Device physics of van der Waals heterojunction solar cells

Heterostructures based on atomically thin semiconductors are considered a promising emerging technology for the realization of ultrathin and ultralight photovoltaic solar cells on flexible substrates. Much progress has been made in recent years on a technological level, but a clear picture of the physical processes that govern the photovoltaic response remains elusive. Here, we present a device model that is able to fully reproduce the current-voltage characteristics of type-II van der Waals heterojunctions under optical illumination, including some peculiar behaviors such as exceedingly high ideality factors or bias-dependent photocurrents. While we find the spatial charge transfer across the junction to be very efficient, we also find a considerable accumulation of photogenerated carriers in the active device region due to poor electrical transport properties, giving rise to significant carrier recombination losses. Our results are important to optimize future device architectures and increase power conversion efficiencies of atomically thin solar cells.


INTRODUCTION
Two-dimensional (2D) semiconductors [1][2][3][4][5] provide a unique opportunity for the realization of ultrathin and ultralight photovoltaic solar cells [6], owing to their strong optical absorption in the solar spectrum region [3,7], high internal radiative efficiencies [8], and favorable band gaps for both single-junction and tandem cells [9]. Theoretical estimates of power conversion efficiencies (PCEs) have predicted efficiency values exceeding 25 % [9], indicating that 2D semiconductors may become competitive with 2 conventional photovoltaic technologies. The suitability of 2D materials for photovoltaic applications was first demonstrated in lateral p-n junctions [10][11][12], defined by split-gate electrodes, and in lateral Schottky junctions [13]. However, those device architectures do not allow for easy scalability of the photoactive area, for which a vertical junction would be desirable. Vertical van der Waals heterostructures [14] can be obtained by manual stacking [15,16] or growth [17][18][19] of different 2D materials in a layered configuration. It has been shown that MoS2 and WSe2, when placed on top of each other, form a type-II heterojunction [20][21][22][23][24], with the lowest-energy conduction band states spatially located in the MoS2 layer and the highest-energy valence band states in WSe2. Relaxation of photogenerated carriers, driven by the conduction and valence band offsets, then results in a charge transfer across the 2D junction and a sizeable photovoltaic effect [20][21][22]. Similar results have been obtained using other material combinations that exhibit type-II band alignment, including MoS2/WS2 [25], MoS2/black phosphorus [26,27], MoTe2/MoS2 [28,29], GaTe/MoS2 [30], MoSe2/WSe2 [31,32], MoS2/carbon nanotubes [33], MoS2/pentacene [34], MoS2/silicon [35,36], and many more. In addition, homojunction architectures have been explored, in which chemical doping is applied to form a vertical p-n junction in the same 2D material. Examples include plasma-induced p-doping of the upper layers in an n-type MoS2 multi-layer crystal [37] and mechanical stacking of few-layer flakes of n-type MoS2:Fe on top of p-type MoS2:Nb [38]. In an optimized ~15 nm thick MoS2/WSe2 heterostructure, an experimental absorbance of >90 %, an external quantum efficiency (EQE; the ratio between collected charge carriers and incident photons) exceeding 50 %, and a (single-wavelength) PCE of 3.4 % have been achieved [39].
The PCE, defined as the fraction of incident optical power #$% that is converted into electricity with output power &' , is the most important parameter describing a photovoltaic device. It is given by the product of open-circuit voltage )* , short-circuit current ,* , and fill factor : Today, fill factors in 2D heterostructure photovoltaic structures are typically in the range 0.3-0.5, only half as large as in conventional silicon solar cells. Closely connected to low fill factors are excessively high (≫2) ideality factors and low short-circuit currents, pointing 3 towards substantial carrier recombination losses. Open-circuit voltages are typically less than 0.6 V, implying a band gap-)* offset larger than 0.8 V [9]. For all these reasons, PCEs in 2D photovoltaic devices have as yet remained below 5 %, much lower then the Shockley-Queisser limit for their band gaps.
Besides technological challenges, the lack of a clear picture of the device physics in 2D heterostructure solar cells hampers further progress. Optimization of device architectures, however, will require an in-depth understanding of exciton dissociation, carrier transport processes, and recombination losses. Here, we address these questions by presenting a systematic experimental study of a MoS2/WSe2 van der Waals heterostructure and, based on the results, a model that reproduces the current-voltage characteristics under optical illumination. While we find the exciton dissociation to be very efficient, we also find a considerable pile-up of photocarriers in the device due to poor electrical transport properties, giving rise to carrier recombination and consequently low , ,* and )* -values. We finally provide guidelines to optimize future device layouts and increase PCEs.

