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The analysis of light trapping and internal quantum efficiency of a solar cell with DBR back reflector
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Solar Energy 83 (2009) 2050–2058 www.elsevier.com/locate/solener
The analysis of light trapping and internal quantum efficiency of a solar cell with DBR back reflector Kuo-Hui Yang a, Jaw-Yen Yang b,* b
a Institute of Applied Mechanics, National Taiwan University, Taipei 10746, Taiwan Center for Quantum Science and Engineering, National Taiwan University, Taipei 10764, Taiwan
Received 18 May 2009; received in revised form 26 July 2009; accepted 6 August 2009 Available online 31 August 2009 Communicated by: Takhir Razykov
Abstract A theoretical analysis of the total internal quantum efficiency (IQE) of a flat-band p–n homo-junction silicon solar cell with back reflector using distributed Bragg reflectors to improve the light trapping is presented and contributions of different regions of the structure to IQEs are simulated. An optical model for the determination of generation profile of the cell is adopted and multiple light passes are considered and compared to previous single light pass approach. It is found that the spatial widths of the cell, the surface recombination velocities, the front surface transmittance and the back reflector have significant impacts on the IQEs. With two light passes and normal incident light, the simulation result shows the IQEs can be increased over the one pass value by 6.34% and with a 60° light reflection angle, the IQEs can be further increased by 9.01% while assuming the reflectance at back structure closed to 100%. The effect on IQEs by back reflectance is more significant than that by front transmittance. Under multiple light passes simulation, up to 51 light trapping passes have been considered at wavelength range 900–1100 nm, the cell IQEs can be enhanced by about 26.98%. Ó 2009 Elsevier Ltd. All rights reserved. Keywords: Solar cells; Homo-junction; Internal quantum efficiency; Light trapping; DBR
1. Introduction The thin-film crystalline silicon (c-Si) solar cell is considered one of the best candidates for the next generation of photovoltaic applications due to its economic cost and easy fabrication, which are ideal for large-scale production. It is well known that crystalline silicon has an indirect band gap, which gives rise to weak absorption of light for the photons in the wavelengths close to the band gap ðkg ¼ 1108 nmÞ. For example, the absorption length increases from just over 10 lm for k = 800 nm to over 1 mm for ðkg ¼ 1108 nmÞ (Herzinger et al., 1998), which * Corresponding author. Address: Center for Quantum Science and Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10764, Taiwan. Tel.: +886 2 33665636; fax: +886 2 23639290. E-mail address: [email protected] (J.-Y. Yang).
0038-092X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2009.08.003
far exceeds the thickness of the core silicon layer of most thin-film cells (Yamamoto et al., 2004) and can result in a very low absorption of photons in this spectral range. However, the range of wavelengths contains 36.2% of the solar photons with energies above the band gap of c-Si (ASTMG 173-03, 2005). Thus, solar cells made from c-Si thin films may fail to absorb a significant number of photons that could otherwise be used to generate power in the cell. Meanwhile, the expense of thicker c-Si wafers with their correspondingly longer diffusion lengths (Rohatgi et al., 1993), drives up costs markedly. Also, the absorption coefficient for monochromatic light in silicon is a strong function of wavelength, so the spectral response of silicon solar cells contains information about their internal operation. But to study the internal operation, it is first necessary to eliminate external optical effects from the spectral response data. This can be accomplished by measuring
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the total hemispherical reflectance of the cell as a function of wavelength. The spectral response can be adjusted to account for the optical reflectance, yielding the internal quantum efficiency (IQE) of the device (Basore, 1993; Green, 1997; Bermel et al., 2007). One of the foremost challenges in designing silicon photovoltaic cells is devising an efficient light-trapping scheme which allows one to create thin yet efficient solar cells, made from c-Si and other, closely related materials, such as nano-crystalline silicon (which has the same band gap and similar absorption characteristics) (Hamakawa, 2004). One of the main approaches to light trapping: geometrical optics which is widely used in current solar cells and wave optics, which represents a novel approach to the problem that just begun to be explored. Several approaches have been proposed to enhance the absorption in the long-wavelength range, including searching alternative inexpensive materials (Nelson, 2003; Green, 1987), enhancing collection efficiency through randomly roughened surfaces (Green, 1987; Campbell and Green, 1987), and light trapping by optical structures (Muller et al., 2004; Heine and Morf, 1995; Brendel et al., 1996; Eisele et al., 2001; Shvarts et al., 2001; Johnson et al., 2005). The light-trapping using optical structures is of particular interest and could be done based on improving the equivalent optical path length by bending the light to a tilted angle through diffraction gratings (Heine and Morf, 1995; Brendel et al., 1996; Eisele et al., 2001; Shvarts et al., 2001) or simply reflecting light back using distributed Bragg reflectors (DBRs) (Shvarts et al., 2001; Johnson et al., 2005). Commonly used approaches to light trapping in solar cells rely on monitoring light paths through geometrical optics. Two main mechanisms to increase these path lengths are (i) the normally incident light can be scattered into an angle at the front interface via surface texturing (Campbell and Green, 1987), and (ii) it can be reflected back into the cell via a back reflector. Solar cells with DBR structure can offer reflection superior to highly absorptive aluminum reflectors. Photonic crystal enhanced solar cells in which the metallic back reflector is replaced by a distributed Bragg reflector DBR consisting of a onedimensional dielectric superlattice have been considered (Bermel et al., 2007; Feng et al., 2007; Zhou and Biswas, 2008). The distributed Bragg reflectors (DBR) with high index contrast can reflect light over a broad range of incident angles and wavelengths, for either or both polarizations. The use of DBR stacks Si/Si3N4 (n1/n2 = 3.5/2.0) or Si/SiO2 (n1/n2 = 3.5/1.46) can achieve reflectivity 99.8% for k between 800 and 1100 nm with just a few quarter-wave pairs. In contrast, when light is incident from Si to an Al backside reflector commonly used in solar cells, the reflectivity is <80%. The high reflectivity of DBR guarantees that almost no light can transmit through the backside (Zeng et al., 2006; Venkatasubramanian et al., 2003). Combining perfect random scattering at the front with a lossless reflector in the back can theoretically enhance the effective
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path length by a factor of 4n2, where n is the index of refraction, corresponding to a factor of about 50 for Si, and 30 for TiO2 (Yablonovitch and Cody, 1982). In this study, a theoretical analysis of the internal quantum efficiency (IQE) of a flat-band p–n homo-junction silicon solar cell with optical back reflector, as shown in Fig. 1, is presented. An improved light-trapping approach using DBR back structures is considered and current contributions from all cell regions including emitter, space charge region, base and DBR are all accounted for. A multiple-pass model is adopted which allows the analysis of multiple passes of unabsorbed light through the cell. The IQEs from two light passes and multiple passes up to 51 light passes are analyzed. The effect of back angled reflection is also considered in the model. We compare the IQEs of the present model to that of a baseline single pass IQEs analysis without back reflector (Yang et al., 2008). This paper is organized as follows. In Section 2, the analysis of cell efficiency based on semiconductor theory is described. In Section 3, the theoretical derivation of a solar cell with distributed Bragg back reflector is presented. The IQEs of each region of the cell structure is explicitly derived. The results are given in Section 4. The effects of the number of light paths, incident light angle, and variable
Fig. 1. Schematic of a crystalline silicon solar cell with DBR. Light incident onto the cell is partially reflected at the front surface with Rf . Electron/hole generation attenuates the light during its first transmitted pass ðl ¼ 1Þ and second reflected pass ðl ¼ 2Þ. Transmitted light with angle h is reflected at the back with reflectance RDBR with angle u and reflected back to the cell at the front surface with transmittance t. Similarly, the light intensity is further reduced by the second and all subsequent passes through the cell.
