Simple Analytic Description of Collection Efficiency in Organic

We derive two analytic relationships for carrier collection efficiency in organic photovoltaics .... organic solar cells? Biswajit Ray , Muhammad A. A...
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Simple Analytic Description of Collection Efficiency in Organic Photovoltaics Brett M. Savoie,* Bijan Movaghar, Tobin J. Marks, and Mark A. Ratner Argonne-Northwestern Solar Energy Research Center, Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60201, United States S Supporting Information *

ABSTRACT: The collection of charge carriers is a fundamental step in the photovoltaic conversion process. In disordered organic films, low mobility and disorder can make collection the performance-limiting step in energy conversion. We derive two analytic relationships for carrier collection efficiency in organic photovoltaics that account for the presence or absence of carrier-selective electrodes. These equations directly include drift and diffusive carrier transport in the device active layers and account for possible losses from Langevin and Shockley−Read−Hall recombination mechanisms. General relationships among carrier mobility, contact selectivity, recombination processes, and organic photovoltaic figures of merit are established. Our results suggest that device collection efficiency remains mobility-limited for many materials systems, and a renewed emphasis should be placed on materials’ purity. SECTION: Energy Conversion and Storage; Energy and Charge Transport

O

It is of widespread general interest to identify relationships between device observables like JSC, VOC, and FF, and synthetically accessible materials’ properties like carrier mobility and purity. Efforts to establish such connections have largely relied on continuum differential methods12,17,18 or Monte Carlo19,20 approaches to collection. Whereas these numerical methods are powerful, they also require expertise and computational resources that limit their widespread application. To date, when researchers have required a useful analytic approach for conceptually rationalizing the collection features of a data set, they have turned to the so-called “Hecht” equation.3,12,21,22 Unfortunately, Hecht’s model assumes driftonly collection and perfectly selective electrodes (inherent to the drift assumption). These considerations render it inappropriate for describing collection within the OPV operating window, where both carrier diffusion18,20,23 and contact selectivity24,25 are known to strongly influence performance. Here we report two new analytic descriptions of carrier collection in two key OPV operating regimes with or without selective electrodes. Among the challenges to developing an analytic model to account for these scenarios is the need to account for both drift and diffusive transport in the deviceoperating window and the role of electrode selectivity in thin devices. Here we provide an analytic treatment capable of addressing both finite size effects and electrode selectivity. The

rganic semiconductors are widely employed as active materials in organic photovoltaics (OPVs), thin-film transistors (OTFTs), and light-emitting diodes (OLEDs). However, they have relatively low mobilities (commonly 10−5 to 10−1 cm2 V−1 s−1) and high structural disorder, owing to the weak interactions organizing the π structures.1,2 In OPVs, low mobility limits the distance carriers can move before recombining, and carriers must be collected at the correct electrode before encountering a complementary carrier (Langevin recombination) or a Shockley−Read−Hall (SRH) recombination/trapping center.3 The low mobility is partially compensated by strong optical absorption in most organic chromophores, permitting relatively thin devices with short average carrier collection lengths (50−100 nm).4 Nevertheless, collection-related losses are still thought to severely limit the device fill factor (FF),5 open circuit voltage (VOC),6−9 and short circuit current (JSC).10,11 The fact that optimum active layer thicknesses are typically below that required to absorb all of the impinging light convincingly argues that current OPVs are collection-limited.4,12 Whereas organic semiconductors exhibit relative insensitivity to impurity-related performance losses,13 at certain levels impurities seriously limit OPV performance. Thus, Sun reported that the excellent performance of small molecule solution-processed systems is severely compromised by impurity addition at the 1% level,14 and similar results are noted15,16 when impurities were deliberately introduced into bulk-heterojunction (BHJ) blends. Such impurities degrade mobility and represent potential recombination centers that reduce carrier collection efficiency and fill factor. © 2013 American Chemical Society

Received: December 22, 2012 Accepted: February 6, 2013 Published: February 6, 2013 704