RESULTS
Van der Waals heterostructure solar cell. Fig. 1a shows a schematic illustration of the MoS2/WSe2 heterostructure investigated in this work. Both layers exhibit monolayer thickness, as verified by Raman spectroscopy [40,41]. An optical micrograph of the device can be found in Supplementary  As reported previously [20], the electrical characteristics can be controlled by electrostatic doping via a back-gate voltage F , applied to the silicon substrate (Fig. 1c). For a large positive F we find resistive n-n behavior, whereas for an appropriate choice of F an atomically thin p-n junction is formed and the device current as a function of external bias voltage displays diode-like rectification behavior (inset in Fig. 1c where O is the dark generation current, the elementary charge, A Boltzmann's constant, and the temperature. F denotes the photogenerated current and GJ is the ideality factor whose value depends on the type of recombination mechanism: GJ = 1 for direct bimolecular (Langevin) recombination and GJ = 2 for trap-mediated  [20,21]. However, if we plot equation (1) with GJ = 1.6 in Fig. 2a (dash-dotted line; #$% = 16 nW) we find very poor agreement 5 with the experimental data. The Shockley equation strongly overestimates short-circuit current, fill factor, and forward current. This is in contrast to lateral 2D semiconductor p-n junctions defined by split-gate electrodes, that can be well described by the Shockley model [10,11].
To obtain better modeling of solar cells a series (contact) resistance is often taken into account. However, as shown in see Supplementary Fig. S3, an extended Shockley model does not either fit our data. Particularly, the strong illumination dependence of the forward current and the interception of the ( ) curves cannot be explained. Another mechanism that can affect the electrical characteristics of solar cells is the build-up of space charge regions at the contacts or in the bulk, as a result of strongly unbalanced electron and hole transport [44]. In order to explore this mechanism, we plot in Fig [45], that can result in GJ > 2. We rule out this possibility for the following reasons. First, the forward current in atomically thin p-n junctions is of different origin than in conventional diodes; it is governed by tunneling-mediated interlayer recombination, rather than carrier injection over a potential barrier [20,21]. Second, sample inhomogeneities cannot explain the illumination dependent device behavior.
In the following we will instead argue that the photovoltaic response is transport-limited.
It is thus inappropriate to employ Shockley's model, as it does not account for the impact of charge transport (it assumes infinitely large conductivities for electrons and holes). In order to obtain better modelling of the ( ) characteristics we follow the approach by Würfel et al., initially developed for organic solar cells [46]. In brief, carrier accumulation due to poor transport properties leads to a quasi-Fermi level splitting Gd% = 7,& − 7,9 in the electron and hole transport layers that differs from the externally applied voltage:  [46] to use Gd% in (2) instead of . A closed form approximation of the ( ) curves can then be derived [47], that has been shown to reproduce the results of full drift-diffusion simulations for a wide range of parameters: with a dimensionless figure of merit , that is a direct measure of non-ideal device behavior as a result of insufficient carrier extraction. It is given by where G denotes the intrinsic electrical conductivity in the dark (see Supplementary Note S1 and Ref. 47). If we fit equation (3) to the experimental data (solid lines in Figs. 2a and b) we find excellent agreement for all illumination intensities. The photogenerated current F scales linearly with #$% , as expected (see Supplementary Fig. S4). The ideality parameter varies over a wide range and reaches values as high as ~78. Insufficient carrier extraction, described by large values, leads to high carrier densities and consequently to a large quasi-Fermi level splitting Gd% , even under short-circuit conditions. As the interlayer recombination is governed by Gd% , this results in recombination losses and ,* < F . Under forward bias, the accumulated charge enhances the conductivity, resulting in the experimentally observed crossing of dark and illuminated ( ) curves. As shown in Supplementary Note S2, expression (4) can be further simplified to yield where denotes the effective carrier mobility in the electron and hole transport layers, L&v is the interlayer recombination coefficient, and all other physical constants and geometry factors have been lumped into the prefactor . The expression predicts a square-root dependence of on the optical power, which is indeed observed experimentally (Fig. 3a). 7 The question that remains to be addressed is why the carrier extraction in our device is inefficient, given the rather high mobility of 2D semiconductors (typically 10-100 cm 2 /Vs).
In high-mobility materials, such as crystalline silicon, approaches zero and the impact of charge transport is negligible. In organic materials, on the other hand, is extremely low and transport-limitations are expected. From the dark ( )s in Fig. 1c it is apparent that varying the sample temperature provides us with an opportunity to tune G (∝ JKLM ) in the p-n regime over almost three orders of magnitude. In Fig. 3b (blue symbols) we depict its temperature variation, and we find that it can be described by a thermally activated transport model [48], G ( ) ∝ exp(− K / A ), with an activation energy K ≈ 80 meV (dashed line). Next, we record the temperature dependent photovoltaic properties ( Fig. 2c). From that we determine ( ) and then, with equation (4) Interlayer charge transfer. So far, we have assumed an ultrafast interlayer charge transfer (exciton dissociation), and subsequent charge transport in the ETL and HTL on a longer time scale. We now substantiate this claim by presenting PC autocorrelation measurements, a powerful technique to study the carrier dynamics in 2D materials [52,53] and heterostructures [54]. It exploits the non-linearity of the photoresponse to infer the underlying dynamics of the system. Using this technique, the device is optically excited

