cdot \mathbf{R}}.\tag{5}$$

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

<sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup> <sup>≈</sup> ð Þ for <sup>a</sup><sup>ξ</sup> <sup>≫</sup> <sup>1</sup> <sup>1</sup>

<sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup> <sup>¼</sup> <sup>π</sup>=4.

< 0:02

> 0:02

8 >>><

>>>:

masses and Eg. Thus for nanoparticles with radii a > 2 nm, the first regime holds only close to Eg (less than the 2% distance to limiting Eg), whereas the second regime holds in the rest of the

<sup>&</sup>gt; <sup>0</sup>:<sup>5</sup> � <sup>10</sup>�<sup>9</sup>

<sup>&</sup>lt; <sup>0</sup>:<sup>5</sup> � <sup>10</sup>�<sup>9</sup>

widens considerably (to ca. 50% relative distance to Eg), but holds only for ultrasmall size of nanoparticles (a < 0:5 nm). For larger nanospheres, e.g., with a > 10 nm, the second regime is

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

The number of photons of the ω e-m wave with electric field component amplitude E<sup>0</sup> in the

<sup>¼</sup> 4 4ð Þ ffiffiffi 2 p 3

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 a growth, as expressed by Eq. (10).

εE<sup>2</sup> 0V 8πħω

� � <sup>ħ</sup><sup>ω</sup> � Eg

� �, hence the probability of single photon absorption by the semicon-

μ<sup>5</sup>=<sup>2</sup>e<sup>2</sup> m<sup>∗</sup><sup>2</sup>

� �<sup>3</sup>=<sup>2</sup>

<sup>p</sup> ωεħ<sup>3</sup> <sup>ħ</sup><sup>ω</sup> � Eg

� �<sup>3</sup>=<sup>2</sup>

½ � m for

aξ ð∞ 0

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

d xa ð Þ <sup>ξ</sup> sin <sup>2</sup>ð Þ xa<sup>ξ</sup>

p

ð Þ xa<sup>ξ</sup> <sup>2</sup> <sup>¼</sup> <sup>π</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ð Þ <sup>ħ</sup>ω�Eg <sup>m</sup><sup>∗</sup>

< 0:5

> 0:5

, and this range weakly depends on effective

ħω � Eg Eg

½ � <sup>m</sup> for <sup>ħ</sup><sup>ω</sup> � Eg Eg

<sup>n</sup>þm<sup>∗</sup> ð Þ<sup>p</sup>

<sup>ħ</sup> , we see that

121

, the first region

: (11)

, (12)

<sup>2</sup>a<sup>ξ</sup> ,

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

ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup> dx ffiffiffiffiffiffiffiffiffiffiffiffiffi

ħω � Eg Eg

With regard to two limiting cases, aξ ≪ 1 or aξ ≫ 1, ξ ¼

½ � <sup>m</sup> for <sup>ħ</sup><sup>ω</sup> � Eg Eg

<sup>δ</sup>w<sup>0</sup> <sup>¼</sup> <sup>4</sup> ffiffiffi

0V 8πħω

q<sup>0</sup> ¼ δw<sup>0</sup>

2 p 3

ductor per time unit, attains the form in the ordinary photo effect [8],

εE<sup>2</sup> 0V 8πħω � ��<sup>1</sup>

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

μ<sup>5</sup>=<sup>2</sup>e<sup>2</sup> m<sup>∗</sup><sup>2</sup> <sup>p</sup> ωεħ<sup>3</sup>

½ � m for

ð1 0

whereas in the former one, Ð <sup>1</sup>

8 >>><

>>>:

thus dominating.

probability is different [8],

volume <sup>V</sup> equals to, <sup>ε</sup>E<sup>2</sup>

<sup>&</sup>gt; <sup>2</sup> � <sup>10</sup>�<sup>9</sup>

<sup>&</sup>lt; <sup>2</sup> � <sup>10</sup>�<sup>9</sup>

ω domain. For comparison, a≃ 1=ξ≃

a ≃1=ξ≃

dx sin <sup>2</sup>ð Þ xa<sup>ξ</sup> ð Þ xa<sup>ξ</sup> <sup>2</sup>

can be found analytically by a direct integration, which gives the formula (q ¼ k<sup>1</sup> � k2),

$$<\mathbf{k}\_1|w^+|\mathbf{k}\_2> = \frac{-1}{\left(2\pi\right)^3} \frac{e\mathbf{e}^{i\alpha}}{\varepsilon} D\_0 \cos\Theta(2\pi) \int\_a^\mathbf{\kappa} d\mathbf{R} \frac{1}{q} \frac{d}{d\mathbf{R}} \frac{\sin q\mathbf{R}}{q\mathbf{R}} = \frac{1}{\left(2\pi\right)^2} \frac{e\mathbf{e}^{i\alpha}}{\varepsilon} \frac{\mathbf{D}\_0 \cdot \mathbf{q}}{q^2} \frac{\sin qa}{qa}.\tag{6}$$

