Spin-Density Localization in Graphene at Boundaries and at Vacancy

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C: Physical Processes in Nanomaterials and Nanostructures

Spin-Density Localization in Graphene at Boundaries and at Vacancy Defects Tamal Goswami, Anirban Panda, and Douglas J. Klein J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11736 • Publication Date (Web): 18 Mar 2019 Downloaded from http://pubs.acs.org on March 18, 2019

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The Journal of Physical Chemistry

Spin-Density Localization in Graphene at Boundaries and at Vacancy Defects Tamal Goswami,†,1,* Anirban Panda,‡ and Douglas J. Klein† †MARS,

Texas A&M University at Galveston, Galveston, Texas 77554, United States

‡Department

of Chemistry, J. K. College, Purulia, Purulia, West Bengal, 723101, India Email: [email protected]

Abstract The degree of localization of non-bonding orbitals (and the related spin-density) in defected or decorated graphenic materials is addressed. For a defect consisting of a translationally symmetric boundary, such orbitals (if they occur) appear to be exponentially localized at the boundary, with the range correlating with the “mobility gap” at the corresponding wave-vector. For a defect consisting of vacancies of one or more nearby sites, such orbitals (if they occur) appear to be localized around the defect with a power-law decay away from the defect. Moreover, a simple criterion for such defect localized orbitals is noted – as also are some characteristics and possible consequences of the non-bonding defect-localized orbitals.

Introduction Conjugated-carbon structures are considered as of great relevance in the development of nanotechnology. There has been a wealth of research on all kinds of conjugated-carbon nanostructures, viz. polymers1,2 carbon cages,3,4 carbon nano-tubes,5,6 graphene,7 graphinic strips.8,9 During the past 3 decades there have been 3 Nobel prizes in this area. Such graphenic materials offer a plethora of interesting properties for diversified applications in electronics, spintronics and optoelectronics – under the influence of different decorations (or defects). Boundaries (edges) and vacancy defects (of different numbers of sites) are two very natural types of decorations or defects, whereat rather often defect-localized orbitals arise. In fact often such defect localized orbitals turn out to be non-bonding, thereby giving rise to more active behaviors, possibly with unpaired electrons. Vacancy defects in graphene have drawn special interest, e.g., in connection with the generation of magnetic moments10,11,12 localized around the defects. Indeed some general results have been proposed, and expressed in terms of counts of sites of various degrees (in terms of C—C  -bonds present in the  -network). Present Address: Department of Chemistry, Raiganj University, Raiganj, Uttar Dinajpur, West Bengal, 733134, India 1

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Besides possibilities for new nanotechnological uses, the characterization of defected graphenic structures in coal, and more degraded species (peat and humic acids) is of interest. These materials have extensive uses in the energy sector, and serve as sources of conjugated-carbon species. It is known13,14,15,16 that carbon in coal has turbostratic structure of graphite and amorphous carbon structures, with graphite-like stacking and poly-aromatic structures as its main constituents.17,18,19 Defective and defect-free graphene surfaces are considered20,21,22 to represent the structural heterogeneity of the organic matrix of coal-like complex chemical systems. Moreover, defects are often introduced in graphene for suitable applications to enhance different properties. It has been shown23 that the presence of vacancies reduces the clustering of metal atoms and strongly attaches metal nanoparticles onto graphene. The role of defect sites in the interaction between defected graphene sheets as a model of coal and CO2 has also been investigated24, 25 to find26,27,28 that, graphene with monovacancy defect has four times larger CO2 physisorption compared to perfect graphene. Large scale ab initio computation also suggest that graphenic materials decorated with defects can lead to high tunability of band structures that further facilitates numerous applications of these materials in spintronics, optoelectronics, light detection, molecular sensing.12,29 Thence, defects have a major role on the properties of graphene – and graphene with controlled defects (i.e., decorations or functionalizations) can lead to new developments for novel materials. Graphene is an archetypal bipartite system, in which the single honey-comb layer of sp2 hybdridized carbon atoms can be partitioned into starred and unstarred sets so that every starred site finds only unstarred sites as immediate neighbors. And the behavior of graphene and various derived defects or fragments turns out to depend crucially on this “alternancy” feature – as was early emphasized by Coulson and Rushbrooke.30 For the case of vacancy defects (of 1 or more sites) in extended graphene, the number of defect-localized non-bonding molecular orbitals is identified as the number of unpaired electrons. And this number has been argued31 to be given as 𝑈 = |(2𝑁1 ⋆ + 𝑁2 ⋆ ) ― (2𝑁1 ∘ + 𝑁2 ∘ )| 3

