Yaroslav B.
Martynov
a,
Rashid G.
Nazmitdinov
*bcd,
Andreu
Moià-Pol
b,
Pavel P.
Gladyshev
d,
Alexey R.
Tameev
e,
Anatoly V.
Vannikov
e and
Mihal
Pudlak
f
aState Scientific-Production Enterprise “Istok”, 141120 Fryazino, Russia
bDepartament de Fsica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
cBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia. E-mail: rashid@theor.jinr.ru
dDubna State University, 141982 Dubna, Moscow region, Russia
eFrumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 119071 Moscow, Russia
fInstitute of Experimental Physics, 04001 Kosice, Slovak Republic
First published on 30th June 2017
Organometal triiodide perovskites are promising, high-performance absorbers in solar cells. Considering the perovskite as a thin film absorber, we solve transport equations and analyse the efficiency of a simple heterojunction configuration as a function of electron–hole diffusion lengths. We found that for a thin film thickness of ≃1 μm the maximum efficiency of ≃31% could be achieved at the diffusion length of ∼100 μm.
It is noteworthy, however, that our aim is to determine the power-conversion efficiency of perovskite solar cells when steady conditions for carrier transport are already achieved. In other words, the question of how early time dynamics would affect our estimation is of minor importance, and we do not take it into account. Moreover, the efficiency can be further improved by purification of perovskite semiconductors or/and by decreasing crystal lattice defects, that will, however, increase the cost of solar cells. Evidently, the higher the purification, the lesser the importance of the above mentioned mechanisms.
In order to attain a high efficiency, a semiconductor has to fulfill some basic requirements. In particular, an ideal solar cell requires: (i) a short absorption length (A); (ii) large recombination life-times τ for carriers; and (iii) high carrier mobilities μ. In total, the following condition should hold for the diffusion length
![]() | (1) |
Available data indicate that the diffusion length for perovskite semiconductors can vary from 1 up to 175 (μm).5–7 These lengths correspond to the carrier lifetimes from a few nanoseconds to a few hundreds microseconds. It has already been mentioned above that such a variation may be determined by material purity, i.e., by different densities of light trapping center assisted non-radiative recombination, i.e., the Shockley–Read–Hall (SRH) recombination (see details in ref. 8). Evidently, the power-conversion efficiency of perovskite solar cells could be increased when non-radiative recombination pathways are eliminated.9 Indeed, a stand-alone radiative recombination yields very large carrier lifetimes of the order of a few hours (see ref. 10, p. 301). The question arises: what degree of the purity could be enough to reach the Shockley–Queisser limit for perovskite solar cells? In this contribution we shall attempt to elucidate this question and determine the maximal efficiency that corresponds to this limit in perovskite solar cells.
We consider the following set of equations for electron/hole density η = n/p:
![]() | (2) |
![]() | (3) |
![]() | (4) |
![]() ![]() ![]() | (5) |
![]() | (6) |
Here, T(R) is the photo- or impact ionization (the recombination) rate of carriers per unit volume, Na/Nd is the ionized acceptor/donor density, EC(EV) is the edge energy of the conduction (valence) band, and ε is the permittivity. Evidently, eqn (4) transforms to the Poisson's equation, since the electric field strength is related to the scalar potential ϕ.
Note that the mobility and the diffusion coefficient depend on mean carrier energy εη. Therefore, to complete the scheme we extend the considered approach by adding the continuity equations for the energy density η = η·εη of electron–hole plasma
![]() | (7) |
![]() | (8) |
A few remarks are in order. In a low electric field we use constant mobility (low-field mobility). With the increase of the electric field the carrier velocity increases asymptotically towards a maximal value, i.e., the saturation velocity vsη. As is mentioned above, in our approach the carrier mobilities are functions of mean carrier energies and are determined by the following conditions:
![]() | (9) |
![]() | (10) |
Standard drift-diffusion models either overestimate drift velocities in the case of constant carrier mobilities or underestimate them (see discussion in ref. 13) in the case of the field dependent mobilities such as (9). Note that the correct description of the drift velocities at early time dynamics is especially useful in the simulation of the transient current while modelling the real-time current–voltage measurements. In addition, this description is important for reliable calculations of capacitance frequency dispersion of the PVE, i.e., the frequency dependence of the permittivity for perovskite solar cells.
