Excitonics: A Set of Gates for Molecular Exciton Processing and

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Excitonics: A Set of Gates for Molecular Exciton Processing and Signaling Nicolas P. D. Sawaya, Dmitrij Rappoport, Daniel P. Tabor, and Alán Aspuru-Guzik ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.8b00584 • Publication Date (Web): 19 Jun 2018 Downloaded from http://pubs.acs.org on June 26, 2018

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Excitonics: A Set of Gates for Molecular Exciton Processing and Signaling Nicolas P. D. Sawaya,†,‡ Dmitrij Rappoport,† Daniel P. Tabor,† and Alán Aspuru-Guzik∗,†,¶ †Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA ‡Intel Labs, Santa Clara, California 95054, USA ¶Senior Fellow, Canadian Institute for Advanced Research, Bioinspired Solar Energy Program, Toronto, ON M5G 1Z8, Canada E-mail: [email protected]

Abstract Regulating energy transfer pathways through materials is a central goal of nanotechnology, as a greater degree of control is crucial for developing sensing, spectroscopy, microscopy, and computing applications. Such control necessitates a toolbox of actuation methods that can direct energy transfer based on user input. Here we introduce a proposal for a molecular exciton gate, analogous to a traditional transistor, for regulating exciton flow in chromophoric systems. The gate may be activated with an input of light or an input flow of excitons. Our proposal relies on excitation migration via the second excited singlet (S2 ) state of the gate molecule. It exhibits the following features, only a subset of which are present in previous exciton switching schemes: picosecond-timescale actuation, amplification/gain behavior, and a lack of molecular rearrangement. We demonstrate that the device can be used to produce

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universal binary logic or amplification of an exciton current, providing an excitonic platform with several potential uses, including signal processing for microscopy and spectroscopy methods that implement tunable exciton flux. Keywords: excitonics, excited states, organic dyes, microscopy, transistor, circuit, binary logic.

Devising ways to reliably control energy transfer on the nanoscale allows for the further development of nanoscopic devices for spectroscopy, microscopy, photocatalysis, and sensing. 1–8 Here we propose a design for an elementary excitonic gate (or excitonic transistor) that uses multiply-excited states to control the flow of excitons in organic molecules, resulting in a scheme that provides advantages over previous methods of controlling exciton movement. The concept of a gate is essential in many areas of engineering and science—it allows one to regulate the movement of energy, mass, or charge, leading to useful applications. Ubiquitous gate-like devices include field effect transistors for computing, relays for larger electronics, and valves for modulating fluid flow in civil engineering. By controlling the direction and magnitude of exciton migration (equivalently, of excitonic energy migration), the device proposed in this work aims for such gate-enabled applications on the nanoscale, offering improvements relative to other excitonic gating designs. Besides potential microscopy applications, 3–8 it has also been suggested that exciton-based logic might eventually become a low-power alternative to energy-intensive modern computation, perhaps even allowing us to continue following Moore’s Law longer than expected. 9–11 In this theoretical study, we introduce an exciton gate for controlling the rate of single exciton transfer, with the actuation input provided by light or by the input of auxiliary excitons. The gate itself is a molecule considered to be off in its ground singlet state (S0 ) and on in its first excited singlet state (S1 ). As described below, when the gate is turned on, excitons migrate through the second excited singlet state (S2 ), while no excitons (or “excitonic current”) flows when the gate is off. A successful experimental implementation of 2


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this S2 exciton gate involves overcoming competing energy transfer and decay processes. Several previously proposed schemes use nanoscale excitonic energy transfer to perform logical operations or to regulate exciton flow. The state of the art can be placed into four broad categories. First, the earliest examples are logical devices based on DNA nanotechnology, in which single stranded DNA (ssDNA) labeled with chromophores is added to a solution as a logical input. 12–16 After the ssDNA binds to complementary base pairs on an existing DNA-based structure, the resulting changes are usually probed optically. Because the movement of ssDNA is diffusion-limited, a single logical operation within this scheme takes seconds to minutes. The second category for exciton gating uses incident light to induce molecular rearrangement, especially via light-induced isomerization reactions. 9,17–19 When one configuration’s properties allow Förster resonance energy transfer (FRET) to occur while the other does not, this system can be used as an on/off switch for excitonic flow. In the third design, specific chromophores are excited into an optically inaccessible (“dark”) state, which changes the path that excitons take through a chromophoric logic circuit. 10,20 These circuits take seconds to minutes to reset because of the long-lived dark states. Finally, a fourth theoretical proposal, explicitly motivated by the need for low-power computation, uses one-dimensional semiconductor channels and quantum dots to perform logical operations, 11 intended for integration into traditional silicon chips. The Supplementary Information (SI) Section S1 includes comprehensive analyses of these methods’ strengths and limitations in comparison with our proposal. Also notable are the developments in other DNA-based components that might be used to build nanoscopic excitonic circuits, including memory storage 21 and excitonic wires. 22–25 Our S2 exciton gate proposal exhibits the following functionalities, while only a subset of these features are present in each of the excitonic switching devices summarized above: 1. Gate actuation timescales are < 1 ps. 2. A single logical operation can be performed in < 10 ps.