(1)

where 𝑁𝛿 ⋆ is the number of degree-𝛿 starred sites, and 𝑁𝛿 ∘ is the number of degree-𝛿 unstarred sites. For the case of translationally symmetric boundaries a rather similar formula has also been to apply, for the number of boundary-localized non-bonding orbitals per unit-cell of boundary. This associated number of unpaired electrons per unit-cell of edge is argued32

𝑢 = |(2𝑛1 ⋆ + 𝑛2 ⋆ ) ― (2𝑛1 ∘ + 𝑛2 ∘ )| 3

(2)

where 𝑛𝛿 ⋆ and 𝑛𝛿 ∘ are the respective numbers per unit cell of boundary of degree-𝛿 starred and unstarred sites. Indeed these formulas arise from a simple resonance-theoretic argument, but have been checked within the context of Hückel theory. The consilience of resonance and Hückel theory suggests some reliability for these conclusions. But also the Hückel-theoretic results may be 2 ACS Paragon Plus Environment

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readily extended to SCF solutions for Hubbard and PPP models. Moreover, these formulas are consilient with Lieb’s theorem33 for the ground-state spin of the half-filled Hubbard model for finite alternant systems. Indeed Lieb’s result was earlier noted to apply34,35,36 for the nearest neighbor Pauling-Wheland (or Heisenberg) model. Yet further equation (2) for boundaries is in agreement with some ab-initio quantum chemical computations,37,38,39,40,41,42 or even experiment.43,44,45,46 But a further point concerns the degree of localization of the unpaired electrons. For the case of the boundaries, resonance theory suggests that the localization is such that the unpairedelectron density falls off exponentially fast with the distance from the boundary, and Hückel theory checks this in a qualitative fashion, but with the exponent changing with wave-number 𝑘. In fact, one might naturally expect the fall off to be slower as the energy splitting to the nearest delocalized orbitals (of the same 𝑘-value) decreases. That is, the range 𝑟𝑘 for (exponential) decay of an edgelocalized orbital should correlate with the inverse of the energy gap ∆𝑘 for delocalized orbitals at the same 𝑘. Here the gap ∆𝑘 might be termed a “mobility gap” (for the direction normal to the overall boundary direction). One anticipates that as the gap collapses to 0 the range of exponential fall-off diverges toward ∞. Now for the vacancy defects, resonance theory suggests localization, the argument does not suggest exponential localization. And to check the possibility of localization with Hückel theory when there is no translational symmetry to help the computation, a natural approach is to perform computations on large finite fragments with a missing site (or sites) near the center of the fragment. Then the consequent associated gap for nearby delocalized orbitals is close to 0, whence the exponential range of decay might plausibly be expected to be very large, perhaps → ∞ (on an exponential scale) as the gap closes → 0. Earlier Hückel-theoretic computations on fragments of up to ≈ 1500 sites indicated localization, though it was not concluded to be exponential localization. Thus here the aim is to elucidate the manner of defect localization of any non-bonding MOs – or (within a resonance theoretic frame) to elucidate the manner of localization of unpaired spin density. First, for 3 types of boundaries with non-bonding MOs localized near these boundaries, we consider correlations between delocalized-orbital gap and decay range of defectlocalized MOs. Second, for 3 groupings of vacancy defects in large graphene fragments (of up to >100,000 sites), we look for sub-exponential (power-law) localization, treating different vacancy defects. A unified picture emerges.

Observations and Correlations for Boundaries The 3 types of boundaries to be (explicitly) considered are shown in Figure 1. Within the context of the Huckel model, it turns out that each has a number of edge-localized non-bonding MOs as given by eqn. (1). This may be derived, or easily checked for a strip of some reasonable



width – showing flat non-bonding bands covering a wave-number range 2𝜋 and (c), or |𝑘| ≤ 3 for 1(b).

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2𝜋

3 ≤ |𝑘| ≤ 𝜋 for 1(a)

(a)

(b)

(c) Figure 1. The 3-types of graphene boundaries considered.