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reflectance and transmittance are studied. Some concluding remarks are given in Section 5. 2. Analysis of cell efficiency Based on the semiconductor theory (Hamakawa, 2004; Wurfel, 2005), the current in a cell always includes the photo-generated current and the dark diode current, with the former being proportional to the incident light intensity and the latter being independent of it. Thus, the conversion efficiency relies on both the photo-generated current and the diode property. Meanwhile, the photo-generated current is strongly related to the quantum efficiency (QE). In theory, the QE corresponds to the spectral response (SR) which determines the spectral distribution of the short circuit current I SC . By calculating the QE, both the SR and the contributions to the I SC of different wavelengths can be determined, helping one analyze quantum yields from the different cell regions. Therefore, it is useful to calculate the QE for finding the performance of a cell. The QE involves both the external quantum efficiency (EQE) and internal quantum efficiency (IQE), with the latter defined as the number of minority carriers contributing to the short circuit current divided by the number of photons entering the cell. To increase the conversion efficiency one needs to raise the QE. In experiment, the IQE can be obtained through the light reflectance and the SR. Thus, the SR can be defined as the ratio of the measured cell current I SC ðkÞ to the light intensity I light ðkÞ (Yang et al., 2008) SRðkÞ ¼
I SC ðkÞ : I light ðkÞ
ð1Þ
i¼e;scr;b;s
A photon’s contribution to the quantum efficiency of the region i is given by IQEi ¼ Ai gi =ð1 RÞ;
ð4Þ
where i = e, scr, b, s, if this light quantum is actively absorbed in region i with a probability Ai and then collected at the junction with collection efficiency gi . A photon is called as being actively absorbed if an electron/hole pair is generated, while parasitic absorption processes such as by free carriers or optical losses at the back surface mirror do not contribute to the active absorption Ai . The factor ð1 RÞ normalizes the QE to the fraction of photons that are not reflected by the cell. The relative contribution Ai , from different cell region i to the total active absorption A is R g ðxÞ dx Ai region i i R ¼ : ð5Þ A gðxÞ dx total cell Here, gi ðxÞ ¼
gðxÞ; for x in region i; 0;
elsewhere:
ð6Þ
The collection efficiency of region i,
The IQE can be derived and is given by 1 I SC ðkÞ=e IQEðkÞ ¼ 1 RðkÞ I light ðkÞ=ðhc=kÞ 1 hc ¼ SRðkÞ; 1 RðkÞ ek
gap wavelengths where photo-generation is uniform throughout the device (Basore, 1993). A theoretical model for the internal quantum efficiency must consider contributions of the emitter e, the space charge region scr, the base region b, and the substrate region s and can be expressed as (Brendel et al., 1996): X IQEi : ð3Þ IQE ¼
jðgi Þ gi ¼ R q region i gi ðxÞ dx ð2Þ
where RðkÞ is the reflectance, h the Planck constant, c the speed of light and e the unit charge. The electrical model of the solar cell considered in this work is a simple one. The base region of the cell is a quasi-neutral region in low-level injection, characterized by a uniform minoritycarrier diffusion length (L) and diffusivity (D), and a surface recombination velocity (S) at the back surface. The collecting junction is assumed to be located at the front surface. The wavelength range considered is restricted to k > 800 nm so that the influence of recombination in the thin emitter region can be neglected. Despite this simple model, an analytical expression valid over the entire wavelength range could be extraordinarily complex, because the photo-generation profile in the device is affected by light trapping at the longer wavelengths and many light passes are considered. However, simple expressions can be obtained in two limiting cases: (i) for near infrared wavelengths unaffected by light trapping, and (ii) at near band
ð7Þ
depends on the short circuit current density jðgi Þ from the device for (the artificial) generation profile g; and is normalized to the cumulative generation in region i; the symbol q denotes the elementary charge. Inserting (5) and (7) into (4) yields IQEi ¼
A jðgi Þ R : 1 R q total cell gðxÞ dx
ð8Þ
Eq. (8) describes the properties of IQE of the cell, which we need to calculate the IQE in Eq. (3). Here, R and A are the total reflectance and active absorption (Brendel et al., 1996). 3. Theoretical derivation A crystalline silicon solar cell with light-trapping structure using distributed Bragg reflector (DBR), as shown in Fig. 1, has recently been experimentally demonstrated for a very thick cell (e.g., 675 lm) (Feng et al., 2007). In Fig. 1, light incident onto the cell is partially reflected at the front surface with Rf and transmits with angle h
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through the emitter, space charge region and base and reflected back into the cell by the DBR with reflectance RDBR with reflected angle u and then it reaches the front surface with transmittance t and partially reflected back to the cell again. Electron/hole generation attenuates the light during its first transmitted pass and second reflected pass and all subsequent passes through the cell until the light is completely absorbed. Here, we will present a comprehensive study of this cell structure and its effectiveness for thin-film or thick cell applications. A multiple-pass model is adopted which allows the analysis of multiple passes of unabsorbed light through the cell. Two light passes and multiple passes up to 51 light passes are analyzed. We will demonstrate that the cell efficiency can be significantly improved through the optimization process of the DBR. Then we solve the diffusion equation with a generation profile gðxÞ that follows from the multiple-pass optical model. The effect of back angled reflection is also considered in the model. Special attention is paid to the base and substrate region because the minority carrier concentrations in both regions are interdependent via the boundary conditions at the base substrate interface. With the transport model, we calculate the minority density nðxÞ, and finally the quantum efficiency IQEi . 3.1. Back structure using DBR The DBR as part of the back reflector by itself can enhance the optical path length up to twice according to the ray theory. The design of DBRs is rather simple. The thicknesses of the DBR pairs are set to be the quarterwavelength of the materials. The only parameter that needs to be determined is the center wavelength kd (Feng et al., 2007). The DBR structure generated is a one-dimensional photonic crystal with high index contrast. Superior to any other high quality mirrors, DBR has a wide stop band expanding several hundred nanometers with nearly 100% reflectivity (Zhou and Biswas, 2008). Referring to the cell structure depicted in Fig. 1 and following the approach similar to Brendel et al. (1996), we have the following generation profile ð1Þ
ð2Þ
ðnÞ
n
n
gDBR ðxÞ ¼ RDBR ðkÞ ð1 tÞ a k l F ð1 Rf Þ ! ! n X k l H expðak l xÞ; exp a
where k l ¼ 1= cosðhl Þ is the light trapping coefficient, and hl is the light trapping angle of the lth path. For the case of ð3Þ only two passes ðH 0; gDBR ðxÞ 0Þ we have k l (l = 1, 2) equal to 1, i.e., hl 0, and the generation profile is ð1Þ
ð2Þ
gDBR ðxÞr¼2 ¼ gDBR ðxÞ þ gDBR ðxÞ ¼ aF ð1 Rf Þ expðaxÞ RDBR ðkÞ aF ð1 Rf Þ expðaH Þ expðaðH xÞÞ:
ð12Þ
Here, F is the flux of photons onto the front cell surface, Rf the front surface reflectance, and a is absorption coefficient. The subscript r represents the light trapped in rth pass. Now, r equals to two in Eq. (12), which means ½gDBR ðxÞr is consisted of contribution of one pass and two passes, i.e., ð1Þ ð2Þ ½gDBR ðxÞr¼2 ¼ gDBR ðxÞ þ gDBR ðxÞ. Here, we first consider one light pass and two passes, where pass one is light trapped from top to the bottom back reflector and pass two is light trapped from the back reflector to the top, which representing transmission and reflection in Fig. 1, respectively. If the value of RDBR ðkÞ is set equal to 0, then Eq. (12) reduces to ð1Þ ½gDBR ðxÞr¼2 ¼ gDBR ðxÞ, which has the form similar to Yang et al. (2008). The differential equations of the excess minority carrier density ½Dnb r¼2 and ½Dne r¼2 for the base and emitter are governed by the following diffusion equation with appropriate boundary conditions at interface between regions. If we consider steady-state situation, the diffusion equation (Sze, 1981) can be derived simply as d 2 ½Dnb ðxÞr¼2 ½Dnb ðxÞr¼2 ½g ðxÞr¼2 ; ¼ DBR Db L2b ðxÞ dx2
D½n0b ðxÞr¼2 ¼
Db ½Dnb ðxÞr¼2 : aF ð1 RÞ
ð9Þ
ð1 þ RDBR ðkÞ expð2aH þ 2axÞÞ: ð15Þ
ð1Þ
gDBR ðxÞ ¼ ak 1 F ð1 Rf Þ expðak 1 xÞ;
The general solution of Eq. (15) can be found and is given by
ð2Þ
ð3Þ
and
ð14Þ
d 2 ½Dn0b ðxÞr¼2 ½Dn0b ðxÞr¼2 ¼ expðaxÞ L2b ðxÞ dx2
ðlÞ
gDBR ðxÞ ¼ RDBR ðkÞð1 tÞak 3 F ð1 Rf Þ expða ðk 1 þ k 2 Þ H Þ expðak 3 xÞ;
ð13Þ
where Lb is diffusion length and Db is the minority carrier diffusion coefficient of the base. To simplify the calculations, a reduced excess minority carrier density is defined. We set
where the gDBR ðxÞ; ðl ¼ 1; 2; . . . ; nÞ are given by
gDBR ðxÞ ¼ RDBR ðkÞak 2 F ð1 Rf Þ expðak 1 H Þ expðak 2 ðH xÞÞ;
ð11Þ
1
With this substitution, the diffusion equation becomes
ð3Þ
gDBR ðxÞr¼n ¼ gDBR ðxÞ þ gDBR ðxÞ þ gDBR ðxÞ þ þ gDBR ðxÞ;
ðnÞ
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ð10Þ
0 Dnb ðxÞ r¼2 ¼ Ab exp
x x þ Bb exp Lb Lb
expðaxÞL2b ; ð1 þ RDBR ðkÞ expð2aðH þ xÞÞÞ 2 2 a Lb 1
ð16Þ
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where Ab and Bb are coefficients to be determined by the boundary conditions. The boundary conditions are the same as that for ½Dnb r¼2 at the interface between space charge region-base and at the back surface, respectively, and are given by
It is noted that for multiple-pass analysis, we need to solve the diffusion equations with generation profiles many times and the expressions can be very tedious and we have to rely on Matlab to solve the equations.
