Self-Consistent Drift-Diffusion Analysis of Intermediate Band Solar Cell (IBSC): Effect of Energetic Position of IB on Conversion Efficiency

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1 Self-onsistent Drift-Diffusion Analysis of ntermediate Band Solar ell (BS): Effect of Energetic Position of B on onversion Efficiency Katsuhisa Yoshida 1,, Yoshitaka Okada 1,, and Nobuyuki Sano 3 1 raduate School of Engineering, The niversity of Tokyo, Hongo, Bunkyo-ku, Tokyo , JAPAN Recearch enter for Advanced Technology, The niversity of Tokyo, Komaba, Meguro-ku, Tokyo , JAPAN 3 nsititute of Applied Physics, niversity of Tsukuba, Tennodai, Tsukuba, baraki , JAPAN ABSTRAT The intermediate band solar cell (BS) has been intensively investigated both eperimentally and theoretically. Numerical analyses based on the detailed balance method are performed to search for the best suitable candidates of material combination for BS and the operation conditions. Analytical treatment of driftdiffusion equations has also been reported under limited approimations. However, to study the device characteristics, self-consistent treatments of both the carrier continuity equations and the Poisson equation are required. n this work, we report on the dependence of conversion efficiency on energetic position of B and on the concentration by using 1-D self-consistent driftdiffusion simulator which we developed for aas based solar cell with nas quantum dots. The dependence of the efficiency on energetic position of B above the midgap of aas was calculated for 1, 1, 1 and 1 suns conditions with and without doping in B region. The optimal B position shifted to lower energies with increase of concentration in the case of intrinsic B region. While, in the case of doped B region, the optimal B position was almost fied at.95e. f, however, the B was set in the middle of the energy gap of aas, efficiencies showed lower values with higher concentrations. This is because, according to our present model, very few photons contribute to the optical transition (generation) in B-B and most photons are absorbed in B-B transitions such that there is a large mismatch in the generation-recombination rates in these transition paths. NTRODTON The intermediate band solar cell (BS) [1] has been intensively investigated both eperimentally [] and theoretically. BS utilizes photons, which have lower energies than the bandgap energy of host material and do not contribute to carrier generation in single-junction solar cells, by two-step photogeneration via B. Thus, optimal combination of host material bandgap and intermediate band gives about 63% conversion efficiency under full concentration. Eperimentally, quantum dot based structure [] is one of the candidates for realizing the BS operation principle. Numerical analyses based on the detailed balance method have been performed to search for the best suitable candidates of material combination for BS and the operation conditions [1]. Analytical treatments of drift-diffusion equation are also reported under limited approimations [3,4]. nfortunately, the selfconsistency of both the carrier continuity equations and the Poisson equation is seldom achieved. From the viewpoint of a realistic analysis, a self-consistent treatment would be of most importance. Recently, Lin et al. [5] have reported BS device operations based on the selfconsistent drift-diffusion method by etending the approach proposed by Schmeits et al. [6]. n this work, we report on the dependence of conversion efficiency on energetic position of B by using 1-D self-consistent driftdiffusion simulator developed appropriate for aas based solar cell with nas quantum dots (QDs), and on the concentration. n the following section, treatments of the Self- onsistent Drift-Diffusion Method with B Balance Equation and modeling of optical generation rates via B are presented. By using our method, optimal combinations of aas bandgap and energy position of B under 1, 1, 1 and 1 concentrated the sun light were investigated. NMERAL METHOD The self-consistent drift-diffusion method is usually adopted to analyze the device characteristics. This method is based on the basic device equations in semiconductors, namely, the Poisson equation and carrier continuity equations for electrons in conduction band (B) and holes in valence band (B). n the case of BS, there Figure 1 llustration of energy states E, E and E and of quasi Fermi states, and, Relations of absorption specta used in the simulation, respectively. Figure alculated solar cell structure. is the end of p emitter layer, L is the length of B region and L is total length /1/$6. 1 EEE 711