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3. Amplification/gain of excitonic flow is possible, by varying the intensity of incident light or an auxiliary flow of excitons. We argue that the S2 exciton gate is the first single exciton device to exhibit this functionality (see also SI Section S1). 4. The design is biocompatible and usable in aqueous solutions. 5. Precise “clocking” is not required in order to perform logic, simplifying circuit design and perhaps making it more robust to stochastic errors. 6. Molecular rearrangements are not required for gating functionality, while they are intrinsic to the isomerization switches. This reduces the possibility of side-reactions. Also, chemical systems which do not change shape in difficult-to-control ways may be easier to build into larger designs. It is worth noting that there are additional nanoscopic transistor-like devices, other than those used for single exciton gating. Perhaps the most well-studied are single-molecule electronic transistors and gates, 26,27 in which an applied voltage regulates electronic current flow through a single molecule. Though molecular electronics is a mature field with many proposed applications, fabrication and integration remain challenging. It is not clear how to introduce single-molecule electrical transistors biologically/aqueously or how to probe them optically, since the applied voltage is introduced via solid-state electrodes. Other intriguing related concepts worth mentioning, though not used for single exciton processing, include electrothermal transistors 28 and optically regulated bulk excitons. 29,30 Together, the binary logic and gain functionalities of the S2 exciton gate provide for an exciton processing platform with several potential applications. We anticipate that the S2 exciton gate may (a) improve on previously proposed single exciton processing schemes and (b) lead to the further development of spectroscopy tools. In improving on previous proposals, one advantage of our design is that it may produce faster switching times than some current nanosocopic logic methods, for example in logicbased biomarkers. Schemes based on binary logic have been used in the past to optically 4


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Figure 1: Schematic of the S2 exciton gate, a transistor-like excitonic device. The vertical direction denotes energy. C, G, and E denote organic molecules. States g and e correspond to electronic singlet ground and first excited states, respectively. States 0, 1, and 2 are respectively the ground, first excited singlet, and second excited singlet states of molecule G. The exciton gate is OFF (top) when G is in its ground state G0. In this case, the energy levels forbid an exciton on C from migrating through G. Conversely, the exciton gate is ON (bottom) when G is in state G1. Because the energy gap [G2] − [G1] is less than or on resonance with [Ce] − [Cg], energy can now flow through G via the G2 state. probe the presence of multiple species or processes in solution. 4,15 In addition to usage in biocompatible environments, the S2 exciton gate could provide an alternate route for integrating single exciton logic into a low-energy high-gate-density solid platform, as has been proposed for related excitonic devices. 9–11 Regarding advanced tools for spectroscopy, we anticipate that tunable gain behavior may be useful for delivering arbitrary rates of excitonic flux to specific chromophores. For example, to aid in the spectroscopic study of photosynthetic complexes, 31,32 one could use S2 exciton gates to direct different amounts of flux to different chromophores. One might be able to probe specific energy transfer pathways through a photosynthetic complex, using multiple S2 exciton gates to simultaneously vary the relative quantity of excitonic flux flowing to different molecules. This would be especially useful if two chromophores are energetically similar, since simply varying light intensity would not allow you to tune relative energy flux rates to the two chromophores. A similar strategy might eventually be useful in photocatalysis, if different photoactivated species require different amounts of energy flux. Additionally, tunable rates of local heating could be achievable, which may be useful in

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probing some temperature-sensitive processes. In this paper, we begin by describing the S2 exciton gate’s functionality and its design constraints, before summarizing preliminary virtual screening results indicating that it straight-forward to meet these constraints. We then discuss details regarding gate actuation and gain/amplification behavior, and show how all binary operations can be performed. Next, we discuss gate fidelity and error rates with the aid of simulation. Finally, we outline fabrication strategies and discuss experimental viability.