We have introduced in eqn (7) the photogenerated rate of the energy density of free carriers at the depth x from a semiconductor surface:
![]() | (11) |
The photoionization rate T(x) is defined in a similar way
![]() | (12) |
Δ(x,ν) = α(ν)F(ν)exp[−α(ν)x], | (13) |
R = (np − ni2)/(τpn + τnp). | (14) |
We consider a typical planar heterojunction architecture (see Fig. 1), where the following parameters are used. The minimal width of ZnO and CuI layers (≃0.2 μm) has been taken to ensure that it is large enough than the charged layers on the boundaries with perovskites. The parameters for the contacts and the perovskites for ZnO/CH3NH3PbI3/CuI (see ref. 14 and 15) are as follows: (a) the relative dielectric constant is 9/1000/6.25; the effective electron mass is 0.275/1/0.3 (×me); the effective hole mass is 0.59/1/2.4 (×me); the electron mobility is 150/50/50 (cm2 (V s)−1); the hole mobility is 50/50/44 (cm2 (V s)−1); all energy relaxation times are 10−12 (s); the carrier lifetime (τn = τp) is 10−9/τx/10−9 (s). In our analysis we vary τx from a few nanoseconds to a few hundreds of microseconds. Here, me is a free electron mass. We would like to mention one more point that deserves attention: the permittivity of perovskite solar cells may increase 1000 times under 1 sun illumination compared to dark conditions (see Fig. 2d in ref. 3 and the following discussion). In addition, we have also checked that the decrease of the permittivity (chosen in our calculations to be equal to 1000 for perovskite solar cells) on two orders of magnitude does not affect our results.
![]() | ||
Fig. 1 Band profile of the CH3NH3PbI3 p–i–n solar cell. The following data are taken from ref. 14 and 15 for ZnO, the perovskite, and CuI, respectively: (a) band gap energies are 3; 1.6; 3.1 (eV); (b) electron affinity EEA = 4; 3.9; 2.1 (eV). The built-in potential Vbi ≃ 1.17 V. The width of ZnO and CuI layers is 0.2 μm, while the width of CH3NH3PbI3 is varied, in order to find the optimal thickness of the perovskite structure (see Section 3). |
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Fig. 2 (a) Absorption coefficient α as a function of the photon energy: (i) experimental values are connected by solid (black) line; (ii) auxiliary values = 10![]() ![]() ![]() ![]() |
To investigate the efficiency of the planar heterojunction architecture we have to solve the set of eqn (2)–(14) using the following boundary conditions. At the left boundary, in heavily doped ZnO the majority carrier concentration is set to the value n = Nd = 1019 (cm−3), while the minority one is set to the value p = ni2/Nd. Here, ni2 is a carrier concentration in the intrinsic semiconductor (see for details ref. 16). The opposite situation occurs over heavily doped CuI (the right boundary), where the majority carrier concentration is set to the value p = Na = 1019 (cm−3), while the minority one is set to the value n = ni2/Na. We assume that at the boundaries electron and hole subsystems have average energies εn = εp = 3kBT0/2, i.e., they are in thermal equilibrium.
In order to define the current–voltage characteristics (CVC) of the considered system, one has to extract the contact potential difference from the applied voltage to the p–i–n structure by means of the standard procedure.16 At a given voltage applied to the solar cell under the incident light, we are able to define the current over contacts. The numerical approach to the solution of the set of equations is discussed in detail in ref. 18. The results of CVC calculations of the PVE are displayed in Fig. 3, while the PVE power conversion efficiency in Fig. 4.
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Fig. 4 The efficiency of the p–i–n solar cell as a function of the perovskite film thickness for different diffusion lengths ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
The maximal efficiency of the p–i–n system is developed at the perovskite film thickness L = 0.5 μm at the diffusion length D = 1 μm. The increase of the length values 10 <
D < 150 μm leads to the shift of the maximal efficiency to the thickness ∼1 μm. A further increase of the thickness L decreases the efficiency of the p–i–n system. Below we provide a few arguments that allow us to shed light on this result.
In fact, the output voltage U of the considered system at the maximal efficiency could be approximated by a simple expression
U = ![]() | (15) |
To gain a better insight into the nature of this resistance let us focus on the flux of energy density, dissipated by electron and hole currents in the semiconductor lattice, and defined formally by the last term Γ in eqn (7). Once the steady solutions for electron/hole densities are obtained, one can calculate this term. Fig. 6 displays the dissipated power density at the CVC point that corresponds to the maximum efficiency of the p–i–n solar cell (as an example of the CVC, see Fig. 3 at the diffusion length D = 150 μm). Evidently, the amount of dissipated energy increases with the increase of the thickness of the perovskite film for all diffusion lengths. We observe, however, that at the diffusion length
D = 1 μm the dissipation reaches the maximal value at the thickness L = 1.5 μm. A further increase of the thickness above this value yields a slight decrease of dissipation. In this case the thickness exceeds the diffusion length, and, consequently, the current decreases due to the SRH recombination, which decreases the dissipation. The efficiency of such PVEs is much lower, and decreases much faster in comparison with those with
D > 1 μm (see Fig. 4).
