## Abstract

The use of localized surface plasmon (LSP) interaction for significantly enhancing InGaN absorption near its band edge and the overall efficiency of an InGaN-based solar cell by embedding Ag nanoparticles (NPs) in the InGaN absorbing layer is numerically demonstrated. The generation of LSP resonance on the embedded Ag NPs and the NP scattering can produce a field distribution in the InGaN layer for enhancing absorption. It is shown that the embedded Ag NPs do not significantly affect the transport of the photo-generated carriers. The distortion of static electrical stream lines in the solar cell due to the embedded Ag NP leads to a decrease of photocurrent by only a few percents. Based on the material parameter values we use, unless the surface recombination velocity at the interface between the Ag NP and surrounding InGaN is extremely high, Ag NP embedment in the absorbing layer of an InGaN-based solar cell can enhance its efficiency by up to 27%. Such an increase is significantly larger than that achieved by depositing metal NP on the top surface of a solar cell.

©2010 Optical Society of America

## 1. Introduction

For the development of concentrator-type solar cell, although the relatively smaller band gaps of crystalline Si, GaAs, and other compound semiconductors, which have been proven useful for solar cell fabrication, can lead to effective absorption in the major portion of the solar spectrum, the problems of effective surface absorption and excess carrier energy generation in utilizing the UV-green spectral range of sunlight still limit the efficiency improvement of a solar cell based on such a material [1–5]. To solve such problems, multi-junction solar cells using more than one material have been proposed and fabricated for achieving efficiencies higher than 30% [6–8] and a record of ~41% [9]. For fabricating such a multi-junction device, the combination of InGaN with the aforementioned smaller band-gap materials can be a good choice. The effective absorption and photo-current generation of In_{x}Ga_{1-x}N (x < 0.4) in the spectral range of 380-600 nm can significantly enhance the use efficiency of sunlight in this spectral range.

Because of the advantages of broad absorption spectral coverage (350-1800 nm) [10], relatively easier multi-junction fabrication (simply changing indium content) [11], high absorption coefficient (~10^{5} cm^{−1}) [12], high carrier mobility [13], and mature fabrication techniques based on the development of light-emitting diode, InGaN/GaN heterostructures have caught much attention for solar cell application. Also, because InGaN has high durability under high-energy particle bombardment, it has been considered for space solar cell application. Although InGaN/GaN-based solar cells have been successfully fabricated [14–19], the indium contents of the used InGaN compounds were quite low (resulting in a large band gap), leading to the effective sunlight absorption of only a small spectral range. To cover a larger absorption spectral range for increasing solar cell efficiency, the indium content of InGaN must be increased. However, with higher indium content (say, 25%), the InGaN critical thickness of high crystal quality becomes limited (<30 nm) [20,21]. Beyond the critical thickness, the heterostructure-induced strain in an InGaN/GaN structure is relaxed and defects of high density are formed. In this situation, the thickness of an InGaN layer of high quality for effective sunlight absorption can be too small. Although the absorption coefficient of InGaN can be as large as ~10^{5} cm^{−1} for effective sunlight absorption in the spectral range well above the band gap, the lower absorption coefficient near the band edge still limits the effective spectral range of sunlight use in an InGaN-based solar cell. If certain methods can be used for increasing InGaN absorption near its band edge, the effective sunlight absorption range of InGaN of limited indium content and limited thickness can be increased.

