Faraday Rotation in Graphene Quantum Dots - American Chemical

Article pubs.acs.org/JPCC

Faraday Rotation in Graphene Quantum Dots: Interplay of Size, Perimeter Type, and Functionalization Jarkko Vaḧ ak̈ angas,* Perttu Lantto,* and Juha Vaara* NMR Research Group, Department of Physics, University of Oulu, P.O. Box 3000, Oulu FIN-90014, Finland S Supporting Information *

ABSTRACT: Nanometer-sized graphene systems have optical properties that can be tuned in the visible range to enable new optoelectronic device applications. For such purposes it is of critical importance to fundamentally understand the behavior that is specific for the size, shape, and composition of the system. Recently, graphene has gained attention due to its capability to rotate the plane of polarization of linearly polarized light up to 6 degrees at 7 T magnetic field, which is a massive rotation for a single sheet of atoms. We present a computational study that contributes to understanding of this Faraday optical rotation (FOR) for graphene quantum dots (GQDs) of different size, perimeter structure, and composition. Based on first-principles calculations we predict FOR characterized by the Verdet constant, for a systematically growing series of hexagonal GQDs in the visible frequency range. We show evidence for the independence of FOR of the type of the perimeter, zigzag or armchair, in these hexagonal GQDs. In addition, we show how FOR is drastically changed at different levels of hydrogenation, leading to complete or partial sp3 hybridization of the GQD. While FOR is a global property for a particular molecular system, the recently proposed technique based on optical rotation by polarized nuclear spins (nuclear spin optical rotation, NSOR) characterizes the system with atomic resolution. Here we demonstrate the capability of NSOR to distinguish between GQDs of specific size and edge structure.

1. INTRODUCTION The Faraday effect is a phenomenon where the plane of polarization of a linearly polarized light (LPL) beam rotates when it travels through a material that is exposed to an externally applied magnetic field, directed along the direction of beam propagation.1 The Faraday optical rotation (FOR) has been applied in the optoelectronic technology for a rather long time. Lately it has been seen that the effect can also be used for molecular spectroscopy2−6 and, in materials science, for mobility and quantum Hall effect measurements.7−9 In the Faraday effect, the different indices of refraction for left- and right-polarized light result in an optical rotation (OR), which is dependent on the chemical composition of the molecules of the medium.1 From the spectroscopic point of view, a possible way to gain atomic resolution by the detection of OR is by the method dubbed nuclear spin optical rotation (NSOR).2−6,10 It is based on the same phenomenon as FOR, but the magnetic field causing the NSOR is due to spin-polarized nuclei. The external field-induced FOR is characterized by the Verdet constant,1 V, which parametrizes the rotation angle as ΦFOR = V(ω)Bl, where B is the field in the direction of the beam and l is the optical path length. In the language of quantum chemistry the OR associated with the macroscopic spin magnetization in the NSOR effect is enhanced by the hyperfine interaction between nuclear spins and virtual optical excitations of the electron cloud.2,4,5,11 In contrast, in the case of FOR, the rotation arises from the orbital Zeeman interaction.1,12,13 © 2014 American Chemical Society

Graphene with its massless charge carriers and a linear relationship between energy and momentum has inspired researchers to suggest a wide range of applications.14 Recently, graphene has gained attention due to its potential in paving the way to fast, tunable, ultrathin magneto-optic devices in the infrared (IR), owing to its unprecedented, massive FOR.9,15 The giant rotation by single-layer graphene was demonstrated to arise from inter-Landau-Level transitions at the proximity of a transition frequency in the IR region of the spectrum.9 Recently, collective electron oscillations, molecular plasmons, have been suggested to occur and be tunable by manipulating the charge state of nanometer-sized molecules consisting of finite arrangements of aromatic rings.16−18 Such molecules can be regarded as graphene quantum dots (GQDs) due to the observed quantum confinement effects.19−21 GQDs have many of the special physical properties of graphene due to their similar atomic structure. They feature, by means of electrical doping, longer-lifetime plasmons than conventional metalbased plasmonic materials. Due to the finite size of the GQDs, molecular plasmons occur in the visible rather than the IR part of the electromagnetic spectrum.16,22 Specific interactions of light with the quantum-confined GQDs enable the electrooptical tunability, and the optical application capabilities of Received: August 5, 2014 Revised: September 11, 2014 Published: September 19, 2014 23996