Results/Discussion S2 exciton gate The minimal S2 exciton gate consists of three molecules, denoted C, G, and E. G will be called the gate molecule, while C and E are the input and output molecules, respectively. Below, g or 0 denotes the ground state, e or 1 denotes the first excited singlet state, and 2 denotes the second excited singlet state. denotes the energy of a state, and µ[Aia ] (with norm µ ˆ[Aia ]) denotes the transition dipole between molecule A’s electronic states a and i. V [Aia , Bbj ] denotes the excitonic dipole–dipole interaction between molecule A’s a → i transition and molecule B’s b → j transition. The exciton transfer is driven by a down-hill energy cascade. The upper panel of Figure 1 shows that the device is OFF when G is in its ground electronic state G0. Because the excitation on C (in state Ce) does not have sufficient energy to populate and migrate through the G1 state, no appreciable transfer to the output molecule E occurs. Conversely, the device is in its ON state (lower panel), when G is in its first excited singlet state G1. In this case, an excitation in state Ee has enough energy to populate the G2 state, after which the exciton migrates further to the Ee state. In summary, when the gate is turned on, the initial state is (Ce, G1, Eg), the intermediate state is (Cg, G2, Eg), and the final state is (Cg, G1, Ee). Because the migration occurs only through the S2 state of G, we call the 6


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device the S2 exciton gate. We simplify the design’s constraints by requiring that it operates in the incoherent Förster resonance energy transfer (FRET) regime, an approximation that is further discussed below. Certain design advantages are immediately apparent. The first is that, for most molecules, the S1 – S2 energy gap tends to be smaller than the S0 – S1 energy gap. The second advantage is the enormous molecular space 33 from which one can choose dyes with appropriate properties, including energy gaps, transition dipole strengths, and exciton decay times. Though our proposed gating concept is conceptually very simple, we think that the reason it has not been previously proposed is that, at first glance, two roadblocks to a viable design appear to exist. The first apparent roadblock is that the vast majority of organic molecules have shortlived S2 states (0.1 to 10 ps 34,35 ), as internal conversion normally causes fast decay to the first excited (S1 ) state. This behavior is related to Kasha’s rule, 36 which states that fluorescence tends to be observed only from the S1 state. In demonstrating that non-ideal behavior related to short-lived states can be overcome, we show below that (a) for small excitonic circuits, relatively unremarkable excited-state lifetimes are sufficient for the design to operate as intended, and (b) there are many known molecules with very long-lived S2 states. These two points are discussed at length in this paper. The second apparent roadblock is the problem of suppressing undesired exciton transfer pathways. As one example, consider an exciton migrating directly from the gate molecule’s G1 state to the output Ee state (Figure 2), even though the input state Ce was not populated. This “leakage” can be thought of as a false positive, since an ideal device would not have resulted in an output exciton Ee unless there had been an input exciton on Ce. Hence, among other processes, the transfer from G1 to Ce must be suppressed. In the FRET regime, there are two chief parameters (other than distance) that determine transfer rates: the overlap integral between the donor emission and the acceptor absorption

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spectra, and the relative orientation of the two transition dipoles. 1 Crucially, within the dipole-dipole approximation, exciton transfer will be completely suppressed when the socalled orientation factor κ is equal to zero (see SI Section S2). κ is defined as

κ=µ Â · µ ˆB − 3(ˆ µA · rˆ)(ˆ µB · rˆ)

(1)

where µ Â and µ ˆB are unit vectors of two molecules’ transition dipole moments for specific transitions (e.g. S0 → S1 or S1 → S2 ), and rˆ is the unit displacement vector between them. As the formula includes only unit vectors, it is clear that κ is purely dependent on orientations. κ varies from –2 to +1, and there is a manifold of orientations that yield κ=0, effectively suppressing transfer between two transitions of interest. Hence to solve the second roadblock, one needs to choose two dyes with negligible absorption-emission overlap or to orient two dyes such that κ = 0. In this work, we demonstrate that (a) it is often straightforward to choose sets of dyes such that relevant spectral overlaps are zero, and (b) when nonzero spectral overlap necessitates the use of orientational constraints, the necessary precision in molecular placement is likely within reach of modern nanofabrication methods.

Figure 2: Schematic of some competing rates in the S2 exciton gate, where k’s denote transfer rate and γ’s denote decay rates. An optimal gate design minimizes k1E and all γ.

In order to synthesize an ideal exciton gate, several energy gaps must be aligned and multiple transfer processes must be suppressed. In summary, these are the design criteria: 1. [Ce] − [Cg] < [G1] − [G0]. C’s S0 – S1 energy gap must be smaller than G’s S0 – S1 gap. 8


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2. [Ce] − [Cg] ≥ [G2] − [G1]. C’s S0 –S1 energy gap must be greater than or on resonance with G’s S1 – S2 gap. 3. [G2] − [G1] ≥ [Ee] − [Eg]. G’s S1 –S2 energy gap must be greater than or on resonance with E’s S0 – S1 gap. 4. V [Ceg , G10 ] = 0. Suppress transfer from G1 to Ce. 5. V [G10 , Ege ] = 0. Suppress transfer from G1 to Ee. 6. V [Cge , G21 ] 6= 0. Appreciable transfer between state (Ce, G1) and state (Cg, G2). 7. V [G21 , Ege ] 6= 0. Appreciable transfer between state (G2, E0) and state (G1, Ee). Note that for choice of gate sets where appropriate absorption/emission overlaps involving ˆ[G21 ]. G are not negligible, an implicit condition is that µ ˆ[G10 ] 6= µ Below we discuss the likelihood that all these constraints can be met in a laboratory design, after further describing the basic functionality of the S2 exciton gate.

