**2.1. Surface plasmon polaritons**

from various active inorganic materials, including amorphous and polycrystalline silicon, GaAs, CuIn*x*Ga1−*x*Se2 and CdTe, hybrid lead halide perovskites, as well as organic semicon‐ ductors. As the thickness of the absorbing semiconductor is decreased, the absorption naturally reduces at energies close to the band gap of the semiconductor. This is particularly a problem for thin film silicon solar cells. Thus, the compromise between enhancing the absorbance of broader solar lights and reducing the usage of narrower band gap semicon‐ ductor materials has to be taken in a thin film solar cell. And novel designs of thin film solar cells in which broadband light can be trapped inside to increase the absorption are highly

In the past decade, a few light‐trapping techniques have been investigated, among which a typical example is using a pyramidal surface texture [2]. However, such surface texture is intended for active light‐harvesting layers, which are thicker than the wavelength of sunlight in the visible and near‐infrared regions. The improved light trapping is balanced by the surface roughness that is almost the same order as the film thickness and by the increased surface recombination due to the larger surface area. Recently, the use of metallic nanostructures, which support surface plasmons (SPs) [3], has been regarded as an efficient way for enabling light trapping inside the active layer of a thin film solar cell and has consistently drawn an increasing amount of attention. SPs are coherent electron oscillations that propagate along the interface between a metal and a dielectric or semiconductor material. And SPs cause the electromagnetic field strongly confined at the metal/dielectric or semiconductor interface, with their intensity having an exponential dependence on the distance away from the interface. Thus near‐field electromagnetic field enhancement and the enhanced scattering cross section (SCS) can be obtained through excitation of SPs. The larger electrical field means a stronger absorption, and a larger scattering cross section redirects more incident sunlight into the absorbing layer, resulting in a much larger light absorption in a much thinner semiconductor layer. Hence, both localized surface plasmons (LSPs) [4] excited in metallic nanoparticles and surface plasmon polaritons (SPPs) [5] propagating at the periodic metal/semiconductor interfaces have been so far widely investigated with great interests in designing high‐efficient

In early work using plasmonic structures to improve the light absorption of photovoltaic devices, Au or Ag nanoparticles [5, 10] and nanograting [11] have been introduced into the front side of solar cells [5, 10–12]. Such efforts have common disadvantages that resonances can only occur at certain wavelengths, and the use of metallic nanostructures directly on top of solar cells will block a fairly large amount of total incident light. And then a layer of antireflection coating was combined into the surface metallic grating, to reduce the reflected light and thus to improve sunlight absorption [13]. The fractal‐like pattern of Ag nano cuboids with several feature sizes [14] was employed to simultaneously excite low‐index and high‐ index SP modes along the silicon‐silver interface to achieve broadband absorption. On the other hand, Wang et al. [15] achieved a broadband and polarization‐insensitive absorption enhancement by placing a metallic nanograting at the bottom of the optically active layer. In such design, planar waveguide modes, the Fabry‐Pérot (FP) resonance, and the SPP resonance were effectively coupled, and photons blocking by the surface nanostructures can be avoided.

needed to break the compromise balance.

138 Nanostructured Solar Cells

thin film solar cells [6–9].

It has been more than a century since the electrons in solids were first regarded as hot dense plasma to explain some natural phenomenon like the color of metals and the temperature‐ dependent conductivity. Decades later since then, the word 'Plasmon' is carried out to describe a quantum of plasma oscillations, which are longitude density fluctuations that propagate through the volume of metal. The so‐called volume plasmons have an eigen frequency <sup>=</sup> 2 0, when *n* is the electron density, at the order of 10 eV. They can be excited by both free electron beams and ultraviolet photons. In the 1950s, this was a fascinating phenom‐ enon and has been well‐studied theoretically and experimentally with electron‐loss spectro‐ scopy.

Probably as a result of the well investigations, in 1957, Rufus Ritchie [25] first predicted the existence of surface plasmons, which made up the other half of plasmon physics and is referred to as 'Plasmonics' nowadays. Surface plasmons are indeed the electron charges that perform coherent fluctuations on the metal boundary and are localized in the normal direction of the boundary within the Thomas‐Fermi screening length. Polarized light source is frequently used to efficiently and conveniently excite a surface plasmon wave. Therefore, surface plasmon can also be treated as a collective set of surface plasmon polaritons (SPPs). Surface plasmon polariton is thus the elementary 'particle' of this unique surface phenomenon and will be investigated in the following.