In this paper, we report the simulation results of using the interaction of metal nanoparticle (NP) induced localized surface plasmon (LSP) with an InGaN absorbing layer for enhancing the efficiency of an InGaN/GaN-based solar cell. Three-dimensional (3-D) numerical simulations are performed to demonstrate the increased absorption and overall solar cell efficiency. The absorption increase is implemented by embedding Ag NPs in InGaN for inducing LSP. The effective LSP absorption and NP scattering lead to the enhancement of InGaN absorption. The use of surface plasmon (SP) interaction with the absorbing material of a solar cell for enhancing its efficiency has been widely reported [22–28]. However, because the used metal nanostructures were fabricated on device surfaces, the significantly large distances between metal nanostructures and absorbing layers make the effects of SP interactions weak unless the top layer is extremely thin or is used for sunlight absorption, such as in the case of an amorphous-Si solar cell [24]. With metal nanostructures on the surface, in some cases, the enhanced forward scattering is used to produce small increases of solar cell efficiency [28]. By embedding metal NPs within or near the absorbing layer, the LSP interaction can be significantly enhanced. Embedding Ag NPs in GaN near an InGaN/GaN quantum well for enhancing the efficiency of a light-emitting diode has been reported [29]. The embedment of metal NPs in semiconductor may affect carrier transport. However, we will show that such an effect is small unless the surface recombination velocity at the interface between a metal NP and surrounding InGaN is extremely high. In section 2 of this paper, the simulated solar cell structure is described and the used simulation methods and theories are discussed. The results of absorption and overall solar cell efficiency enhancements are shown in section 3. Then, the effects of Ag NP embedment on carrier transport on solar cell efficiency are discussed in section 4. Finally, the discussions and conclusions are presented in section 5.

## 2. Solar cell structure and simulation methods

For demonstrating the effects of embedded Ag NP, a solar cell structure of n-GaN/i-InGaN/p-GaN is considered. A periodical Ag NP array embedded in the absorbing layer (i-InGaN) of 100 nm in period is assumed. The inset of Fig. 1 shows a period for 3-D simulation in such a solar cell of the periodical structure. An indium-tin-oxide (ITO) layer of 80 nm in thickness is assumed at the top of the solar cell structure for current conduction. The n-GaN, i-InGaN, and p-GaN layers are assumed to be 200, 100, and 120 nm, respectively, in thickness. The bottom metal layer can serve as the p-type contact and the back reflector. When the metal layer is thicker than 100 nm, the reflectivity is close to 100%. In this situation, it can be assumed to be infinitely thick in simulation. Note that the thicknesses of the n-, i-, and p-type layers are chosen for locating the absorbing InGaN layer around an intensity peak of the Fabry-Perot pattern in the whole semiconductor layer such that the sunlight absorption is maximized. The whole solar cell structure is similar to a vertical light-emitting diode after the laser lift-off process for removing sapphire substrate [30]. For enhancing absorption, an Ag NP of spherical shape with the diameter of 40 nm is embedded at the center of the InGaN region in the simulation window. The LSP resonance wavelength of such an Ag NP surrounded by InGaN is around 605 nm. The sunlight is assumed to be normally incident onto the solar cell as indicated by the arrows in the inset of Fig. 1. The incident light is assumed to be linearly polarized along the x-direction. Since under the considered condition of problem geometry, the effects of periodical NP array, including the LSP coupling between neighboring NPs and the grating diffraction, are weak, the observed solar cell behaviors originate mainly from the absorption and scattering of isolated spherical Ag NPs. Therefore, the obtained results are independent of the assignment of linearly polarized direction under the condition of normal incidence. In other words, the absorption and efficiency enhancement percentages of a concerned solar cell, as to be shown below, are still correct for the case of normally incident sunlight of un-polarized nature.

The main body of Fig. 1 shows the used InGaN absorption spectrum for simulation. It was obtained from the transmission measurement of a 25-nm InGaN thin film grown on a 2-μm GaN layer, which was deposited on a double-polished c-plane sapphire substrate with metalorganic chemical vapor deposition. The InGaN thin film was then covered by a GaN cap layer of 100 nm in thickness. The growth temperatures of the top and bottom GaN layers were 800 and 1000 °C, respectively. That of the InGaN layer was 675 °C, leading to the fully-strained indium content of about 27% and the absorption edge around 560 nm. Because of the indium composition fluctuation in the InGaN layer, a band tail in the absorption spectrum can be clearly seen. Following the Sellmeir dispersion relation [31], the dispersive refractive index of InGaN is assumed to be the same as that of GaN. GaN is expected to be transparent for sunlight transmission. As to the wavelength-dependent permmitivity of Ag, we use the experimental data given in [32].

