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, A, is equal to the output of plasmon oscillation energy (resulting in plasmon damping), one can estimate the corresponding damping rate of plasmon oscillations. Namely, at the damped (lowering in time) plasmon amplitude <sup>D</sup>0ðÞ¼ <sup>t</sup> <sup>D</sup>0e�t=τ<sup>0</sup> , one finds for a total transmitted energy,

$$\mathcal{A} = \beta \int\_0^{\infty} \delta w \hbar \omega dt = \beta \hbar \omega \delta w r'/2 = \begin{cases} \frac{2 \beta \omega r' \mu}{3} \frac{\sqrt{m\_n^\* m\_p^\*} (\hbar \omega - E\_\xi) e^2 D\_0^2}{\hbar^4 \varepsilon^2}, & \text{for } a \xi \ll 1, \\\\ \frac{2 \beta \omega r' \mu^{3/2} \sqrt{2} \sqrt{\hbar \omega - E\_\xi} e^2 D\_0^2}{\hbar \hbar^3 \varepsilon^2}, & \text{for } a \xi \gg 1, \end{cases} \tag{14}$$

where τ<sup>0</sup> is the damping time-rate and β accounts for losses (not included in the model). Comparing the value of A given by the formula (14) with the energy loss of damping plasmon estimated in Ref. [4] (the initial energy of the plasmon oscillations which has been transferred step-by-step to the semiconductor, <sup>A</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup> 0 <sup>2</sup>ε<sup>a</sup>3), one can find

$$\frac{1}{\frac{1}{\tau'}} = \begin{cases} \frac{4\beta\omega\mu\sqrt{m\_n^\* m\_p^\*} (\hbar\omega - E\_\beta) e^2 a^3}{3\hbar^4 \varepsilon}, & \text{for } a\xi \ll 1, \\\\ \frac{4\beta\omega\mu^{3/2}\sqrt{2}\sqrt{\hbar\omega - E\_\beta} e^2 a^2}{3\hbar^3 \varepsilon}, & \text{for } a\xi \gg 1. \end{cases} \tag{15}$$

By τ<sup>0</sup> , we denote here a large damping of plasmons due to energy transfer to the semiconductor substrate highly exceeding the internal damping, characterized by τ, due to scattering of electrons inside the metallic nanoparticle [4] (<sup>1</sup> <sup>τ</sup> ≪ <sup>1</sup> <sup>τ</sup>0). We neglect also the irradiation to farfield upper hemisphere zone of plasmon energy due to the Lorentz friction, which is also smaller than near-field zone energy transfer to the substrate [4].

For example, for nanospheres of Au deposited on the Si layer, we obtain for Mie self-frequency ω ¼ ω1,

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

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

: (13)

qm ¼ βNmδw

3. Damping rate for plasmons in a metallic nanoparticle deposited

2 3 βωτ<sup>0</sup>

8 >>>><

>>>>:

0

3ħ<sup>4</sup>

2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ħω � Eg p e<sup>2</sup>a<sup>2</sup>

3ħ<sup>3</sup>

2 3 βωτ<sup>0</sup>

where τ<sup>0</sup> is the damping time-rate and β accounts for losses (not included in the model). Comparing the value of A given by the formula (14) with the energy loss of damping plasmon estimated in Ref. [4] (the initial energy of the plasmon oscillations which has been transferred

<sup>2</sup>ε<sup>a</sup>3), one can find

� �e<sup>2</sup>a<sup>3</sup>

, we denote here a large damping of plasmons due to energy transfer to the semiconductor substrate highly exceeding the internal damping, characterized by τ, due to scattering of

<sup>τ</sup> ≪ <sup>1</sup>

field upper hemisphere zone of plasmon energy due to the Lorentz friction, which is also

=2 ¼

4βωμ ffiffiffiffiffiffiffiffiffiffiffiffi m<sup>∗</sup> nm<sup>∗</sup> p <sup>p</sup> <sup>ħ</sup><sup>ω</sup> � Eg

4βωμ<sup>3</sup>=<sup>2</sup> ffiffiffi

Assuming that the energy acquired by the semiconductor band system, A, is equal to the output of plasmon oscillation energy (resulting in plasmon damping), one can estimate the corresponding damping rate of plasmon oscillations. Namely, at the damped (lowering in

> 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>

ħ4

aħ<sup>3</sup>

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

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

, one finds for a total transmitted energy,

� �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>,

(14)

(15)

0

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

<sup>τ</sup>0). We neglect also the irradiation to far-

semiconductor medium),

122 Plasmonics

on a top of a semicoductor

A ¼ β ð ∞

By τ<sup>0</sup>

0

time) plasmon amplitude <sup>D</sup>0ðÞ¼ <sup>t</sup> <sup>D</sup>0e�t=τ<sup>0</sup>

step-by-step to the semiconductor, <sup>A</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup>

1 τ<sup>0</sup> ¼

electrons inside the metallic nanoparticle [4] (<sup>1</sup>

8 >>>><

>>>>:

smaller than near-field zone energy transfer to the substrate [4].