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approach exactly treats both drift and diffusive transport and electrode selectivity, which makes it compatible with the operating conditions of an OPV and suitable for connecting the general characteristics of collection to mobility, device thickness, and material purity. The general issue of OPV carrier collection is: if a carrier is generated at some arbitrary point x within an OPV active layer, what is the probability that at some time t later it has been collected? Under operation, only carriers collected at the appropriate electrode contribute to photocurrent. We term selective electrodes those that have been fabricated, possibly using interfacial layers,24,25 to collect only the appropriate carrier; nonselective electrodes exhibit no carrier preference and can parasitically collect the “wrong” carrier. The general formulation of this transport problem is the following master equation dni = −∑ Wijni + dt j

∑ Wjinj − τi−1ni + gi j

(1)

Here ni is the carrier density, τi the carrier lifetime, and gi is the generation rate at site i. Wij is the transfer rate from site i → j. The first and second terms describe transfer away and to the site, respectively; the third term describes recombination, and the fourth term describes generation. Equation 1 represents N coupled equations, where N is the number of modeled sites. The general solution to the system of eqs 1 with initial condition ni(t = 0) = 0 is obtained through Laplace transformation. Using matrix notation, the solution is n̂p = Gĝ, where n̂ and ĝ are vectors of site densities and generation rates, p is the Laplace variable, and G = (pI − H0)−1 are the Green’s functions with

Figure 1. Schematic description of the systems considered. (a) Infinite homogeneous material. W+ and W− indicate site-independent forward and backward transfer rates with electric field biased transfer. (b) Finite homogeneous active layer, with N sites, and selective contacts. The selective contact acts as a barrier between sites 0 and 1, and the absorbing contact is a sink at site N + 1. (c) Finite homogeneous active layer, with N sites, and nonselective contacts. Absorbing contacts are sinks at sites 0 and N + 1. Schematics only depict the dynamics of one carrier.

and 1. For a perfect barrier, the rates go to 0 and the perturbation takes the intuitive form: B01 = (W+|0⟩ − W−| 1⟩)(⟨0| − ⟨1|). Similarly, a sink AN+1 at site N+1 (Figure 1b) has an additional trapping rate −εN+1: AN+1 = −εN+1|N + 1⟩⟨N + 1|. For a perfectly absorbing contact, εN+1 → ∞, resulting in irreversible collection. Applying perturbations B01 and AN+1 transforms the infinite system into a finite chain of length Na with the, now perturbed, Green’s functions G[B01,AN+1] (a barrier on the left and a sink on the right). Similarly, the inclusion of two sinks leads to the nonselective contact problem (Figure 1c) with perturbed Green’s functions G[A0,AN+1] (a sink on both left and right). In each case, the perturbed Green’s functions can be obtained from the unperturbed infinite chain Green’s functions of eq 3 using the T-matrix formalism from scattering theory. A full discussion is given in the Supporting Information and also ref 29. We now define the relevant OPV currents and collection efficiency. The total device current under illumination can be described by a superposition of the dark current JDark and the photocurrent JPH:

H 0 = − (W + + W − + τ −1)|i⟩⟨i| + W +|i⟩⟨i − 1| + W −|i⟩⟨i + 1| (2)

We call the matrix H0 anticipating the formal equivalence to the tight-binding Hamiltonian,26 allowing application of Green’s function machinery. Here the matrix elements Gij contain the complete dynamics of the system, expressing the probability that a carrier generated at site i is found at site j at transformed time p. For convenience, the transfer rates27 with (W+) and against (W − ) the electric field have been considered homogeneous. We treat systems of interest (e.g., finite systems with boundaries and sinks) as perturbations of the known infinite homogeneous chain problem (Figure 1a), described by eqs 1 and 2 with N → ∞. The known Green’s function solution26,28 has diagonal matrix elements Gii = (p + g+ + g−)−1, where g± are “self-energies”, 2g± = −(p + W∓ − W±) + ((p + W∓ − W±)2 + 4pW±)1/2. The general Green’s functions are obtained from the diagonal elements, using operators h± = g± (p + g±)−1, for forward and backward propagation ⎧G h(j − i) for j > i ⎪ ii + Gij = ⎨ ⎪G h (i − j) for i > j ⎩ ii −

J(V ) = JDark (V ) − JPH (V )

(4)