Dn0b ðx ¼ we þ wscr Þ ¼ 0;
d½Dn0b ðxÞr¼2
½Dn0b ðH Þr¼2 ¼ s ; b
dx Lb x¼H
ð17Þ
3.2. Determination of RDBR ðkÞ
ð18Þ
As mentioned in Section 1 the use of DBR back reflector can guarantee that almost no light can transmit thought the backside and thus reflect the light back to the cell layer and increase the absorption of photons. In this study, we use Si/SiO2 pair and four DBR layers with each layer of thickness kd =4, where kd is the center wavelength which we set kd ¼ 0:85 lm in our structure. Fig. 2 shows the reflectance of such structure at the center wavelength 0.85 lm. The value of reflectance RDBR ðkÞ is derived by using transmission matrix method, which optimizing the reflectance between wavelengths 0.8 lm and 1.2 lm and is found to be close to 100%.
with sb ¼ S b Lb =Db and H is the total thickness of the cell, i.e., H ¼ we þ wscr þ wb . Thus, the coefficients Ab and Bb can be obtained and the contribution of the base to the IQE calculated:
d½Dnb ðxÞr¼2
q D
b dx jb ðx ¼ H Þ x¼H ¼ ½IQEb ðaÞr¼2 ¼ q F ð1 Rf Þ q F ð1 Rf Þ
d½Dn0b ðxÞr¼2
¼a ð19Þ
: dx x¼H Similar to the above procedure, the general solution for ½Dn0e ðxÞr¼2 can be obtained, 0 x x Dne ðxÞ r¼2 ¼ Ae exp þ Be exp Le Le expðaxÞL2e 1 þ lRDBR ðkÞ expð2aðH þ xÞÞ 2 2 ; a Le 1 ð20Þ where se ¼ S e Le =De is the reduced front surface recombination velocity and S e denotes the front surface recombination velocity and De the minority carrier diffusion coefficient of the emitter, respectively. With the above boundary conditions, the coefficients Ae and Be can be calculated and the contribution of the emitter to the IQE can be obtained ½IQEe ðaÞr¼2 ¼ a
d½Dn0e ðxÞr¼2 : dx
ð21Þ
With the assumption that recombination in the space charge region can be neglected, the IQE of the scr is given by the product of the two probabilities that a photon reaches the scr and is absorbed in the scr, and IQE of space charge region is derived including pass one and pass two ðr ¼ 2Þ,
4. Results The above formulae can be applied to the flat-band homo-junction solar cells with DBR back reflector universally. For the following discussion the specific structural and electrical cell parameters are taken from Table 1 to calculate the IQEs of the different regions shown in Fig. 1. In Table 1, we use the similar values as those presented in Yang et al. (2008), which defines the cell properties and thickness of emitter, space charge and base regions. The physical properties of the cell structure like diffusion coefficient, diffusion lengths and surface recombination velocity are completely defined. We also examine the geometric size effect of the cell structure on IQEs by varying the thickness of emitter and base region while the thickness of space charge region is fixed. In the following, we denote the IQE results obtained using one light pass and the total reflectance R ¼ 0 as IQER¼0 , results with two light passes and R ¼ 1 as IQER¼1 , and results with the effect of reflected light angle u as IQEu . A typical DBR enhanced cell struc-
½IQEscr r¼2 ¼ expðawe Þ½1 expðawscr Þ þ expðaðH þ wb ÞÞ½1 expðawscr Þ:
ð22Þ
In the above analysis, we assume that normal incident light and transmitted angle h = 0° and the back reflected angle u = h = 0°. We can also take into account the effects of light transmitted angle h and the back reflected angle u, i.e., h = u = 30°. The total IQE is the sum of the contributions of the separate parts IQEtotal ðx; aÞ ¼ IQEe ðx; aÞ þ IQEscr ðx; aÞ þ IQEb ðx; aÞ:
ð23Þ
Fig. 2. Reflectance at designed wavelength 850 nm.
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Table 1 Structural and electrical parameters of the cell. Region
Widths of region (lm)
Diffusion coefficient (cm2/s)
Diffusion lengths (lm)
Surface recombination velocity (cm/s)
Emitter ðwe Þ Space charge ðwscr Þ Base ðwb Þ Total width ðH Þ Reflectance (R)
0.5/1.0/1.5 1.0 98/197.5/297 99.5/199.5/299.5
5
15
1.0E4
30
100
1.0E7
(0.0/0.5/1.0)
ture with we ¼ 0:5 lm; wb ¼ 298 lm, and wscr ¼ 1:0 lm is first considered. With these data, we get the IQEs profiles with single light pass and two light passes as shown in Fig. 3. Three different values of back reflectance R are calculated. For the case R ¼ 0, the present simulated cell is identical to the cell considered in Yang et al. (2008). The case of R ¼ 0:5 is to simulate back reflector which is not perfect. The total quantum efficiency increases as the value R varies from 0 to 1 and the efficiency can be enhanced up to 6.34% more with perfect reflector R ¼ 1 as compared with R ¼ 0 one pass case. 4.1. Impact of DBR on IQEs Here, we investigate the impact of DBR back reflector on the IQEs. The cell structure with we ¼ 0:5 lm; wb ¼ 298 lm, and wscr ¼ 1:0 lm is used. For all the results presented, we assume h = 0° and reflection angle u = 0° except for the case u = 60°, which is denoted as IQEu¼60 . Tables 2–4 show, respectively, the IQEs values of base, space charge and emitter regions at wavelength range 900– 1150 nm. The effect of light reflection angle u = 60° on the IQEs is also shown together. This is to simulate the effect of light bending to a tilted angle similar to that caused by diffraction grating at the back. Table 2 displays the contribution of the base, which indicating that the base contributes dominantly to the IQEs and it can be enhanced up to 6–7% at wavelength 900–1150 nm. The contributions to IQEs by the emitter and scr are much less than that by the base due to the difference in their thickness. In Tables 2–4, IQER¼0 represents the light passed by one path at
Fig. 3. Comparison of IQEs with one and two light passes.
Table 2 Impact on IQEb with second light trapping and back reflector at various wavelengths. Wavelength
900 nm
950 nm
1000 nm
1050 nm
1100 nm
IQEb;R¼0 IQEb;R¼1 IQEb;u¼60
0.7181 0.7181 0.7181
0.5923 0.593 0.5926
0.3758 0.3985 0.391
0.1302 0.1936 0.2202
0.0308 0.0571 0.0796
Table 3 Impact on IQEscr with second light trapping and back reflector at various wavelengths. Wavelength
900 nm
950 nm
1000 nm
1050 nm
1100 nm
IQER¼0 IQER¼1 IQEu¼60
0.0297 0.0297 0.0297
0.0155 0.00155 0.0155
0.0064 0.0065 0.0064
0.0016 0.0022 0.0021
0.00035 0.000634 0.000811
Table 4 Impact on IQEe with second light trapping and back reflector at various wavelengths. Wavelength
900 nm
950 nm
1000 nm
1050 nm
1100 nm
IQEe;R¼0 IQEe;R¼1 IQEe;u¼60
0.0145 0.0145 0.0145
0.0075 0.0075 0.0075
0.003 0.0031 0.0031
0.000777 0.0011 0.0011
0.000167 0.000302 0.000411
the cell, which means the reflectance of DBR is close to 0. When the reflectance of DBR is close to one, the equation needs to consider the reflected light path, because the reflection light would increase IQEs again. By Eqs. (12)–(19) we can get the IQER¼0 and IQER¼1 of base if we substitute different reflection coefficient of DBR. The value of IQEu¼60 is calculated by the same Eqs. (12)–(19) with h1 = 0° and u = h2 = 60°. The IQE of emitter is governed by the same process, which can be obtained by substituting the corresponding diffusion length and diffusion coefficient of the emitter into the above set of equations. As we can see on Table 2, the IQEb for the normal light incident case (h = 0°) is larger than the case of IQEu¼60 at wavelength 950–1050 nm, which is due to their diffusion length and absorption coefficient. At wavelength range 1050– 1150 nm, the IQEu¼60 is larger than that of (u = 0°) case, due to their longer light path and low absorption coefficient. Overall, with two light passes and DBR back reflector and u = 60°, the best achievable total IQE enhancement is 9.01% more than the value of the cell without light-trapping structure.
