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JOURNAL OF ENVIRONMENTAL SCIENCES ISSN 1001-0742 CN 11-2629/X
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179
www.jesc.ac.cn
Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes Satoru Isoda Advanced Science Research Laboratory, Saitama Institute of Technology, 1690 Fusaiji, Hukaya, Saitama 3690293, Japan. E-mail: [email protected]
Abstract Power conversion efficiency of p-i-n type microcrystalline silicon (µc-Si:H) solar cells has been analyzed in terms of sequential processes of photo-induced electron transfer. The effect of the excitonic state on the charged carrier generation has been studied compared to a conventional scheme in which only charged carriers are taken into account for the operation of the solar cells. A numerical model has been developed to calculate current-voltage characteristics of solar cells on the basis of two types of charged carrier generation processes (exciton process and charged carrier process). The light trapping effect due to a textured back surface reflector (BSR) was embedded in the numerical model by using the effective medium theory in combination with the matrix method in the field of the electromagnetic theory of light. As an application of this modeling, it was found that the reported data of the power conversion efficiency were not explained by the conventional charged carrier process model and that the combined model of the charged carrier process with the exciton process well explains the performance of the p-i-n type µc-Si:H solar cells. In this way, the typical power conversion efficiencies were estimated to be 10.5% for the device (i-layer thickness: 1.8 µm) with the BSR (period: 600 nm; height: 250 nm) and 8.6% for the device with the flat reflector under the condition that the fractions of the exciton process and charged carrier process were 60% and 40%, respectively. Key words: solar cell; thin-film silicon; microcrystalline silicon; p-i-n type; current-voltage characteristics; power conversion efficiency; photo-induced electron transfer; diffusion length; exciton; light trapping
Introduction Although multi-crystalline and single-crystalline silicon solar cells play a main role in the solar cell market at present, thin-film silicon solar cells have been expected as the mainstream of next generation solar cells from the standpoint not only of their high performance but also of their less use of material and energy for manufacturing. Among several types of the thin-film silicon solar cells, the p-i-n type microcrystalline silicon (µc-Si:H) solar cells are considered to be most promising and have been intensively developed in order to increase the performance. The µc-Si:H solar cells have already reached to the commercial level, however, the mechanisms underlying carrier generation, transport and recombination have not been completely understood. Thin-film silicon solar cells generally consist of pin junctions instead of pn junction in order to increase the density of charged carriers produced by light absorption. For the theoretical study on the photovoltaic properties of thin-film silicon pin type solar cells, the simulation model for amorphous silicon solar cells presented by Hack and Shur (Hack and Shur, 1985) has been accepted as a fundamental model. In their model,
however, the exciton effect (Corkish et al., 1996; Kane and Swanson, 1993) as well as the optical interference effect (Pettersson et al., 1999) including the light trapping effect by texturing an electrode surface (Yamamoto et al., 2004) were not taken into consideration. Under these circumstances, this study deals with the numerical model which totally covers sequential rate determining processes involving the dynamics of the energy conversion in solar cells. Especially, the author focused on the modeling of charged carrier generation processes in combination with light trapping.
1 Materials and methods 1.1 Theory Since the author already reported the details of the photoinduced electron transfer processes in thin-film solar cells (Isoda, 2011), only fundamental equations are described in this chapter.
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
1.1.1 Light absorption process and modeling of light trapping effect The internal optical electric field in layer j at position x at normal incidence (Q(x)) is expressed by Pettersson et al. (1999) 2 Q j (x) = 12[cε0 α j η j E j (x) ] 4πη ,,2 = α j T j I0 e−α j x + ρ j ·e−α j (2d j −x) +2 ρ,,j ·e−α j d j · cos( λ j (d j − x) + δ,,j )
(1)
where, c is the speed of light, ε0 is the permittivity of free space, α j is the absorption coefficient of layer j, η j is the real part of the complex index of refraction, T j is the internal intensity transmittance, I0 is the intensity of the incident light, d j is the thickness of layer j, and ρ j ′′ and δ j ′′ are the absolute value and the argument of the complex reflection coefficient. In order to use the optical electric field in the thin-film solar cell as the light intensity in the cell, the textured structure must be treated as a flat effective medium. Figure 1 shows an effective medium model for a textured back surface reflector (BSR) of the solar cell. The complex index of refraction in the effective medium, np , is expressed based on the second-order effective medium theory as literature (Raguin and Morris, 1993), (2)
(0)
n p = n p [1 +
π2 Λ 2 2 (n s − ni )2 1 ( ) f (1 − f 2 ) ]2 3 λ (0)2 np
1.1.2
Photo-excited relaxation microcrystalline-silicon film
processes
ni ni ns
1.1.3 Diffusion process of initial carrier The initial carrier (electron and hole in the charged carrier process; exciton in the exciton process) density n(Z) at depth, z is expressed by the equation of continuity as, dn di n d2 n n =− + Q(z) − = D 2 + Q(z) − dt dz τ τ dz
in
np
Textured Back Surface Reflector Effective Medium Fig. 1 Effective medium for textured back surface reflector.
(3)
where, Q(z) is the internal optical electric field in position z (Eq. (1)), i is the carrier current (i = −d dn dz ), D is the diffusion constant and τ is the volume recombination lifetime of the carrier. At the steady-state condition, dn/dt = 0, Eqation (3) has the general solution of, n(z) =
The generation process of charged carriers is determined by localized band-tail states and deep defects. In this paper, the effect of excitonic state on the charged carrier generation has been studied compared to a conventional scheme in which only charged carriers are taken into account for the operation of a solar cell. Here, the author would like to propose the scheme for photo-excited relaxation processes in a µc-Si:H thin film by summarizing reported experimental results (Stachowitz et al., 1998; Singh and Oh, 2005; Boehme et al., 2005; Fuhs, 2008) as shown in Fig. 2. As shown in Fig. 2, two processes are considered to be feasible for charged carrier generation process; one is a charged carrier process in which charged carriers are
ns
generated directly by way of a kng process and the other is an exciton process in which excitons are generated by way of a kg1 process at the initial stage of photoexcitation. In the charged carrier process, electrons and holes generated in i-layer diffuse to i/n junction and to p/i junction, respectively. Electrons and holes finally reach to thermal equilibrium state based on the continuity equation under the bias voltage condition. In the exciton process, excitons diffuse to p/i and i/n junctions, and electrons and holes are generated at p/i and i/n junctions by charge separation.
(2)
where, np (0) is the zeroth-order approximation of the complex index of refraction ( [ f n2s +(1− f )n2i ]1/2 ), f is the filling factor ( b/Λ), b is the volume fraction of grating, Λ is the grating period, and λ is the wavelength.
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[ ] αT N 4πη A · e−βz + B · eβz + e−αz + C1 · eαz + C2 · cos( (d − z) + δ,, ) , λ D(β2 − α2 )
(4)
√ where, β = 1/ Dτ, i.e., the reciprocal of the initial carrier diffusion length, A and B are constants obtained by the boundary conditions, C1 and C2 are expressed as (Pettersson et al., 1999): −2αd C1 = ρ,,2 j e
C2 =
(β2 −α2 ) 2ρ,, 4πη (β2 +( λ )2 )
(5)
· e−αd .
Charged carrier process (1) electron diffusion in p-layer The boundary conditions are given by n(z = d) = 0 and dn dz z=0 = 0, where d is the layer thickness. Then, the two constants A and B in Eq. (4) can be solved as: Si* exciton
kng
kg1 exciton
knr
kg2
G kr1
knr
kr2
electron, hole
knr
knr
Si Fig. 2 Photo-excited relaxation processes in µc-Si:H film. Si: ground state (valence band); Si*: photo-excited state (conduction band); G: photo-excitation; kng: electron and hole generation process; kg: exciton generation process; knr: non-radiative thermal relaxation; kr: radiative relaxation.