The relevant device photocurrent JPH is naturally defined as the net photocurrent emerging from the device (i.e., JPH = JN→N+1 − J1→0) and the steady-state photocurrent collection efficiency ηColl is the ratio of JPH to the total absorbed photon-current JIN

(3)

Equation 3 represents the exact solution for the infinite homogeneous chain. This description is inappropriate for OPVs, which have electrodes and finite collection lengths. Our modification for OPVs introduces a barrier (selective electrode) and a sink (collecting electrode). As a perturbation of H0, a barrier B01 between sites 0 and 1 (Figure 1b) is introduced to alter transfer rates between sites 0

ηColl =

JPH JIN

N

=

N

∑i = 1 giW +GiN − ∑i = 1 giW −Gi1 N

∑i = 1 gi

(5)

Equation 5 is a general result but requires specific knowledge of the generation profile gi, which requires optical modeling. For the following discussion, we assume a constant generation profile g, which divides out of eq 5, allowing direct evaluation. 705

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Figure 2. (a,b) Plots of bulk lifetime τb and carrier concentration n as a function of trap volume percentage and varying carrier mobility across the curves. (c−f) Plots of collection efficiency with varying trap concentration and mobility (c,d) for the selective electrode result (eq 6) and (e,f) for the nonselective electrode result (eq 7). Blue curves show effects of varying mobility in the absence of traps. Gold curves show effect of varying the trap concentration at a constant mobility (0.1 cm2 V−1 s−1). d = 200 nm, a = 1 nm, and g = 3.4 × 1021 cm−3 s−1 used in all simulations. Shaded area in panel c shows the electric field range within the operating window of a conventional ITO/Al device. Plots showing JV curve generation for selective contacts using eqs 4 and 6 are shown for (g) varying mobility while holding the trap concentration at 0 and (h) varying trap concentration while holding the mobility at 0.1 cm2 V−1 s−1.

For the present figure-of-merit analysis, we approximate the carrier lifetime with the bulk lifetime τb obtained from direct solution of the continuity equations for holes and electrons in the absence of collection (see Supporting Information for explicit derivation)

Substituting the perturbed Green’s functions G[B01,AN+1] into eq 5 directly yields the collection efficiency for the selective electrode case. N

ηColl =

∑ GiN [B01, AN + 1]W +N −1 i=1

k τb = − SRH + 2klangg

N

−

∑ Gi1[B01, AN + 1]W −N −1 i=1

(6)

N

∑ GiN [A 0, AN + 1]W +N −1 i=1

N

−

∑ Gi1[A 0, AN + 1]W −N −1 i=1

(8)

Here kSRH and kLang are the standard Shockley−Read−Hall and Langevin rates. The detrapping SRH process, much slower than recombination, is neglected. Traps here thus refer to sites sufficiently deep in energy to irreversibly bind a carrier. The bulk lifetime and carrier densities are shown in Figure 2a,b as functions of SRH trap concentration NT and mobility. The bulk carrier density nb is obtained as nb = gτb, and the trap volume percentage is calculated as Nt/a−3 × 100, where a−3 is the site density (1 × 1027 m−3).7,30,31 The expressions for collection efficiency in the selective and nonselective regimes are plotted in Figures 2c,d and Figures 2e,f. Parameters used for evaluation are shown in the caption; for each case, the effects of mobility and trap density are plotted. In the high-field limit, the two formulas are exact because contact selectivity is enforced by the drift condition, with carriers moving in only one direction. Low field differs: the nonselective current approaches zero, reflecting equal likelihood of carrier collection at each electrode. In the selective case, diffusive collection efficiency is possible although highly sensitive to the mobility. The strong dependence of collection efficiency on diffusional collection and electrode identify clearly demonstrates the need to account for these features in a model of collection efficiency. We now illustrate the usefulness of these analytic expressions for establishing general relationships between the device figures of merit, carrier mobility, and deep trap concentration. We have conducted a figure-of-merit analysis of a P3HT:PCBM OPV system, where eq 4 is evaluated using the selective boundary result of eq 6. The incident photon flux is obtained by integrating the above bandgap (hν > Eg) AM1.5G photons, taking Eg = 2 eV for P3HT and assuming 25% loss from incomplete absorption and exciton dissociation. The dark current in eq 4 is modeled using an ideal diode fit JDark =