dx.doi.org/10.1021/jp507892j | J. Phys. Chem. C 2014, 118, 23996−24005

The Journal of Physical Chemistry C

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Figure 1. Atomic structures of the inspected graphene quantum dot (GQD) systems and their Verdet constants (V) as well as the anisotropies ΔV = V⊥ − V∥, where V⊥ and V∥ are the Verdet constants for the light beam propagating perpendicular to and parallel with, respectively, the plane of the GQD. At the top, atomic structures are illustrated for systems with zigzag (Gn, n = 1−5) and armchair (mcrenG, m = 2, 3) perimeter. In panel (a), the isotropic Verdet constants [in rad/(T m)] are plotted as functions of the light wavelength (nm) for the Gn and mcrenG systems of increasing size. In (b), the corresponding anisotropies are plotted. In the inset of panel (a), V divided by the number of carbon atoms in the corresponding systems is plotted so that the individual curves are shifted to align the wavelength corresponding to the first strong dipole transition in each system with that of the G1 molecule. DFT calculation at the B3LYP/completeness-optimized basis-set (co2-MOR) level.

Figure 2. As Figure 1 but for graphane quantum dots with zigzag (HGn, n = 1−4) and armchair (mcrenHG, m = 2−3) perimeter. In the inset, the curves are shifted to align the wavelength corresponding to the first strong dipole transition in each system with that of the HG1 molecule.

GQDs cover quantum plasmonics,17,18 as well as light harvesting21 and light manipulation.23 The importance of the size and perimeter geometry of the GQDs for absorption, fluorescence, and photoluminescence have been studied and are well-known.21,24 In contrast, their OR capability is still unknown. However, a giant FOR in the visible range has been observed in mesogenic organic molecules.25 In this context, the GQDs may also be used to create a corresponding FOR due to the fact that they have strong self-assembly tendency; they form disc-like supramolecules in solution and at liquid−solid interfaces.26 Nowadays defect-free and identical GQDs can be routinely chemically synthesized in the bottomup manner. They can furthermore also be imparted to dissolve into water by functionalizing the edges of the structures with carboxylic acid moieties.21,26 FOR and NSOR offer novel ways to characterize GQDs. The present study gives for the first time a survey of how FOR depends on the size and the perimeter structure of GQDs, as well as how NSOR can characterize different types of GQDs with atomic information.

In this article, we provide a computational demonstration, via first-principles nonlinear response theory calculations, of the Verdet constants of FOR and the nucleus-specific NSOR constants, VK, for GQDs of increasing size, consisting of concentric layers of benzene and cyclohexane rings, with both zigzag and armchair -type perimeter structure. All the studied systems are characterized by an equal number of atoms in each of the two interpenetrating, triangular carbon sublattices, A and B.27 Thus, the systems possess no zero-energy states, in contrast to case of GQDs where molecular plasmons occur due to the finite total spin S in the ground state, arising from an unequal number of atoms in the two sublattices.16,27 The armchair perimeter leads to a repeating pattern of conjugation causing a different type of bond length alternation as compared to the zigzag-edge GQDs.28,29 By inspecting GQDs of different size and edge structure we can demonstrate the sensitivity of FOR to these parameters, as well as the dependence of NSOR on the local chemical environment of the nuclei. It is a well-known fact that the 23997



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electronic structure of GQD systems depends crucially on the geometry of the edge region.16,28,30−32 In addition, we demonstrate how different amounts of hydrogen adatoms in the originally entirely sp2-hybridized GQD can be monitored with the aid of FOR and NSOR. Specifically, we present calculations of V and VK for structures containing up to 150 carbon atoms (see Figures 1 and 2) with the laser frequency ranging from the visible to the near-IR region of the spectrum. The molecules are treated as static and isolated. In the calculations we used third-order time-dependent perturbation theory at the density-functional theory (DFT) level, with a hybrid functional employed for the exchange and correlation energy, along with the so-called completeness-optimized (co) Gaussian basis sets33 (see Computational Details for a detailed description). A formal description of the NSOR effect has been presented in several studies4,11,34 and the first-principles method for its calculation was developed in ref 4, and further used in refs 5 and 35−37. Laser-induced nuclear magnetic resonance (NMR) splitting, an observable with its origin in the same molecular property as NSOR, was investigated for coronene and circumcoronene in an earlier study.38 In addition, the normal NMR parameters were predicted for the innermost carbons at the limit of large graphene nanoflakes and their hydrogenated and fluorinated counterparts in ref 39. The characteristic NMR spectral patterns of finite GQDs were predicted in ref 29.