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179 / Satoru Isoda
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A pn =
B pn =
[
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2
4πη βd λ e
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
β(eβd + e−βd )
[
αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2
4πη −βd λ e
] 4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(2) hole diffusion in p-layer boundary conditions: p(z = 0) = 0 and A pp =
B pp =
[
−αe−αd − βeβd + C1 (αeαd − βeβd ) + C2
4πη λ
d p dz z=d
[
] 4πη sin(δ,, ) − βeβd cos( λ d + δ,, )
4πη λ
(7)
Bnp =
(8)
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
β(eβd + e−βd )
−αe−αd − βeβd + C1 (αeαd − βeβd ) + C2
[
4πη λ
] 4πη sin(δ,, ) − βeβd cos( λ d + δ,, )
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2
[
4πη λ
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
β(eβd + e−βd )
(16)
(17)
According to the above consideration, the electron current Je and the hole current Jh can be expressed as:
=0
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2
(6)
Anp =
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Jh (cathode) = D dn dx x=0 ] [ 4πη 4πη N = αT −βA + βB pp − α + αC1 + C2 λ sin( λ d + δ,, ) pp (β2 −α2 )
(18)
(9)
Je (anode) = −D dn dx x=d [ ] N −βd − B βeβd + αe−αd − C αeαd − C 4πη sin(δ,, ) = αT nn 1 2 λ 2 2 Ann βe
(19)
(β −α )
(3) electron diffusion in i-layer boundary conditions: n(z = d) = 0 and Ain =
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2
[
4πη βd λ e
dn dz z=0
=0
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
(10)
β(eβd + e−βd )
Jh = D
] [ 4πη dn αT N 4πη sin( d + δ,, ) = −βAip + βBip − α + αC1 + C2 dx x=0 (β2 − α2 ) λ λ
(20)
[
Bin =
4πη αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2 λ e−βd
]
4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(4) hole diffusion in i-layer boundary conditions: p(z = 0) = 0 and
d p dz z=d
(11)
] [ dn αT N 4πη sin(δ,, ) (21) = βAin e−βd − βBin eβd + αe−αd − αC1 eαd − C2 dx x=d (β2 − α2 ) λ
Exciton process (1) exciton diffusion in p-layer boundary conditions: n(z = d) = 0 and
=0
[
Aip =
Bip =
] 4πη 4πη −αe−αd − βeβd + C1 (αeαd − βeβd ) + C2 λ sin(δ,, ) − βeβd cos( λ d + δ,, )
Je = −D
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2 β(eβd
[
4πη λ
Ap =
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
+ e−βd )
(5) electron diffusion in n-layer boundary conditions: n(z = d) = 0 and Ann =
Bnn =
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2 β(eβd
[
4πη βd λ e
dn dz z=0
[
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
4πη −βd λ e
] 4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(6) hole diffusion in n-layer boundary conditions: P(z = 0) = 0 and
d p dz z=d
(13)
=0
Bp =
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2
[
4πη βd λ e
=0
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
β(eβd + e−βd )
αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2
[
4πη −βd λ e
] 4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(22)
(23)
(2) exciton diffusion in i-layer boundary conditions: n(z = 0) = 0 and n(z = d) = 0
=0
+ e−βd )
αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2
(12)
dn dz z=0
(14)
(15)
Ai =
Bi =
[ ] 4πη (eβd − e−αd ) + C1 (eβd − eαd ) + C2 eβd cos( λ d + δ,, ) − cos(δ,, )
(24)
(e−βd − eβd )
[ ] 4πη (e−βd − e−αd ) + C1 (e−βd − eαd ) + C2 e−βd cos( λ d + δ,, ) − cos(δ,, )
(25)
(e−βd − eβd )
(3) exciton diffusion in n-layer boundary conditions: n(z = 0) = 0 and
dn dz z=d
=0
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
] 4πη 4πη −αe−αd − βeβd + C1 (αeαd − βeβd ) + C2 λ sin(δ,, ) − βeβd cos( λ d + δ,, )
the charge distribution is obtained from the self-consistent numerical solution using the Poisson equation as
[
An =
Bn =
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2
[
4πη λ
(26) q(p(z) − n(z)) dE(z) = dz ε0 εr
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
β(eβd + e−βd )
(27)
According to the above consideration, the exciton current can be expressed as i(p) = −D
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[ ] αT N 4πη dn A p βe−βd − B p βeβd + αe−αd − C1 αeαd − C2 sin(δ,, ) (28) = dx x=d (β2 − α2 ) λ
i(p/i) = D dn [dx x=0 ] 4πη 4πη N ,, = αT −βA i + βBi − α + αC1 + C 2 λ sin( λ d + δ ) (β2 −α2 )
Jn = µn n(z)qE(z) + µn kT
dn(z) dz
(35)
J p = µ p p(z)qE(z) − µ p kT
d p(z) dz
(36)
J = Jn + J P
(37)
1.1.5 Charge injection process at electrodes As for the mechanism of the charge injection, the diode equation with an ideality factor of 2 is considered to be applicable to the case of p-i-n type thin film solar cell as ( (30)
i(n) = D dn dx x=0 ] [ 4πη 4πη αT N = − 2 2 An β − Bn β + α − C1 α − C2 λ sin( λ d + δ,, ) (β −α )
where, εr is the relative permittivity of the film. At the thermal equilibrium, carriers flow in the forms of the drift current and the diffusion current as
(29)
i(i/n) = −D dn [ dx x=d ] 4πη N βA e−βd − βBi eβd + αe−αd − αC1 eαd − C2 λ sin(δ,, ) = αT i (β2 −α2 )
(34)
(31)
Thus, excitons diffuse to p/i and i/n junctions, then excitons change to electrons and holes at p/i and i/n junctions by charge separation. 1.1.4 Properties of photo-induced charged carriers The steady-state charge distribution within the film is expressed by the continuity equations using electron and hole number densities, n and p, as d2 n(z) dn(z) dn(z) dE(z) 1 = Gn + µn E(z) + µn n(z) + µn kT dt dz dz q dz2
(32)
d p(z) d2 p(z) d p(z) dE(z) 1 = G p − µ p E(z) − µ p p(z) + µ p kT dt dz dz q dz2
(33)
where, Gn and Gp are the production rates of electrons and holes, µn and µp are the carrier mobility, q is the electronic charge, and E(z) is the electric field strength. Since the continuity equations involve the electric field strength which depends on the charge distribution itself,
J ≈ T3 exp
V − Eµ 2kT
) (38)
where, Eµ is the mobility gap and T is the temperature. 1.2 Experimental data for solar cells n-i-p substrate-type µc-Si:H cells (Sai et al., 2008) were adopted as the numerical model for the device structure: Al substrate/Ag (100 nm)/ZnO (40 nm)/n-layer of µc-Si:H/ilayer of µc-Si:H/p-layer of µc-Si:H/ITO (75 nm)/Ag grid. The textured structure for light trapping is a back surface reflector (BSR) formed on Al substrate. The effective medium of the textured structure due to the BSR is considered to be formed by two materials; one is blended material composed of Ag and ZnO (volume fraction Ag:ZnO = 5:2) and the other is µc-Si:H. The complex index of refraction for a µc-Si:H film was quoted from the paper (Lioudakisa et al., 2006). From the restriction of reading the data, the wavelength region was taken to be in the range between 0.3 and 0.9 µm. The complex indices of refraction for a ZnO film and an Ag film were quoted from the papers (Postava et al., 2000; Keast, 2005). The diffusion length of charged carriers in the µc-Si:H film was taken to be 0.6 µm by applying the grain size of 0.3 µm (Shah et al., 2003) to the relationship between diffusion length and the grain size (Joshi and Srivastava, 1986). The value of 0.6 µm for the diffusion length of charged carriers is considered to be a reasonable value compared with the experimental data in the range between
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179 / Satoru Isoda
0.15 and 0.8 µm (Shah et al., 2003). Since the ratio of the diffusion length of exciton to electron was reported to be 3.3 for a single-crystalline silicon film (Zhang et al., 1998), the value of 2.0 µm was used as the diffusion length of excitons in the µc-Si:H film. The mobility gap ( Eµ ) for p-layer, i-layer, and n-layer were reported to be 1.18, 1.18, and 1.75 eV, respectively (Pieters, 2008). Hole mobility (µp ) and electron mobility (µn ) of the µc-Si:H film were reported to be to be 1.5×10−3 and 5.0×10−3 m2 /(V·S), respectively (Pieters, 2008).