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4.2. Impact of base and emitter widths on IQEs To study the impact of the base width on the IQEs and the total IQE, we consider cell structure with fixed we ¼ 0:5 lm; wscr ¼ 1:0 lm and vary only the base width for three values wb = 98.0 lm, 198.0 lm, and 298.0 lm. As one can see in Fig. 4, when the base width increases, the contribution of the base to the IQEb also increases, which means the IQEs are function of the base width while the contributions of the emitter and space charge region are almost unchanged. The base thickness varies from 98 lm to 298 lm and affects the IQE from wavelength 800 nm to 1200 nm and the total IQE can be enhanced up to 8.84% at wavelength 950 nm. Following the same procedure, we are going to examine the effect of emitter width on IQEs and the total IQE. We have cell structure with wscr ¼ 1:0 lm and vary the thickness of emitter and consider three cases with we ¼ 0:5 lm; 1:0 lm and 1:5 lm with corresponding base thickness wb ¼ 298:0 lm; 297:5 lm and 297 lm, respectively, to yield same total cell thickness equal to 299.5 lm. The other parameters are specified as those in Table 1, which are unchanged including diffusion coefficient ðDe ; Db Þ, diffusion length ðLe ; Lb Þ and surface recombination velocity ðS e ; S b Þ. Given the emitter thickness variation in the range of 0.5–1.5 lm, we can see in Fig. 5 that the IQEscr and IQEb decrease as emitter thickness increases. The lowest IQEs take place at the emitter thickness equal to 1.5 lm, and the IQEs would be enhanced if we make the emitter thickness thinner. The total IQE of the case we ¼ 0:5 lm is about 14.28% more than that of the case we ¼ 1:5 lm at wavelength range 300–750 nm. As we can see in Figs. 4 and 5, the effect of small variation of emitter thickness (0.5–1.5 lm) on IQE is about 15% while the effect of large variation of base thickness (98– 298 lm) on IQE is about 9%. So the IQEs are more readily impacted by the emitter width than by the base width and for application purpose a solar cell should be made so as to have a smaller emitter width and a larger base width from this point of view. Also, the contribution of thicker base on
In the above, we consider only normal incident light thus with normal transmitted light angle h = 0°. Here, we consider non-normal incident and with a transmitted light angle h and the angle of reflected light at back reflector is also u = h. With the same cell parameters from Table 1, we set wb ¼ 298 lm; wscr ¼ 1:0 lm and we ¼ 0:5 lm. We consider two light trapping passes and three different transmitted light angles h = 0°, 30° and 60°. The effect of angled transmitted light propagation in whole cell on the IQEs is shown in Fig. 6. As we can see, the result indicates the cell actually performs better than that at normal incident case, such that the IQEs of each component like emitter, scr and base, are shifted to the right, i.e., higher wavelength ranges, when given a finite h angle. This change of IQE is caused by the increase of light path length in the cell, which directly increasing absorption on weak absorption wavelength between 700 and 1108 nm. The total IQE could be enhanced up to 17.14% at incident angle h = 60° as com-
Fig. 4. Effects of base width variation on IQEs.
Fig. 6. Impact of light incident angle on IQEs with h = 0°, 30°, and 60°.
Fig. 5. Effect of emitter width variation on IQEs.
IQE is huge for the entire weak absorption wavelength if we consider multiple light trapping passes. 4.3. Effect of incident light with angle
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pared with that at 0 incident angle and 0 reflectance. Thus, if we set a larger transmitted light angle, the IQEs would shift more toward the right. 4.4. Impact on IQEs with multiple-pass light trapping Based on the same parameters listed in Table 1, we analyze the effects of multiple-pass light trapping. We consider the cell structure with size wb ¼ 298 lm; we ¼ 0:5 lm and wscr ¼ 1:0 lm and the back reflectance is R ¼ 1 and the front transmittance t ¼ 0. The number of light path considered can be up to 51, which means that in the theoretical model, the generation profiles need to be considered 51 times. Based on our simulation we need to consider at least 51 multiple light paths in presented cells, so that the photon of wavelength 950–1100 nm can be completely absorbed within the cell. The effect of multiple light trapping paths on IQEb is shown in Fig. 7. For 21 light trapping paths the IQEb can be enhanced up to 23.79% more and for 51 light paths 26.98% more as compared with that of one light trapping path. It is observed that the effect of multiple passes on IQEb is more profound at wavelength range 950–1150 nm. Fig. 8 shows the IQEb versus the number of light trapping paths at wavelength 900–1200 nm. For wavelength 900–1000 nm the IQEb increases and becomes constant quickly as the photon can be absorbed completely within three light trapping paths, and for wavelength 1000–1100 nm the IQEb increases with the number of light trapping path until 51 light trapping paths where the photon can be absorbed completely. 4.5. Effects of variable reflectance and transmittance under multiple-pass Lastly, we analyze the effects of back reflectance variation and front transmittance variation on IQEb under the multiple-pass light trapping model. The cell parameters are the same as shown in Table 1. Here, we use cell with size we ¼ 0:5 lm; wscr ¼ 1:0 lm, and wb ¼ 298 lm and consider 51 light paths trapped in the present cell. To study the effect of variation of back reflectance on the IQEb we
Fig. 7. Effects of multiple light trapping paths on IQEb .
Fig. 8. The impact of light trapping paths on IQEb at wavelength 950– 1200 nm.
assume the front transmittance is fixed at t ¼ 0 and the results are shown in Fig. 9 for wavelength range from 850 to 1150 nm. As can be observed in Fig. 9, the variation of back reflectance R does not affect IQEb at wavelength range 850–950 nm due to the higher absorption coefficient and affects the IQEb more significantly from 950 to 1150 nm due to the weak absorption coefficient. Next we study the effect of variation of front transmittance on the IQEb and we fix the back reflectance at R ¼ 1 and the results are shown in Fig. 10. As can be seen in Fig. 10, the variation of t does not affect IQEb at wavelength 850–950 nm due to the higher absorption coefficient and it affects the IQEb quite markedly from 950 to 1100 nm. Specifically, under the same 51 light trapping paths, with t ¼ 0 and at wavelength 1050 nm, we have IQEb equal to 19.95% for R = 0.5 and 25.9% for R = 0.8, respectively. Similarly, under the 51 light trapping paths, with R ¼ 1 and at the same wavelength 1050 nm, we have IQEb equal to 23.85% and 27.71% for t ¼ 0:5 and 0:2, respectively. From these two specific cases, it is evident that the variation of reflectance on IQEb is more effective than the variation of transmittance on IQEb .
Fig. 9. Impact on IQEb with reflectance variation.
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and back reflectance, the impact of back reflectance on IQEs is relatively higher than that of front transmittance. Therefore, the design methods can follow the above findings, and by appropriately choosing these cell parameters, an optimal IQE can be realized. These cell parameters have impacts to different degrees on the IQEs which should be considered while designing a solar cell. Acknowledgments This work is supported by a grant from National Science Council, Taiwan through NSC-97-2212-020-3456 and partially supported by CQSE Subproject #5-97R0066-69. References Fig. 10. Impact on IQEb with transmittance variation.
5. Concluding remarks An optical model for the determination of generation profile of a flat band uniform-doped solar cell with distributed Bragg reflector back structure is presented and the analysis of IQEs for such cell structures is theoretically derived. We applied multiple-pass light trapping model to analyze and simulate the IQEs with variable thickness of cell regions and different incident angles. With this method, the enhancing effect of DBR on IQEs can be assessed clearly and the effects of width variations of different cell regions and incident angle on IQEs are also investigated in this simulation. The present DBR back reflector can achieve reflectivity 99.8% for wavelength between 800 and 1100 nm with just a few quarter-wave pairs. Using perfect back reflectors, the loss of light is mainly caused by the transmission through top front surface and the diffusion length of the cell. For the structure with DBR and with two light passes, the simulation result shows the IQEb can be increased up to 6.34% and with a u = 60° light reflection angle, the IQEb can be further increased up to 9.01% and the IQEb could be enhanced up to 17.14% at incident angle h = 60° with reflection angle u = 60° as compared with that at h = 0° and R = 0. The effect of multiple-pass light-trapping on IQEs is most effective in the wavelength range of 950–1150 nm and the total IQE can be relatively enhanced up to 26.98% when 51 light paths are used as compared to the IQE of one light trapping path ðR ¼ 0Þ. After 51 light-trapping passes, the photon can be absorbed completely, which is consistent with the theoretical enhanced effective path length 4n2 for silicon ðnSi ¼ 3:5Þ. With incident light transmitted angle h = 60° and reflection angle u = 60°, the photon can be fully absorbed within the cell after 25 light passes instead of 51 passes for the case of h = 0° and u = 0°. Under the variation of front surface transmittance
ASTMG 173-03, 2005. Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface, ASTM International, West Conshohocken, Pennsylvania. Basore, P.A., 1993. In: Photovoltaic Specialists Conference, Conference Record of the Twenty Third IEEE. Bermel, P., Luo, C., Zeng, L., Kimerling, L.C., Joannopoulos, J.D., 2007. Opt. Exp. 15, 25. Brendel, R., Hirsch, M., Plieninger, R., Werner, J.H., 1996. IEEE Trans. Electron. Device 43, 1104. Campbell, P., Green, M.A., 1987. J. Appl. Phys. 62, 243. Eisele, C., Nebel, C.E., Stutzmann, M., 2001. J. Appl. Phys. 89, 7722. Feng, N.N., Michel, J., Zeng, L., Liu, J., Hong, C.Y., Kimerling, L.C., Duan, X., 2007. IEEE Trans. Electron. Device 54, 1926. Green, M.A., 1987. Trans. Tech., Aedermannsdorf, Switzerland/Brookfield, VT, p. 70. Green, M.A., 1997. IEEE Optoelectron. Microelectron. Mater. Device Proc., 1. Hamakawa, Y., 2004. In: Thin-Film Solar Cells: Next Generation Photovoltaics and Its Applications. Springer, Berlin. Heine, C., Morf, R.H., 1995. Appl. Opt. 34, 2474. Herzinger, C., Johs, B., McGahan, W., Woollam, J., Paulson, W., 1998. J. Appl. Phys. 83, 3323. Johnson, D.C., Ballard, I., Barnham, K.W.J., Bishnell, D.B., Connolly, J.P., Lynch, M.C., Tibbits, T.N.D., Ekins-Daukes, N.J., Mazzer, M., Airey, R., Hill, G., Roberts, J.S., 2005. Sol. Energy Mater. Sol. Cells 87, 169. Muller, J., Rech, B., Springer, J., Vanecek, M., 2004. Sol. Energy 77, 917. Nelson, J., 2003. The Physics of Solar Cells. Imperial College, London. Rohatgi, A., Weber, E., Kimerling, L.C., 1993. J. Electron. Mater. 22, 65. Shvarts, M.Z., Chosta, O.I., Kochnev, I.V., Lantratov, V.M., Andreev, V.M., 2001. Sol. Energy Mater. Sol. Cells 68, 105. Sze, S.M., 1981. Physics of Semiconductor Devices. Wiley, New York. Venkatasubramanian, R., Siivola, E., Colpitts, T., 2003. In: Proceedings of the Third World Conference on Photovoltaic Energy Conversion, Osaka, Japan, p. 781. Wurfel, P., 2005. Physics of Solar Cell. Wiley–VCH, New York. Yablonovitch, E., Cody, G., 1982. IEEE Trans. Electron. Device ED 29, 300. Yamamoto, K., Nakajima, A., Yoshimi, M., Sawada, T., Fukuda, S., Suezaki, T., Ichikawa, M., Koi, Y., Goto, M., Meguro, T., Matsuda, T., Kondo, M., Sasaki, T., Tawada, Y., 2004. Sol. Energy 77, 939. Yang, W.J., Ma, Z.Q., Tang, X., Feng, C.B., Zhao, W.G., Shi, P.P., 2008. Sol. Energy 82, 106. Zeng, L., Yi, Y., Hong, C., Liu, J., Feng, N., Duan, X., Kimerling, L.C., Alamariu, B.A., 2006. Appl. Phys. Lett. 89, 111111. Zhou, D., Biswas, R., 2008. J. Appl. Phys. 103, 093102.