αϵ′τ(,Bν0 / IK ) = −⟨⟨μϵ ; μτ , hνOZ/PSO⟩⟩ω ,0

where μϵ and μτ are components of the electric dipole moment. The third operator, h, is a static magnetic operator that characterizes the orbital Zeeman (OZ) interaction with the external field in FOR and the orbital hyperfine (paramagnetic nuclear-spin electron orbit, PSO) interaction in NSOR, respectively. The full perturbation operators in question are defined, respectively, as HOZ =

∑ αϵ′τ(,Bν )B0,ν + ∑ αϵ′τ(,Iν )IK ,ν + 6(B03 , IK3 ) ν

ν

hνOZ =

∑ hνPSOIK ,ν K ,ν

(4)

e 2me

hνPSO =

∑ SiO,ν i

SiK , ν eℏ μ0 γK ∑ 3 me 4π riK i (5)

Here, γK is the gyromagnetic ratio of nucleus K and S iO/K = −iℏ(ri − R0/K) × ∇i, the angular momentum of electron i with respect to either the gauge origin R0 or the location RK of the nucleus K. The magneto-optic rotation angle, Φ, per unit of optical path length l, can be written as12,42 Φ 1 ′ ⟩ = ω 5μ0 c Im⟨αXY l 2

(6)

where 5 is the number density of either the molecules (in the FOR case) or the nuclei K (NSOR), providing the link to bulk properties. μ0 and c are the vacuum permeability and speed of light, respectively. The FOR angle is parametrized conventionally as ΦFOR = VB0l, from which we obtain for the Verdet constant 1 e3 1 V = − ω 5μ0 c 2 2me 6

∑

Im⟨⟨rϵ ; rτ , lO , ν⟩⟩ω ,0

ϵτν

(7)

Denoting the degree of nuclear spin polarization along the beam as PK = ⟨IK,Z⟩/IK, where IK is the nuclear spin quantum number, the NSOR angle, Φ(K) NSOR per unit of optical path length, spin polarization, and molar concentration nK = 5/NA (NA is Avogadro’s constant) of the polarized nuclei K, becomes

(1)

where ω is the angular frequency of the light and ϵτν are the Cartesian xyz coordinates in the molecule-fixed frame. In this expression, the terms involving the α′(B0) and α′(IK) coefficients give rise to FOR and NSOR, respectively. Due to molecular tumbling in gaseous or liquid media, the expression of the isotropic rotational average of the antisymmetric polarizability in the laboratory frame becomes, in the case of FOR with the magnetic field in the Z direction of the light beam ′ ⟩= ⟨αXY

HPSO =

where

K

0

∑ hνOZB0, ν ν

2. METHODS 2.1. Theory. Both V and VK are calculated using third-order time-dependent perturbation theory. The central concept in both properties, in the case of an isotropic medium, is the rotationally averaged antisymmetric polarizability ⟨αXY ′ ⟩ (refs 4,11), where XYZ are Cartesian coordinates in the laboratory frame, with the light beam propagating in the Z direction. α′XY can be expanded as a power series in terms of the external magnetic field B0 and the nuclear spin IK, retaining the firstorder terms4,11 αϵ′τ(ω) =

(3)

VK =

K) Φ(NSOR lPK nK

2 1 e 3ℏ μ0 1 = − ωNAcIK γ 2 me 4π K 6

B 1 ′ (,Bz0) + αyz ′ (,Bx0) + αzx ′ (,By0)) B0 ∑ ϵϵτναϵ′τ(,Bν0) = 0 (αxy 6 ϵτν 3

Im

(2)

rϵ ; rτ ,

∑ ϵϵτν ϵτν

lK , ν rK3

ω ,0

(8)

that is reported in experimentally convenient units of 10−6 rad/ (M cm).2,4−6 The expressions 7 and 8 lend themselves to quantum-chemical calculations of the components of the polarizability derivates αϵτ,ν ′ in the molecule-fixed frame. 2.2. Number Densities. Due to the lack of a full series of tabulated liquid-state mass density values of the studied systems, necessary to obtain corresponding number densities (5 = ρNA /M ), we obtained approximate values by a scaling procedure on the basis of the 5 value of benzene. The sizespecific 5 is calculated in the following way:

Here ϵϵτν is the Levi-Civita symbol. The corresponding expression for NSOR can be obtained by replacing αϵτ,ν ′(B0) by (IK) α′ϵτ,ν and the amplitude of the field, B0, by the average spin polarization ⟨IK,Z⟩ in the direction of the propagation of the beam. The conventional electric dipole polarizability can be expressed as a linear response function40 αϵτ(ω) = −⟨⟨μ; μ⟩⟩ω. FOR and NSOR arise from the derivatives of αϵτ(ω) with respect to B0 and IK, respectively, expressed in terms of quadratic response theory40,41 as 23998



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Figure 3. Isotropic Verdet constant (V) values [in rad/(T m)] of graphene and graphane quantum dots as a function of the number of carbon atoms in the corresponding system for standard visible laser wavelengths. (a) Zigzag-type (Gn) and crenellated (mcrenG) graphene systems. (b) Corresponding HGn systems. Solid lines are for the systems with a zigzag perimeter, whereas the dashed lines denote crenellated systems.