The corresponding result for nonselective electrodes is obtained similarly, introducing a sink at site 0 rather than a barrier. The perturbed Green’s functions G[A0,AN+1] when substituted into eq 5 yield the nonselective solution ηColl =

⎞2 ⎛ k SRH ⎟⎟ + 1 ⎜⎜ 2 k g kLangg ⎝ lang ⎠

(7)

Explicit evaluation of the Green’s functions GiN[B01,AN+1], Gi1[B01,AN+1], GiN[A0,AN+1], and Gi1[A0,AN+1] in eqs 6 and 7 are provided in the Supporting Information. Within the drift approximation, both results simplify to the conventional Hecht result.21 All terms in eqs 6 and 7 are experimentally accessible and only depend on mobility μ, carrier lifetime τ, electric field F, and device thickness Na. These results self-consistently relate the experimentally observable collection efficiency to the carrier lifetime, electrode identity, and mobility. To obtain the analytic descriptions, we have assumed (i) an effective medium of identical transfer rates in the active layer, (ii) matched electron and hole collection rates, and (iii) a constant electric field. The effective medium approximation (EMA) has found widespread use in solar cell modeling and is almost universal even in numerical drift-diffusion simulations.30 The latter two assumptions can be lifted through a direct evaluation of Poisson’s equation, although at the expense of the analytic solution. In systems with heavy doping and mismatched electron and hole mobilities, performance may be further affected by space charge effects not considered here.12 706

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J0[exp(qV/nkBT) − 1] to experimental data with a reverse saturation current density J0 = 1.9 × 10−5 A/m2 and an ideality factor n = 1.5, chosen to reproduce experimental JSC and VOC for these OPVs when using realistic mobility and trap density values. Finally, a built-in potential VBI = 0.7 V and a device thickness of 200 nm are assumed.8 These parameters reproduce typical JV behavior for P3HT:PCBM devices.3 This analysis isolates the impact of collection-related losses on the device figures-of-merit while holding other features such as exciton dissociation, absorption efficiency, and dark behavior constant. The results should be interpreted as the general effects of low mobility and lifetime on performance because the only P3HT:PCBM specific features are the carrier generation rate and dark current, which do not greatly affect general photocurrent trends. Figures 2g,h show the effects of varying mobility and trap concentration using the selective contact formulation. As expected, the fill factor from all curves is strongly dependent on mobility and trap concentration. It is also clearly possible to have collection-limited currents at short-circuit, even under the assumption of a built-in field. This behavior manifests itself as an apparently low shunt resistance under illumination that is absent in the dark; the model suggests that this is a characteristic of collection-limited current under combined low mobility or high trapping conditions. In accord with recent reports,10,11 the results suggest caution in ascribing J-V/JSC behavior exclusively to carrier generation. The present analysis indicates that OPV voltage dependence near short-circuit can be explained solely in terms of collection related losses, even when sizable built-in voltages (0.7 V) are present. Within the EMA description, mobilities are assumed to reflect system disorder arising from processing, phase separation, batch quality, transport dimensionality, and so on. This assumption is also in line with the present level of morphological understanding of the OPV active layer.32 Caution is urged when quantitatively comparing BHJ mobilities with measurements on the pristine materials.33,34 More sophisticated models of mobility35,36 accounting for local disorder could be exchanged for the scalar effective mobility employed here with no loss of generality in formulas 6 and 7. Similarly, “traps” refer to sites with sufficiently low energies to irreversibly bind a carrier, without specific assumptions about their structural or compositional nature. Chemical impurities, structural disorder, or nonpercolating pathways could, in principle, all act as recombination centers. From the analytic results, we generate a series of plots, as in Figures 2g,h, as a function of μ and NT from which the figuresof-merit can be readily extracted. The results, summarized in Figures 3a−d, show the strong dependence of all four figuresof-merit on both μ and NT. As expected, FF deteriorates as traps and declining mobility are introduced. It is perhaps surprising that all figures-of-merit exhibit significant sensitivity to both μ and NT. The explanation is that the photocurrent can be collection-limited at short circuit even at modest NT (0.01%) and reasonable μ (0.01 cm2 V−1 s−1). The high sensitivity of performance to the trap concentration as quantified in Figure 3 has important implications for the synthesis of materials and fabrication of devices. Given the relative impurity of organic semiconductors to their inorganic counterparts, it is remarkable that high performance is routinely reported. Even 99.99% purity of the organic components can only be reconciled with high efficiency if most of the remaining impurities are electrically inert. This is likely the case with