In choosing the Ag NP size (40 nm in diameter of our choice), the NP scattering efficiency and LSP resonance wavelength represent two important factors. However, it is difficult to define the scattering efficiency of an NP surrounded by an absorbing medium like the case of our study. To show the variation trend of scattering efficiency with Ag NP size, we assume the condition of transparent InGaN and evaluate the scattering and absorption efficiencies of a single Ag NP as functions of its diameter to give the results in Fig. 2
. In this figure, the scattering and absorption efficiencies are defined as the cross sections of NP scattering and absorption, respectively, at its LSP resonance wavelength divided by the NP geometric cross-section area. In Fig. 2, the absorption and scattering efficiency data points for a particular NP diameter are displayed in a circled group (labeled by the NP diameter). Their horizontal locations correspond to individual LSP resonance wavelengths along the abscissa. Here, one can see that when the NP diameter is larger than 30 nm, NP absorption efficiency starts to decrease and NP scattering efficiency reaches a saturated level. The major reason for us to choose 40 nm for NP diameter is to match its LSP resonance wavelength with the assumed In_{0.27}Ga_{0.73}N band edge because the embedded NP is designed for enhancing sunlight absorption around the In_{0.27}Ga_{0.73}N absorption band tail. It is noted that, with significant absorption in InGaN, the NP LSP resonance actually becomes weak.

The finite-element method with the commercial software COMSOL was used for numerical simulations in this study. As shown in the inset of Fig. 1, a plane wave $\left({\stackrel{\rightharpoonup}{E}}_{i},\text{}{\stackrel{\rightharpoonup}{H}}_{i}\right)$ is normally incident upon the solar cell along the z-direction. The In_{0.27}Ga_{0.73}N absorption layer is characterized by its complex dielectric constant, ${{\epsilon}^{\prime}}_{r}+i{{\epsilon}^{\u2033}}_{r}$. Once the field distribution, $\overrightarrow{E}$, is obtained by solving the Maxwell equations, the total absorbed power in the InGaN layer can be calculated through the following expression:

Here, *ω* is the angular frequency and ${\epsilon}_{0}$ is the permittivity in free space. The volume integral over *V* covers the entire InGaN layer. For a differential spectral interval $d\lambda $ at a certain wavelength, *λ*, the field distribution in the absorption layer is also used for obtaining the carrier generation rate, *G _{λ}*, through the following equation:

Here, ${N}_{p}\left(\lambda \right)$ is the incident photon flux density as a function of wavelength under the condition of AM1.5G. The integrated (total) carrier generation rate, *G _{tot}*, contributed by the entire AM1.5G spectral range of effective absorption (320-800 nm) is then given by

For simulating the carrier transport process in the solar cell, we solve Poisson's equation and the continuity equation, which form a nonlinear equation set with the unknowns of the electron and hole concentrations (*n*, *p*), and the electric potential (*ψ*), as given by

Here, *G* and *R* are the carrier generation rate and carrier recombination rate, respectively. The notation *q* denotes the electron charge, *μ _{n(p)}* represents the carrier mobility,

*D*is the carrier diffusion coefficient, and

_{n(p)}*ε*stands for the low-frequency dielectric constant of In

_{s}_{x}Ga

_{1-x}N, which is given by

*ε*= 10.4(1-x) + 15.3x [33]. Also,

_{s}*N*=

*N*-

_{D}*N*, where

_{A}*N*and

_{D}*N*are the concentrations of donor and acceptor, respectively. Both

_{A}*N*and

_{D}*N*are reasonably assumed to be 5x10

_{A}^{17}cm

^{−3}. Other electronic properties of In

_{0.27}Ga

_{0.73}N are assumed to be the same as those of GaN. It is noted that Eq. (4) can be used for solving the wavelength-dependent (