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0.2 p-layer
n-layer i-layer
Optical electric field
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BSR Flat
0.1
EM-layer
2 Results and discussion
0 2000 1500 500 1000 Distance from ITO/p-layer interface (nm) Fig. 4 Simulated distribution of normarized optical electric field in a ITO (75 nm)/p-layer (20 nm)/i-layer (1800 nm)/n-layer (40nm)/ZnO (40 nm)/Ag (100 nm) device. red line: Flat device; blue line: BSR device; BSR structure: period 600nm; height (thickness of EM-layer) 250nm; filling factor 0.17. 0
2.1 Light trapping effect in light absorption process Figure 3 shows the reflectance spectra of two types of µcSi:H solar cells: one with a flat back reflector (Flat) and the other with the textured structure due to a BSR. As shown in the figure, the reflectance spectra of both devices are virtually identical in the wavelength region below 0.6 µm and take the minimum value at 0.6 µm. On the other hand, in the wavelength region above 0.6 µm, the oscillation due to interference in the thin film appears and the reflectance of the BSR device is much lower than that of the Flat device, which means that the light trapping occurs in the BSR device. The simulated examples of the optical electric field Q j (x) in the µc-Si:H solar cells are shown in Fig. 4. In the case of the simulation for the BSR device, the i-layer is divided into two layers: the first-layer is the i-layer of which thickness is thicker by the thickness of the secondlayer; the second-layer is the effective medium (EM) layer which is formed in the i-layer by the back surface reflector. As shown in the figure, it is evident that the light intensity in the cell does not obey the Beer’s law but is expressed by the optical electric field distribution due to reflections and 1.0 BSR
Reflectance
0.8
Flat
0.6
interferences inside the cell. It is found that the Q(x) value in the µc-Si:H solar cell with the BSR is significantly lower than that of the Flat device. Especially, the Q(x) of the BSR device decreases in the region of the textured structure (EM-layer). These results suggest that the incident light is more effectively absorbed in the BSR device by light trapping effect than in the Flat device. 2.2 Initial carrier diffusion process Figure 5 shows the spectral current response of µc-Si:H solar cells. The spectral current response based on the charged carrier process is calculated by using Eqs. (18), (19), (20), and (21). In the case of the exciton process, Eqs. (28), (29), (30), and (31) are used for the simulation. As shown in Fig. 5, the spectral response of the BSR device is higher than that of the Flat device in the wavelength region above 0.6 µm in both processes, which corresponds to the reflectance spectra and indicates the light trapping effect is caused by the BSR. It is found that the spectral response yield of the charged carrier process is lower than that of the exciton process, whereas the relative contribution of the light trapping effect in the charged carrier process is higher than in the case of exciton process. 2.3 Performance of µc-Si:H solar cells
0.4
0.2
0.0 0.3
0.4
0.5
0.6 0.7 0.8 0.9 λ (μm) Fig. 3 Simulated reflectance of µc-Si:H solar cells. Device structure: Al substrate/Ag (100 nm) /ZnO (40 nm) /µc-Si:H n-i-p layers, (n-layer (40 nm)/i-layer (1800 nm)/p-layer (20 nm))/ITO (75 nm)/Ag grid. Red line: Flat device (flat reflector); blue line: BSR device (back surface reflector.
The current-voltage characteristics can be obtained based on the total photo-induced electron transfer processes with the consideration of the carrier injection process at the electrodes as shown in Fig. 6. It is found that the power conversion efficiency (PCE) based on the exciton process is estimated to be 14.6%. On the other hand, the PCE based on the charged carrier process is estimated to be 3.5%. This result indicates that the reported data of the PCE (Yamamoto et al., 2004; Sai et al., 2008) cannot be explained by the conventional charged carrier process model. Thus, it is strongly suggested that the combined
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
Spectral current response (arbitrary unit)
1.0 BSR Flat
Charged carrier process 0.8 0.6 0.4 0.2 0
0.3
0.4
0.5
0.6 λ (μm)
0.7
0.8
0.9
Spectral current response (arbitrary unit)
1.0 BSR Flat
Exciton process 0.8 0.6 0.4 0.2 0
0.9 0.6 0.7 0.8 λ (μm) Fig. 5 Simulated spectral current response of µc-Si:H solar cells based on the charged carrier process (a) and the exciton process (b). Device structure: Al substrate/Ag(100 nm)/ZnO(40 nm)/µc-Si:H nip layers(nlayer (40 nm)/i-layer (1800 nm)/p-layer (20 nm)/ITO(75 nm)/Ag grid. red line: flat device; blue line: BSR device. 0.3
0.4
0.5
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at the i-layer thickness of 1.8 µm. On the contrary, the PCE based on the charged carrier process decreases with the increase of the i-layer thickness. This result also indicates that the exciton process is the major process for the charged carrier generation. Figure 8 shows the dependence of the PCE on the thickness of effective medium (EM) layer. The PCE increases sharply till the EM-layer thickness of 150 nm and the slope becomes smaller beyond the EM-layer thickness of 150 nm. Although the higher PCE is obtained by the longer thickness of the EM-layer, the appropriate thickness of the EM-layer is considered to be between 150 and 300 nm from the standpoint of manufacturing. Figure 9 shows the current-voltage characteristics which is virtually identical with the reported data (Yamamoto et al., 2004). As shown in the figure, the PCE is estimated to be 10.6% on the condition that the fractions of the charged carrier generation process are 70% for the exciton process and 30% for the charged carrier process, respectively. The PCE of the Flat device is estimated to be 9.8% in this condition, which means the PCE is increased by 0.8% due to the light trapping effect. It should be noted here that the simulated light trapping effect is rather small compared to the reported data. Figure 10 shows the distribution of the carrier density as a function of the µc-Si:H film position within the cell at bias voltages of 0 .43 V which shows the maximum power
200 Exciton process 100
Exciton process
a
14
0 η (%)
Current density (A/m2)
15
Charge process
-100
13
Charged carrier process (η : 3.5%) -200
12 500
1000
-300
1500 2000 Thickness of i-layer (nm)
2500
6.0
Exciton process (η : 14.6%)
Charged carrier process
b
-400 0.1
0.2
0.3
0.4
0.5
0.6
Voltage Fig. 6 Simulated current-voltage curves for a ITO(75 nm)/p-layer(20 nm)/i-layer(1800 nm)/n-layer(40 nm)/ZnO(40 nm)/Ag(100 nm) device with BSR structure (period 600 nm; height: 250 nm; filling factor 0.5).