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

$$
\delta w = \int d^3 k\_1 \int d^3 k\_2 \left[ f\_1(1 - f\_2) w(\mathbf{k}\_1, \mathbf{k}\_2) - f\_2(1 - f\_1) w(\mathbf{k}\_2, \mathbf{k}\_1) \right] \tag{7}
$$

where f <sup>1</sup>, f <sup>2</sup> assign the temperature dependent distribution functions (Fermi-Dirac distribution functions) for initial and final states, respectively. For room temperatures, f <sup>2</sup> ≃ 0 and f <sup>1</sup> ≃1, which leads to,

$$
\delta w = \int d^3 k\_1 \int d^3 k\_2 \cdot w(\mathbf{k}\_1, \mathbf{k}\_2). \tag{8}
$$

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

$$\begin{split} \delta w &= \frac{4}{3} \frac{\mu^2 \left( m\_n^\* + m\_p^\* \right) 2 \left( \hbar \omega - E\_\xi \right) e^2 D\_0^2}{\sqrt{m\_n^\* m\_p^\*} 2 \pi \hbar^5 \varepsilon^2} \int\_0^1 d\mathbf{x} \, \frac{\sin^2(\mathbf{x} \mathbf{a} \boldsymbol{\xi})}{\left( \mathbf{x} \mathbf{a} \boldsymbol{\xi} \right)^2} \sqrt{1 - \mathbf{x}^2} \\ &= \frac{4}{3} \frac{\mu^2}{\sqrt{m\_n^\* m\_p^\*}} \frac{e^2 D\_0^2}{2 \pi \hbar^3 \varepsilon^2} \xi^2 \int\_0^1 d\mathbf{x} \, \frac{\sin^2(\mathbf{x} \mathbf{a} \boldsymbol{\xi})}{\left( \mathbf{x} \mathbf{a} \boldsymbol{\xi} \right)^2} \sqrt{1 - \mathbf{x}^2}, \end{split} \tag{9}$$

according to assumed band dispersions, m<sup>∗</sup> <sup>n</sup> and m<sup>∗</sup> <sup>p</sup> denote the effective masses of electrons and holes, <sup>μ</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup> nm<sup>∗</sup> p m<sup>∗</sup> nþm<sup>∗</sup> p is the reduced mass, the parameter ξ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ð Þ <sup>ħ</sup>ω�Eg <sup>m</sup><sup>∗</sup> <sup>n</sup>þm<sup>∗</sup> ð Þ<sup>p</sup> p <sup>ħ</sup> . In limiting cases for a nanoparticle radius a, we finally obtain,

$$\delta w = \begin{cases} \frac{4}{3} \frac{\mu \sqrt{m\_n^\* m\_p^\*} (\hbar \omega - E\_\S) e^2 D\_0^2}{\hbar^5 \varepsilon^2}, & \text{for } a \xi \ll 1, \\\\ \frac{4}{3} \frac{\mu^{3/2} \sqrt{2} \sqrt{\hbar \omega - E\_\S} e^2 D\_0^2}{a \hbar^4 \varepsilon^2}, & \text{for } a \xi \gg 1. \end{cases} \tag{10}$$

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

$$\int\_0^1 d\mathbf{x} \frac{\sin^2(\mathbf{x}a\underline{\xi})}{\left(\mathbf{x}a\underline{\xi}\right)^2} \sqrt{1-\mathbf{x}^2} \approx \left(\text{for } a\underline{\xi} \gg 1\right) \frac{1}{a\underline{\xi}} \int\_0^\infty d\left(\mathbf{x}a\underline{\xi}\right) \frac{\sin^2(\mathbf{x}a\underline{\xi})}{\left(\mathbf{x}a\underline{\xi}\right)^2} = \frac{\pi}{2a\underline{\xi}}.$$

whereas in the former one, Ð <sup>1</sup> <sup>0</sup> dx ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup> <sup>¼</sup> <sup>π</sup>=4.