Solar Energy 83 (2009) 2050–2058 www.elsevier.com/locate/solener
The analysis of light trapping and internal quantum efficiency of a solar cell with DBR back reflector Kuo-Hui Yang a, Jaw-Yen Yang b,* b
a Institute of Applied Mechanics, National Taiwan University, Taipei 10746, Taiwan Center for Quantum Science and Engineering, National Taiwan University, Taipei 10764, Taiwan
Received 18 May 2009; received in revised form 26 July 2009; accepted 6 August 2009 Available online 31 August 2009 Communicated by: Takhir Razykov
Abstract A theoretical analysis of the total internal quantum efficiency (IQE) of a flat-band p–n homo-junction silicon solar cell with back reflector using distributed Bragg reflectors to improve the light trapping is presented and contributions of different regions of the structure to IQEs are simulated. An optical model for the determination of generation profile of the cell is adopted and multiple light passes are considered and compared to previous single light pass approach. It is found that the spatial widths of the cell, the surface recombination velocities, the front surface transmittance and the back reflector have significant impacts on the IQEs. With two light passes and normal incident light, the simulation result shows the IQEs can be increased over the one pass value by 6.34% and with a 60° light reflection angle, the IQEs can be further increased by 9.01% while assuming the reflectance at back structure closed to 100%. The effect on IQEs by back reflectance is more significant than that by front transmittance. Under multiple light passes simulation, up to 51 light trapping passes have been considered at wavelength range 900–1100 nm, the cell IQEs can be enhanced by about 26.98%. Ó 2009 Elsevier Ltd. All rights reserved. Keywords: Solar cells; Homo-junction; Internal quantum efficiency; Light trapping; DBR
1. Introduction The thin-film crystalline silicon (c-Si) solar cell is considered one of the best candidates for the next generation of photovoltaic applications due to its economic cost and easy fabrication, which are ideal for large-scale production. It is well known that crystalline silicon has an indirect band gap, which gives rise to weak absorption of light for the photons in the wavelengths close to the band gap ðkg ¼ 1108 nmÞ. For example, the absorption length increases from just over 10 lm for k = 800 nm to over 1 mm for ðkg ¼ 1108 nmÞ (Herzinger et al., 1998), which * Corresponding author. Address: Center for Quantum Science and Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10764, Taiwan. Tel.: +886 2 33665636; fax: +886 2 23639290. E-mail address: [email protected] (J.-Y. Yang).
0038-092X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2009.08.003
far exceeds the thickness of the core silicon layer of most thin-film cells (Yamamoto et al., 2004) and can result in a very low absorption of photons in this spectral range. However, the range of wavelengths contains 36.2% of the solar photons with energies above the band gap of c-Si (ASTMG 173-03, 2005). Thus, solar cells made from c-Si thin films may fail to absorb a significant number of photons that could otherwise be used to generate power in the cell. Meanwhile, the expense of thicker c-Si wafers with their correspondingly longer diffusion lengths (Rohatgi et al., 1993), drives up costs markedly. Also, the absorption coefficient for monochromatic light in silicon is a strong function of wavelength, so the spectral response of silicon solar cells contains information about their internal operation. But to study the internal operation, it is first necessary to eliminate external optical effects from the spectral response data. This can be accomplished by measuring
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the total hemispherical reflectance of the cell as a function of wavelength. The spectral response can be adjusted to account for the optical reflectance, yielding the internal quantum efficiency (IQE) of the device (Basore, 1993; Green, 1997; Bermel et al., 2007). One of the foremost challenges in designing silicon photovoltaic cells is devising an efficient light-trapping scheme which allows one to create thin yet efficient solar cells, made from c-Si and other, closely related materials, such as nano-crystalline silicon (which has the same band gap and similar absorption characteristics) (Hamakawa, 2004). One of the main approaches to light trapping: geometrical optics which is widely used in current solar cells and wave optics, which represents a novel approach to the problem that just begun to be explored. Several approaches have been proposed to enhance the absorption in the long-wavelength range, including searching alternative inexpensive materials (Nelson, 2003; Green, 1987), enhancing collection efficiency through randomly roughened surfaces (Green, 1987; Campbell and Green, 1987), and light trapping by optical structures (Muller et al., 2004; Heine and Morf, 1995; Brendel et al., 1996; Eisele et al., 2001; Shvarts et al., 2001; Johnson et al., 2005). The light-trapping using optical structures is of particular interest and could be done based on improving the equivalent optical path length by bending the light to a tilted angle through diffraction gratings (Heine and Morf, 1995; Brendel et al., 1996; Eisele et al., 2001; Shvarts et al., 2001) or simply reflecting light back using distributed Bragg reflectors (DBRs) (Shvarts et al., 2001; Johnson et al., 2005). Commonly used approaches to light trapping in solar cells rely on monitoring light paths through geometrical optics. Two main mechanisms to increase these path lengths are (i) the normally incident light can be scattered into an angle at the front interface via surface texturing (Campbell and Green, 1987), and (ii) it can be reflected back into the cell via a back reflector. Solar cells with DBR structure can offer reflection superior to highly absorptive aluminum reflectors. Photonic crystal enhanced solar cells in which the metallic back reflector is replaced by a distributed Bragg reflector DBR consisting of a onedimensional dielectric superlattice have been considered (Bermel et al., 2007; Feng et al., 2007; Zhou and Biswas, 2008). The distributed Bragg reflectors (DBR) with high index contrast can reflect light over a broad range of incident angles and wavelengths, for either or both polarizations. The use of DBR stacks Si/Si3N4 (n1/n2 = 3.5/2.0) or Si/SiO2 (n1/n2 = 3.5/1.46) can achieve reflectivity 99.8% for k between 800 and 1100 nm with just a few quarter-wave pairs. In contrast, when light is incident from Si to an Al backside reflector commonly used in solar cells, the reflectivity is <80%. The high reflectivity of DBR guarantees that almost no light can transmit through the backside (Zeng et al., 2006; Venkatasubramanian et al., 2003). Combining perfect random scattering at the front with a lossless reflector in the back can theoretically enhance the effective
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path length by a factor of 4n2, where n is the index of refraction, corresponding to a factor of about 50 for Si, and 30 for TiO2 (Yablonovitch and Cody, 1982). In this study, a theoretical analysis of the internal quantum efficiency (IQE) of a flat-band p–n homo-junction silicon solar cell with optical back reflector, as shown in Fig. 1, is presented. An improved light-trapping approach using DBR back structures is considered and current contributions from all cell regions including emitter, space charge region, base and DBR are all accounted for. A multiple-pass model is adopted which allows the analysis of multiple passes of unabsorbed light through the cell. The IQEs from two light passes and multiple passes up to 51 light passes are analyzed. The effect of back angled reflection is also considered in the model. We compare the IQEs of the present model to that of a baseline single pass IQEs analysis without back reflector (Yang et al., 2008). This paper is organized as follows. In Section 2, the analysis of cell efficiency based on semiconductor theory is described. In Section 3, the theoretical derivation of a solar cell with distributed Bragg back reflector is presented. The IQEs of each region of the cell structure is explicitly derived. The results are given in Section 4. The effects of the number of light paths, incident light angle, and variable
Fig. 1. Schematic of a crystalline silicon solar cell with DBR. Light incident onto the cell is partially reflected at the front surface with Rf . Electron/hole generation attenuates the light during its first transmitted pass ðl ¼ 1Þ and second reflected pass ðl ¼ 2Þ. Transmitted light with angle h is reflected at the back with reflectance RDBR with angle u and reflected back to the cell at the front surface with transmittance t. Similarly, the light intensity is further reduced by the second and all subsequent passes through the cell.