5.5 η (%)
0
5.0
4.5
processes of charged carrier with exciton exist in the photoexcited relaxation processes in the µc-Si:H solar cells. Figure 7 shows the dependence of the PCE on the thickness of the i-layer for two types of charged carrier generation processes. As shown in the figure, the PCE based on the exciton process increases with the increase of the thickness of the i-layer and reaches to the maximum
4.0 300
400
500 600 700 800 Thickness of i-layer (nm) Fig. 7 Dependence of the power conversion efficiency on the i-layer thickness. BSR structure (period 600 nm; height: 250 nm; filling factor 0.5).
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179 / Satoru Isoda
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Vol. 25
6
10
Hole Electron
Carrier density (1017/m3)
η 䠄䠂䠅
9 8 7 6
0
300 v 400 v 500 v 200 v Thickness of EM-layer (nm) Fig. 8 Dependence of the power conversion efficiency on the EM-layer thickness. i-layer thickness: 1800 nm; p-layer thickness: 20 nm; n-layer thickness: 40 nm BSR structure (period 600 nm; filling factor 0.17; fraction of exciton process: 50%; fraction of charged carrier process: 50%. v
BSR η : 10.6% at 0.43 V Jsc : 28.6 (mA/cm2) Voc : 0.55 V F.F. : 67% Flat η : 9.8%
0
2000 1600 800 1200 Position (nm) Fig. 10 Carrier density vs. µc-Si:H film position for a ITO (75 nm)/player(20 nm)/ i-layer(1800 nm)/n-layer(40 nm)/ZnO(40 nm)/Ag(100 nm) device. BSR structure (period 600 nm; height: 250 nm; filling factor 0.5); exciton process: 70%; charged carrier process: 30%; applied voltage: 0.43 V (maximum power output). 0
-200
400
100
Current density (A/m2)
Current density (A/m2)
-100
-300
2
100
100
0
4
Flat BSR
0
η : 8.6% η : 10.5% at 0.43 V Jsc : 28.4 mA/cm2 Voc : 0.55 V F.F. : 67%
-100
-200 a
0.1
0.2
0.3 0.4 0.5 0.6 Voltage Fig. 9 Simulated current-voltage curves for a ITO(75 nm)/p-layer (20 nm)/i-layer (1800 nm)/n-layer(40 nm)/ZnO(40 nm)/Ag(100 nm) device. BSR structure (period 600 nm; height: 250 nm; filling factor 0.5); fractions of charged carrier generation: exciton process 70%; charged carrier process 30%.
output in Fig. 9. In the simulation of Figs. 9 and 10, the diffusion length of charged carrier is taken to be 0.6 µm as described in 1.2. Shah et al. (2003) reported that the maximum value of the diffusion length of the charged carrier is 0.8 µm. Accordingly, the simulation using 0.8 µm as the diffusion length of the charged carrier was performed in order to estimate the effect of the diffusion length on the light trapping effect. The simulated current-voltage curves using 0.8 µm as the diffusion length of the charged carrier is shown in Fig. 11. As shown in Fig. 11, the light trapping effect is found to be improved in this simulation condition. According to the study on the relationship between the morphology of µc-Si:H films and the performance of µcSi:H solar cells, the highest power conversion efficiency was obtained for the film deposited at the onset of the phase transition from crystalline to mixed phase (crystalline + amorphous) (Dingemans et al., 2008). This transition was observed as plane view. The TEM observation of the
-300 0.0
100
Current density (A/m2)
0
0.1
Flat BSR
0
0.2
0.3 Voltage
0.4
0.5
0.6
η : 7.5% η : 10.2% at 0.41 V Jsc : 29.3 mA/cm2 Voc : 0.53 V F.F. : 66%
-100
-200 b -300 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Voltage
Fig. 11 Simulated current-voltage curves for µc-Si:H solar cells. BSR structure (period 600nm; height: 250nm; filling factor 0.17); diffusion length of charged carrier: 0.8 µm. (a) i-layer thickness: 1.8 µm; fractions of charged carrier generation: exciton process 60%; charged carrier process 40%; (b) i-layer thickness: 1.3 µm; fractions of charged carrier generation: exciton process 55%; charged carrier process 45%.
cross section of the µc-Si:H film shows first an amorphous zone, followed by a seeding zone that leads into a mixedphase region and thereafter fully microcrystalline region (Shah et al., 2003; Vallat-Sauvain et al., 2000). Ferlauto et
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
al. evaluated compositional depth profiles in mixed-phase silicon films based on microcrystalline cone growth model incorporated into amorphous silicon (Ferlauto et al., 2004). Thus, the optimum morphology of the µc-Si:H film for the performance of the solar cells is not a homogeneous one-phase structure but a heterogeneous two-phase structure composed of cone-like formed crystallites and their surrounding amorphous region. Therefore, it is suggested that the fractions of charged carrier generation processes depend on the two-phase morphology of the µc-Si:H film. Since the diffusion length is also strongly dependent on the morphology, the relationship between the morphology and the performance of the solar cells remains to be further studied in order to realize the higher power conversion efficiency.
3 Conclusions The author has developed a numerical model to predict current-voltage characteristics of µc-Si:H solar cells on the basis of two types of charged carrier generation processes (exciton process and charged carrier process) including light trapping effect due to a textured back surface structure. It was found that the reported data of the power conversion efficiency cannot be explained by the conventional charged carrier process model and that the combined state model of charged carrier with exciton well explains the performance of the p-i-n type µc-Si:H solar cells. In this way, the typical power conversion efficiencies, under 1 sun AM1.5 solar illumination, were estimated to be 10.5% for the device (i-layer thickness: 1.8 µm) with the BSR (period: 600 nm; height: 250 nm) and 8.6% for the device with the flat reflector under the condition that the fractions of the exciton process and the charged carrier process were 60% and 40%, respectively. The performance of µc-Si:H solar cells depends strongly on the film morphology which determines the fractions of charged carrier generation processes and the diffusion length of carriers. Accordingly, the further improvement of the performance remains to be done. Acknowledgments The author would like to thank Dr. S. Yagi for his encouragement to this study. The author is also grateful to Mr. O. Watanabe for his precious advice on the programming of the numerical model.