Gate actuation

Figure 3: Bottom Left: Symbolic representation of a bipolar junction transistor, with emitter (E), base (B), and collector (C) electrodes. Top left: Symbolic representation of an S2 exciton gate. Center: Three-molecule construction, with incident light (denoted by a blue line and asterisk) controlling the gate molecule G. Right: Four-molecule construction, with exciton flow from molecule B providing control of the gate. B may be pumped by a light source, analogously to the pumping that is performed in a three-level laser to achieve population inversion in the first excited state. 9


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Figure 4: Excitonic current response curves resulting from five sets of design parameters. The horizontal axis represents the rate at which B is excited, by pumping with external light or by introducing a flow of excitons. The vertical axis is the normalized excitonic current response flowing from C to E. The ratio k/γ=5 is kept constant for each curve. The darkest curve behaves much like a binary on/off switch, while the shallower curves may be used for gain or amplification, in which the excitonic current can be tuned by modulating the external input. Here, we draw an analogy between the S2 exciton gate and the NPN bipolar junction transistor (BJT), where the transistor itself is like molecule G, the collector terminal is like C, and the emitter terminal is like E (Figure 3). In the BJT, the base (B) terminal is the input that regulates current flow from the collector to the emitter. One may activate the gate molecule by introducing light with frequency on resonance with G’s S0 – S1 gap. In this case, the light source behaves as the base (B) terminal. One drawback of using light to directly activate the gate is that it is not possible to achieve population inversion (a steady-state population greater than 50%) in the excited state G1 by directly pumping the state with light. 37 To overcome this low-population limitation on populations of ON states (i.e., to achieve population inversion), the base terminal may instead be mimicked by a fourth molecule, molecule B, that feeds excitons into G. The exciton migrates from the Be state (B’s S1 state) to the G1 state. B may itself be activated by an external light source, or by an exciton flowing from yet another set of chromophores. In other words, in the four-molecule construction the B molecule would be continually excited, in the same way that a three-level laser is “pumped” to provide population inversion.

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Figure 5: Schematics of the exciton binary logic gates AND, OR, and NOT, and an energy up-converter. The blue line and asterisk denote input of light. The OUT terminal may connect to a fluorescing molecule, or to an additional binary operation as part of a larger circuit. Universal computation is achievable with AND and NOT gates alone, or with OR and NOT gates alone. For each binary operation, Type A mimics one particular construction (of many) of a traditional electronic transistor-based operation, while Type B is a schematic that is unique to excitonic computation and requires fewer resources. The up-converter, in which E is populated from the G20 transition instead of the G21 , increases an excitonic bit’s energy, allowing energy funneling to continue in deeper circuits.

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This fourth molecule necessitates additional constraints. In addition to those outlined above, these criteria are necessary: 8. [Be] − [Bg] ≥ [G1] − [G0]. B’s S0 – S1 energy gap must be greater than or on resonance with G’s S0 – S1 gap. 9. V [Bge , G21 ] = 0. Suppress transfer from Be to G2. 10. V [Bge , Cge ] = 0. Suppress transfer from Be to Ce. 11. V [Bge , Ege ] = 0. Suppress transfer from Be to Ee. 12. V [Bge , G10 ] 6= 0. Appreciable transfer needed between states Be and G1. We note that, though these added constraints may appear daunting, we performed a virtual screening procedure to show that there are several choices of four molecules (B, C, E, G) that allow the 12 constraints to be met. We used both experimental spectral data and timedependent density functional theory (TD-DFT) results in our screening procedure, summarized in SI Sections S8 & S9. Approximate calculations suggest that undesired transfer processes can be suppressed almost entirely simply by choosing dyes with near-zero spectral overlaps for appropriate transitions. Additional details on suppressing undesired transfer are discussed below. Notably, functionality analogous to a traditional electronic transistor’s current gain is possible. Instead of the binary ON/OFF behavior to which previous gating schemes are restricted (see Introduction and SI Section S1), one can regulate the quantity of exciton flux by modifying the intensity of light incident on the B molecule. The gain mechanism functions because the decay rate of state G1 competes with the energy transfer from Be. As the population on Be is fed faster, B is able to replenish the G1 state more quickly. Using results from rate equations that describe this process, Figure 4 shows normalized “response curves,” i.e. the exciton transfer rate from C to E as a function of pumping rate on B. The ratio k/γ is arbitrarily set to 5 and is kept constant for each 12


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simulation, and we note that the unnormalized maximum currents for all simulations are within a factor of five of each other. A smaller γ yields a response curve approximating a step function, ideal for binary operations. On the other hand, a larger γ yields a shallower response curve, allowing a user to regulate flow by modifying the input rate. Finally, because each component is driven by an energetically down-hill process, it is useful to have an up-converter that increases the energy of an excitonic bit (Figure 5); otherwise the energy would soon become so low as to prohibit exciton processing. We note that the up-converter’s energy level alignment is similar to the AND gate of the previously proposed quantum dot-based quantum platform. 11 The up-converter’s constraints are similar to those of the standard S2 exciton gate, except that the middle (G) molecule is continually pumped to its S1 state to prepare for an incoming bit, and the output (E) molecule is higher energy and is populated from the S2 →S0 transition of the middle dye. With this description of multiple (light and exciton) actuation methods and basic gain functionality, we now turn to higher-level constructs.

Universal logic gate sets The implementation of excitonic logic gates allows for binary logic and what one might call “excitonic signal processing,” potentially leading to the further development of spectroscopic tools. Many of the previously designed exciton gate platforms mentioned above can be used for binary logic; some of the gates proposed here are conceptually similar to some proposed in the dark state pre-charge circuits. 10 As noted previously, the advantages of our design include faster gate times, no dependence on molecular rearrangement, and no need for clocking. Figure 5 shows how one would create the AND, OR, and NOT gates, from which all other binary operators can be made. To connect the gates, the output exciton from one S2 exciton gate may be used as input to the next exciton gate. In other words, one gate molecule can serve as either the C or B molecule of the next gate. With two of these three operations one can implement universal binary logic, which can 13


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in turn be used for computations ranging from solving Boolean satisfiability problems to performing arithmetic. We first designed binary operations that are the direct analogues of standard transistorbased implementations of AND, OR, and NOT operations. We call these Type A constructions. Next we show alternative constructions, which are tailored to the S2 exciton gate, called Type B constructions. The Type B constructions use fewer molecules, and hence require fewer geometric constraints. While the Type A gates treat the 3-molecule construct as a unit, the Type B gates take advantage of the flexibility of excitonic interactions. Unlike the other five constructs, the Type B NOT operator is itself a redesigned threemolecule exciton gate, with an entirely different set of seven design requirements for relative energy levels and couplings (see SI Section S4). We envision that these digital constructs will be useful in microscopy applications, similarly to how previous FRET-based logic has been used in experiments. 15

Errors and orientational constraints As mentioned previously, we used a virtual screening procedure to produce several sets of molecules that meet the design constraints (SI Sections S8 & S9). Once this set of molecules has been chosen, many factors determine the difficulty with which the final S2 exciton gate can be fabricated, one of which is the number of available geometric degrees of freedom. A real implementation is easier to achieve with more available degrees of freedom. Consider constraints 4 and 5 (V [Ceg , G10 ]=0 and V [G10 , Ege ]=0). In the unlikely worst case that neither of these two dipole pairs is detuned (zero overlap), then κ must be near zero for both pairs. This is achievable even in a planar configuration. Because tuning κ for various pairs depends on orientations but not distances, the number of degrees of freedom (DOFs) available for placing the molecules is NDOF = 4m−5−NC (see SI Section S3), where m > 1 is the number of molecules and NC is the number of constraints (i.e. the number of pairs for which we must set κ = 0). 14


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The three-molecule construction (of C, G, and E) has a worst case of NDOF ≥ 5 (after applying NC ≤ 2). Adding molecule B for the four-molecule construction yields a worst-case NDOF ≥ 6 (with NC ≤ 5). Clearly even in the worst case of there existing spectral overlap in all undesired transfer processes, there are many available degrees of freedom; there is a very large “space” of viable geometric configurations. Somewhat surprisingly, the results of our virtual screening procedure (see SI Section S9) demonstrate the ease with which one can find four-molecule constructs for which many undesired transfer processes are suppressed solely as a result of zero spectral overlap, and for which the remaining approximate undesired overlaps are a small fraction of the approximate desired overlaps. But when considering more complex excitonic circuits, or when other design considerations limit the dye choice, the undesired transfer rate ku might demand orientation constraints. Under the pessimistic assumption that the least optimal orientation yields a ku that is one-tenth of the desired transfer rate k, and that the molecular pair is restricted to lie on a plane, we find that the ranges of angles that produce a negligible undesired transfer rate (ku ≤ 1%) add up to ∼ 50 degrees of the unit circle, a permissive window (see SI Section S2). This provides confidence that adequate orientations are achievable in a laboratory, as further discussed below). We note that classical error correction, 38 an indispensable subfield of computer science, may be useful in mitigating imperfections in excitonic logic, especially for complex circuits. To help determine how robust an excitonic circuit would be to errors, we performed numerical simulations of rate equations describing error-prone S2 exciton gates. We consider (a) the transient case, in which the system is given initial excitations and propagated, and (b) the steady-state case, in which there is a continuous input of light. Simulation details are given in Methods. For the transient case, we used two initial conditions: (Cg, G1) and (Ce, G1). In an ideal S2 exciton gate, the former yields an output signal of OFF (False) while the latter yields

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Figure 6: Results for a transient numerical rate model of the three-molecule construction of an S2 exciton gate. ku is the rate of undesired transfer processes, γ is the decay rate of the excitations on the gate molecule, and k is the rate of desired transfers. Two initial conditions were simulated: (Cg, G1), which for an ideal gate would produce a negative (zero output) signal, and (Ce, G1), which would ideally produce a positive signal. This simulation is of a transient process in the sense that there is no input of light after the initial condition is set. Note that greater ku increases the false positive rate, and greater γ increases the false negative rate. ON (True). Results for the transient case are shown in Figure 6, where the heat map plots the rates of the single-exciton output yielding true positives, true negatives, and so on. A higher decay rate γ naturally leads to fewer positive signals, which adversely affects the true negative rate. Conversely, the probability of a true positive is relatively robust to decay rate, at the expense of an increased number of false negatives. The undesired transfer rate ku has the opposite effect, as true negatives are quite robust to ku while true positives are not. Intuitively, this is because the undesired transfer increases the population on E. In the steady-state analysis (Figure 7), we simulated a Type A AND operation, with steady light input or excitonic input provided for the two logical inputs x and y. We chose this operation as a worst-case study, because it is the most complex of the six binary operations. The purpose of this steady-state analysis is to determine whether an experiment would be able to differentiate between an OFF result and an ON result, where x and y are inputs to the

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Figure 7: Even highly imperfect S2 exciton gates can be useful, as long as the difference in intensity between OFF and ON outputs can be easily distinguished. Here we perform a steady state simulation of an error-prone AND gate (Type A). Binary variables x and y represent steady exciton input or light input. γ is the decay rate on the gate molecules, ku is the rate of undesired transfer, and k is the rate of the desired transfer. Top: The steady-state population on the output chromophore is plotted. This can be interpreted as the fluorescence intensity from an output molecule, I(x,y) . Bottom: The quotient of transferred population, between (x, y) inputs (0,1) and (1,1), is plotted. For most values shown, the difference in intensity would make it easy to distinguish between true positives and false positives.

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AND operation. Effectively, this is related to a signal-to-noise ratio, with a larger difference between the two output implying that it is easier to differentiate between an intended OFF and ON output. Because the largest errors occur in the case of (x, y)=(0,1), for a worstcase comparison we plot the quotient of results from (x, y)=(0,1) and (1,1). I(x,y) can be interpreted as the intensity of fluorescence from the output chromophore or simply the rate of exciton flow from the output chromophore, and the smaller the ratio of I(0,1) /I(1,1) , the easier it is to distinguish the signals. For most of the range of parameters shown, the two signals would be distinguishable because they would emit substantially different intensities. As a foray into analyzing more complex circuits, SI Section S5 includes a steady-state analysis of a universal NAND gate with discussion of circuit depth. Our numerical results show that an S2 exciton gate can function even when γ, ku , and (desired transfer rate) k are of similar orders of magnitude. Now it is necessary to consider these simulation results in the context of fabricating S2 exciton gates.

Fabrication and experimental viability Before fabricating a viable excitonic device, many experimental details must be considered. These include supramolecular design methods, expected desired/undesired transfer rates, S2 decay times, and dye pair compatibility. These factors and their interplay are discussed below. To show that the design constraints (regarding energy level alignment, desired transfer, and undesired transfer) can be met, we performed an approximate virtual screening procedure (SI Section S9). Taking experimental data for candidates of B, C, and E, and using computational results for a set of candidates for molecule G, we found several four-molecule sets to be good candidates for a viable S2 exciton gate, based on energy alignment and approximate spectral overlaps. In an initial experimental realization, such a set of dyes will need to be placed in proximity to each other on the nanoscale. Until relatively recently, it would have been virtually impossible to place molecules in 18


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arbitrary positions and orientations. But modern advancements in synthetic methods have made possible relatively precise nanoscopic control. DNA nanotechnology, 8 also called DNA origami, has been used in previous versions of excitonic gate designs 9,10,16,17,20 and for funneling excitonic energy via FRET, 23,39–44 though there are other promising methods for spatially organizing chromophores, 45 including metal-organic frameworks, 46,47 covalent-organic frameworks, 48,49 and configurations similar to donor-bridge-acceptor devices. 50 We focus on DNA nanotechnology for the remainder of this paper, because of its demonstrated versatility and success in nanoscopic precision placement. Not only has DNA been used to produce a variety of arbitrary shapes and patterns, 51,52 there are also many attachment sites available. 45,53 This is a crucial point. One can implement “internal stacking” by replacing a sugar unit, replacing a base, or inserting a dye into the polymer backbone. Alternatively, one may append molecules to the outside of the DNA structure (“outside stacking”), by connecting to one of the various atoms of a nucleobase or to an atom on the sugar unit. Of course, multiple atoms on the dye itself may be used as attachment points. Finally, DNA origami methods on their own can tune distances with Bohr radius precision 54 and high angular precision. 55,56 Each of these many choices enforces a different orientation on the molecule, and the sheer number of options suggests it will be possible to produce orientations that produce a sufficiently low κ2 , especially considering the relatively permissible theoretical window of allowed orientations mentioned previously (SI Section S2). It is useful to consider prototypical examples of dye-functionalized DNA origami, to garner an understanding of achievable experimental parameters, which can then be considered in relation to our simulations. One early work 40 synthesized several DNA origami constructs with covalently attached dyes, demonstrating down-hill energy funneling behavior. Pairs of dyes (pyrene and Cy3) were spaced as closely as 1.8 nm, with the fastest FRET transfer times experimentally determined to be ∼30 ps. A computational study analyzing a similar DNA origami construct calculated that a pyrene-Cy3 pair spaced at 1.7 nm yields a transfer time of ∼8 ps. 41 In another recent experimental work, 44 a Cy3-Cy3 pair spaced at 1.3 nm

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on a DNA nanostructure showed a FRET transfer time of ∼1 ps. These complexes were not necessarily designed for maximizing transfer rates or dye density, suggesting that these values do not represent bounds on DNA-based constructs. Arguably, the limit of linear chromophore density is 0.32 nm, the spacing between two adjacent DNA base pairs, though with such a dense spacing it would probably be very difficult to limit quantum effects and undesired transfer rates. It is interesting to note that a conservative molecular spacing of 2 nm, assuming 3 molecules per excitonic gate, would represent a planar density of 0.083 gates nm−2 , or 8.3×1010 gates mm−2 . Modern silicon processors have a transistor density of 107 to 108 mm−2 . 57 Another key parameter is the excited-state decay time γ. Our numerical simulations (Figures 6 and 7) suggest that a viable S2 exciton gate is possible even when the decay rate is of a similar order of magnitude as the desired transfer rate. Since S2 → S1 decay rates are usually (though not always) reported to range from 0.1 to 10 ps, 34,35 dyes might be chosen whose decay times are comparable to the picosecond-scale FRET time scales that have been demonstrated on DNA origami in the past. In other words, because our simulations imply that gate behavior is acceptable when γ is similar in magnitude to k (Figures 6 & 7), it is likely that even non-exotic dyes can find viable use as G molecules. If a designed device is meant to either (a) show gain-type behavior for regulating excitonic flow to a specific area or (b) implement a short logical circuit, then these unremarkable values would likely suffice. However, γ need not be limited to a few picoseconds, since some dyes are known to have much longer S2 lifetimes. Especially notable are azulene and its derivatives, which can have lifetimes greater than 1 ns, at the expense of a shorter-lived S1 state. 58–62 With longer-lived excited states, higher circuit depths would be possible. The SI Sections S6 & S7 discuss viable classes of molecules with long higher excited state lifetimes, that could be used as G molecules with longer excited-state lifetimes. This includes the possible use of a rationally designed heterodimer. 63 We note that using the FRET approximation is justified only when vibrational relaxation

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occurs on a timescale shorter than the energy transfer. 1,37 Typical vibrational relaxation rates are reported to be about 0.1 ps, 34,35 shorter than the transfer rates considered above. Though FRET transfer times can easily be made to lie within 1 to 10 ps, 34 even faster energy transfer rates would probably arise from the quantum coherent regime, under which the energy levels may change. It might be possible to extend our model to this quantum regime, where complex quantum effects and delocalization are relevant, but the resulting design constraints may become more complex. This summarized experimental evidence strongly supports the notion that fabricating an S2 exciton gate is possible using current DNA-based experimental methods, and our virtual screening results (SI Sections S8 & S9) demonstrate the ease with which one can find viable G candidates and choose four-molecule constructs with favorable approximate spectral overlaps.

Conclusions We have designed a transistor-like molecular device capable of controlling exciton transfer through organic chromophoric systems, and shown how it may be used to perform binary logic operations and exhibit excitonic current gain. Features including picosecond-scale computation, lack of atomic rearrangement, biocompatibility, and current gain behavior are all inherent in our design, whereas only a subset of these functions are available in previously proposed exciton gating proposals. Using arguments based on theory and on published experimental data, we determined that the two apparent implementation obstacles—shortlived excited state lifetimes and enforcement of particular molecular orientations—can be overcome. With careful choice of gate molecules as well as placement of the relative orientations of the species, an experimental demonstration is possible, likely based on DNA nanotechnology. This design could be used as a building block for applications in microscopy, photocatalyis, and chemical sensing, improving on the progress in these areas made from

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previous exciton-based platforms. For instance, binary logic makes it possible to probe multiple interacting species or processes in a compact way. 4 Advancing tools for multiplexing 6,7 may also be possible, since N parallel gates lead to 2N unique outputs. Finally, directed energy flow, with arbitrary tunable flux introduced to specific regions of an experimental sample, may be used for probing the nanoscale with more precision than before.

Methods Numerical model We use a rate model to study the effect of decay and transfer rates. The model includes three processes: exciton decay, pumping to the excited state (via light or an influx of excitons), and exciton transfer. Transfer is assumed to take place via incoherent Förster hopping, though in the future it would be possible to extend the model to the quantum regime, using an open quantum systems formalism such as the Lindblad equation. 64 Our state space must allow for conditional probabilities. For example, there need to be two separate matrix elements that distinguish between (Ce, G1) → (Cg, G2) and (Ce, G0) → (Cg, G2). Hence a Hilbert-like space is required in order to properly design operators. For the three-molecule construction, the state space would be defined as       0 Eg  Cg    ⊗ x= ⊗   1     Ce Ee 2

(2)

d x = Qx dt

(3)

Our rate equation is

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where Q is an operator that includes exciton transfer, pumping, and decay terms within this Hilbert-like space.

Transient state (true/false positives/negatives) In the transient simulations, Bayes’ theorem was used to determine the likelihood that a true positive, false positive, true negative, or false positive has occurred. Consider our threemolecule exciton gate system consisting of molecules C, G, and E. We consider the effects of G’s decay rate γ, set to the same value for G2 → G1 and G1 → G0, as well as ku , which is the rate of undesired transfer from G1 to Ee, arising due to imperfect geometric constraints. These are studied in terms of ratios relative to k, which is the rate of each of the desired energy transfer processes. In our simple gate example, a positive signal (“+”) denotes molecule E fluorescing (or transferring its exciton to the next molecule in a circuit), and a negative (“−”) denotes no signal from molecule E. Assuming an ideal gate, the “correct” result would either be ON (“1”) or OFF (“0”). We consider two initial states, both with E in the ground state. Initial state (Ce, G0) ought to produce a negative, or “0” result, while initial state (Ce, G1) ought to produce a positive, or “1.” P (1|+) denotes the probability of a true positive, i.e. the probability that E would have fluoresced in the ideal case, given that the observed signal was positive. For true positives, for instance, Bayes theorem yields:

P (1|+) =

P (+|1)P (1) . P (+|1)P (1) + P (+|0)P (0)

(4)

P (0|+), P (1|−), and P (0|−) denote probabilities for a false positive, false negative, and true negative, respectively. We assume a uniform prior such that P (0) = P (1) = 12 . P (+|0), P (+|1), P (−|0), and P (−|1) are determined from simulations. In order to monitor how much population has reached E, no decay term is present on Ee. After equilibrium is reached, population in state Ee is a positive (+) result, and population in state Eg is a negative (−)

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result.

Steady state In the steady-state analysis, we simulated a Type A AND operation, with steady light input provided for the two logical inputs, and undesired transfer included for each G molecule’s S1 state to its output molecule’s S1 state. This numerical experiment provides some insight into how robust a logical operation in a true excitonic circuit is, with respect to errors. Results are shown in Figure 7. Binary inputs x and y are set to “1” by pumping the appropriate molecule with light, while the molecule directly above the first gate molecule (AND Type A in Figure 5) is always pumped. To determine steady-state populations, normalization is imposed (population of all the states on a single molecule sum to unity), pumping (opposite of decay terms) terms are added to the equation for inputs of light, and Equation 3 is set to zero. In all the simulations we set kmigr = k, where kmigr is the rate at which the exciton migrates on to the next molecule in the exciton circuit, after the AND operation. For the molecules that are pumped, the excitation rate and decay rate are conservatively (but arbitrarily) set to 0.4 and 0.2, respectively.

Acknowledgements This work was funded by the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science and Office of Basic Energy Sciences, under Award Number DE-SC0001088. This work was also supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319. A. A.-G. is very thankful to the Canadian Institute for Advanced Research (CIFAR) for their generous support and collaborations. We are grateful for useful conversations with Thomas Markovich, Semion Saikin, Doran Bennett, Marc Baldo, and Roland Lindh.

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Supporting Information The supplementary document includes information and analysis related to the orientation factor and degrees of freedom. Results of simulating a universal NAND gate are given, with comments on their relation to circuit depth. The NOT Type B gate is elaborated on. Also included is a discussion of the classes of molecules that would be most promising as gate molecules, including the use of a designed heterodimer as a gate molecule. Finally, a virtual screening procedure for discovering gate molecules and for finding B-C-E-G sets is presented, and its implementation provides promising candidates for these sets of molecules.

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Excitonics: A Set of Gates for Molecular Exciton Processing and

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