5(GQD) = 5(C6H6 , l) ×

6 NC

3. RESULTS AND DISCUSSION 3.1. Faraday Optical Rotation in Graphene Quantum Dots. In Figure 1, isotropic Verdet constants as well as their anisotropies, V and ΔV, respectively, are plotted as functions of the wavelength of incoming radiation for different graphene molecules. ΔV is calculated as V⊥ − V∥, where V⊥ is the component in the direction perpendicular to the plane of the carbon sheet of the system, and V∥ is the component in the direction of the plane. A normal dispersion, i.e., an increase with the laser frequency for these quantities, can be seen, in accordance with earlier studies.4,35,36,43 That happens for all the present molecules, pure graphene systems (G) and fully hydrogenated graphene [i.e., graphane (HG)] fragments alike, regardless of the perimeter type. In the case of HG, the structures and the corresponding V and ΔV are illustrated in Figure 2. Figures S1 and S2 in the Supporting Information also display the V⊥ and V∥ components individually for the G and HG systems. A closer inspection of Figures 1 and 2 reveals a striking difference in the magnitude of V between G and HG systems. For G systems the Verdet constant is not only about an order of magnitude larger at a given wavelength, but also red-shifted with an increasing G system size. The latter phenomenon is consistent with results of the studies of molecular plasmons in G disks.22,56 Hardly any red shift occurs for the HG molecules. The difference reflects the delocalized character of the electronic states in the G systems, as well as the related fact that the excitation energies decrease from the ultraviolet into the visible range when the system size increases (see dashed vertical lines marking excitations for particular G systems in Figure 6 and also Tables S2 and S3 in the Supporting Information). The diverging behavior of the dispersion curves, visible for the G molecules in Figure 1, is partially due to the inadequacy of the perturbation theory in close proximity to the excitation energies. Concerning this technical issue, it was recently shown37 within the complex polarization propagator framework,57,58 which is convergent at resonances, that the strongly enhanced dispersion in the vicinity of optical excitations is a physical fact.25 In all the HG models, the excitation energies change relatively little with the system size. In contrast to graphene, the

(9)

where NC is the number of carbon atoms in the corresponding system. The used 5 of benzene is 6.737 × 1027 m−3, obtained with mass density value of benzene,43 ρ(C6H6, l) = 873.8 kg/ m3. The scaled 5 values are tabulated in the Supporting Information (Table S1). 2.3. Computational Details. Geometry optimization was done with the Turbomole software44 at the DFT (PBE45/def2TZVP46) level. All structures in xyz-format are given in the Supporting Information. Both types of optimized GQD structures, with zigzag- and armchair-perimeter structure, are in good agreement with previous geometry analyses.28,32,39 The quadratic response functions for V and VK were calculated on the Dalton program47 at the hybrid DFT B3LYP48,49 level, containing a 20% amount of the exact Hartree−Fock exchange. The overall suitability of the B3LYP functional for magnetooptic properties of carbon nanosystems has been tested for the planar hydrocarbons and fullerene in studies of light-induced NMR splittings38 and NSOR,4 as well as magnetic dichroism.50 In any case, the present systems are too large for a comparative survey of the results with several DFT functionals. These response functions are challenging for conventional energyoptimized basis sets, because not only accurate valence (electric dipole) but also core (orbital magnetic hyperfine) properties are required. It would require an unmanageably large traditional basis set for our biggest systems, to obtain converged results. For this reason the concept of completeness-optimized (co) basis sets33 was used presently. By the co paradigm it is possible to construct compact basis sets to produce molecular properties with a controlled deviation from the basis-set limit. The functionality of the concept in calculating hyperfine properties has been verified recently in several studies.4,35,36,38,39,51−53 The topic of co basis sets is detailed further in the Supporting Information. Sets of primitive co-bases for H and C were generated with the ERKALE code.54,55 The generated co2MOR set, (5s4p) for hydrogen and (12s7p3d) for carbon, feature a less than 5% deviation from the basis-set limit for V, VC, and VH, and were used throughout the study for all systems. 23999



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Figure 4. Atomic structures of graphene quantum dot systems consisting of concentric layers of hexagons (G4), with different levels of partial hydrogenation (top) and their different (a) isotropic Verdet constants (V) and (b) the anisotropies of Verdet constants (ΔV). For the sake of clarity, the carbon atoms in the sp2-domain are marked with red.