References Boehme C, Friedrich F, Ehara T, Lips K, 2005. Recombination at silicon dangling bonds. Thin Solid Films, 487: 132–136. Corkish R, Chan D S-P, Green M A, 1996. Excitons in silicon diodes and solar cells: A three particle theory. Journal of Applied Physics, 79(1): 195–203. Dingemans G, van den Donker M N, Hrunski D, Gordijin A, Kessels W M M, van de Sanden M C M, 2008. The atomic hydrogen
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flux to silicon growth flux ratio during microcrystalline silicon solar cell deposition. Applied Physics Letter, 93: 111914-1–3. Ferlauto A S, Ferreira G M, Koval R J, Pearce J M, Wronski C R, Collins R W et al., 2004. Evaluation of compositional depth profiles in mixed-phase (amorphous + crystalline) silicon films from real time spectroscopic ellipsometry. Thin Solid Films, 455-456: 665–669. Fuhs W, 2008. Recombination and transport through localized states in hydrogenated amorphous and microcrystalline silicon. Journal of Non-Crystalline Solids, 354: 2067–2078. Hack M, Shur M, 1985. Physics of amorphous silicon alloy p-i-n solar cells. Journal of Applied Physics, 58(2): 997–1020. Isoda S, 2011. Photo-induced electron transfer processes in thinfilm solar cells. Journal of Environmental Sciences, 23(Suppl.): S18–S25. Joshi D P, Srivastava R S, 1986. Theoretical study of the photovoltaic properties of polycrystalline silicon. Journal of Applied Physics, 59: 2849-2858. Kane D E, Swanson R M, 1993. The effect of excitons on apparent band gap narrowing and transport in semiconductors. Journal of Applied Physics, 73(3): 1193–1197. Keast V J, 2005. Ab initio calculations of plasmons and interband transitions in the low-loss electron energy-loss spectrum. Journal of Electron Spectroscopy and Related Phenomena, 143: 97–104. Koi Y, Goto M, Meguro T, Matsuda T, Kondo M, Sasaki T et al., 2004. A high efficiency thin film silicon solar cell and module. Solar Energy, 77: 939–949. Lioudakisa E, Nassiopoulou A, Othonos A, 2006. Ellipsometric analysis of ion-implanted polycrystalline silicon films before and after annealing. Thin Solid Films, 496: 253–258. Pettersson L A A, Roman L S, Inganas O, 1999. Modeling photocurrent action spectra of photovoltaic devices based on organic thin films. Journal of Applied Physics, 86(1): 487–496. Pieters B E, 2008. Characterization of Thin-Film Silicon Materials and Solar Cells through Numerical Modeling. Ph. D. Thesis Delft University of Tecnology. Postava K, Sueki H, Aoyama M, Yamaguchi T, Ino Ch, Igasaki Y, 2000. Spectroscopic ellipsometry of epitaxial ZnO layer on sapphire substrate. Journal of Applied Physics, 87(11): 7820– 7824. Raguin D H, Morris G M, 1993. Antireflection structured surfaces for the infrared spectral region. Applied Optics, 32(7): 1154–1167. Sai H, Fujiwara H, Kondo M, Kanamori Y, 2008. Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern. Applied Physics Letter, 93: 143501-1–3. Shah A V, Meier J, Vallat-Sauvain, Wyrsch E N, Kroll U, Droz C et al., 2003. Material and solar cell research in microcrystalline silicon. Solar Energy Materials & Solar Cells, 78: 469–491. Singh J, Oh I K, 2005. Radiative lifetime of excitonic photoluminescence in amorphous semiconductor. Journal of Applied Physics, 97: 063516-1 – 063516-14. Stachowitz R, Schubert M, Fuhs W, 1998. Non-radiative distant pair recombination in amorphous silicon. Journal of NonCrystalline Solids, 227-230: 190–196. Vallat-Sauvain E, Kroll U, Meier J, Wyrsch N, Shah A, 2000. Microstructure and surface roughness of microcrystalline silicon prepared by very high frequency-glow discharge using hydrogen dilution. Journal of Non-Crystalline Solids, 266-269: 125–130. Yamamoto K, Nakajima A, Yoshimi M, Sawada T, Fukuda S, Suezaki T et al., 2004. A high efficiency thin film silicon solar cell and module. Solar Energy, 77: 939–949. Zhang Y, Mascarenhas A, Deb S, 1998. Effects of excitons on solar cells. Journal of Applied Physics, 84: 3966–3971.
JOURNAL OF ENVIRONMENTAL SCIENCES ISSN 1001-0742 CN 11-2629/X
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179
www.jesc.ac.cn
Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes Satoru Isoda Advanced Science Research Laboratory, Saitama Institute of Technology, 1690 Fusaiji, Hukaya, Saitama 3690293, Japan. E-mail: [email protected]
Abstract Power conversion efficiency of p-i-n type microcrystalline silicon (µc-Si:H) solar cells has been analyzed in terms of sequential processes of photo-induced electron transfer. The effect of the excitonic state on the charged carrier generation has been studied compared to a conventional scheme in which only charged carriers are taken into account for the operation of the solar cells. A numerical model has been developed to calculate current-voltage characteristics of solar cells on the basis of two types of charged carrier generation processes (exciton process and charged carrier process). The light trapping effect due to a textured back surface reflector (BSR) was embedded in the numerical model by using the effective medium theory in combination with the matrix method in the field of the electromagnetic theory of light. As an application of this modeling, it was found that the reported data of the power conversion efficiency were not explained by the conventional charged carrier process model and that the combined model of the charged carrier process with the exciton process well explains the performance of the p-i-n type µc-Si:H solar cells. In this way, the typical power conversion efficiencies were estimated to be 10.5% for the device (i-layer thickness: 1.8 µm) with the BSR (period: 600 nm; height: 250 nm) and 8.6% for the device with the flat reflector under the condition that the fractions of the exciton process and charged carrier process were 60% and 40%, respectively. Key words: solar cell; thin-film silicon; microcrystalline silicon; p-i-n type; current-voltage characteristics; power conversion efficiency; photo-induced electron transfer; diffusion length; exciton; light trapping
Introduction Although multi-crystalline and single-crystalline silicon solar cells play a main role in the solar cell market at present, thin-film silicon solar cells have been expected as the mainstream of next generation solar cells from the standpoint not only of their high performance but also of their less use of material and energy for manufacturing. Among several types of the thin-film silicon solar cells, the p-i-n type microcrystalline silicon (µc-Si:H) solar cells are considered to be most promising and have been intensively developed in order to increase the performance. The µc-Si:H solar cells have already reached to the commercial level, however, the mechanisms underlying carrier generation, transport and recombination have not been completely understood. Thin-film silicon solar cells generally consist of pin junctions instead of pn junction in order to increase the density of charged carriers produced by light absorption. For the theoretical study on the photovoltaic properties of thin-film silicon pin type solar cells, the simulation model for amorphous silicon solar cells presented by Hack and Shur (Hack and Shur, 1985) has been accepted as a fundamental model. In their model,
however, the exciton effect (Corkish et al., 1996; Kane and Swanson, 1993) as well as the optical interference effect (Pettersson et al., 1999) including the light trapping effect by texturing an electrode surface (Yamamoto et al., 2004) were not taken into consideration. Under these circumstances, this study deals with the numerical model which totally covers sequential rate determining processes involving the dynamics of the energy conversion in solar cells. Especially, the author focused on the modeling of charged carrier generation processes in combination with light trapping.
1 Materials and methods 1.1 Theory Since the author already reported the details of the photoinduced electron transfer processes in thin-film solar cells (Isoda, 2011), only fundamental equations are described in this chapter.
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
1.1.1 Light absorption process and modeling of light trapping effect The internal optical electric field in layer j at position x at normal incidence (Q(x)) is expressed by Pettersson et al. (1999) 2 Q j (x) = 12[cε0 α j η j E j (x) ] 4πη ,,2 = α j T j I0 e−α j x + ρ j ·e−α j (2d j −x) +2 ρ,,j ·e−α j d j · cos( λ j (d j − x) + δ,,j )
(1)
where, c is the speed of light, ε0 is the permittivity of free space, α j is the absorption coefficient of layer j, η j is the real part of the complex index of refraction, T j is the internal intensity transmittance, I0 is the intensity of the incident light, d j is the thickness of layer j, and ρ j ′′ and δ j ′′ are the absolute value and the argument of the complex reflection coefficient. In order to use the optical electric field in the thin-film solar cell as the light intensity in the cell, the textured structure must be treated as a flat effective medium. Figure 1 shows an effective medium model for a textured back surface reflector (BSR) of the solar cell. The complex index of refraction in the effective medium, np , is expressed based on the second-order effective medium theory as literature (Raguin and Morris, 1993), (2)
(0)
n p = n p [1 +
π2 Λ 2 2 (n s − ni )2 1 ( ) f (1 − f 2 ) ]2 3 λ (0)2 np
1.1.2
Photo-excited relaxation microcrystalline-silicon film
processes
ni ni ns
1.1.3 Diffusion process of initial carrier The initial carrier (electron and hole in the charged carrier process; exciton in the exciton process) density n(Z) at depth, z is expressed by the equation of continuity as, dn di n d2 n n =− + Q(z) − = D 2 + Q(z) − dt dz τ τ dz
in
np
Textured Back Surface Reflector Effective Medium Fig. 1 Effective medium for textured back surface reflector.