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reflectance and transmittance are studied. Some concluding remarks are given in Section 5. 2. Analysis of cell efficiency Based on the semiconductor theory (Hamakawa, 2004; Wurfel, 2005), the current in a cell always includes the photo-generated current and the dark diode current, with the former being proportional to the incident light intensity and the latter being independent of it. Thus, the conversion efficiency relies on both the photo-generated current and the diode property. Meanwhile, the photo-generated current is strongly related to the quantum efficiency (QE). In theory, the QE corresponds to the spectral response (SR) which determines the spectral distribution of the short circuit current I SC . By calculating the QE, both the SR and the contributions to the I SC of different wavelengths can be determined, helping one analyze quantum yields from the different cell regions. Therefore, it is useful to calculate the QE for finding the performance of a cell. The QE involves both the external quantum efficiency (EQE) and internal quantum efficiency (IQE), with the latter defined as the number of minority carriers contributing to the short circuit current divided by the number of photons entering the cell. To increase the conversion efficiency one needs to raise the QE. In experiment, the IQE can be obtained through the light reflectance and the SR. Thus, the SR can be defined as the ratio of the measured cell current I SC ðkÞ to the light intensity I light ðkÞ (Yang et al., 2008) SRðkÞ ¼
I SC ðkÞ : I light ðkÞ
ð1Þ
i¼e;scr;b;s
A photon’s contribution to the quantum efficiency of the region i is given by IQEi ¼ Ai gi =ð1 RÞ;
ð4Þ
where i = e, scr, b, s, if this light quantum is actively absorbed in region i with a probability Ai and then collected at the junction with collection efficiency gi . A photon is called as being actively absorbed if an electron/hole pair is generated, while parasitic absorption processes such as by free carriers or optical losses at the back surface mirror do not contribute to the active absorption Ai . The factor ð1 RÞ normalizes the QE to the fraction of photons that are not reflected by the cell. The relative contribution Ai , from different cell region i to the total active absorption A is R g ðxÞ dx Ai region i i R ¼ : ð5Þ A gðxÞ dx total cell Here, gi ðxÞ ¼
gðxÞ; for x in region i; 0;
elsewhere:
ð6Þ
The collection efficiency of region i,
The IQE can be derived and is given by 1 I SC ðkÞ=e IQEðkÞ ¼ 1 RðkÞ I light ðkÞ=ðhc=kÞ 1 hc ¼ SRðkÞ; 1 RðkÞ ek
gap wavelengths where photo-generation is uniform throughout the device (Basore, 1993). A theoretical model for the internal quantum efficiency must consider contributions of the emitter e, the space charge region scr, the base region b, and the substrate region s and can be expressed as (Brendel et al., 1996): X IQEi : ð3Þ IQE ¼
jðgi Þ gi ¼ R q region i gi ðxÞ dx ð2Þ
where RðkÞ is the reflectance, h the Planck constant, c the speed of light and e the unit charge. The electrical model of the solar cell considered in this work is a simple one. The base region of the cell is a quasi-neutral region in low-level injection, characterized by a uniform minoritycarrier diffusion length (L) and diffusivity (D), and a surface recombination velocity (S) at the back surface. The collecting junction is assumed to be located at the front surface. The wavelength range considered is restricted to k > 800 nm so that the influence of recombination in the thin emitter region can be neglected. Despite this simple model, an analytical expression valid over the entire wavelength range could be extraordinarily complex, because the photo-generation profile in the device is affected by light trapping at the longer wavelengths and many light passes are considered. However, simple expressions can be obtained in two limiting cases: (i) for near infrared wavelengths unaffected by light trapping, and (ii) at near band
ð7Þ
depends on the short circuit current density jðgi Þ from the device for (the artificial) generation profile g; and is normalized to the cumulative generation in region i; the symbol q denotes the elementary charge. Inserting (5) and (7) into (4) yields IQEi ¼
A jðgi Þ R : 1 R q total cell gðxÞ dx
ð8Þ
Eq. (8) describes the properties of IQE of the cell, which we need to calculate the IQE in Eq. (3). Here, R and A are the total reflectance and active absorption (Brendel et al., 1996). 3. Theoretical derivation A crystalline silicon solar cell with light-trapping structure using distributed Bragg reflector (DBR), as shown in Fig. 1, has recently been experimentally demonstrated for a very thick cell (e.g., 675 lm) (Feng et al., 2007). In Fig. 1, light incident onto the cell is partially reflected at the front surface with Rf and transmits with angle h
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through the emitter, space charge region and base and reflected back into the cell by the DBR with reflectance RDBR with reflected angle u and then it reaches the front surface with transmittance t and partially reflected back to the cell again. Electron/hole generation attenuates the light during its first transmitted pass and second reflected pass and all subsequent passes through the cell until the light is completely absorbed. Here, we will present a comprehensive study of this cell structure and its effectiveness for thin-film or thick cell applications. A multiple-pass model is adopted which allows the analysis of multiple passes of unabsorbed light through the cell. Two light passes and multiple passes up to 51 light passes are analyzed. We will demonstrate that the cell efficiency can be significantly improved through the optimization process of the DBR. Then we solve the diffusion equation with a generation profile gðxÞ that follows from the multiple-pass optical model. The effect of back angled reflection is also considered in the model. Special attention is paid to the base and substrate region because the minority carrier concentrations in both regions are interdependent via the boundary conditions at the base substrate interface. With the transport model, we calculate the minority density nðxÞ, and finally the quantum efficiency IQEi . 3.1. Back structure using DBR The DBR as part of the back reflector by itself can enhance the optical path length up to twice according to the ray theory. The design of DBRs is rather simple. The thicknesses of the DBR pairs are set to be the quarterwavelength of the materials. The only parameter that needs to be determined is the center wavelength kd (Feng et al., 2007). The DBR structure generated is a one-dimensional photonic crystal with high index contrast. Superior to any other high quality mirrors, DBR has a wide stop band expanding several hundred nanometers with nearly 100% reflectivity (Zhou and Biswas, 2008). Referring to the cell structure depicted in Fig. 1 and following the approach similar to Brendel et al. (1996), we have the following generation profile ð1Þ
ð2Þ
ðnÞ
n
n
gDBR ðxÞ ¼ RDBR ðkÞ ð1 tÞ a k l F ð1 Rf Þ ! ! n X k l H expðak l xÞ; exp a
where k l ¼ 1= cosðhl Þ is the light trapping coefficient, and hl is the light trapping angle of the lth path. For the case of ð3Þ only two passes ðH 0; gDBR ðxÞ 0Þ we have k l (l = 1, 2) equal to 1, i.e., hl 0, and the generation profile is ð1Þ
ð2Þ
gDBR ðxÞr¼2 ¼ gDBR ðxÞ þ gDBR ðxÞ ¼ aF ð1 Rf Þ expðaxÞ RDBR ðkÞ aF ð1 Rf Þ expðaH Þ expðaðH xÞÞ:
ð12Þ
Here, F is the flux of photons onto the front cell surface, Rf the front surface reflectance, and a is absorption coefficient. The subscript r represents the light trapped in rth pass. Now, r equals to two in Eq. (12), which means ½gDBR ðxÞr is consisted of contribution of one pass and two passes, i.e., ð1Þ ð2Þ ½gDBR ðxÞr¼2 ¼ gDBR ðxÞ þ gDBR ðxÞ. Here, we first consider one light pass and two passes, where pass one is light trapped from top to the bottom back reflector and pass two is light trapped from the back reflector to the top, which representing transmission and reflection in Fig. 1, respectively. If the value of RDBR ðkÞ is set equal to 0, then Eq. (12) reduces to ð1Þ ½gDBR ðxÞr¼2 ¼ gDBR ðxÞ, which has the form similar to Yang et al. (2008). The differential equations of the excess minority carrier density ½Dnb r¼2 and ½Dne r¼2 for the base and emitter are governed by the following diffusion equation with appropriate boundary conditions at interface between regions. If we consider steady-state situation, the diffusion equation (Sze, 1981) can be derived simply as d 2 ½Dnb ðxÞr¼2 ½Dnb ðxÞr¼2 ½g ðxÞr¼2 ; ¼ DBR Db L2b ðxÞ dx2
D½n0b ðxÞr¼2 ¼
Db ½Dnb ðxÞr¼2 : aF ð1 RÞ
ð9Þ
ð1 þ RDBR ðkÞ expð2aH þ 2axÞÞ: ð15Þ
ð1Þ
gDBR ðxÞ ¼ ak 1 F ð1 Rf Þ expðak 1 xÞ;
The general solution of Eq. (15) can be found and is given by
ð2Þ
ð3Þ
and
ð14Þ
d 2 ½Dn0b ðxÞr¼2 ½Dn0b ðxÞr¼2 ¼ expðaxÞ L2b ðxÞ dx2
ðlÞ
gDBR ðxÞ ¼ RDBR ðkÞð1 tÞak 3 F ð1 Rf Þ expða ðk 1 þ k 2 Þ H Þ expðak 3 xÞ;
ð13Þ
where Lb is diffusion length and Db is the minority carrier diffusion coefficient of the base. To simplify the calculations, a reduced excess minority carrier density is defined. We set
where the gDBR ðxÞ; ðl ¼ 1; 2; . . . ; nÞ are given by
gDBR ðxÞ ¼ RDBR ðkÞak 2 F ð1 Rf Þ expðak 1 H Þ expðak 2 ðH xÞÞ;
ð11Þ
1
With this substitution, the diffusion equation becomes
ð3Þ
gDBR ðxÞr¼n ¼ gDBR ðxÞ þ gDBR ðxÞ þ gDBR ðxÞ þ þ gDBR ðxÞ;
ðnÞ
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ð10Þ
0 Dnb ðxÞ r¼2 ¼ Ab exp
x x þ Bb exp Lb Lb
expðaxÞL2b ; ð1 þ RDBR ðkÞ expð2aðH þ xÞÞÞ 2 2 a Lb 1
ð16Þ
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where Ab and Bb are coefficients to be determined by the boundary conditions. The boundary conditions are the same as that for ½Dnb r¼2 at the interface between space charge region-base and at the back surface, respectively, and are given by
It is noted that for multiple-pass analysis, we need to solve the diffusion equations with generation profiles many times and the expressions can be very tedious and we have to rely on Matlab to solve the equations.