*G*=

*G*) or wavelength-integrated (

_{λ}*G*=

*G*) behaviors. In either case, the same carrier recombination rate,

_{tot}*R*, is used. The Schockley-Read-Hall expression was used to describe the carrier recombination process in bulk materials with the carrier recombination rate

*R*given by

Here, *τ _{n(p)}* is the carrier lifetime and

*n*is the intrinsic carrier concentration. It is noted that

_{i}*n*is determined by the band gap energy of the semiconductor material. The carrier generation rate,

_{i}*G*, can be obtained directly from the photon absorption rate assuming that every absorbed photon is converted into an electron-hole pair.

The symmetric boundary conditions are used for the four boundaries in the x-y directions based on the periodic nature of the structure. The Dirichlet boundary conditions are used for Poisson's equation at the contacts, as shown in the following equations [34]:

at the p-contact andat the n-contact. Here,*V*

_{0}

*and*

_{p}*V*

_{0}

*are the potentials in thermal equilibrium at the p- and n-contact, respectively.*

_{n}*V*is the device applied voltage. The Nuemann boundary conditions are used for the continuity equation at the contacts, given by [34]for hole andfor electron. These boundary conditions state that the outward carrier flux

_{a}*f*through the contact is proportional to the excess minority carrier times the effective surface recombination velocity,

_{p(n)}*S*, of the contact. Here,

_{c}*p*(

_{0}*n*) is the hole (electron) concentration in thermal equilibrium. It is noted that before the use of Eqs. (6) and (7), the potential and carrier concentration distributions in thermal equilibrium need to be first calculated through iteration to reach the result of zero output current under the conditions of zero applied voltage and zero light illumination intensity.

_{0}As to the interface between the metal NP and InGaN, at which surface defects can be formed for generating surface recombination, we use the following boundary condition:

Here, ${R}_{s}$ is the effective surface recombination rate. The surface recombination rate can also be described within the Schockley-Read-Hall framework [34], given by:

Here, *S _{n}* and

*S*denote the surface recombination velocities of electron and hole, respectively. We further assume that

_{p}*S*is equal to

_{n}*S*and is denoted by

_{p}*S*in our simulations. The used value of

*S*is on the order of several tens to thousands m/s [35]. The used semiconductor parameters are listed in Table 1 . The cited references for those parameter numbers are also shown in Table 1 [36–38].

## 3. Enhancements of solar cell absorption and overall efficiency

Figure 3
shows the photon absorption rates (the left ordinate) of the InGaN layer and Ag NP and the short-circuit current densities (the right ordinate) of the solar cell as functions of wavelength for the cases with and without Ag NP when the surface recombination velocity, *S*, is 10 m/s. For comparison, the incident photon flux under the condition of AM1.5G (the curve labeled by “incidence”) is also plotted in Fig. 3. Here, one can see the oscillatory spectral behaviors in the two InGaN absorption curves, i.e., absorption (without NP) and absorption (with NP). The significant absorption enhancement on the long-wavelength side by embedding the Ag NP can be clearly seen. In Fig. 3, the curve labeled by “metal dissipation” represents the part of photons absorbed by the Ag NP and turned into dissipation heat. The embedment of the Ag NP results in an increase of integrated photon absorption rate by 28.44%. The two curves of short-circuit current density (per unit spectral width), i.e., J_{SC} (without NP) and J_{SC} (with NP), which essentially follow the oscillatory behaviors of the two curves for InGaN absorption, in Fig. 3 show the enhancement of photo-generated current. With the embedded Ag NP, the integrated J_{SC} is increased by 27.87%.