2 are electrons in B states. The electrostatic potential ( is given by the Poisson equation, d e n n N N D (1) A d where e is elementary charge, is the permittivity of aas, is the hole density, n ( is electron density in B, n ( is electron density in B, N + D ( is the ionized donor density, and N - A ( is the ionized accepter density. n this work, accepters and donors are assumed completely ionized, i.e. N + D ( = N D ( and N - A ( = N A ( where N D ( and N A ( are the donor and the accepter concentrations. The relations between carrier densities and their quasi Fermi energies (or a constant chemical potential in thermal equilibrium) are approimately described by, n N N n N E ep () kbt E ep (3) kbt 1 E ( ) ( ) ep 1 (4) kbt where N and N are the effective density of states for B and B, N is the effective density of state for B, and assumed as twice the density of QDs and, and are quasi Fermi energy for B, B and B, respectively. n the steady state, these carriers satisfy carrier continuity equations. For electrons in B, We call this equation as B balance equation in this work. This equation decides the electron density in B. J and J are given by drift-diffusion equations shown below. d dn J e en ed (8) e d d d d J e h ed (9) h d d where e and h are the carrier mobility of electrons in B and holes, D e and D h are the diffusion coefficients, respectively. The optical generation rate for is given by, F ( E)e de (1) E where is the absorption coefficient of transition and is taken to be 1 4 cm -1, E is the bandgap of aas, 1.43e at 3K. F (E) is AM solar spectrum epressed by a 58K black-body radiation, F ( E) Xf S E (11) 3 h c e E / k T ) 1 where X is the amplitude of concentration, f S epresses the angular rage of the sun, h is Planck s constant, c is the speed of light, k B is Boltzmann s constant, and T S is temperature of the sun, 58K. Eq. (1) implicitly assumes that occupation rates of electrons in B and holes in B are negligible because their rates are usually very low. While, the occupation rate of B affects the generation rate via B strongly. Thus, these generation rates are epressed by the following equations. B S 1 dj e d (5) F ( E)ep f ( ') d' f de (1) E E And for holes in B, 1 dj e d n Eq. (5) and (6), ij is the optical generation rate, ij is the radiateve recombination rate where subscript ij=,, epresses B-B, B-B and B-B transition respectively, J is the electron current density in B and J is the hole current density. For electrons in B, if we assume the carrier flu in B is negligible, then, (6) (7) E E F ( E) ep 1 f ( ' ) d' (13) 1 f de where and are the absorption coefficients of and transitions and both are taken to be 1 4 cm -1, E and E are the bandgaps of and B, f ( is the occupation rate of B, and is the end of p emitter layer. These absorption coefficients are assumed to be nonoverlapping. The relations of absorption coefficients and energy gaps are shown in Fig.1 and, and the device structure assumed is presented in Fig., respectively. The treatment of the occupation rate of B is assumed that B state is a constant with wavenumbers in the reciprocal space /1/$6. 1 EEE 71

3 For the treatment of radiative recombination rates, we adopt the same treatment as shown in Ref. [5]. So, these rates are epressed by, ( n p n ) (14) i p i n p n - n ep 1 kbt ep 1 kbt (15) (16) where ij is a coefficient following by Ref. [5] and T is the temperature of the cell, 3K. For the boundary conditions, we assume ideal ohmic contacts. Then, applied bias is epressed as the difference of quasi Fermi energies between () and (L), Figure 3 alculated band diagrams in equilibrium condition. ase 1. ase, respectively. e ( ) ( L) (17) We also assume infinite surface recombination velocities. Thus, the contacts are kept in the equilibrium which means ) () and ( L) ( L). ( n our present method, the device characteristics of BS is given by solving Eq.(1), (6), (7) and (8) selfconsistently. The mobility models of e and e depend on the cell temperature, the impurity and the electric field, the negative differential resistance for e and the velocity saturation for h. The material parameters ecept for absorption coefficients and N are presented in Ref.[7]. N is set 1 17 cm -3. n p-type emitter region, N A( is cm -3. n n-type base region, N D( is cm -3. For B region, we simulate the two conditions: intrinsic (ase 1) or donor doping N D(=N / (ase ). Energy position of B (E ) is set from the middle of E to band edge of (E ). oncentrations X are set 1, 1, 1 and 1 suns. Figure 4 alculated band diagrams in short circuit condition. ase 1. ase, respectively. RESLTS n Figure 3, calculated band diagrams are presented for both cases when E =1.e in equilibrium. n Fig.3, there is a electric field in B region. While in Fig.3, the center of B region is almost flat and B occupation rate is about.5. n both cases, owing to the self-consistent treatment of the basic device equations and B balance equation, B occupation rates have position dependent values. When sun light enters the cell from the top of p emitter layer, band diagrams change from Fig.3 to Fig.4 by optical generation rates. The band diagram is strongly affected by the incident photons in Fig.4. The B occupation rate is increased in B region and are negative charged. This is because that B region is intrinsic in ase 1. However, half-filled condition is not satisfied. On the other hand, the band diagram is not so changed in Fig.4 in ase Figure 5 urrent voltage characteristics for ase 1, and ase and for without B structure under 1 sun, respectively. because of eisting donor atoms in B region. Moreover, the half-filled condition is almost achieved in Fig.4 assisted by optical generation via B. n Figure 5, current-voltage characteristics of ase 1, ase and simple p-i-n single-junction solar cell (without B state case) are plotted. n ase 1 and ase, shortcircuit current is enhanced by the introduction of B state. However, the difference of this enhancement between ase 1 and ase is the difference of occupation rate of B. n ase 1, there are clear splits between and E near /1/$6. 1 EEE 731