Figure 3. (a−d) Figure-of-merit analysis showing the effect of varying trap concentration and mobility for selective electrodes. JV curves were generated as described in the text, with figures-of-merit extracted and plotted as a function of mobility and trap concentration. Note that for low-efficiency devices, fill factor becomes a poor descriptor of performance.

residual solvent and processing additives.37 It seems likely, however, that further improvements can be gained from a renewed emphasis on purity. The predicted overall figure-of-merit improvement with mobility also indicates that the reduced carrier density resulting from rapid collection has the overall effect of suppressing recombination. Whereas it is likely that in situations of electron and hole mobility mismatch, increasing the mobility of one carrier may hurt performance, the present analysis shows that rapid collection at higher mobilities outweighs the increase in Langevin recombination in the matched scenario. Analogous results for nonselective contacts (Supporting Information, Figure S4) show overall lower performance even in the high mobility limit, where the nonselective limiting PCE is ∼0.5% lower than for the selective case, and the limiting FF is several percentage points lower. These results reveal the significant role played by diffusional processes throughout the operating window of the device. Although intuitive (the selective contact case should be higher performing), this result is difficult to explain when considering only the drift limit, as is commonly done in JV curve interpretation. In summary, we provide two new descriptions of OPV carrier collection efficiency that are formally exact within the EMA at steady-state. These formulas are also the first analytic models for collection specifically tailored to OPVs and capable of treating contact selectivity. This approach should be valuable to the organic electronics community because it is compatible with all recombination processes and even finite-size effects that are unsuitable for continuum differential methods. The present descriptions also account for both drift and diffusive transport 707