proportional to NC, is canceled due to simultaneously decreasing 5 . In addition, V in graphane GQDs is both much smaller in magnitude and significantly more isotropic [smaller ΔV, c.f. Figures 1b and 2b] than for the sp2-systems. The strong nonlinearity of the plots in Figure 3a at the shorter wavelengths can be partially rationalized by the proximity of the optical resonance for the biggest sp2-systems, while the transitions are still far away for the sp3-systems. The figure also illustrates the fact that the perimeter type has little effect on FOR. This is in contrast to the study of molecular plasmonic effects in polycyclic aromatic hydrocarbons, where it was found that the perimeter type determines the potential for having lowenergy plasmon states.16 Figures S4 and S5 in the Supporting Information show the dependence of the individual V⊥ and V∥ components of the Verdet constant on the system size (radius) in graphene and graphane systems, respectively. Figure S4 reveals that in G systems the enhancement of V is due to the V⊥ component, which increases very rapidly with the system size. That figure reveals also that V∥ is roughly similar in magnitude with the value appropriate to the corresponding value in HG systems. 3.2. Impurity Effects on Faraday Optical Rotation. Because pure G and HG systems have very different capabilities of rotating the polarization in FOR experiments, it is interesting to study how different levels of hydrogen adatoms affect this property. For this purpose we examined a series of differently hydrogenated systems of the size of the G4 molecule. We confine our attention to such cases where the entire concentric carbon ring is hydrogenated. The systems are labeled as follows. The case where the carbons in the innermost ring are hydrogenated is denoted as HG1G4, the case of two innermost rings hydrogenated is HG2G4, and the one with three innermost rings is HG3G4. In a corresponding way, another set of partially hydrogenated systems of similar size was obtained by removing hydrogens from the concentric rings of the HG4 system, starting from the innermost carbon 6-ring. Such systems are labeled successively as G1HG4, G2HG4, and G3HG4, with one, two, and three concentric carbon rings in the sp2 hybridization. All these atomic arrangements are illustrated in Figure 4, where their V and ΔV are also plotted. The dispersion curves of V in these systems reflect the systematic change of the hydrogenation level. There is a red shift of the location of the onset of the rapid pretransitional

energy gap between the occupied and unoccupied molecular orbitals (HOMO−LUMO gap) does not tend to zero in graphanes when system size increases (see also Supporting Information Table S3 and Figure S2). Figures 1a and 2a, as well as the insets in those figures, where V is divided by the number of carbon atoms NC in each of the G and HG systems, show clearly the distinct characters of these systems. In the HG systems the extensive nature of the underlying antisymmetric polarizability α′, scaling with the number of carbon atoms NC (ref 35), becomes apparent with the V dispersion curves overlapping completely, suggesting the value of the infinite homogeneous HG systems. The Verdet constant is proportional to the number density 5 of molecules, which, in turn, is inversely proportional to NC. This results in a cancellation of the size dependences of α′ and 5 for the HG systems. However, the anisotropy ΔV does feature a non-negligible system size dependence. A similar overlap of the dispersion curves of V does not occur in the case of the G systems due to the above-mentioned rapid decrease of excitation energies with the system size. Hence, α′ is not an extensive property for graphene quantum dots that have delocalized electronic structure. Instead, α′ possesses a superextensive character. Concerning the G systems, it appears that the energy gap determines the form and magnitude of the dispersion curve of V, not the edge geometry or the Clar structure, which can be used to characterize the level of localization of the π-electrons in sp2-systems.59 The Clar structures of the zigzag- and armchair-edged systems are different.59−61 For the former, Gn class of systems, aromaticity is homogeneous, rapidly approaching the infinite-size π-system, a graphene sheet. In contrast, armchair systems (mcrenG) belong to fully benzenoid hydrocarbons60,61 that have a localized π-system.59 The incapability of FOR to distinguish between the edge geometries of graphene systems can be rationalized with the global character of the orbital Zeeman (OZ) operator by which the molecular electron cloud is coupled with the magnetic field (see eqs 4−6 above, in the Theory subsection). Figure 3, where V is plotted as a function of the number of carbon atoms NC in a corresponding system, reveals that FOR increases monotonically and nonlinearly for both Gn and mcrenG systems. In contrast, in the HGn and mcrenHG systems, V is practically constant as a function of the system size. This verifies that an increasing antisymmetric polarizability α′ that is, in turn, 24000



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Figure 5. Nuclear spin optical rotation constants for the nuclei at the center of graphene and graphane quantum dot systems. (a) VC [10−6 rad/(M cm)] for zigzag- (Gn) and armchair-perimeter (mcrenG) graphene and (b) the corresponding graphane systems (HGn and mcrenHG). (c) VH for HG systems. In the inset of the (c) panel, the dispersion curve for HG1 is given, while the main panel illustrates the deviation of the HG2, HG3, and HG4 data from the HG1 results. Calculations at the B3LYP/completeness-optimized basis-set (co2-MOR) level.