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

trons from the conduction and valence bands, respectively, Eg is the forbidden gap).

can be found analytically by a direct integration, which gives the formula (q ¼ k<sup>1</sup> � k2),

ð∞ a dR<sup>1</sup> q d dR

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

where f <sup>1</sup>, f <sup>2</sup> assign the temperature dependent distribution functions (Fermi-Dirac distribution functions) for initial and final states, respectively. For room temperatures, f <sup>2</sup> ≃ 0 and f <sup>1</sup> ≃1,

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

2 ħω � Eg � �e<sup>2</sup>D<sup>2</sup>

ε2

� �wð Þ� <sup>k</sup>1; <sup>k</sup><sup>2</sup> <sup>f</sup> <sup>2</sup> <sup>1</sup> � <sup>f</sup> <sup>1</sup>

0

dx sin <sup>2</sup>ð Þ xa<sup>ξ</sup> ð Þ xa<sup>ξ</sup> <sup>2</sup>

<sup>n</sup> and m<sup>∗</sup>

� �e<sup>2</sup>D<sup>2</sup>

0

<sup>ε</sup><sup>2</sup> , for <sup>a</sup><sup>ξ</sup> <sup>≫</sup> <sup>1</sup>:

0

<sup>ε</sup><sup>2</sup> , for <sup>a</sup><sup>ξ</sup> <sup>≪</sup> <sup>1</sup>,

ð1 0

dx sin <sup>2</sup>ð Þ xa<sup>ξ</sup> ð Þ xa<sup>ξ</sup> <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup> ,

p

ð Þ 2π 3 ð d3 R e ε2i e <sup>i</sup><sup>α</sup>n<sup>b</sup> � <sup>D</sup><sup>0</sup>

D<sup>0</sup> cos Θð Þ 2π

k<sup>2</sup> f <sup>1</sup> 1 � f <sup>2</sup>

δw ¼ ð d3 k1 ð d3

<sup>n</sup> <sup>þ</sup> <sup>m</sup><sup>∗</sup> p � �

> ffiffiffiffiffiffiffiffiffiffiffiffi m<sup>∗</sup> nm<sup>∗</sup> p p 2πħ<sup>5</sup>

e<sup>2</sup>D<sup>2</sup> 0 2πħ<sup>3</sup> ε2 ξ2 ð1 0

is the reduced mass, the parameter ξ ¼

ħ5

aħ<sup>4</sup>

μ ffiffiffiffiffiffiffiffiffiffiffiffi m<sup>∗</sup> nm<sup>∗</sup> p <sup>p</sup> <sup>ħ</sup><sup>ω</sup> � Eg

μ<sup>3</sup>=<sup>2</sup> ffiffiffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ħω � Eg p e<sup>2</sup>D<sup>2</sup>

2m<sup>∗</sup> p � Eg, Enð Þ¼ <sup>k</sup> <sup>ħ</sup><sup>2</sup>

1 R2 e

sinqR qR <sup>¼</sup> <sup>1</sup> ð Þ 2π 2 ee<sup>i</sup><sup>α</sup> ε

k2 2m<sup>∗</sup> n

� �wð Þ <sup>k</sup>2; <sup>k</sup><sup>1</sup> � �, (7)

k<sup>2</sup> � wð Þ k1; k<sup>2</sup> : (8)

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup><sup>2</sup> <sup>p</sup>

<sup>p</sup> denote the effective masses of electrons

<sup>n</sup>þm<sup>∗</sup> ð Þ<sup>p</sup>

<sup>ħ</sup> . In limiting cases

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ð Þ <sup>ħ</sup>ω�Eg <sup>m</sup><sup>∗</sup>

(indices n, p refer to elec-

�ið Þ� <sup>k</sup>1�k2 <sup>R</sup>: (5)

D<sup>0</sup> � q q2

sinqa

qa : (6)

(9)

(10)

ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>=<sup>2</sup> <sup>e</sup><sup>i</sup>k�R�iEn pð Þð Þ <sup>k</sup> <sup>t</sup>=<sup>ħ</sup>, Epð Þ¼� <sup>k</sup> <sup>ħ</sup>2k<sup>2</sup>