δwħωdt ¼ βħωδwτ<sup>0</sup>

$$\frac{1}{\pi^\* \omega\_1} = \begin{cases} 44.092 \beta \left(\frac{a|nm|}{1|nm|}\right)^3 \frac{\mu}{m} \frac{\sqrt{m\_n^\* m\_p^\*}}{m}, & \text{for } a \xi \ll 1, \\\\ 13.648 \beta \left(\frac{a|nm|}{1|nm|}\right)^2 \left(\frac{\mu}{m}\right)^{3/2}, & \text{for } a \xi \gg 1, \end{cases} \tag{16}$$

for light(heavy) carriers in Si, m<sup>∗</sup> <sup>n</sup> <sup>¼</sup> <sup>0</sup>:19 0ð Þ :<sup>98</sup> <sup>m</sup>, <sup>m</sup><sup>∗</sup> <sup>p</sup> ¼ 0:16 0ð Þ :52 m, m is the bare electron mass, <sup>μ</sup> <sup>¼</sup> <sup>m</sup><sup>∗</sup> nm<sup>∗</sup> p m<sup>∗</sup> nþm<sup>∗</sup> p and Eg ¼ 1:14 eV, ħω<sup>1</sup> ¼ 2:72 eV. For these parameters and nanospheres with the radius a in the range of 5 � 50 nm, the lower case of Eq. (16) applies (at ω ¼ ω1). The parameter β fitted from the experimental data [4, 14] equals to ca 0.001.

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, <sup>ω</sup><sup>1</sup> <sup>¼</sup> <sup>ω</sup><sup>p</sup> ffiffi 3 <sup>p</sup> , the frequency of plasma oscillation equals to the driven electric field 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 semiconductor gap Eg=ħ.

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 and this plasmon is damping with the rate <sup>1</sup> <sup>τ</sup><sup>0</sup> (15). This damping causes a red shift of a 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 shaped by <sup>f</sup>ð Þ¼ <sup>ω</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 1�ω<sup>2</sup> ð Þ<sup>2</sup> <sup>þ</sup>4ω2=τ0<sup>2</sup> <sup>q</sup> : The extremum of red-shifted resonance is attained at

ω<sup>m</sup> ¼ ω<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>2</sup> <sup>ω</sup>1τ<sup>0</sup> ð Þ�<sup>2</sup> q with corresponding amplitude � τ<sup>0</sup> = 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 <sup>1</sup> � <sup>τ</sup>0�<sup>2</sup> � � q . The red shift is proportional to 1<sup>=</sup> <sup>ω</sup>1τ0<sup>2</sup> � �. In the case of the described energy transfer balance, one obtains according to Eq. (10),

$$q\_m = \begin{cases} \beta \mathbb{C}\_0 \frac{128}{9} \pi^2 a^3 \frac{\mu^4 \sqrt{\mu\_n^\* \mu\_p^\*}}{m^2} (\hbar \omega - \mathbb{E}\_8) \frac{e^6 n\_\epsilon^2 \omega}{\hbar^4 \varepsilon^3} f^2(\omega), & \text{for } a \xi \ll 1, \\\beta \mathbb{C}\_0 \frac{128}{9} \sqrt{2} \pi^2 a^2 \frac{\mu^{3/2}}{m^2} \sqrt{\hbar \omega - \mathbb{E}\_8} \frac{e^6 n\_\epsilon^2 \omega}{\hbar^3 \varepsilon^3} f^2(\omega), & \text{for } a \xi \gg 1, \end{cases} \tag{17}$$

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

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

where N is the number of incident photons and q<sup>0</sup> and qm are the probabilities of single photon absorption in the ordinary photo effect [8] and of single photon absorption mediated by the presence of metallic nanospheres, respectively, as derived in the previous paragraph; <sup>A</sup> <sup>¼</sup> <sup>τ</sup><sup>n</sup>

for carriers [the time of traversing the distance between electrodes]). From the above formulae, it