Consider the simplest geometry sustaining SPPs, that is, a single and flat interface between a non‐absorbing dielectric space with positive real dielectric constant *ε*<sup>2</sup> and a conductive space with complex dielectric function *ε*1(*ω*). Maxwell's equations of macroscopic electromagnetism read:

$$\nabla \times \overline{E}\left(\bar{r}, t\right) + \mu \frac{\partial \overline{H}\left(\bar{r}, t\right)}{\partial t} = 0 \tag{1a}$$

$$\nabla \cdot \left( \mu \overline{H} \left( \bar{r}, t \right) \right) = 0 \tag{1b}$$

$$
\nabla \times \overline{H}\left(\bar{r}, t\right) - \varepsilon\_0 \varepsilon(\bar{r}) \frac{\partial \overline{E}\left(\bar{r}, t\right)}{\partial t} = 0 \tag{1c}
$$

$$\nabla \cdot \left( \varepsilon\_0 \varepsilon(\bar{r}) \overline{E} \left( \bar{r}, t \right) \right) = 0 \tag{1d}$$

There are two fundamental sets of solutions, which are termed as transverse magnetic (TM) and transverse electric (TE), respectively. Let us first look into TE solutions:

where *z* > 0:

$$E\_{\gamma} \begin{pmatrix} z \\ \end{pmatrix} = A\_2 e^{i\beta \times} e^{-k\_2 z} \tag{2a}$$

$$iH\_{\times} \left( z \right) = -iA\_2 \frac{1}{\alpha \mu\_0} k\_2 e^{i\beta \chi} e^{-k\_2 z} \tag{2b}$$

$$H\_{\varepsilon} \left( z \right) = A\_2 \frac{\beta}{a \mu\_0} e^{i \beta \chi} e^{-k\_2 z} \tag{2c}$$

And where *z* < 0:

$$E\_{\circ} \left( z \right) = A\_{\text{i}} e^{i\beta \cdot x} e^{k \cdot z} \tag{3a}$$

Plasmonic Thin Film Solar Cells http://dx.doi.org/10.5772/65388 141

$$iH\_{\times} \left( z \right) = iA\_{\parallel} \frac{1}{\alpha \mu\_0} k\_{\parallel} e^{\prime \beta \times} e^{k\_{\parallel} z} \tag{3b}$$

$$H\_{\pm} \left( \mathbf{z} \right) = A\_1 \frac{\beta}{\alpha \mu\_0} e^{i \beta \mathbf{x}} e^{k\_{\parallel} \mathbf{z}} \tag{3c}$$

Here the perpendicular component of wave vector *ki* (*z*) is denoted as *ki* , with *i* = 1, 2 for short. Boundary condition requires the continuity of *Ey* and *Hx* at the interface *z* = 0, thus we have

$$A\_1 = A\_2 \tag{4a}$$

$$A\_1(k\_1 + k\_2) = 0\tag{4b}$$

Since *ki* is positive, the only solution of Eq. (4) is *A*1 = *A*2 = 0. Hence, there are no possible surface modes for TE polarization. Now let us examine the result for TM polarization. Similarly, respective expressions for the field components in TM solutions are:

where *z >* 0:

Consider the simplest geometry sustaining SPPs, that is, a single and flat interface between a non‐absorbing dielectric space with positive real dielectric constant *ε*<sup>2</sup> and a conductive space with complex dielectric function *ε*1(*ω*). Maxwell's equations of macroscopic electromagnetism

> ( ) ( , ) , 0 *H rt*

¶ Ñ´ + =

m

Ñ× = (m

( ) ( ) <sup>0</sup> , , () 0 *E rt*

¶

There are two fundamental sets of solutions, which are termed as transverse magnetic (TM)

( ) <sup>2</sup> 2 *i x k z E z Ae e <sup>y</sup>* b

( ) <sup>2</sup> 2 2 0 <sup>1</sup> *i x k z H z iA k e e <sup>x</sup>*

wm

( ) <sup>2</sup> 2 0 *i x k z Hz A ee <sup>z</sup>* b b

wm

( ) <sup>1</sup> 1 *i x k z E z Ae e <sup>y</sup>* b

b

*H rt r*

Ñ ×(e e

and transverse electric (TE), respectively. Let us first look into TE solutions:

e e

Ñ´ - <sup>=</sup> ¶

*t*

*t*

uuv v uv v (1a)

uv v uuv v v (1c)

<sup>0</sup> () , 0 *rE rt* ( )) <sup>=</sup> v uv v (1d)




<sup>=</sup> (3a)