(3)
where, Q(z) is the internal optical electric field in position z (Eq. (1)), i is the carrier current (i = −d dn dz ), D is the diffusion constant and τ is the volume recombination lifetime of the carrier. At the steady-state condition, dn/dt = 0, Eqation (3) has the general solution of, n(z) =
The generation process of charged carriers is determined by localized band-tail states and deep defects. In this paper, the effect of excitonic state on the charged carrier generation has been studied compared to a conventional scheme in which only charged carriers are taken into account for the operation of a solar cell. Here, the author would like to propose the scheme for photo-excited relaxation processes in a µc-Si:H thin film by summarizing reported experimental results (Stachowitz et al., 1998; Singh and Oh, 2005; Boehme et al., 2005; Fuhs, 2008) as shown in Fig. 2. As shown in Fig. 2, two processes are considered to be feasible for charged carrier generation process; one is a charged carrier process in which charged carriers are
ns
generated directly by way of a kng process and the other is an exciton process in which excitons are generated by way of a kg1 process at the initial stage of photoexcitation. In the charged carrier process, electrons and holes generated in i-layer diffuse to i/n junction and to p/i junction, respectively. Electrons and holes finally reach to thermal equilibrium state based on the continuity equation under the bias voltage condition. In the exciton process, excitons diffuse to p/i and i/n junctions, and electrons and holes are generated at p/i and i/n junctions by charge separation.
(2)
where, np (0) is the zeroth-order approximation of the complex index of refraction ( [ f n2s +(1− f )n2i ]1/2 ), f is the filling factor ( b/Λ), b is the volume fraction of grating, Λ is the grating period, and λ is the wavelength.
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[ ] αT N 4πη A · e−βz + B · eβz + e−αz + C1 · eαz + C2 · cos( (d − z) + δ,, ) , λ D(β2 − α2 )
(4)
√ where, β = 1/ Dτ, i.e., the reciprocal of the initial carrier diffusion length, A and B are constants obtained by the boundary conditions, C1 and C2 are expressed as (Pettersson et al., 1999): −2αd C1 = ρ,,2 j e
C2 =
(β2 −α2 ) 2ρ,, 4πη (β2 +( λ )2 )
(5)
· e−αd .
Charged carrier process (1) electron diffusion in p-layer The boundary conditions are given by n(z = d) = 0 and dn dz z=0 = 0, where d is the layer thickness. Then, the two constants A and B in Eq. (4) can be solved as: Si* exciton
kng
kg1 exciton
knr
kg2
G kr1
knr
kr2
electron, hole
knr
knr
Si Fig. 2 Photo-excited relaxation processes in µc-Si:H film. Si: ground state (valence band); Si*: photo-excited state (conduction band); G: photo-excitation; kng: electron and hole generation process; kg: exciton generation process; knr: non-radiative thermal relaxation; kr: radiative relaxation.
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179 / Satoru Isoda
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A pn =
B pn =
[
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2
4πη βd λ e
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
β(eβd + e−βd )
[
αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2
4πη −βd λ e
] 4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(2) hole diffusion in p-layer boundary conditions: p(z = 0) = 0 and A pp =
B pp =
[
−αe−αd − βeβd + C1 (αeαd − βeβd ) + C2
4πη λ
d p dz z=d
[
] 4πη sin(δ,, ) − βeβd cos( λ d + δ,, )
4πη λ
(7)
Bnp =
(8)
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
β(eβd + e−βd )
−αe−αd − βeβd + C1 (αeαd − βeβd ) + C2
[
4πη λ
] 4πη sin(δ,, ) − βeβd cos( λ d + δ,, )
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2
[
4πη λ
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
β(eβd + e−βd )
(16)
(17)
According to the above consideration, the electron current Je and the hole current Jh can be expressed as:
=0
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2
(6)
Anp =
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Jh (cathode) = D dn dx x=0 ] [ 4πη 4πη N = αT −βA + βB pp − α + αC1 + C2 λ sin( λ d + δ,, ) pp (β2 −α2 )
(18)
(9)
Je (anode) = −D dn dx x=d [ ] N −βd − B βeβd + αe−αd − C αeαd − C 4πη sin(δ,, ) = αT nn 1 2 λ 2 2 Ann βe
(19)
(β −α )
(3) electron diffusion in i-layer boundary conditions: n(z = d) = 0 and Ain =
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2
[
4πη βd λ e
dn dz z=0
=0
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
(10)
β(eβd + e−βd )
Jh = D
] [ 4πη dn αT N 4πη sin( d + δ,, ) = −βAip + βBip − α + αC1 + C2 dx x=0 (β2 − α2 ) λ λ
(20)
[
Bin =
4πη αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2 λ e−βd
]
4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(4) hole diffusion in i-layer boundary conditions: p(z = 0) = 0 and
d p dz z=d
(11)
] [ dn αT N 4πη sin(δ,, ) (21) = βAin e−βd − βBin eβd + αe−αd − αC1 eαd − C2 dx x=d (β2 − α2 ) λ
Exciton process (1) exciton diffusion in p-layer boundary conditions: n(z = d) = 0 and
=0
[
Aip =
Bip =
] 4πη 4πη −αe−αd − βeβd + C1 (αeαd − βeβd ) + C2 λ sin(δ,, ) − βeβd cos( λ d + δ,, )
Je = −D
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2 β(eβd
[
4πη λ
Ap =
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
+ e−βd )
(5) electron diffusion in n-layer boundary conditions: n(z = d) = 0 and Ann =
Bnn =
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2 β(eβd
[
4πη βd λ e
dn dz z=0
[
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
4πη −βd λ e
] 4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(6) hole diffusion in n-layer boundary conditions: P(z = 0) = 0 and
d p dz z=d
(13)
=0
Bp =
−αeβd − βe−αd + C1 (αeβd − βeαd ) + C2
[
4πη βd λ e
=0
] 4πη sin( λ d + δ,, ) − β cos(δ,, )
β(eβd + e−βd )
αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2
[
4πη −βd λ e
] 4πη sin( λ d + δ,, ) + β cos(δ,, )
β(eβd + e−βd )
(22)
(23)
(2) exciton diffusion in i-layer boundary conditions: n(z = 0) = 0 and n(z = d) = 0
=0
+ e−βd )
αe−βd − βe−αd − C1 (βeαd + αe−βd ) − C2
(12)
dn dz z=0
(14)
(15)
Ai =
Bi =
[ ] 4πη (eβd − e−αd ) + C1 (eβd − eαd ) + C2 eβd cos( λ d + δ,, ) − cos(δ,, )
(24)
(e−βd − eβd )
[ ] 4πη (e−βd − e−αd ) + C1 (e−βd − eαd ) + C2 e−βd cos( λ d + δ,, ) − cos(δ,, )
(25)
(e−βd − eβd )
(3) exciton diffusion in n-layer boundary conditions: n(z = 0) = 0 and
dn dz z=d
=0
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
] 4πη 4πη −αe−αd − βeβd + C1 (αeαd − βeβd ) + C2 λ sin(δ,, ) − βeβd cos( λ d + δ,, )
the charge distribution is obtained from the self-consistent numerical solution using the Poisson equation as
[
An =
Bn =
β(eβd + e−βd )
αe−αd − βe−βd − C1 (αeαd + βe−βd ) − C2
[
4πη λ
(26) q(p(z) − n(z)) dE(z) = dz ε0 εr
] 4πη sin(δ,, ) + βe−βd cos( λ d + δ,, )
β(eβd + e−βd )
(27)
According to the above consideration, the exciton current can be expressed as i(p) = −D
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[ ] αT N 4πη dn A p βe−βd − B p βeβd + αe−αd − C1 αeαd − C2 sin(δ,, ) (28) = dx x=d (β2 − α2 ) λ
i(p/i) = D dn [dx x=0 ] 4πη 4πη N ,, = αT −βA i + βBi − α + αC1 + C 2 λ sin( λ d + δ ) (β2 −α2 )