Figure 4(a) shows the distribution of electrical field magnitude at 580 nm in the region around the Ag NP by assuming that the magnitude of the incident field is unity. The coordinate of z = 0 is set at the interface between the ITO and n-GaN layer. Here, one can see that strong near field is generated around the Ag NP. Also, certain backscattered field is distributed in the n-GaN layer. The strong near field is absorbed by the surrounding InGaN material, leading to absorption enhancement. Figure 4(b) shows the stream line distribution of static electric field inside the solar cell. Here, one can see the distortion of static electric field by the embedded Ag NP. Some stream lines pass through the Ag NP. For those electron-hole pairs generated on such stream lines, an electron or hole may drift into the Ag NP. In this situation, another electron or hole is expected to be emitted from the other end of the Ag NP. Therefore, the overall photocurrent is affected by the embedded Ag NP mainly through the surface recombination around the metal/semiconductor interface. However, with the embedded Ag NP, the carrier transport paths become longer such that the carrier annihilation probability is increased and the solar cell efficiency is reduced. Nevertheless, in the next section, we will demonstrate that this effect is quite weak (only a few %).

Figure 5
shows the integrated current densities (the left ordinate) and the output power densities (the right ordinate) as functions of applied voltage for the cases with NP and without NP corresponding to the results shown in Fig. 3. The first and second numbers in the parentheses represent the voltage for the maximum output power (in V) and the maximum output power density (in mW/cm^{2}). Here, one can see that the open-circuit voltage of ~1.51 V is not affected by the embedment of the Ag NP. The integrated current density, J, is significantly increased from 8.08 to 10.17 mA/cm^{2} by embedding the Ag NP. The maximum output powers of both cases are achieved at 1.4 V in applied voltage. The maximum output power is increased from 10.59 to 13.53 mW/cm^{2} (roughly from 10.59 to 13.53% in efficiency), corresponding to an increase of 27.76%. The fill factors of the cases with and without NP are 0.85 and 0.84, respectively. The embedment of the Ag NP does not significantly affect the fill factor either.

## 4. Effects of metal nanoparticle embedment on carrier transport

A major concern of embedding metal NP in semiconductor is the generation of surface defects at the interface between the metal NP and semiconductor. Such defects will produce surface recombination to reduce solar cell efficiency. Figure 6
shows the integrated current densities as functions of applied voltage with different surface recombination velocities, *S*, for comparing with the case without NP. Here, one can see that by increasing *S* to 1000 m/s, the short-circuit current density is still maintained at a level around that (10 mA/cm^{2}) when *S* is 10 m/s. This level is significantly higher than that of the case without NP (8 mA/cm^{2}). By increasing S, the open-circuit voltage is reduced. This result is reasonable because when the applied voltage is increased, the built internal field for driving electrons and holes into the n- and p-layer, respectively, is more significantly cancelled by the applied electric field. Therefore, with a larger *S*, the photocurrent is more reduced compared to that of a smaller *S* at high applied voltage, effectively leading to a smaller open-circuit voltage. The fill factors of the cases without NP, S = 10, 100, and 1000 m/s are 0.84, 0.85, 0.83, and 0.82, respectively. The fill factor is not significantly changed. Figure 7
shows the output power densities as functions of applied voltage under various conditions the same as those in Fig. 6. Again, the first and second numbers in the parentheses represent the voltage for the maximum output power (in V) and the maximum output power density (in mW/cm^{2}). Here, one can see that both the voltage for the maximum output power and the maximum output power density decrease with increasing *S*. However, even with an *S* value as large as 1000 m/s, the maximum output power density at 11.6 mW/cm^{2} is still higher than that in the case without embedded NP (10.59 mW/cm^{2}). The level of surface recombination around the metal NP is a key parameter for using embedded metal NPs in semiconductor for enhancing solar cell efficiency.

Because of the carrier loss in embedding the Ag NP in InGaN, a parameter of great concern is the percentage of photo-generated carriers for contributing to photocurrent. From the results in Fig. 3, one can evaluate the integrated incident photon rate (AM1.5G) in the spectral range between 320 and 800 nm over the illumination area of the simulation window (100 x 100 nm^{2}) to give 4.10 x 10^{7} s^{−1}. The integrated photon absorption rates by InGaN over the same illumination area in the cases with NP and without NP are 6.47 x 10^{6} and 5.04 x 10^{6} s^{−1}, respectively. Figure 8
shows the output electron rates (photocurrent divided by electron charge) as functions of applied voltage under various conditions the same as those in Figs. 6 and 7. One can see that the reductions of output electron rate are mainly observed at the high applied voltage end, similar to what observed in Fig. 6.