DISCUSSION
2D semiconductors contain a significant amount of electronic band tail states [49][50][51]. As shown in the previous section, these defects trap charge and adversely affect the electronic transport and, consequently, the PCE. We conclude that the performance of van der Waals heterojunction solar cells is transport-limited and we suggest that equation (3) should be used to describe the photovoltaic response, instead of the commonly used Shockley

equation. An investigation of the role of defects in the limitation of the open-circuit voltage
is beyond the scope of this article, but we note that it is well understood that defects not only reduce and z{ , but also |{ [57].
To further test out model, we analyzed some exemplary cases from literature. The symbols in Fig. 5a show data points extracted from Ref. 21, along with a fit of equation (3) as solid line. This work employed a device structure similar to the one presented in this article, but with more favorable (unintentional) doping of the MoS2 and WSe2 layers. This resulted in a comparably high despite the high illumination intensity used in the experiment (diamond-shaped symbol in Fig. 3c). We believe that transport-limitations also occur in device architectures that employ optically transparent (graphene or indium-tin-oxide) electrodes for vertical carrier extraction. There, the charge transport occurs on much shorter length scales, but also with extremely low out-of-plane mobilities (~10 -2 cm 2 /Vs) [54]. In Figs. 5b and c we present two examples taken from Refs. 39 and 37, respectively.
The data can again be well fitted by equation (3) and the results are summarized as yellow symbols in Fig. 3c. Interestingly, lateral p-n junctions, based on split-gate electrodes, typically show better photovoltaic properties ( |{ > 0.85 V [10], > 0.7 [58], ideality factor < 2 [11]) than van der Waals structures. This is explained by the higher electrical conductivities in these devices, because of independent doping of the p-and n-type regions.
Based on our results, we finally provide guidelines that might allow to avoid charge pile-up in future device architectures and harness the true potential of 2D materials in photovoltaic applications. We suggest (i) to increase the device conductivity, e.g. by elimination of charge traps or by (chemical) doping, and (ii) to make the active region as thin/short as possible to facilitate more efficient charge extraction before recombination.   b, Arrhenius plot of (normalized) conductivity G , as extracted from dark-current    (3). a, Lee et al. [21]. b, Wong et al. [39]. c, Wi et al. [37].