Jn = µn n(z)qE(z) + µn kT
dn(z) dz
(35)
J p = µ p p(z)qE(z) − µ p kT
d p(z) dz
(36)
J = Jn + J P
(37)
1.1.5 Charge injection process at electrodes As for the mechanism of the charge injection, the diode equation with an ideality factor of 2 is considered to be applicable to the case of p-i-n type thin film solar cell as ( (30)
i(n) = D dn dx x=0 ] [ 4πη 4πη αT N = − 2 2 An β − Bn β + α − C1 α − C2 λ sin( λ d + δ,, ) (β −α )
where, εr is the relative permittivity of the film. At the thermal equilibrium, carriers flow in the forms of the drift current and the diffusion current as
(29)
i(i/n) = −D dn [ dx x=d ] 4πη N βA e−βd − βBi eβd + αe−αd − αC1 eαd − C2 λ sin(δ,, ) = αT i (β2 −α2 )
(34)
(31)
Thus, excitons diffuse to p/i and i/n junctions, then excitons change to electrons and holes at p/i and i/n junctions by charge separation. 1.1.4 Properties of photo-induced charged carriers The steady-state charge distribution within the film is expressed by the continuity equations using electron and hole number densities, n and p, as d2 n(z) dn(z) dn(z) dE(z) 1 = Gn + µn E(z) + µn n(z) + µn kT dt dz dz q dz2
(32)
d p(z) d2 p(z) d p(z) dE(z) 1 = G p − µ p E(z) − µ p p(z) + µ p kT dt dz dz q dz2
(33)
where, Gn and Gp are the production rates of electrons and holes, µn and µp are the carrier mobility, q is the electronic charge, and E(z) is the electric field strength. Since the continuity equations involve the electric field strength which depends on the charge distribution itself,
J ≈ T3 exp
V − Eµ 2kT
) (38)
where, Eµ is the mobility gap and T is the temperature. 1.2 Experimental data for solar cells n-i-p substrate-type µc-Si:H cells (Sai et al., 2008) were adopted as the numerical model for the device structure: Al substrate/Ag (100 nm)/ZnO (40 nm)/n-layer of µc-Si:H/ilayer of µc-Si:H/p-layer of µc-Si:H/ITO (75 nm)/Ag grid. The textured structure for light trapping is a back surface reflector (BSR) formed on Al substrate. The effective medium of the textured structure due to the BSR is considered to be formed by two materials; one is blended material composed of Ag and ZnO (volume fraction Ag:ZnO = 5:2) and the other is µc-Si:H. The complex index of refraction for a µc-Si:H film was quoted from the paper (Lioudakisa et al., 2006). From the restriction of reading the data, the wavelength region was taken to be in the range between 0.3 and 0.9 µm. The complex indices of refraction for a ZnO film and an Ag film were quoted from the papers (Postava et al., 2000; Keast, 2005). The diffusion length of charged carriers in the µc-Si:H film was taken to be 0.6 µm by applying the grain size of 0.3 µm (Shah et al., 2003) to the relationship between diffusion length and the grain size (Joshi and Srivastava, 1986). The value of 0.6 µm for the diffusion length of charged carriers is considered to be a reasonable value compared with the experimental data in the range between
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179 / Satoru Isoda
0.15 and 0.8 µm (Shah et al., 2003). Since the ratio of the diffusion length of exciton to electron was reported to be 3.3 for a single-crystalline silicon film (Zhang et al., 1998), the value of 2.0 µm was used as the diffusion length of excitons in the µc-Si:H film. The mobility gap ( Eµ ) for p-layer, i-layer, and n-layer were reported to be 1.18, 1.18, and 1.75 eV, respectively (Pieters, 2008). Hole mobility (µp ) and electron mobility (µn ) of the µc-Si:H film were reported to be to be 1.5×10−3 and 5.0×10−3 m2 /(V·S), respectively (Pieters, 2008).
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0.2 p-layer
n-layer i-layer
Optical electric field
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BSR Flat
0.1
EM-layer
2 Results and discussion
0 2000 1500 500 1000 Distance from ITO/p-layer interface (nm) Fig. 4 Simulated distribution of normarized optical electric field in a ITO (75 nm)/p-layer (20 nm)/i-layer (1800 nm)/n-layer (40nm)/ZnO (40 nm)/Ag (100 nm) device. red line: Flat device; blue line: BSR device; BSR structure: period 600nm; height (thickness of EM-layer) 250nm; filling factor 0.17. 0
2.1 Light trapping effect in light absorption process Figure 3 shows the reflectance spectra of two types of µcSi:H solar cells: one with a flat back reflector (Flat) and the other with the textured structure due to a BSR. As shown in the figure, the reflectance spectra of both devices are virtually identical in the wavelength region below 0.6 µm and take the minimum value at 0.6 µm. On the other hand, in the wavelength region above 0.6 µm, the oscillation due to interference in the thin film appears and the reflectance of the BSR device is much lower than that of the Flat device, which means that the light trapping occurs in the BSR device. The simulated examples of the optical electric field Q j (x) in the µc-Si:H solar cells are shown in Fig. 4. In the case of the simulation for the BSR device, the i-layer is divided into two layers: the first-layer is the i-layer of which thickness is thicker by the thickness of the secondlayer; the second-layer is the effective medium (EM) layer which is formed in the i-layer by the back surface reflector. As shown in the figure, it is evident that the light intensity in the cell does not obey the Beer’s law but is expressed by the optical electric field distribution due to reflections and 1.0 BSR
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interferences inside the cell. It is found that the Q(x) value in the µc-Si:H solar cell with the BSR is significantly lower than that of the Flat device. Especially, the Q(x) of the BSR device decreases in the region of the textured structure (EM-layer). These results suggest that the incident light is more effectively absorbed in the BSR device by light trapping effect than in the Flat device. 2.2 Initial carrier diffusion process Figure 5 shows the spectral current response of µc-Si:H solar cells. The spectral current response based on the charged carrier process is calculated by using Eqs. (18), (19), (20), and (21). In the case of the exciton process, Eqs. (28), (29), (30), and (31) are used for the simulation. As shown in Fig. 5, the spectral response of the BSR device is higher than that of the Flat device in the wavelength region above 0.6 µm in both processes, which corresponds to the reflectance spectra and indicates the light trapping effect is caused by the BSR. It is found that the spectral response yield of the charged carrier process is lower than that of the exciton process, whereas the relative contribution of the light trapping effect in the charged carrier process is higher than in the case of exciton process. 2.3 Performance of µc-Si:H solar cells
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0.6 0.7 0.8 0.9 λ (μm) Fig. 3 Simulated reflectance of µc-Si:H solar cells. Device structure: Al substrate/Ag (100 nm) /ZnO (40 nm) /µc-Si:H n-i-p layers, (n-layer (40 nm)/i-layer (1800 nm)/p-layer (20 nm))/ITO (75 nm)/Ag grid. Red line: Flat device (flat reflector); blue line: BSR device (back surface reflector.