Figure 9 shows the variation of the conversion efficiency of absorbed photons into current (CEAPC) with applied voltage under various conditions. The CEAPC is defined as

Here, *h* is the Planck constant, *c* is the speed of light in vacuum, and *A* is the p-contact area of a solar cell. Also, *J _{p}* is the output current density at the p-contact and

*P*(

*λ*) is the absorption power spectrum of the solar cell. The CEAPC at the applied voltages for the individual maximum output power densities in the cases without NP, S = 10, 100, and 1000 m/s are 93.0, 93.6, 93.0, and 92.2%, respectively. The difference of CEAPC from 100% is due to carrier loss during their transport (recombination or defect). Here, one can see that the embedment of the Ag NP does not much affect the conversion efficiency from photo-generated carriers into photocurrent.

The enhancement of solar cell efficiency can also be understood by calculating the spectral variation of external quantum efficiency (EQE). Figure 10 shows the comparison of wavelength-dependent EQE between the cases with and without embedded Ag NP under the condition the same as those in Fig. 3, i.e., S = 10 m/s. Here, one can see that the EQE is increased by several tens percents in the spectral range between 470 and 720 nm. The oscillatory behaviors are due to the Fabry-Perot effect in the device.

## 5. Discussions and conclusions

It is noted that in our simulation, we do not consider the effects of piezoelectric field and spontaneous polarization in the InGaN layer. Normally, the spontaneous polarization in an InGaN/GaN heterostructure is quite weak and can be neglected [39]. The piezoelectric field in an InGaN/GaN quantum well has been estimated to give an order of magnitude of 10^{6} V/cm [40]. The piezoelectric field in an InGaN/GaN heterostructure can be assumed to have the same order of magnitude. The direction of the piezoelectric field is opposite to that of the built-in electric field in a p-i-n structure of the conventional growth procedure (n-type GaN grown first on sapphire) with MOCVD (Ga polarity). The piezoelectric field strength is at the same order of magnitude as the built-in electric field in such a p-i-n structure. However, because the piezoelectric field can be screened by the photo-generated carriers, it is expected that the cancellation effect of the built-in field by the piezoelectric field is small. Such a cancellation effect, if significant, will lower solar cell efficiency. This problem can be solved by growing an n-i-p structure with Ga polarity or a p-i-n structure with N polarity.

In summary, we have numerically demonstrated the use of LSP interaction for significantly enhancing the absorption near the InGaN band edge and the overall efficiency of an InGaN/GaN-heterostructure solar cell by embedding Ag NPs in the absorbing InGaN layer. The generation of LSP resonance on the embedded Ag NPs and the NP scattering produced a strong field distribution around the NP for enhancing InGaN absorption. It was shown that the existence of the embedded Ag NPs did not significantly affect the transport of the photo-generated carriers. The distortion of electrical lines in the solar cell due to the embedded Ag NPs led to only a small reduction in carrier-current conversion efficiency. Unless the surface recombination velocity at the interface between the metal NP and semiconductor was extremely high, metal NP embedment in the absorbing layer of an InGaN/GaN-heterostructure solar cell could significantly enhance its efficiency.

## Acknowledgement

This research was supported by National Science Council, The Republic of China, under the grants of NSC 97-2120-M-002-005, NSC 97-2622-E-002-011-CC1, NSC 96-2628-E-002-044-MY3, NSC 97-2221-E-002-044, by the Excellent Research Projects of National Taiwan University, and by US Air Force Scientific Research Office under the contracts AOARD-07-4010 and AOARD-09-4117.

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