*H rt* ( , 0 )) uuv v (1b)

¶

*E rt*

read:

140 Nanostructured Solar Cells

where *z* > 0:

And where *z* < 0:

$$H\_{\gamma} \left( z \right) = A\_2 e^{i \beta \chi} e^{-k\_1 z} \tag{5a}$$

$$iE\_x\left(z\right) = iA\_z \frac{1}{\alpha \varepsilon\_0 \varepsilon\_2} k\_2 e^{i\beta \chi} e^{-k\_1 z} \tag{5b}$$

$$E\_z\left(z\right) = -A\_2 \frac{\beta}{\alpha \varepsilon\_0 \varepsilon\_2} e^{l\beta \times} e^{-k\_2 z} \tag{5c}$$

and where *z* < 0:

$$H\_{\gamma} \left( z \right) = A\_{\mathfrak{r}} e^{i \ell \times} e^{\mathfrak{b} \cdot z} \tag{6a}$$

$$E\_{\times} \left( z \right) = -i A\_{\rm l} \frac{1}{\
cos \varepsilon\_0 \varepsilon\_1} k\_{\rm l} e^{i \beta \times} e^{\hbar \varepsilon} \tag{6b}$$

$$E\_z\left(z\right) = -A\_l \frac{\beta}{\alpha \varepsilon\_0 \varepsilon\_1} e^{i\beta \cdot x} e^{k\_l z} \tag{6c}$$

Boundary condition requires the continuity of *Hy* and *εi Ez* at the interface *z* = 0, thus we have

$$A\_1 = A\_2 \tag{7a}$$

$$\frac{k\_1}{k\_2} = -\frac{\varepsilon\_1}{\varepsilon\_2} \tag{7b}$$

Different from the TE case, we can expect some non‐null solution from Eq. (7). Note that the wave equation for TM modes reads:

$$\frac{\partial^2 H\_y}{\partial \boldsymbol{\varpi}^2} + \left(k\_o^2 \boldsymbol{\varepsilon} - \boldsymbol{\beta}^2\right) H\_y = 0 \tag{8}$$

To fulfill Eq. (8) with the expression for *Hy* shown in Eqs. (5a) and (6a), we thus have

$$k\_1^2 = \mathcal{J}^2 - k\_0^2 \varepsilon\_1 \tag{9a}$$

$$k\_2^2 = \mathcal{J}^2 - k\_0^2 \varepsilon\_2 \tag{9b}$$

Combing this and Eq. (7), the solution at the boundary *z* = 0 is finally reached with propagating constant being:

$$
\beta = k\_0 \sqrt{\frac{\varepsilon\_1 \varepsilon\_2}{\varepsilon\_1 + \varepsilon\_2}} \tag{11}
$$

Note that the metallic character of *ε*1(*ω*) requires that *ε*1 should be complex and its real part Re(*ε*1) < 0. Therefore, it is obvious that *β* is also complex, and its imaginary part indicates strong attenuation while propagating. By expanding *ε*1(*ω*) into 1 ′ + 1 ″ and assuming 1 ′ ≫ 1 ″ as most metals do in reality, we further have

$$\beta = \beta' + \mathbf{i} \cdot \beta'' = k\_0 \sqrt{\frac{\varepsilon\_1' \varepsilon\_2}{\varepsilon\_1' + \varepsilon\_2}} + ik\_0 \frac{\varepsilon\_1''}{2\left(\varepsilon\_1'\right)^2} \left(\frac{\varepsilon\_1' \varepsilon\_2}{\varepsilon\_1' + \varepsilon\_2}\right)^{3/2} \tag{12}$$

Here, *β*″ determines the attenuation of SPPs. On the other hand, one can say the reason for SPPs being surface mode is that SPPs will quickly vanish once it 'leaves' the surface. The energy is tightly bounded to the interface and travels only for a small amount of length, typically in the order of 1–10 wavelengths. This behavior is in nature similar to the well‐known evanes‐ cence wave, and in fact, SPPs can be efficiently excited by the impact of evanescence wave when the momentum match between SPPs and the evanescent wave is satisfied. Unlike the old method using electron beam, optical way to excite SPPs is of much more convenience and efficiency and has paved the way for plasmonics into application.