A check may be made of the expectation of a correlation between the inverse of orbital1 density decay range 𝑟 of a defect-localized non-bonding MO, and the gap ∆𝑘 between the 𝑘 nearest delocalized MOs (at the particular wave-vector 𝑘). First, we consider a zig-zag boundary, 2𝜋 such as depicted in Figure 1(a). Here32 the non-bonding MOs occur at 𝑘-values with 3 < |𝑘| < 𝜋, where the asymptotic gap between delocalized MOs is

|

𝑘

∆𝑘 = 2 ― 4cos (

|

2)

(3)

And the non-bonding MOs amplitudes decay by a factor ―(1 + 𝑒𝑖𝑘) in proceeding to the next site of the same type (starred/unstarred) 2 steps along away from the boundary. Thus the associated


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―2

orbital density changes by a factor 𝑒 Information for details) 𝑟𝑘 = ―

1

𝑟𝑘

with 𝑟𝑘 the range of decay given as (see Supporting

ln (2 + 2cos 𝑘) = ―

1 𝑘

ln (4𝑐𝑜𝑠22)

(4)

which is to say that the non-bonding MO amplitude at n steps from the zig-zag boundary is ―𝑛

proportional to 𝑒

𝑟𝑘

(again with

2𝜋

3 < |𝑘| < 𝜋). Thence we have

𝑟𝑘 = ―

1

(

2 ∙ ln 1 ―

∆𝑘

)

(5)

2

which is seen to go from near 0 when ∆𝑘→ 2 as 𝑘 → π, to → infinity when ∆𝑘→ 0 as 𝑘 → 2π/3 1 from above. A plot of ∆𝑘 vs. 𝑟 is given in Figure 2 where ∆𝑘 is reported in multiple of Hückel 𝑘 parameter β.

Figure 2. Plot of mobility gap and inverse of the orbital density decay range for the zigzag edged graphene ribbon.



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A similar analysis may be made for the whiskered boundary of Figure 1(b), where analytic solutions are also available.47 Indeed, the results are closely related to those for a zig-zag boundary, though there are 2/3 of an unpaired electron per unit-cell of boundary, and the non-bonding 2𝜋 2𝜋 boundary-localized MOs appear in the range ― 3 100000 atoms, while



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the purple and yellow (almost invisible) lines correspond to a smaller graphene chunk of < 8000 atoms. Our numerical results are summarized in Figure 4, giving log 𝐷𝑑 vs log 𝑑 for different frontier orbitals for each of our types of vacancy defects. In the first plot 4(a), results are displayed for two cases: a single-site vacancy (curve in red) and for an adjacent pair of site vacancies (in green) – it being seen that the frontier orbital for the single-site case is notably localized, with a 1 slope of ≈ –2 indicating a 𝐷𝑑~ decay with distance d from the defect – this non-bonding MO 𝑑2 manifests an (approximate) A-symmetry of the approximate 𝒞3𝑣 symmetry of the single-site vacancy. The 2-site vacancy case (in green) in 4(a) shows little in the way of localization. In Figure 4(b), there is a plot for the frontier orbital for an allyl-triple of vacant sites, again 1 showing a 𝐷𝑑~ localization. In this case the non-bonding MO turns out to manifest an 𝑑2 approximate (local) 𝒞2𝑣 B-symmetry. In Figure 4(c), for a trimethylene-methyl quartet of vacant sites, there are two non-bonding MOs (indicated as NBMO-1 and NBMO-2), and it turns out that they manifest what is approximately an E-representation of the approximate local 𝒞3𝑣 symmetry of the TMM vacancy defect. In 4(c) we report a single orbital-density decay for the pair – again the localization is found 1 to manifest a ~ power law decay. For comparison we have diagonalized adjacency matrix of 𝑑2 a smaller graphene chunk with a TMM defect at the center of the graphene sheet consisting of less than 8000 atoms. The density decay plots of the NBMOs of this smaller graphene sheet are also plotted in Figure 4(c). It can be seen that the density decays at a similar slope (actually ~ –1.9) for both the bigger and smaller sheet – and much like that for the single-site methyl defect or the triplesite allyl defect.

Conclusions Evidence is here provided that the exponential range of localization for our graphenic defect localized MOs varies inversely with the relevant mobility gap – and that further there can be defect-localization of a weaker (power-law) form when this mobility gap collapses → 0. In particular, exponential localization applies with translationally symmetric boundaries when there are predicted (via equation 2) to be a multiplicity of non-bonding MOs. When at a given wave number 𝑘 this band gap is > 0, the decay is exponential – and the bigger this band gap, the faster the exponential decay. With vacancy defects (possibly of several sites), inverse power-law localization ~