<sup>&</sup>lt; <sup>k</sup>1∣wþ∣k<sup>2</sup> <sup>&</sup>gt;<sup>¼</sup> <sup>1</sup>

ee<sup>i</sup><sup>α</sup> ε

ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>

simplicity), <sup>Ψ</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>

120 Plasmonics

The matrix element,

<sup>&</sup>lt; <sup>k</sup>1∣wþ∣k<sup>2</sup> <sup>&</sup>gt;<sup>¼</sup> �<sup>1</sup>

δw ¼ ð d3 k1 ð d3

<sup>δ</sup><sup>w</sup> <sup>¼</sup> <sup>4</sup> 3

> ¼ 4 3

according to assumed band dispersions, m<sup>∗</sup>

for a nanoparticle radius a, we finally obtain,

δw ¼

4 3

8 >>>>><

> 4 3

>>>>>:

nm<sup>∗</sup> p m<sup>∗</sup> nþm<sup>∗</sup> p μ<sup>2</sup> m<sup>∗</sup>

μ2 ffiffiffiffiffiffiffiffiffiffiffiffi m<sup>∗</sup> nm<sup>∗</sup> p

p

transition probability we have,

which leads to,

and holes, <sup>μ</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup>

With regard to two limiting cases, aξ ≪ 1 or aξ ≫ 1, ξ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ð Þ <sup>ħ</sup>ω�Eg <sup>m</sup><sup>∗</sup> <sup>n</sup>þm<sup>∗</sup> ð Þ<sup>p</sup> p <sup>ħ</sup> , we see that a ≃1=ξ≃ <sup>&</sup>gt; <sup>2</sup> � <sup>10</sup>�<sup>9</sup> ½ � <sup>m</sup> for <sup>ħ</sup><sup>ω</sup> � Eg Eg < 0:02 <sup>&</sup>lt; <sup>2</sup> � <sup>10</sup>�<sup>9</sup> ½ � m for ħω � Eg Eg > 0:02 8 >>>< >>>: , and this range weakly depends on effective

masses and Eg. Thus for nanoparticles with radii a > 2 nm, the first regime holds only close to Eg (less than the 2% distance to limiting Eg), whereas the second regime holds in the rest of the

$$\mu \text{ domain. For comparison, } a \simeq 1/\xi \simeq \begin{cases} > 0.5 \times 10^{-9} [m] \text{ for } \frac{\hbar \omega - E\_{\xi}}{E\_{\xi}} < 0.5\\ < 0.5 \times 10^{-9} [m] \text{ for } \frac{\hbar \omega - E\_{\xi}}{E\_{\xi}} > 0.5 \end{cases} \text{, the first region, } \mu \text{ for } \frac{\hbar \omega - E\_{\xi}}{E\_{\xi}} > 0.5$$

widens considerably (to ca. 50% relative distance to Eg), but holds only for ultrasmall size of nanoparticles (a < 0:5 nm). For larger nanospheres, e.g., with a > 10 nm, the second regime is thus dominating.

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],

$$
\delta w\_0 = \frac{4\sqrt{2}}{3} \frac{\mu^{5/2} e^2}{m\_p^{\*2} \alpha \varepsilon \hbar^3} \left(\frac{\varepsilon E\_0^2 V}{8\pi \hbar \omega}\right) \left(\hbar \omega - E\_\S\right)^{3/2}.\tag{11}
$$

The number of photons of the ω e-m wave with electric field component amplitude E<sup>0</sup> in the volume <sup>V</sup> equals to, <sup>ε</sup>E<sup>2</sup> 0V 8πħω � �, hence the probability of single photon absorption by the semiconductor per time unit, attains the form in the ordinary photo effect [8],

$$q\_0 = \delta w\_0 \left(\frac{\varepsilon E\_0^2 V}{8\pi \hbar \omega}\right)^{-1} = \frac{4(4)\sqrt{2}}{3} \frac{\mu^{5/2} e^2}{m\_p^{\*2} \omega \varepsilon \hbar^3} \left(\hbar \omega - E\_\mathrm{g}\right)^{3/2},\tag{12}$$

(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 a growth, as expressed by Eq. (10).