#### **2.2. Localized surface plasmons**

( ) <sup>1</sup> 1

> 1 1 2 2

e

e

Different from the TE case, we can expect some non‐null solution from Eq. (7). Note that the

*y*

*k k*

( ) <sup>2</sup> 2 2 <sup>2</sup> <sup>0</sup> <sup>0</sup> *<sup>y</sup>*

*k H <sup>z</sup>* e b

To fulfill Eq. (8) with the expression for *Hy* shown in Eqs. (5a) and (6a), we thus have

2 22 <sup>1</sup> 0 1 *k k* = b

2 22 <sup>2</sup> 0 2 *k k* = b

 e

 e

Combing this and Eq. (7), the solution at the boundary *z* = 0 is finally reached with propagating

1 2

e e

e e

1 2

Note that the metallic character of *ε*1(*ω*) requires that *ε*1 should be complex and its real part Re(*ε*1) < 0. Therefore, it is obvious that *β* is also complex, and its imaginary part indicates strong

0

*k*

b

attenuation while propagating. By expanding *ε*1(*ω*) into 1

most metals do in reality, we further have

+- =

*H*

¶

wave equation for TM modes reads:

142 Nanostructured Solar Cells

constant being:

we e

0 1 *i x k z Ez A ee <sup>z</sup>* b

Boundary condition requires the continuity of *Hy* and *εi Ez* at the interface *z* = 0, thus we have

b

= - (6c)

*A A* 1 2 <sup>=</sup> (7a)

= - (7b)

¶ (8)

(9a)

(9b)

″ and assuming 1

′ ≫ 1 ″ as

<sup>=</sup> <sup>+</sup> (11)

′ + 1

If the interface supporting SPPs that we discussed above shrinks to the scale of nanometer and forms a closed surface like sphere or ellipsoid, there would be no SPPs existing as the dispersive relationship and boundary condition have changed. However, by simply assuming the electrons in such a tiny metal object to be a neutral plasma, one should expect that an intrinsic resonance similar to the volume plasma resonance still existed for it. In that case, electrons would also collectively oscillate with the impact photons. In fact, this behavior is reasonable and termed 'localized surface plasmon' (LSP). LSP should be in nature different from SPP because SPP can propagate along the interface while LSP is totally bounded and cannot propagate at all. In general, LSP does not consist of 'polaritons' as SPP does, hence we will call LSP directly in the following text.

Considering a metal sphere with radius *R,* which is much smaller than the incident light's wavelength, we could treat the incident electromagnetic field as a static electric field. Hence, under this static field approximation, the eigen modes for LSP can be solved from Laplace equations. The electrostatic potential from Laplace equations thus reads [26]:

$$\phi\_{\prec}(r,\theta,\phi) = \sum\_{l=0}^{\ll} \sum\_{m=-l}^{l} a\_{lm} r^{l} Y\_{lm} \left(\theta,\phi\right) \tag{13a}$$

where 0 ≤ *r* ≤ *R* and

$$\phi\_{\succ}(r,\theta,\phi) = \sum\_{l=0}^{\@\infty} \sum\_{m=-l}^{\@\infty} b\_{lm} \frac{1}{r^{l+1}} Y\_{lm} \left(\theta,\phi\right) \tag{13b}$$

Here *Ylm*(*θ*,*φ*) is the spherical harmonics, *alm* and *blm* are the coefficients. At the sphere surface, boundary condition requires *ϕ* and *ε*∂*ϕ*/∂*r* to be continuous, thus we arrive at the dispersive relation for LSP:

$$\frac{\text{Re}\,\mathcal{E}\left(\text{o}\_{LSp}\right)}{\varepsilon\_0} + \frac{l+1}{l} = 0\tag{14}$$

Assuming *ε*(*ωLSP*) has a Drude form, i.e., = 1 <sup>−</sup> 2 2 thus Eq. (14) turns into:

$$\alpha o\_l = \alpha o\_p \left[ \frac{l}{\varepsilon\_0 \left(l+1\right) + l} \right] \tag{15}$$

where *l* stands for the angular momentum index. For a small enough sphere where the static field approximation is well satisfied, *l* thus equals to 1, which means that the dipole excitation is mainly responsible for LSP. With the increase in radius, the interaction between multipoles is becoming more and more important. Eventually at infinite radius, the frequency of LSP reaches that of SPPs at a semi-infinite metal-dielectric interface.

Similar to SPPs, LSPs exhibit an impressive local field enhancement as all energy is bounded. However, LSPs exist much more common than SPPs, as LSP does not require a specific polarization direction of the incident electromagnetic wave. A lot of applied techniques thus are developed on the basis of LSP enhancement, such as surface enhanced Raman spectroscopy (SERS), tip enhanced fluorescence spectroscopy (TEFS), and tip enhanced Raman spectroscopy (TERS).