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(9) Li, Z.; Gao, F.; Greenham, N. C.; McNeill, C. R. Comparison of the Operation of Polymer/Fullerene, Polymer/Polymer, and Polymer/ Nanocrystal Solar Cells: A Transient Photocurrent and Photovoltage Study. Adv. Funct. Mater. 2011, 21, 1419−1431. (10) Dibb, G. F. A.; Kirchartz, T.; Credgington, D.; Durrant, J. R.; Nelson, J. Analysis of the Relationship between Linearity of Corrected Photocurrent and the Order of Recombination in Organic Solar Cells. J. Phys. Chem. Lett. 2011, 2, 2407−2411. (11) Koster, L. J. A.; Kemerink, M.; Wienk, M. M.; Maturová, K.; Janssen, R. A. J. Quantifying Bimolecular Recombination Losses in Organic Bulk Heterojunction Solar Cells. Adv. Mater. 2011, 23, 1670− 1674. (12) Kirchartz, T.; Agostinelli, T.; Campoy-Quiles, M.; Gong, W.; Nelson, J. Understanding the Thickness Dependent Performance of Organic Bulk Heterojunction Solar Cells: The Influence of Mobility, Lifetime and Space Charge. J. Phys. Chem. Lett. 2012, 3, 3470−3475. (13) Hopkins, R. H.; Rohatgi, A. Impurity Effects In Silicon for HighEfficiency Solar-Cells. J. Cryst. Growth 1986, 75, 67−79. (14) Sun, Y.; Welch, G. C.; Leong, W. L.; Takacs, C. J.; Bazan, G. C.; Heeger, A. J. Solution-Processed Small-Molecule Solar Cells with 6.7% Efficiency. Nat. Mater. 2012, 11, 44−48. (15) Matsuo, Y.; Ozu, A.; Obata, N.; Fukuda, N.; Tanaka, H.; Nakamura, E. Deterioration of Bulk Heterojunction Organic Photovoltaic Devices by a Minute Amount of Oxidized Fullerene. Chem. Commun. 2012, 48, 3878−3880. (16) Leong, W. L.; Hernandez-Sosa, G.; Cowan, S. R.; Moses, D.; Heeger, A. J. Manifestation of Carrier Relaxation Through the Manifold of Localized States in PCDTBT:PC60BM Bulk Heterojunction Material: The Role of PC84BM Traps on the Carrier Transport. Adv. Mater. 2012, 24, 2273−2277. (17) MacKenzie, R. C. I.; Kirchartz, T.; Dibb, G. F. A.; Nelson, J. Modeling Nongeminate Recombination in P3HT:PCBM Solar Cells. J. Phys. Chem. C 2011, 115, 9806−9813. (18) Koster, L.; Smits, E. C. P.; Mihailetchi, V. D.; Blom, P. W. M. Device Model for the Operation of Polymer/Fullerene Bulk Heterojunction Solar Cells. Phys. Rev. B 2005, 72. (19) Marsh, R. A.; Groves, C.; Greenham, N. C. A Microscopic Model for the Behavior of Nanostructured Organic Photovoltaic Devices. J. Appl. Phys. 2007, 101, 083509. (20) Nelson, J. Diffusion-Limited Recombination in PolymerFullerene Blends and Its Influence on Photocurrent Collection. Phys. Rev. B 2003, 67, 155209. (21) Hecht, K. Zum Mechanismus Des Lichtelektrischen Primarstromes in Isolierenden Kristallen. Z. Phys. A 1932, 77, 235−245. (22) Ng, T. N.; Wong, W. S.; Lujan, R. A.; Street, R. A. Characterization of Charge Collection in Photodiodes under Mechanical Strain: Comparison between Organic Bulk Heterojunction and Amorphous Silicon. Adv. Mater. 2009, 21, 1855−1859. (23) Mihailetchi, V. D.; Blom, P.; Hummelen, J.; Rispens, M. Cathode Dependence of the Open-Circuit Voltage of Polymer:Fullerene Bulk Heterojunction Solar Cells. J. Appl. Phys. 2003, 94, 6849−6854. (24) Murray, I. P.; Lou, S. J.; Cote, L. J.; Loser, S.; Kadleck, C. J.; Xu, T.; Szarko, J. M.; Rolczynski, B. S.; Johns, J. E.; Huang, J.; et al. Graphene Oxide Inter layers for Robust, High-Efficiency Organic Photovoltaics. J. Phys. Chem. Lett. 2011, 2, 3006−3012. (25) Hains, A. W.; Liu, J.; Martinson, A. B. F.; Irwin, M. D.; Marks, T. J. Anode Interfacial Tuning via Electron-Blocking/Hole-Transport Layers and Indium Tin Oxide Surface Treatment in BulkHeterojunction Organic Photovoltaic Cells. Adv. Funct. Mater. 2010, 20, 595−606. (26) Movaghar, B. On the Theory of Diffusion Limited Recombination and Relaxation in Amorphous Systems. J. Phys. C 1980, 13, 4915−4931. (27) The transfer rate W is self-consistently related to the diffusivity D as Wa2, where a is the lattice spacing, and to the mobility μ through the generalized Einstein relation

limited by SRH and Langevin recombination that are neglected in other formulations.38,39 Using these models, the general features of mobility-limited performance, the impact of traps, and contact identity on photocurrent generation in thin organic devices have been investigated. The results show that poor OPV FFs and low shunt resistances upon illumination can be explained solely in terms of collection dynamics without invoking voltage-dependent generation mechanisms. Whereas voltage-dependent generation cannot be generally ruled out,7 arguments based solely on JV curve interpretation should be considered inconclusive. The results suggest that OPV efficiency remains mobility limited, as is also evidenced by the thin optimized device thicknesses commonly employed. Even for low trap concentrations (0.001%), we predict collection losses throughout the operating window and emphasize the need for materials purity. We also suggest that higher mobilities will be required to avoid collection losses in thicker devices (>100 nm), as required for both complete light absorption and typical roll-to-roll fabrication.40,41

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ASSOCIATED CONTENT

S Supporting Information *

Stepwise derivation of the main collection results, the bulk lifetime, and the figure-of-merit analysis for nonselective electrodes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported as part of the ANSER Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0001059. B.M.S. thanks the Northwestern MRSEC for a predoctoral fellowship.

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