NSOR spectroscopy, although this would be conceivable using either localized optical excitations or selective nuclear magnetization.4,37 Here we demonstrate the potential usefulness of future NSOR experiments for the present G and HG systems. In Figure 5, the NSOR coupling constants are plotted as functions of laser wavelength for the innermost carbon nuclei, labeled as VC, for both G and HG systems. In the latter case, the NSOR constant is also plotted for the hydrogen atoms (VH) attached to the innermost carbons. Two clear differences are immediately observed between the G and HG systems. First, the dispersion curves for the sp2 and sp3 carbons have the opposite phasespositive and increasing with frequency for the Gn systems, as compared to negative and increasing in magnitude with frequency for the HG systems. The second observation is that for the HGn system, the VC dispersion curves have a systematic, albeit very small dependence on the system size, whereas for the Gn systems a characteristic red shift occurs, on account of the rapidly decreasing excitation energies, similar to the case of FOR. The only significantly different dispersion curve occurs for the smallest system, cyclohexane (HG1). The fact that all the larger HGn systems feature a very similar behavior in both VC and VH reflects the local nature of both the antisymmetric polarizability, α′(IK), responsible for NSOR and the localized nature of the electronic states in graphane systems. The fact that HG1 is different from the rest of the HGn dots has its counterpart in the smallest present sp2-systems, benzene (G1), which also features a dispersion behavior distinct from that of the bigger Gn systems. Indeed, from the results of these smallest systems one can appreciate the importance of the proximity to the GQD boundary effects for NSOR. HG1 features a clearly smaller VC at all wavelengths as compared to the other HGn dispersion curves, whereas G1 starts to deviate from the curves of G2 and 2crenG at wavelengths smaller than 700 nm. The fact that the Gn dispersion curves also remain distinct from G2 onward, in contrast to the HGn systems, is both due to the less complete localization of electronic states in the sp2 systems and the (related) red shift of the excitation energies in the Gn systems. The opposite phases of the VC in the Gn and HGn systems arises from the opposite signs of the antisymmetric polarizabilities α′(IK) of the sp2 and sp3 carbons. Figure 5c indicates that the VH constants do not show significant differences between the various HGn systems, which reveals that the local environment of the CH bond is practically similar in all of them. Supporting Information Figure S9 illustrates the anisotropies ΔVK = VK,⊥ − VK,∥ corresponding to the NSOR constants of Figure 5. The off-plane VC,⊥ components are very

increase, which follows the decrease of the HOMO−LUMO gap, starting from the widest-gap HG4 system and ending to HG3G4 with the narrowest gap. In Supporting Information Figure S6 the off-plane and in-plane components of V are plotted. As was the case with G systems, also in the defected systems the off-plane component is much larger than the inplane component. All the GnHG4 systems with increasing size of the sp2-hybridized core region show a monotonic behavior with a rapidly increasing magnitude of the V⊥ component. In the case of HGnG4 systems, there is a trend of increasing V, this time with the size of the sp3-hybridized core region instead, and the largest V and ΔV are in fact obtained for HG3G4, where the sp3 core is surrounded by just one layer of sp2 carbons. In this series the behavior is nonmonotonic; however, as the HG2G4 system deviates from the trend to a small value of both V and ΔV. In the case of the much smaller in-plane component, different trends are observed. A systematic change in the density of states of these defectlike systems can be seen in Supporting Information Figure S7. These plots reveal that in GnHG4 systems an increasing region with sp2 hybridization in the core region systematically decreases the energy gap. Instead, in the case of HGnG4 the increasing size of the sp3-hybridized core creates defect states increasingly deeper in the HOMO−LUMO gap. The smallest gap is in the HG3G4 systems where only the perimeter carbons are sp2-hybridized. It is these outermost carbons that possess all the edge states according to the study of Stein and Brown.30 These authors observed that the striking difference between zigzag- and armchair-edged systems is that the π-electron in the HOMO is concentrated at the perimeter in the zigzag-edged, large polyaromatic hydrocarbons, whereas the distribution is uniform in the corresponding armchair-edged systems. The DOS images of G4 and 3crenG systems are shown in Supporting Information Figure S8. 3.3. Nuclear Spin Optical Rotation in Graphene Quantum Dots. NSOR involves an orbital hyperfine operator (the PSO operator), which is proportional to the inverse cube of the distance between the electrons and the nucleus. Therefore, NSOR probes the electronic structure in a very localized manner, similarly to the chemical shift of the conventional NMR spectroscopy. As an experimental spectroscopic tool NSOR is still in its infancy; there are no application studies on complex systems yet. It is, however, promising that different NSOR angles have been measured for different molecular liquids, demonstrating the existence of a magnetooptic chemical shift4 between different molecules.5 In contrast, there are no experimental reports yet of atomic-resolution 24001