The probability of energy absorption in the semiconductor via mediation of surface plasmons per single photon incident on the metallic nanospheres, qm, equals to the product of δw (given by Eq. (10)) and the number, Nm, of metallic nanoparticles divided by photon density with additional phenomenological factor β responsible for all effects not directly accounted for (as deposition separation and surface properties reducing the coupling strength, as well as energy losses due to electron scattering and irradiation to far-field zone (Lorentz friction [9]) into upper hemisphere, if the metallic nanoparticle is not completely embedded in the substrate semiconductor medium),

$$\eta\_m = \beta N\_m \delta w \left(\frac{\varepsilon E\_0^2 V}{8\pi \hbar \omega}\right)^{-1}.\tag{13}$$

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

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

ffiffiffiffiffiffiffiffiffiffiffiffi m<sup>∗</sup> nm<sup>∗</sup> p

and Eg ¼ 1:14 eV, ħω<sup>1</sup> ¼ 2:72 eV. For these parameters and nanospheres with

<sup>p</sup> , the frequency of plasma oscillation equals to the driven electric field

<sup>m</sup> , for <sup>a</sup><sup>ξ</sup> <sup>≪</sup> <sup>1</sup>,

<sup>p</sup> ¼ 0:16 0ð Þ :52 m, m is the bare electron

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

<sup>τ</sup><sup>0</sup> (15). This damping causes a red shift of a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 <sup>1</sup> � <sup>τ</sup>0�<sup>2</sup> � � q

. The red shift is

(16)

123

, for aξ ≫ 1,

p

μ m � �<sup>3</sup>=<sup>2</sup>

<sup>1</sup>½ � nm � �<sup>3</sup> <sup>μ</sup> m

<sup>1</sup>½ � nm � �<sup>2</sup>

<sup>n</sup> <sup>¼</sup> <sup>0</sup>:19 0ð Þ :<sup>98</sup> <sup>m</sup>, <sup>m</sup><sup>∗</sup>

the radius a in the range of 5 � 50 nm, the lower case of Eq. (16) applies (at ω ¼ ω1). The

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 ω e-m wave is transferred to the semiconductor via plasmons, and we deal with the stationary behavior of a driven and damped oscillator for plasmons. Even though the free undamped plasmon has the Mie self-resonance

frequency, ω, of the incident e-m wave of photons. Because of an instant leakage of the plasmon energy in near-field to semiconductor substrate, this large damping of plasmon causes a red-shift and widening of the resonance, as for every damped and driven oscillator. The widened resonance enables the energy transfer from plasmons to electrons to embrace also frequencies lower or larger than Mie frequency but limited from below by the semicon-

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 ω > Eg=ħ of light interfered with intensity distribution of sunlight spectrum. To model this behavior, it is necessary to consider separately each single monochromatic e-m mode, i.e., a Fourier component ω. Its electric field excites plasmon with this frequency

resonance and reduces the resonance amplitude, which in turn allows for the accommodation to the balance of energy transfer to the semiconductor with incident sunlight e-m wave energy intensity (defined by it electric field amplitude E0) at the frequency ω. Within this damped and driven oscillator model, the amplitude of plasmon oscillations D0ð Þ ω is constant in time and

proportional to 1<sup>=</sup> <sup>ω</sup>1τ0<sup>2</sup> � �. In the case of the described energy transfer balance, one obtains

with corresponding amplitude � τ<sup>0</sup>

<sup>q</sup> : The extremum of red-shifted resonance is attained at

= 2

44:092β a nm½ �

13:648β a nm½ �

parameter β fitted from the experimental data [4, 14] equals to ca 0.001.

ω ¼ ω1,

mass, <sup>μ</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup>

frequency, <sup>ω</sup><sup>1</sup> <sup>¼</sup> <sup>ω</sup><sup>p</sup>

ductor gap Eg=ħ.

ω<sup>m</sup> ¼ ω<sup>1</sup>

q

according to Eq. (10),

1 τ0 ω1 ¼

for light(heavy) carriers in Si, m<sup>∗</sup>

ffiffi 3

and this plasmon is damping with the rate <sup>1</sup>

ω2 1�ω<sup>2</sup> ð Þ<sup>2</sup>

<sup>þ</sup>4ω2=τ0<sup>2</sup>

shaped by <sup>f</sup>ð Þ¼ <sup>ω</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>2</sup> <sup>ω</sup>1τ<sup>0</sup> ð Þ�<sup>2</sup>

nm<sup>∗</sup> p m<sup>∗</sup> nþm<sup>∗</sup> p 8 >>>><

>>>>:
