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Quantum units from the topological engineering of molecular graphenoids

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Science  29 Nov 2019:
Vol. 366, Issue 6469, pp. 1107-1110
DOI: 10.1126/science.aay7203

Controlling quantum defects in graphene

The development of quantum technologies relies on the ability to fabricate and engineer materials with robust quantum properties. The controlled introduction of defects in semiconductors is one of the most promising platforms under development. With the capability to precisely position point defects (five-membered rings) in the graphene honeycomb lattice, Lombardi et al. explored recent theoretical work suggesting that such defects should display enhanced quantum properties (see the Perspective by von Kugelgen and Freedman). The spin-bearing properties of the defects and the engineered control of their interactions open up exciting possibilities for graphene-based spintronics and quantum electronics.

Science, this issue p. 1107; see also p. 1070

Abstract

Robustly coherent spin centers that can be integrated into devices are a key ingredient of quantum technologies. Vacancies in semiconductors are excellent candidates, and theory predicts that defects in conjugated carbon materials should also display long coherence times. However, the quantum performance of carbon nanostructures has remained stunted by an inability to alter the sp2-carbon lattice with atomic precision. Here, we demonstrate that topological tailoring leads to superior quantum performance in molecular graphene nanostructures. We unravel the decoherence mechanisms, quantify nuclear and environmental effects, and observe spin-coherence times that outclass most nanomaterials. These results validate long-standing assumptions on the coherent behavior of topological defects in graphene and open up the possibility of introducing controlled quantum-coherent centers in the upcoming generation of carbon-based optoelectronic, electronic, and bioactive systems.

Current hopes of developing radically new technologies (1, 2) in computation, communications, security, and sensing rely on the quantum manipulation of charges (3), spins (4), or photons (5). One of the main approaches is defect engineering (1, 2), which has produced robust quantum systems in diamond and silicon carbide. Conjugated sp2-carbon nanomaterials would, in principle, be extremely appealing for quantum applications because they can be integrated into engineered devices (6) and possess intriguing mechanical (7) and transport properties (8). Methods to add spins to carbon nanomaterials include encaging heteroatoms inside fullerenes (9), confining electrons into carbon nanotubes (10), and functionalizing graphene nanoribbons (Fig. 1) (11). The manipulation of the honeycomb lattice (12, 13) using topological defects (14) is the approach that has seen most theoretical attention and could provide robustness against decoherence (15), single-photon optical control (16), and spintronic manipulation (17). On the other hand, the difficulty of reliably engineering point defects leaves it largely unexplored.

Fig. 1 Strategies toward obtaining aromatic quantum units.

Quantum spin properties are introduced by heteroatom inclusion for endohedral fullerenes (red), Coulomb charging for carbon-nanotubes (magenta), and side-functionalization in graphene nanoribbons (orange). Coherence times refer to room temperature, except for carbon nanotubes (mK). Topological stabilization of magnetic centers is obtained by rational synthetic tailoring of the lattice with atomic precision at preconceived sites (blue). The result is akin to a sequence (right) of introducing a Stone-Wales defect (purple and green), followed by propagation and trimming down.

The desired level of control of the graphene lattice has become possible only recently, with the synthesis of molecules containing many fused rings in an sp2-carbon framework (18), where pentagonal rings can be introduced reliably at precise positions (19). It is useful to relate these structures, obtained with bottom-up synthesis, to their equivalents on a graphene lattice (Fig. 1). After two (1,0) dislocations form (Stone-Wales defect), the heptagon-pentagon pairs can migrate. The lattice can then be trimmed down along the graphene stripe containing the dislocations to the desired geometry. Chemical stabilization by resonance will still occur, but the resonance structure with the most disjoint benzene-like moieties is the most relevant (Clar’s π-sextet rule) (16). The resulting extended open-shell molecule, with singlet and triplet states separated (20) by an energy gap 2J, is, in essence, one small graphene quantum dot with topology and defect positions shaped with atomic precision. Although the associated chemistry is seeing a veritable explosion for optoelectronic, biological, and energy applications (21), the potential of these molecules for quantum devices remains unexplored and untapped.

Our molecule of choice is a saddle-shaped diindeno-fused bischrysene (1) with highly stable open-shell biradical feature (22) (Fig. 2A). It contains a conjugated aromatic backbone and two pentagonal rings; as compared with the perfect heptagon-pentagon pairs, the difference is an unformed bond in each heptagonal ring. It is synthesized from the 11,11′-dibromo-5,5′-bichrysene (S1) in five steps (Fig. 2A and supplementary materials). The synthesis of such radicaloids has one often-overlooked feature: Incomplete dehydrogenation in the final step can lead to open-shell monoradical species (1b) at impurity concentrations, which are hard to identify by structural characterization methods. For instance, because 1b differs by one single hydrogen, it cannot be completely removed from 1 and is undetectable by mass spectrometry. For our purposes, 1b is useful because it allows for determining the behavior of single pentagonal elements in the honeycomb lattice—that is, a positive disclination—which is particularly relevant at graphene edges (23). Although the signal from 1b is overshadowed by 1 at room temperature, it is selectively addressable at low temperature, T, where 1 is completely in the singlet state.

Fig. 2 Polycyclic aromatic radicaloids.

(A) Synthesis of 1, including the monoradical 1b, produced by incomplete dehydrogenation in the final step. (B) EPR spectrum of 1 (blue) and simulation (black). B, static magnetic field. (C) Temperature-dependence of the integrated EPR intensity (circles), fitted to a Bleaney-Bowers equation (black line). The bottom panel displays the signal fraction ξ produced, at every temperature, by 1 (blue) and by 1b (green). The error bars arise from uncertainties in the quality factor of the resonator.

The room-temperature electron paramagnetic resonance (EPR) spectrum shows a single peak of width 0.8 mT in a magnetic field and electron Landé factor g = 2.0027 ± 0.0002 (Fig. 2B). This matches the expected signal for completely delocalized unpaired electrons in graphene, where dipolar and hyperfine couplings are weak (24). The linewidth (<1 mT) is incompatible with metal ions and analogous to the signals reported for radicaloids (2022). Simulation with a spin S = 1 in the high-exchange limit provides excellent agreement. Interestingly, no half-field signal is observed, nor any fine structure, possibly indicating a curvature-induced spin-orbit coupling higher than our accessible energy scale (25), as is the case for the curvature displayed by 1. The integrated EPR signal decreases rapidly on lowering T and levels off at ~90 K, below which temperature 1b is selectively addressable (Fig. 2C). Fitting with the Bleaney-Bowers equation (11) plus a paramagnetic species indicates that ~2% of the molecules are 1b and that 1 has an antiferromagnetic 2J = 50 ± 2meV.

The quantum evolution of a spin is often visualized as a movement over the Bloch sphere: Zenith positions indicate pure |1/2 and |1/2 states, and any possible quantum state |σ=cos(ϑ2)|12+eiϕsin(ϑ2)|12 is represented by a point on the spherical surface. The spin-flip time, T1, represents vertical displacement (variations of ϑ), whereas the evolution of the quantum phase ϕ is described by the azimuthal movement and the associated time T2. We measure T1 with inversion recovery (26) and a lower bound of T2 that also contains spin- and spectral-diffusion effects that are absent in single-molecule measurements, called Tm, by the Hahn-echo sequence (Fig. 3A). The coherence times of the two species, when discernible, are hereby labeled Tm1 and Tm1b. We fit the spin recovery via a biexponential function (Fig. 3B and supplementary materials) and the Hahn-echo decay with the function Y(τ)=Y0[ξ1e(2τTm1)x+ξ1be(2τTm1b)x]Ξ, where Y(τ) is the echo signal, Y0 = Y(τ = 0), and ξ1 and ξ1b = 1 − ξ1 are the weights of 1 and 1b, from Fig. 2C. Ξ=[1+k1sin(2ωτ+ϕ1)+k2sin(4ωτ+ϕ2)] does not affect the extracted decoherence, describing the modulation at a nucleus-specific frequency ω/2π = 14.8 MHz for 1H and 2.4 MHz for 2H, with amplitudes k1 and k2 and phases ϕ1 and ϕ2 for first- and second-order effects (Fig. 3C). Good agreement is always found with stretching parameter 0.9 < x < 1 (27).

Fig. 3 Spin-lattice and coherence times.

(A) Pulse sequence used for the detection of the spin-lattice (azure) and coherence (green) times, together with a Bloch sphere representation. tinv, recovery time. (B) Example of signal recovery, from which T1 is extracted (T = 100 K; line is fit to the data, see text). (C) Hahn-echo intensity versus delay time, from which Tm is extracted (T = 100 K; line is fit to the data, see text). (D) Temperature dependence of the inverse spin-lattice relaxation time (top) and of the spin coherence time (bottom) in powders (blue full circles), toluene (green half-filled circles), d-toluene (blue open symbols), and CS2 (green open symbols). Circles represent 1, and pentagons represent 1b. Arrows and full hexagons represent values with nuclear decoupling (Fig. 4). Errors are smaller than the symbols. Lines for T1 are fits to the data (see text), with the different dynamic regimes shaded. Lines for Tm are guides to the eye. Vertical dashed lines indicate the freezing temperatures of toluene and CS2.

The spin environment strongly affects both T1 and T2: For example, the coherence time of anionic nitrogen-vacancy-pair defects is severely suppressed when in close proximity to the diamond surface (28). Hereafter, we thus assess the behavior of 1 and 1b in crystalline powders and toluene, deuterated-toluene, and carbon disulfide (CS2) solutions (Fig. 3D).

For powders, T1 increases from 1 μs at room temperature to 100 μs at 5 K, in overall agreement with semiconductors (29, 30). Both spectral diffusion and intermolecular electronic effects, such as π-stacking interactions, likely limit T1 in such a closely packed arrangement in the solid state, and dissolution into solvents produces a 1000-fold increase in T1, up to 1 s at 5 K. In solutions, T1 is limited by molecular tumbling and increases only slightly on lowering T. At lower temperatures (170 K for toluene and 160 K for CS2), the solvents turn into a glassy matrix and a two-phonon Raman process becomes dominant down to 15 K, below which direct processes dominate the spin-flip mechanism. Good agreement (Fig. 3D) is obtained with the expression T11=AdirT+ ARam(TϑD)90ϑDTx8ex/(ex1)2dx, where Ai are weights for the two processes and ϑD is the Debye temperature (11). In the solid state, Ξ=1 and Y(τ) is monoexponential, yielding Tm ≈ 300 ns in the whole T range. Dipolar and hyperfine interactions cannot be solely responsible for the decoherence mechanism; were this true, Tm would increase and approach the CS2 solution value below 80 K, where only 1b contributes, and a modulation of the echo decay would be observed (e.g., as in Fig. 3C). Decoherence is likely driven by electron-electron scattering along the π-stacks, which are broken up by solvation. In this sense, these molecular systems behave differently from very-large-bandgap semiconductors—for example, diamond—and rational chemical design eliminating the π-stacking interactions could improve the solid-state coherence.

The suppression of stacking by solvation increases Tm more than 30-fold, and reduction of the solvent nuclear bath by deuteration and by CS2 produces a further increase. Several Tm-limiting mechanisms can be identified. Above the solvent freezing point, Ξ=1 and Y(τ) is monoexponential, because molecular tumbling limits both Tm and T1. Upon freezing, Ξ reveals hyperfine modulation by 1H and 2H and no modulation for CS2, showing dominant solvent hyperfine coupling. In toluene, Y(τ) is always monoexponential, whereas in d-toluene and CS2, the suppression of solvent 1H-hyperfine interactions allows for resolving both Tm1 and Tm1b, and Y(τ) is biexponential wherever 1 and 1b coexist. Tm1b displays a maximum, with the low-T behavior dominated by decoherence via intramolecular hyperfine coupling and modulated by the progressive blocking of the methyl rotational motion (31). In d-toluene, Tm1 is found to rise steadily up to 28 μs at 80 K, and in CS2, the same trend is found but with much improved times, with Tm1 reaching 0.1 ms at T = 90 K.

To verify that the spins can be initialized into an arbitrary superposition of states, we performed nutation experiments (Fig. 4A), detecting Rabi oscillation decays (Fig. 4B) (32). Fourier analysis confirms the quantum behavior, with the Rabi frequency proportional to the square root of the applied power (Fig. 4C). Because 2J ~ 10 THz is much higher than the 10- to 100-MHz driving, which is in turn much higher than the axial spin anisotropy, no unusual evolution of the Rabi is expected (33), as is indeed observed. This analysis also indicates how to improve coherence: The power-independent peak at 14.8 MHz corresponds to the 1H Larmor frequency, meaning that microwave initialization pulses can also drive and decouple the nuclei—for example, by a train of π-pulses, with interpulse intervals τπ that are multiples of the inverse of the nuclear Larmor frequency, 1/νL (Fig. 4D) (34). In d-toluene, decoupling from the solvent nuclei yields a fivefold improvement, producing times comparable to those in CS2 (e.g., Tm1b = 38 μs and Tm1 = 260 μs at 80 K). The role of the intramolecular hyperfine interactions is revealed by decoupling from the molecular hydrogens in CS2: We observe a threefold increase of the coherence, up to Tm1 = 290 μs at 80 K. At room temperature, the decoupling allows for reaching Tm = 2 μs, close to the maximum attainable limit 2T1 = 4.5 μs.

Fig. 4 Rabi oscillations and nuclear demodulation.

(A) Pulse sequence for the measurement of Rabi oscillations. The nutation pulse length Tp is tuned so as to vary the azimuthal position on the Bloch sphere, followed by detection. (B) Echo intensity versus Tp, at different pulse powers (T = 80 K). Black lines are fits to the data. (C) Spectral composition of the time-domain data showing the quadratic dependence of the Rabi frequency on the microwave power (dashed) and the power-independent 1H frequency. (D) Pulse sequence used to progressively cancel the dephasing effect of all nuclei (orange) with Larmor precession time τπ, leading to multiple echoes. (E) Echo signal decays at T = 120 K without (brown represents d-toluene, and orange represents CS2) and with nuclear decoupling, with interpulse spacing τπ = 840 ns for d-toluene (blue) and τπ = 680 ns for CS2 (green). The corresponding coherence times are reported beside the curves.

These observations confirm experimentally the possibility of superior quantum performance in carbon-based nanostructures. The coherence times, although still below those of defects implanted deep into bulk semiconductors (1, 2, 35) and semiconducting quantum dots at millikelvin temperatures (36), outshine the latter at high temperatures and show overall agreement with predictions for graphene quantum dots with >10 nuclear spins (37). Although the quantum performance already beats the quantum behavior of diamonoids and shallowly implanted defects (25), there is ample room for optimization: The measured room-temperature values are limited by tumbling in the solvent, and very basic optimization—for example, by immobilization in an oriented diamagnetic matrix or on surfaces—is likely to produce large improvements. These results can now be used to reconsider the quantum magnetic states of graphene devices, where spins are introduced by similar defects (38). Chemical inclusion into conducting nanostructures, such as nanoribbons and graphene sheets, or fusing several molecules into double and multiple quantum dots, opens the path to using quantum effects in the next generation of optoelectronic, electric, and bioactive systems. When considering the body of work already dedicated to the integration of similar molecules into electronic devices and biologically relevant environments (18), these outcomes appear well within grasp. Optical detection and manipulation are particularly appealing—for example, via the observed single-photon emission (16)—and we may anticipate that these systems will evolve soon into synthetic analogs to optically active quantum centers in semiconductors. In this sense, it is crucial that these observations offer a rational synthetic pathway to add any desired functionality to a graphene quantum unit, opening up an unprecedented multitude of options for the optics and magnetism of quantum nanomaterials.

Supplementary Materials

science.sciencemag.org/content/366/6469/1107/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

References (4043)

References and Notes

Acknowledgments: We thank A. Ardavan for useful discussions. Funding: This work was funded by the European Union (ERC-StG-338258-OptoQMol, ERC-CoG-773048-MMGNRs, ERC-CoG-819698-T2DCP, Graphene Flagship-Core2-696656, and European Social Fund); the Royal Society (University Research Fellow and URF grant); UK-EPSRC EP/L011972/1; German DFG (Excellence Cluster CFAED and EnhanceNano-391979941); and the Max Planck Gesellschaft and Saxony ESF-Project-GRAPHD. Author contributions: F.L. and A.L. performed the EPR measurements, and M.S. and W.K.M. assisted them. J.M. synthesized the compounds and performed the chemical characterization, for which J.L. and X.F. provided supervision. F.L. and L.B. performed the data analysis. L.B. coordinated the experiments and wrote the paper. Competing interests: The authors declare no competing interests. Data and materials availability: All data are available in the main text or the supplementary materials. All raw data and scripts are stored in the computer center of the University of Oxford. All datasets are freely available at the Oxford University Research Archive at https://ora.ox.ac.uk/objects/uuid:99b46501-e3c5-4410-91ac-0e643939a4a6 (39).

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