The current-voltage characteristics can be obtained based on the total photo-induced electron transfer processes with the consideration of the carrier injection process at the electrodes as shown in Fig. 6. It is found that the power conversion efficiency (PCE) based on the exciton process is estimated to be 14.6%. On the other hand, the PCE based on the charged carrier process is estimated to be 3.5%. This result indicates that the reported data of the PCE (Yamamoto et al., 2004; Sai et al., 2008) cannot be explained by the conventional charged carrier process model. Thus, it is strongly suggested that the combined
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
Spectral current response (arbitrary unit)
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0.9 0.6 0.7 0.8 λ (μm) Fig. 5 Simulated spectral current response of µc-Si:H solar cells based on the charged carrier process (a) and the exciton process (b). Device structure: Al substrate/Ag(100 nm)/ZnO(40 nm)/µc-Si:H nip layers(nlayer (40 nm)/i-layer (1800 nm)/p-layer (20 nm)/ITO(75 nm)/Ag grid. red line: flat device; blue line: BSR device. 0.3
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at the i-layer thickness of 1.8 µm. On the contrary, the PCE based on the charged carrier process decreases with the increase of the i-layer thickness. This result also indicates that the exciton process is the major process for the charged carrier generation. Figure 8 shows the dependence of the PCE on the thickness of effective medium (EM) layer. The PCE increases sharply till the EM-layer thickness of 150 nm and the slope becomes smaller beyond the EM-layer thickness of 150 nm. Although the higher PCE is obtained by the longer thickness of the EM-layer, the appropriate thickness of the EM-layer is considered to be between 150 and 300 nm from the standpoint of manufacturing. Figure 9 shows the current-voltage characteristics which is virtually identical with the reported data (Yamamoto et al., 2004). As shown in the figure, the PCE is estimated to be 10.6% on the condition that the fractions of the charged carrier generation process are 70% for the exciton process and 30% for the charged carrier process, respectively. The PCE of the Flat device is estimated to be 9.8% in this condition, which means the PCE is increased by 0.8% due to the light trapping effect. It should be noted here that the simulated light trapping effect is rather small compared to the reported data. Figure 10 shows the distribution of the carrier density as a function of the µc-Si:H film position within the cell at bias voltages of 0 .43 V which shows the maximum power
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5.5 η (%)
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processes of charged carrier with exciton exist in the photoexcited relaxation processes in the µc-Si:H solar cells. Figure 7 shows the dependence of the PCE on the thickness of the i-layer for two types of charged carrier generation processes. As shown in the figure, the PCE based on the exciton process increases with the increase of the thickness of the i-layer and reaches to the maximum
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500 600 700 800 Thickness of i-layer (nm) Fig. 7 Dependence of the power conversion efficiency on the i-layer thickness. BSR structure (period 600 nm; height: 250 nm; filling factor 0.5).
Journal of Environmental Sciences 2013, 25(Suppl.) S172–S179 / Satoru Isoda
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η 䠄䠂䠅
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300 v 400 v 500 v 200 v Thickness of EM-layer (nm) Fig. 8 Dependence of the power conversion efficiency on the EM-layer thickness. i-layer thickness: 1800 nm; p-layer thickness: 20 nm; n-layer thickness: 40 nm BSR structure (period 600 nm; filling factor 0.17; fraction of exciton process: 50%; fraction of charged carrier process: 50%. v
BSR η : 10.6% at 0.43 V Jsc : 28.6 (mA/cm2) Voc : 0.55 V F.F. : 67% Flat η : 9.8%
0
2000 1600 800 1200 Position (nm) Fig. 10 Carrier density vs. µc-Si:H film position for a ITO (75 nm)/player(20 nm)/ i-layer(1800 nm)/n-layer(40 nm)/ZnO(40 nm)/Ag(100 nm) device. BSR structure (period 600 nm; height: 250 nm; filling factor 0.5); exciton process: 70%; charged carrier process: 30%; applied voltage: 0.43 V (maximum power output). 0
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0.3 0.4 0.5 0.6 Voltage Fig. 9 Simulated current-voltage curves for a ITO(75 nm)/p-layer (20 nm)/i-layer (1800 nm)/n-layer(40 nm)/ZnO(40 nm)/Ag(100 nm) device. BSR structure (period 600 nm; height: 250 nm; filling factor 0.5); fractions of charged carrier generation: exciton process 70%; charged carrier process 30%.
output in Fig. 9. In the simulation of Figs. 9 and 10, the diffusion length of charged carrier is taken to be 0.6 µm as described in 1.2. Shah et al. (2003) reported that the maximum value of the diffusion length of the charged carrier is 0.8 µm. Accordingly, the simulation using 0.8 µm as the diffusion length of the charged carrier was performed in order to estimate the effect of the diffusion length on the light trapping effect. The simulated current-voltage curves using 0.8 µm as the diffusion length of the charged carrier is shown in Fig. 11. As shown in Fig. 11, the light trapping effect is found to be improved in this simulation condition. According to the study on the relationship between the morphology of µc-Si:H films and the performance of µcSi:H solar cells, the highest power conversion efficiency was obtained for the film deposited at the onset of the phase transition from crystalline to mixed phase (crystalline + amorphous) (Dingemans et al., 2008). This transition was observed as plane view. The TEM observation of the
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η : 7.5% η : 10.2% at 0.41 V Jsc : 29.3 mA/cm2 Voc : 0.53 V F.F. : 66%
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Fig. 11 Simulated current-voltage curves for µc-Si:H solar cells. BSR structure (period 600nm; height: 250nm; filling factor 0.17); diffusion length of charged carrier: 0.8 µm. (a) i-layer thickness: 1.8 µm; fractions of charged carrier generation: exciton process 60%; charged carrier process 40%; (b) i-layer thickness: 1.3 µm; fractions of charged carrier generation: exciton process 55%; charged carrier process 45%.
cross section of the µc-Si:H film shows first an amorphous zone, followed by a seeding zone that leads into a mixedphase region and thereafter fully microcrystalline region (Shah et al., 2003; Vallat-Sauvain et al., 2000). Ferlauto et
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Modeling the current-voltage characteristics of thin-film silicon solar cells based on photo-induced electron transfer processes
al. evaluated compositional depth profiles in mixed-phase silicon films based on microcrystalline cone growth model incorporated into amorphous silicon (Ferlauto et al., 2004). Thus, the optimum morphology of the µc-Si:H film for the performance of the solar cells is not a homogeneous one-phase structure but a heterogeneous two-phase structure composed of cone-like formed crystallites and their surrounding amorphous region. Therefore, it is suggested that the fractions of charged carrier generation processes depend on the two-phase morphology of the µc-Si:H film. Since the diffusion length is also strongly dependent on the morphology, the relationship between the morphology and the performance of the solar cells remains to be further studied in order to realize the higher power conversion efficiency.
3 Conclusions The author has developed a numerical model to predict current-voltage characteristics of µc-Si:H solar cells on the basis of two types of charged carrier generation processes (exciton process and charged carrier process) including light trapping effect due to a textured back surface structure. It was found that the reported data of the power conversion efficiency cannot be explained by the conventional charged carrier process model and that the combined state model of charged carrier with exciton well explains the performance of the p-i-n type µc-Si:H solar cells. In this way, the typical power conversion efficiencies, under 1 sun AM1.5 solar illumination, were estimated to be 10.5% for the device (i-layer thickness: 1.8 µm) with the BSR (period: 600 nm; height: 250 nm) and 8.6% for the device with the flat reflector under the condition that the fractions of the exciton process and the charged carrier process were 60% and 40%, respectively. The performance of µc-Si:H solar cells depends strongly on the film morphology which determines the fractions of charged carrier generation processes and the diffusion length of carriers. Accordingly, the further improvement of the performance remains to be done. Acknowledgments The author would like to thank Dr. S. Yagi for his encouragement to this study. The author is also grateful to Mr. O. Watanabe for his precious advice on the programming of the numerical model.
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