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Figure 6. Nuclear spin optical rotation (NSOR) constant VK in increasing-size graphene quantum dot systems. VC [10−6 rad/(M cm)] for different nuclei in zigzag-perimeter systems, (a) G2, (b) G3, (c) G4, and (d) G5; and armchair-perimeter systems, (e) 2crenG and (f) 3crenG. For G2, G3, and 2crenG, the NSOR constant is calculated for all the distinct sites of carbon nuclei. Two lowest excitation wavelengths to singlet excited states with the transition dipole moment in the direction of the molecular plane (with large oscillator strength) are indicated with dashed vertical lines. Calculations at the B3LYP/completeness-optimized basis-set (co2-MOR) level.

contrast, the corner nucleus labeled as 4d in Figure 6c causes a negative rotation in the G4 system. A very different π-electron localization is predicted59 for the zigzag- and armchair-edged systems.60,61 The VC dispersion curves of the nuclei in the inner parts of the GQDs start to resemble each other in the bigger Gn systems, and at size G5 there is the group of practically indistinguishable signals reflecting a homogeneous graphenelike environment with delocalized π-electrons. In the case of the larger of the two armchair-edged systems (3crenG), VC dispersion curves of the inner atoms fall into two distinguishable groups, instead. Inspection of the anisotropy ΔVC for these nuclei (Supporting Information Figure S11) illuminates the difference between Gn and mcrenG systems even more clearly. These observations are generally consistent with the findings for the spectral patterns of conventional NMR,29 the aromatic character, and the C−C bond alternation, of these systems.28,32,59 Figure 7 illustrates VC and VH caused by the different nuclei in the sp3-hybridized HG2 and HG3 systems. In both cases, two groups of carbon signals appear; some of the perimeter atoms have clearly smaller signals than the other core or perimeter atoms. In 2crenHG (Supporting Information Figure S12), the VC dispersion curves differ substantially from those of the zigzag-edged HG2 and HG3 only for perimeter nuclei. The same also occurs for VH; only the dispersion curve of the perimeter (axial) hydrogen atoms differs from those of the others. This is expected from a highly localized sp3 system. Figure 8 illustrates the carbon NSOR constants VC of the defected HG1G4 and G3HG4 systems. We chose to display signals arising from the nuclei in a line from center to the perimeter of the systems along the C″2 -rotation axis. The lowest excitation energy to the first singlet, dipole-allowed state is smaller for HG1G4 than for the nondefected G4. For this reason, the VC dispersion curves appear red-shifted toward

large for the Gn systems, resulting in large positive anisotropy. In contrast, in the graphane molecules the NSOR anisotropies of both the innermost carbon atoms and the hydrogens bonded to them, are much smaller. Figure 6 illustrates VC for the different carbon nuclei in the graphene systems. All zigzag-edged GQDs (Figure 6a−d) have VC dispersion curves in their characteristic frequency domain due to the above-mentioned red shift of excitation energies with increasing system size. In these systems, the signals of some of the peripheral carbon atoms bound to hydrogen start to differ strongly from the dispersion curves of the inner nuclei, in the vicinity of resonance. This further underlines the importance of the boundary effects for this localized molecular property. For the two largest systems (G4 and G5), one of the edge carbon atoms even gives an oppositely signed NSOR signal throughout the entire wavelength range, as compared to the other systems. This tells about a special electronic environment of such corner carbon nuclei. Supporting Information Figure S10 illustrates for the G4 and HG4 systems the different localization and amplitude of the density difference between the first five magnetically excited states and the ground state in the symmetry species where the PSO hyperfine Hamiltonian operates in the corresponding point groups. In G4, large amplitudes are hosted in the perimeter region for excitations in the molecular plane, whereas large amplitudes are seen in the inner region of the HG4 molecule. The nuclei in the armchair-edged 2crenG and 3crenG systems (Figure 6e,f), feature dispersion curves of VC that are distinct from the corresponding nuclei in the zigzag-edged G3 and G4, which are of roughly the same respective sizes. These differences again underline the sensitivity of NSOR to the edge electronic structure and geometry. The differences are enhanced in the bigger systems; e.g., in 3crenG the corner nucleus gives a practically vanishing VC elsewhere but in the vicinity of resonance, where a positive value is obtained. In 24002