Dn0b ðx ¼ we þ wscr Þ ¼ 0;
d½Dn0b ðxÞr¼2
½Dn0b ðH Þr¼2 ¼ s ; b
dx Lb x¼H
ð17Þ
3.2. Determination of RDBR ðkÞ
ð18Þ
As mentioned in Section 1 the use of DBR back reflector can guarantee that almost no light can transmit thought the backside and thus reflect the light back to the cell layer and increase the absorption of photons. In this study, we use Si/SiO2 pair and four DBR layers with each layer of thickness kd =4, where kd is the center wavelength which we set kd ¼ 0:85 lm in our structure. Fig. 2 shows the reflectance of such structure at the center wavelength 0.85 lm. The value of reflectance RDBR ðkÞ is derived by using transmission matrix method, which optimizing the reflectance between wavelengths 0.8 lm and 1.2 lm and is found to be close to 100%.
with sb ¼ S b Lb =Db and H is the total thickness of the cell, i.e., H ¼ we þ wscr þ wb . Thus, the coefficients Ab and Bb can be obtained and the contribution of the base to the IQE calculated:
d½Dnb ðxÞr¼2
q D
b dx jb ðx ¼ H Þ x¼H ¼ ½IQEb ðaÞr¼2 ¼ q F ð1 Rf Þ q F ð1 Rf Þ
d½Dn0b ðxÞr¼2
¼a ð19Þ
: dx x¼H Similar to the above procedure, the general solution for ½Dn0e ðxÞr¼2 can be obtained, 0 x x Dne ðxÞ r¼2 ¼ Ae exp þ Be exp Le Le expðaxÞL2e 1 þ lRDBR ðkÞ expð2aðH þ xÞÞ 2 2 ; a Le 1 ð20Þ where se ¼ S e Le =De is the reduced front surface recombination velocity and S e denotes the front surface recombination velocity and De the minority carrier diffusion coefficient of the emitter, respectively. With the above boundary conditions, the coefficients Ae and Be can be calculated and the contribution of the emitter to the IQE can be obtained ½IQEe ðaÞr¼2 ¼ a
d½Dn0e ðxÞr¼2 : dx
ð21Þ
With the assumption that recombination in the space charge region can be neglected, the IQE of the scr is given by the product of the two probabilities that a photon reaches the scr and is absorbed in the scr, and IQE of space charge region is derived including pass one and pass two ðr ¼ 2Þ,
4. Results The above formulae can be applied to the flat-band homo-junction solar cells with DBR back reflector universally. For the following discussion the specific structural and electrical cell parameters are taken from Table 1 to calculate the IQEs of the different regions shown in Fig. 1. In Table 1, we use the similar values as those presented in Yang et al. (2008), which defines the cell properties and thickness of emitter, space charge and base regions. The physical properties of the cell structure like diffusion coefficient, diffusion lengths and surface recombination velocity are completely defined. We also examine the geometric size effect of the cell structure on IQEs by varying the thickness of emitter and base region while the thickness of space charge region is fixed. In the following, we denote the IQE results obtained using one light pass and the total reflectance R ¼ 0 as IQER¼0 , results with two light passes and R ¼ 1 as IQER¼1 , and results with the effect of reflected light angle u as IQEu . A typical DBR enhanced cell struc-
½IQEscr r¼2 ¼ expðawe Þ½1 expðawscr Þ þ expðaðH þ wb ÞÞ½1 expðawscr Þ:
ð22Þ
In the above analysis, we assume that normal incident light and transmitted angle h = 0° and the back reflected angle u = h = 0°. We can also take into account the effects of light transmitted angle h and the back reflected angle u, i.e., h = u = 30°. The total IQE is the sum of the contributions of the separate parts IQEtotal ðx; aÞ ¼ IQEe ðx; aÞ þ IQEscr ðx; aÞ þ IQEb ðx; aÞ:
ð23Þ
Fig. 2. Reflectance at designed wavelength 850 nm.
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Table 1 Structural and electrical parameters of the cell. Region
Widths of region (lm)
Diffusion coefficient (cm2/s)
Diffusion lengths (lm)
Surface recombination velocity (cm/s)
Emitter ðwe Þ Space charge ðwscr Þ Base ðwb Þ Total width ðH Þ Reflectance (R)
0.5/1.0/1.5 1.0 98/197.5/297 99.5/199.5/299.5
5
15
1.0E4
30
100
1.0E7
(0.0/0.5/1.0)
ture with we ¼ 0:5 lm; wb ¼ 298 lm, and wscr ¼ 1:0 lm is first considered. With these data, we get the IQEs profiles with single light pass and two light passes as shown in Fig. 3. Three different values of back reflectance R are calculated. For the case R ¼ 0, the present simulated cell is identical to the cell considered in Yang et al. (2008). The case of R ¼ 0:5 is to simulate back reflector which is not perfect. The total quantum efficiency increases as the value R varies from 0 to 1 and the efficiency can be enhanced up to 6.34% more with perfect reflector R ¼ 1 as compared with R ¼ 0 one pass case. 4.1. Impact of DBR on IQEs Here, we investigate the impact of DBR back reflector on the IQEs. The cell structure with we ¼ 0:5 lm; wb ¼ 298 lm, and wscr ¼ 1:0 lm is used. For all the results presented, we assume h = 0° and reflection angle u = 0° except for the case u = 60°, which is denoted as IQEu¼60 . Tables 2–4 show, respectively, the IQEs values of base, space charge and emitter regions at wavelength range 900– 1150 nm. The effect of light reflection angle u = 60° on the IQEs is also shown together. This is to simulate the effect of light bending to a tilted angle similar to that caused by diffraction grating at the back. Table 2 displays the contribution of the base, which indicating that the base contributes dominantly to the IQEs and it can be enhanced up to 6–7% at wavelength 900–1150 nm. The contributions to IQEs by the emitter and scr are much less than that by the base due to the difference in their thickness. In Tables 2–4, IQER¼0 represents the light passed by one path at
Fig. 3. Comparison of IQEs with one and two light passes.
Table 2 Impact on IQEb with second light trapping and back reflector at various wavelengths. Wavelength
900 nm
950 nm
1000 nm
1050 nm
1100 nm
IQEb;R¼0 IQEb;R¼1 IQEb;u¼60
0.7181 0.7181 0.7181
0.5923 0.593 0.5926
0.3758 0.3985 0.391
0.1302 0.1936 0.2202
0.0308 0.0571 0.0796
Table 3 Impact on IQEscr with second light trapping and back reflector at various wavelengths. Wavelength
900 nm
950 nm
1000 nm
1050 nm
1100 nm
IQER¼0 IQER¼1 IQEu¼60
0.0297 0.0297 0.0297
0.0155 0.00155 0.0155
0.0064 0.0065 0.0064
0.0016 0.0022 0.0021
0.00035 0.000634 0.000811
Table 4 Impact on IQEe with second light trapping and back reflector at various wavelengths. Wavelength
900 nm
950 nm
1000 nm
1050 nm
1100 nm
IQEe;R¼0 IQEe;R¼1 IQEe;u¼60
0.0145 0.0145 0.0145
0.0075 0.0075 0.0075
0.003 0.0031 0.0031
0.000777 0.0011 0.0011
0.000167 0.000302 0.000411
the cell, which means the reflectance of DBR is close to 0. When the reflectance of DBR is close to one, the equation needs to consider the reflected light path, because the reflection light would increase IQEs again. By Eqs. (12)–(19) we can get the IQER¼0 and IQER¼1 of base if we substitute different reflection coefficient of DBR. The value of IQEu¼60 is calculated by the same Eqs. (12)–(19) with h1 = 0° and u = h2 = 60°. The IQE of emitter is governed by the same process, which can be obtained by substituting the corresponding diffusion length and diffusion coefficient of the emitter into the above set of equations. As we can see on Table 2, the IQEb for the normal light incident case (h = 0°) is larger than the case of IQEu¼60 at wavelength 950–1050 nm, which is due to their diffusion length and absorption coefficient. At wavelength range 1050– 1150 nm, the IQEu¼60 is larger than that of (u = 0°) case, due to their longer light path and low absorption coefficient. Overall, with two light passes and DBR back reflector and u = 60°, the best achievable total IQE enhancement is 9.01% more than the value of the cell without light-trapping structure.
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4.2. Impact of base and emitter widths on IQEs To study the impact of the base width on the IQEs and the total IQE, we consider cell structure with fixed we ¼ 0:5 lm; wscr ¼ 1:0 lm and vary only the base width for three values wb = 98.0 lm, 198.0 lm, and 298.0 lm. As one can see in Fig. 4, when the base width increases, the contribution of the base to the IQEb also increases, which means the IQEs are function of the base width while the contributions of the emitter and space charge region are almost unchanged. The base thickness varies from 98 lm to 298 lm and affects the IQE from wavelength 800 nm to 1200 nm and the total IQE can be enhanced up to 8.84% at wavelength 950 nm. Following the same procedure, we are going to examine the effect of emitter width on IQEs and the total IQE. We have cell structure with wscr ¼ 1:0 lm and vary the thickness of emitter and consider three cases with we ¼ 0:5 lm; 1:0 lm and 1:5 lm with corresponding base thickness wb ¼ 298:0 lm; 297:5 lm and 297 lm, respectively, to yield same total cell thickness equal to 299.5 lm. The other parameters are specified as those in Table 1, which are unchanged including diffusion coefficient ðDe ; Db Þ, diffusion length ðLe ; Lb Þ and surface recombination velocity ðS e ; S b Þ. Given the emitter thickness variation in the range of 0.5–1.5 lm, we can see in Fig. 5 that the IQEscr and IQEb decrease as emitter thickness increases. The lowest IQEs take place at the emitter thickness equal to 1.5 lm, and the IQEs would be enhanced if we make the emitter thickness thinner. The total IQE of the case we ¼ 0:5 lm is about 14.28% more than that of the case we ¼ 1:5 lm at wavelength range 300–750 nm. As we can see in Figs. 4 and 5, the effect of small variation of emitter thickness (0.5–1.5 lm) on IQE is about 15% while the effect of large variation of base thickness (98– 298 lm) on IQE is about 9%. So the IQEs are more readily impacted by the emitter width than by the base width and for application purpose a solar cell should be made so as to have a smaller emitter width and a larger base width from this point of view. Also, the contribution of thicker base on
In the above, we consider only normal incident light thus with normal transmitted light angle h = 0°. Here, we consider non-normal incident and with a transmitted light angle h and the angle of reflected light at back reflector is also u = h. With the same cell parameters from Table 1, we set wb ¼ 298 lm; wscr ¼ 1:0 lm and we ¼ 0:5 lm. We consider two light trapping passes and three different transmitted light angles h = 0°, 30° and 60°. The effect of angled transmitted light propagation in whole cell on the IQEs is shown in Fig. 6. As we can see, the result indicates the cell actually performs better than that at normal incident case, such that the IQEs of each component like emitter, scr and base, are shifted to the right, i.e., higher wavelength ranges, when given a finite h angle. This change of IQE is caused by the increase of light path length in the cell, which directly increasing absorption on weak absorption wavelength between 700 and 1108 nm. The total IQE could be enhanced up to 17.14% at incident angle h = 60° as com-
Fig. 4. Effects of base width variation on IQEs.
Fig. 6. Impact of light incident angle on IQEs with h = 0°, 30°, and 60°.
Fig. 5. Effect of emitter width variation on IQEs.
IQE is huge for the entire weak absorption wavelength if we consider multiple light trapping passes. 4.3. Effect of incident light with angle
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pared with that at 0 incident angle and 0 reflectance. Thus, if we set a larger transmitted light angle, the IQEs would shift more toward the right. 4.4. Impact on IQEs with multiple-pass light trapping Based on the same parameters listed in Table 1, we analyze the effects of multiple-pass light trapping. We consider the cell structure with size wb ¼ 298 lm; we ¼ 0:5 lm and wscr ¼ 1:0 lm and the back reflectance is R ¼ 1 and the front transmittance t ¼ 0. The number of light path considered can be up to 51, which means that in the theoretical model, the generation profiles need to be considered 51 times. Based on our simulation we need to consider at least 51 multiple light paths in presented cells, so that the photon of wavelength 950–1100 nm can be completely absorbed within the cell. The effect of multiple light trapping paths on IQEb is shown in Fig. 7. For 21 light trapping paths the IQEb can be enhanced up to 23.79% more and for 51 light paths 26.98% more as compared with that of one light trapping path. It is observed that the effect of multiple passes on IQEb is more profound at wavelength range 950–1150 nm. Fig. 8 shows the IQEb versus the number of light trapping paths at wavelength 900–1200 nm. For wavelength 900–1000 nm the IQEb increases and becomes constant quickly as the photon can be absorbed completely within three light trapping paths, and for wavelength 1000–1100 nm the IQEb increases with the number of light trapping path until 51 light trapping paths where the photon can be absorbed completely. 4.5. Effects of variable reflectance and transmittance under multiple-pass Lastly, we analyze the effects of back reflectance variation and front transmittance variation on IQEb under the multiple-pass light trapping model. The cell parameters are the same as shown in Table 1. Here, we use cell with size we ¼ 0:5 lm; wscr ¼ 1:0 lm, and wb ¼ 298 lm and consider 51 light paths trapped in the present cell. To study the effect of variation of back reflectance on the IQEb we
Fig. 7. Effects of multiple light trapping paths on IQEb .
Fig. 8. The impact of light trapping paths on IQEb at wavelength 950– 1200 nm.
assume the front transmittance is fixed at t ¼ 0 and the results are shown in Fig. 9 for wavelength range from 850 to 1150 nm. As can be observed in Fig. 9, the variation of back reflectance R does not affect IQEb at wavelength range 850–950 nm due to the higher absorption coefficient and affects the IQEb more significantly from 950 to 1150 nm due to the weak absorption coefficient. Next we study the effect of variation of front transmittance on the IQEb and we fix the back reflectance at R ¼ 1 and the results are shown in Fig. 10. As can be seen in Fig. 10, the variation of t does not affect IQEb at wavelength 850–950 nm due to the higher absorption coefficient and it affects the IQEb quite markedly from 950 to 1100 nm. Specifically, under the same 51 light trapping paths, with t ¼ 0 and at wavelength 1050 nm, we have IQEb equal to 19.95% for R = 0.5 and 25.9% for R = 0.8, respectively. Similarly, under the 51 light trapping paths, with R ¼ 1 and at the same wavelength 1050 nm, we have IQEb equal to 23.85% and 27.71% for t ¼ 0:5 and 0:2, respectively. From these two specific cases, it is evident that the variation of reflectance on IQEb is more effective than the variation of transmittance on IQEb .
Fig. 9. Impact on IQEb with reflectance variation.
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and back reflectance, the impact of back reflectance on IQEs is relatively higher than that of front transmittance. Therefore, the design methods can follow the above findings, and by appropriately choosing these cell parameters, an optimal IQE can be realized. These cell parameters have impacts to different degrees on the IQEs which should be considered while designing a solar cell. Acknowledgments This work is supported by a grant from National Science Council, Taiwan through NSC-97-2212-020-3456 and partially supported by CQSE Subproject #5-97R0066-69. References Fig. 10. Impact on IQEb with transmittance variation.
5. Concluding remarks An optical model for the determination of generation profile of a flat band uniform-doped solar cell with distributed Bragg reflector back structure is presented and the analysis of IQEs for such cell structures is theoretically derived. We applied multiple-pass light trapping model to analyze and simulate the IQEs with variable thickness of cell regions and different incident angles. With this method, the enhancing effect of DBR on IQEs can be assessed clearly and the effects of width variations of different cell regions and incident angle on IQEs are also investigated in this simulation. The present DBR back reflector can achieve reflectivity 99.8% for wavelength between 800 and 1100 nm with just a few quarter-wave pairs. Using perfect back reflectors, the loss of light is mainly caused by the transmission through top front surface and the diffusion length of the cell. For the structure with DBR and with two light passes, the simulation result shows the IQEb can be increased up to 6.34% and with a u = 60° light reflection angle, the IQEb can be further increased up to 9.01% and the IQEb could be enhanced up to 17.14% at incident angle h = 60° with reflection angle u = 60° as compared with that at h = 0° and R = 0. The effect of multiple-pass light-trapping on IQEs is most effective in the wavelength range of 950–1150 nm and the total IQE can be relatively enhanced up to 26.98% when 51 light paths are used as compared to the IQE of one light trapping path ðR ¼ 0Þ. After 51 light-trapping passes, the photon can be absorbed completely, which is consistent with the theoretical enhanced effective path length 4n2 for silicon ðnSi ¼ 3:5Þ. With incident light transmitted angle h = 60° and reflection angle u = 60°, the photon can be fully absorbed within the cell after 25 light passes instead of 51 passes for the case of h = 0° and u = 0°. Under the variation of front surface transmittance
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