4 ( [cm 3 s 1 ] onversion Efficiency [%] onversion Efficiency [%] p emitter region. This relation means low occupation rate of B states. n ase, and have almost the same values in whole B region. This high occupation rate helps electron transition from B to B. n ase 1 and ase, their open circuit voltages are not affected constructing voltage of single-junction solar cell. This is because only radiative recombination rates are included in this work. The same behaviors are reported in Ref.[5]. Relations of E position and conversion efficiencies with 1, 1, 1 and 1 suns are presented in Fig.6. n Fig.6, E positions of highest conversion efficiencies under each illuminated condition are shifted to lower energies as concentration is increase. nder one sun illumination, the optimal position is 1.3e. n Fig.7, radiative recombination rates of are plotted. When E =1.e, have positive values in the whole B region which mean they contribute to electron relaation from B to B empty states. When E = 1.3 or 1.4e, have negative values in the B region ecept for near n base, they work as like generation terms. f E is set near the middle of bandgap or E, optical generation rate of or is decreased because of the energy difference of E and E, or E and E is reduced. As the concentration increases, the influence of the effect, mentioned above, around 1.3e decreases. This is because that optical generation rates via B with high concentration relatively decrease the contribution of recombination rates in B balance equation. When E is set near the middle of bandgap, the conversion efficiency is decreased under high concentration. The same behavior is shown in Fig.6, ase. Therefore, more realistic treatments of absorption coefficients are required. n Figure 6, optimal E position is almost fied near.95e. When E =.95e, total number of photons contributing to and transitions have almost the same value. n ase, B occupation rates are about.5 under all concentrations. Thus, the optimal position is not changed by concentration Figure 6 onversion efficiencies dependence on E position under 1, 1, 1 and 1 suns, for ase 1 and ase respectively. [1 +1 ] 1 suns 1 suns 1 suns 1 sun E Position [e] 1 suns 1 suns 1 suns 1 sun E Position [e] ONLSON We investigated BS characteristics by self-consistent drift-diffusion method with B balance equation. ntroduction of B gives enhancements of short circuit current density. n our present model, doping B region (ase ) is better than intrinsic case (ase 1). This indicates that half-filled condition is required to achieve high efficiencies in BS. High occupation rate in B assists optical generations in ase 1. This behavior is also shown in high concentration conditions. However, more realistic treatments of absorption coefficients are necessary and left to our future work. E = 1. e E = 1.3 e E = 1.4 e AKNOWLEDEMENT The authors acknowledge funding from the nc. Administrative Agency New Energy and Technology Development Organization (NEDO) and Ministry of Economy, Trade and ndustry (MET), Japan. 1 Position [] Figure 7 Radiative recombination rates of under 1sun illumination in short-circuit condition of ase /1/$6. 1 EEE 741

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