In Tables 1–3, we list parameters for several semiconductor substrates and for a metallic nanoparticle few materials, which allow for comparison of the ratio qm=q<sup>0</sup> for various material

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 q<sup>0</sup> and the plasmon-mediated absorption with probability qm, respectively. Note also that

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 <sup>4</sup>:<sup>11</sup> � <sup>10</sup><sup>15</sup> 1/s <sup>5</sup>:<sup>2</sup> � 1015 1/s <sup>5</sup>:<sup>7</sup> � 1015 1/s

I 0 <sup>I</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> qm q0

<sup>0</sup> qm <sup>¼</sup> <sup>0</sup> , i.e., the photocurrent without metallic modifications),

<sup>f</sup> is the annihilation time of both sign carriers, tn pð Þ is the drive time

, (19)

<sup>0</sup> ¼ ∣e∣N q<sup>0</sup> þ qm

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

A,

125

f tn <sup>þ</sup> <sup>τ</sup> p f tp

the metallically modified photoactive surface. This photocurrent is given by I

n pð Þ

is the amplification factor (τ

where the ratio qm=q<sup>0</sup> is given by Eq. (18).

configurations by formula (18).

Table 1. Plasmon energies measured in metals.

Table 2. Mie frequency ω<sup>1</sup> to formula (18).

follows that (here I ¼ I

where <sup>f</sup>ð Þ¼ <sup>ω</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 1�ω<sup>2</sup> ð Þ<sup>2</sup> <sup>þ</sup>4ω2=τ0<sup>2</sup> <sup>q</sup> corresponds to amplitude factor for driven damped oscillator

and <sup>D</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>neE04πa<sup>3</sup> <sup>3</sup><sup>m</sup> fð Þ ω (in Eq. (10)); the amplitude of the electric field, E0, in the incident e-m wave is next ruled out from Eq. (17) due to normalization per single photon as in Eq. (13); <sup>C</sup><sup>0</sup> <sup>¼</sup> Nm4=3πa<sup>3</sup> <sup>V</sup> , V is the volume of the semiconductor, Nm is the number of metallic nanospheres. The ratio, qm q0 , revealing the advantage of the plasmon-mediated photo effect over the ordinary

photo effect can be expressed as follows

$$\frac{q\_{\underline{m}}}{q\_0} = \begin{cases} 4\sqrt{2}\pi^2 a^3 \beta \mathbb{C}\_0 \sqrt{m\_n^\* m\_p^\*} \left(m\_p^\*\right)^2 e^4 n\_\varepsilon^2 \omega^2 f^2(\omega) \\ \frac{3\mu^{3/2} m^2 \sqrt{\hbar \omega - E\_\xi \hbar \varepsilon^2}}{3\pi^2 a^2 \beta \mathbb{C}\_0 \left(m\_p^\*\right)^2 e^4 n\_\varepsilon^2 \omega^2 f^2(\omega)}, & \text{for } a\xi \ll 1, \\\frac{8\pi^2 a^2 \beta \mathbb{C}\_0 \left(m\_p^\*\right)^2 e^4 n\_\varepsilon^2 \omega^2 f^2(\omega)}{3\mu m^2 \left(\hbar \omega - E\_\xi\right) \varepsilon^2}, & \text{for } a\xi \gg 1. \end{cases} \tag{18}$$

This ratio turns out to be of order of 104 <sup>β</sup><sup>40</sup> H nm½ � for the surface density of nanoparticles (as in experiment in Ref. [14]), ns � 108 <sup>=</sup>cm2; note that <sup>C</sup><sup>0</sup> <sup>¼</sup> ns4πa<sup>3</sup>=ð Þ <sup>3</sup><sup>H</sup> , <sup>H</sup> is a thickness of the semiconductor layer, which including the phenomenological factor β, and the thickness H (we have confirmed experimentally that the range of the near-field zone exceeds the Mie wavelength), is sufficient to explain the scale of the experimentally observed strong enhancement of absorption rate in semiconductors due to plasmons. The strong enhancement of this transition probability is linked with the allowance of momentum-non-conserved transitions, which is, however, reduced with the radius a growth. The strengthening of the near-field induced interband transitions, in the case of large nanospheres, is, however, still significant as the quenching of oblique interband transitions is partly compensated by � <sup>a</sup><sup>3</sup> growth of the amplitude of dipole plasmon oscillations. The trade-off between these two competing size-dependent factors is responsible for the observed experimental enhancement of light absorption and emission in diode systems mediated by surface plasmons in nanoparticle surface coverings [7, 14–18].