One can readily evaluate the parasitic resistance from the dissipated power density
P = j2 × R. | (16) |
At the maximal efficiency point, for the PVE with the diffusion length D = 150 μm one obtains the resistance R0.5,2 = 12.3(14.8) Ohm cm2 for the thickness L = 0.5(2) μm, respectively. As a result, the parasitic resistance is R = R2 − R0.5 = 2.5 Ohm cm2. Thus, the non-ohmic origin of the parasitic resistance is determined by the dissipated energy of the carriers.
The distribution profile of the dissipation is extremely nonhomogenous along the sample thickness (see Fig. 7). This explains the non-ohmic character of the parasitic resistance. For the sake of illustration, we choose the profile of the dissipation for a particular thickness L = 2 μm at the diffusion length D = 150 μm. The main contributions to the dissipation are brought about by the junction regions. There is a transfer of the carrier energy into the lattice of ZnO, which generates heat (P > 0) in the junction, and the absorption of the energy (P < 0) by the carriers from the perovskite lattice. The latter process is replaced by a slight heating of the perovskite lattice, at some distance away from the junctions, and transforms to a slight cooling in the neighborhood of the right junction. Immediately, after the junction area perovskite-CuI, the heat is generated noticeably by the carriers. In the considered planar heterojunction architecture this phenomenon resembles closely the Peltier effect.
Thus, varying the diffusion length D in the perovskite absorber and its thickness, we have determined the efficiency of the p–i–n system (see Fig. 4). Evidently, the larger the diffusion length, the better the efficiency. However, two features are found. First, all the efficiency curves change the slope once the average thickness approaches the value L ≃ 1 μm, except the case
D = 1 μm where the optimal thickness is L = 0.5 μm. The further increase of the perovskite thickness yields the parasitic resistance. As a result, the efficiency drops down for all values of the diffusion length. Second, the increase of the diffusion length above
D = 150 μm does not increase essentially the efficiency of the p–i–n system. Note that this efficiency is approaching the value of 31%.
Unfortunately, there are no available data for our system. For the sake of illustration of the efficiency of our approach, we compare the results obtained within our model and available experimental data for the system with a similar planar heterojunction architecture studied in ref. 1. Starting from the thickness L = 0.5 μm of this system, we vary the following parameters: the reflection coefficient, the carrier lifetimes, and the parasitic resistance connected in series with the PVE, in order to reproduce the experimental CVC.1 In our calculations we use the experimental behaviour (data) of the absorption coefficient [see Fig. 2(a)]. The reflection coefficient (= 0.18) is determined from the equivalence of the model and experimental values of the short-circuit current. The calculations of the carrier lifetime (obtained from the condition of equivalence of the model and experimental values of the open circuit voltage) yields the value = 0.84 ns that corresponds to the diffusion length D = 0.33 μm. The parasitic resistance (= 6.5 Ohm cm2) is determined from the equivalence of the model and experimental values of the fill factor. The obtained values enable us to reproduce the experimental CVC with a remarkable accuracy (see Fig. 8).
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Fig. 8 Current density as a function of applied voltage: (a) solid (red) circles are used to connect the experimental results;1 (b) solid (green) diamonds are used to connect the model results for ![]() ![]() ![]() ![]() |
Finally, based on our results, let us provide a simple estimation of the energy associated with hot electrons/holes. The incident light, characterised by the Air Mass 1.5 Sun spectrum, carries the power of 100 mW cm−2. Our solar cell with the absorption coefficient (see Fig. 2a) and the thickness L = 2 μm absorbs in one unit of time . The efficiency of the solar cell (see Fig. 4) yields the useful power Puse ≈ 30 mW cm−2. We have to take into account the dissipated energy which is
(see Fig. 6), where d is the length of the heterostructure. Some energy is carried out due to the SRH recombination. It is natural to assume that the minimal lost energy is defined by the energy gap. Knowing the solution of eqn (2)–(14), one can evaluate this lost energy as
. From these estimations we can readily determine the power that heats the electron/hole plasma
Phot = Pabs − Puse − Pdiss − Prec, | (17) |
We have numerically solved transport equations for the carrier and energy densities in the p–i–n system. The obtained results allow us to shed light on dissipation processes in the considered PVE and to trace the evolution of the energy gained by electrons and holes from the sunlight excitation. By varying the diffusion length, we found that there is an optimal thickness of the perovskite absorber L ≃ 1 μm. Once the purity of the perovskite absorber allows reaching the diffusion length D ≤ 150 μm, the considered system is able to attain the efficiency ∼31% in the Air Mass 1.5 spectrum.
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