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be it of the zigzag or armchair type, but it does show a characteristic red shift and become enhanced with increasing sp2-hybridized graphene dot size. In contrast, in hydrogenated GQDs, FOR does not display these features. FOR is very sensitive to the degree of sp2 hybridization in GQDs, and the pure graphene system features a very large FOR in the direction perpendicular to the molecular plane. However, a study of partially hydrogenated GQDs consisting of four layers of carbon hexagons revealed the biggest FOR in the case where just one layer of carbon atoms at the perimeter was sp2hybridized, while the core region of the GQD was hydrogenated. The relatively recently found phenomenon of nuclear spin optical rotation (NSOR) is used here for the first time to study the optical rotation caused by macroscopically polarized nuclear spins in GQDs. Dispersion curves of the 13C NSOR constants, VC, were demonstrated to be different for zigzag- and armchairedged, sp2-hybridized systems. The differences were argued to be related to different π-electron localization. Hence, NSOR may provide a detection method for not only the structure of carbon nanosystems, but also their aromaticity. In a more general vein, the present study reveals that the finite GQDs have interesting properties related to magnetooptic activity in the visible range, reminiscent of the earlier finding of huge FOR of a single graphene sheet in the infrared. There is a sensitivity to the size and composition of the GQDs. The present results related to magneto-optic rotation by GQD systems, caused either by an external magnetic field (FOR) or a net magnetization of nuclear spins (NSOR), may contribute to studies of GQD structure, as well as their self-assembly behavior.26 The strong optical activity of GQDs in the visible spectral range suggests possible applications. Pioneering experimental studies are needed to verify the current predictions.

Figure 7. Nuclear spin optical rotation (NSOR) constants VK for different nuclei in graphane quantum dot systems HG2 and HG3. VC [10−6 rad/(M cm)] in (a) HG2 and (b) HG3; corresponding VH in (c) HG2 and in (d) HG3. Calculations at the B3LYP/completenessoptimized basis set (co2-MOR) level.

longer wavelengths than the corresponding curves in the G4 system. In G3HG4, the corresponding excitation energies are larger; the curves are blue-shifted as referenced to G4. A striking feature is the alternation of the signs of VC. In G4, the nuclei give a positive VC, whereas in HG1G4, both the second carbon from the center and the outermost carbon have negative NSOR constants. In G3HG4, the magnitudes of the VC vary continuously from a positive value at the center to a negative VC at the perimeter, at wavelengths in the proximity of the optical excitation. This demonstrates clearly that, upon chemical modification, the changing energetics drastically affects the magneto-optic characteristics also. This is sensitively reflected in the NSOR constants and their dispersion.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information on the HOMO−LUMO gap values for all molecules, the theory and details of the generated completeness-optimized (co) basis sets, the exponents of the co basis sets, and the structures of the calculated molecules. Additional figures: the atomic structures of HG and defected G4 molecules, density-of-states plots and the difference densities for the excitations of G4 and HG4 molecules, and the completeness profiles of the completeness-optimized basis

4. CONCLUSIONS The present first-principles study focuses in delineating Faraday optical rotation (FOR) in small hexagonal graphene quantum dots (GQD), and the role of the system size, perimeter structure, and the level of hydrogenation therein. We demonstrate that FOR is independent of the edge geometry,

Figure 8. Dependence of the nuclear spin optical rotation constant VK on the nuclear position in differently hydrogenated graphene quantum dots based on G4. (a) VC [10−6 rad/(M cm)] for HG1G4, where the innermost carbon 6-ring is hydrogenated, and (b) for G3HG4, where only the perimeter carbon atoms are hydrogenated. Two lowest excitation wavelengths to singlet excited states with the transition dipole moment in the direction of the molecular plane (with large oscillator strength) are indicated as dashed vertical lines. The sp2-hybridized carbon atoms are marked with red. Calculations at the B3LYP/completeness-optimized basis-set (co2-MOR) level. 24003



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sets. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: jarkko.vahakangas@oulu.fi. *E-mail: perttu.lantto@oulu.fi. *E-mail: juha.vaara@iki.fi. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Academy of Finland, University of Oulu Scholarship Foundation (JVä), Tauno Tönning Fund (JVa), and University of Oulu (JVa), for funding. Computations were partially carried out at CSC-The Finnish IT Center for Science (Espoo, Finland) and the Finnish Grid initiative project.

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