1

is 𝑑2 predicted for a ≠ 0 number of non-bonding MOs as predicted via equation (1). Evidently this power law decay applies when the band gap made by the remaining orbitals (of the same symmetry) is = 0. A further point is that the (approximate) point-group symmetry of defect10 ACS Paragon Plus Environment

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localized non-bonding MO appears to be the same as that of the corresponding non-bonding MO of the π-network structure which has been removed. Thus in the case of the anti-methyl defect, the non-bonding MO manifests the symmetry of the totally symmetric A representation of 𝒞3𝑣, the same as the symmetry of the non-bonding methyl π -electron state. The case of anti-allyl, the nonbonding defect-localized MO manifests a B-symmetry of the 𝒞2𝑣 symmetry, the same as allyl. And the anti-TMM defect case, the 2 non-bonding MO manifest an E-symmetry of the 𝒞3𝑣 group, the same as TMM. There is a simple rationale for this correspondence of non-bonding MO symmetries, if one contemplates the molecule in the center without coupling between the molecule and the defected lattice, then gradually turns the coupling on as a perturbation: this perturbation is totally symmetric, so that corresponding orbital symmetries is what is needed to mix the nonbonding MOs from the molecule and the defect, and give a splitting, lifting the 0-energy nonbonding feature of the MOs for the full graphene fragment. This symmetry restricts where the spin density can show up, and thence magnetic properties, as well as reactivities. A question arises as to the chemical reactivity of the non-bonding MOs from our considered defects. The answer is that although being at the frontier, these MOs are not very reactive (so long as the reactivity is gauged in a local way). For the non-bonding MOs around a vacancy defect, the relatively slow power-law decay leaves little unpaired density on any site – which is to say that there is no site which especially radicaloid. For the case of edge-localized non-bonding MOs, they are localized (sometimes even strongly) in the direction transverse to the edge boundary, but still are delocalized along the longitudinal direction of the boundary – so that any such individual MO has very little density on any site along the boundary. Of course, the number of such edge-localized non-bonding MOs is proportional to the length of the boundary, so that the unpaired electron density due to all these MOs on a single edge site can be substantial – but if a considered reaction is of a local nature involving 1- (or 2-electeron) operators, it is the density of individual orbitals which matters – so that the whole remains non-reactive (so long as the reactions are local). They can however contribute directly to novel magnetic and electrical properties. There is a further point about edge-localized non-bonding MOs, which contrasts with what is often imagined to happen in a Mott transition from a delocalized to a localized insulating state. In this classical scenario what is imagined is that atomic (or local molecular) orbitals may be obtained from a back-Fourier-transform of the orbitals of a flat band (or more modestly just a “narrow” band) – say as for an array of equally spaced H atoms at a sufficiently large spacing – whence singly occupying the localized orbitals is energetically favored (since this avoids admixture of ionic states). Now the point is that our edge-localized non-bonding MOs are robust against such a localization mechanism, because each having a different decay rate away from the edge boundary, a back-Fourier-transform over the flat band does not lead to localization – which is to say they remain delocalized along the length of the boundary. It is believed that our non-pairing equations ((1) and (2)) transcend their origins with resonance or Hückel theory. First, that there is consilience between these two very different chemical approaches is highly suggestive. There are some supporting much higher quality



quantum-chemical computations checking different special cases, while also there is consilience with finite molecule theorems for VB and Hubbard models.36,48,49 The case of separated vacancy defects is interesting, and presumably a simple extension of formula (1) applies. One can apply the formula (1) for each of a pair of vacancies a and b, to obtain 𝑈𝑎 and 𝑈𝑏 for each defect treated individually. But if applied to the pair (as a single 2-part) vacancy defect, the resulting prediction is 𝑈𝑎𝑏 = |𝑈𝑎 ± 𝑈𝑏| with the ± sign depending on the relative signs of the terms (2𝑁1 ⋆ 𝑎 + 𝑁2 ⋆ 𝑎) ― (2𝑁1 ∘ 𝑎 + 𝑁2 ∘ 𝑎) and (2𝑁1 ⋆ 𝑏 + 𝑁2 ⋆ 𝑏) ― (2𝑁1 ∘ 𝑏 + 𝑁2 ∘ 𝑏) used in computing 𝑈𝑎 and 𝑈𝑏. If the two vacancy defects are in concert, and their total number of unpaired electrons adds, then it is clear what is expected. With the unpaired 1 spin-density decreasing ~ from each defect, the coupling between such separated defects 𝑑2 should decrease with their separation. The symmetry of the non-bonding MOs can lead to orthogonalities in some cases, which then also influence the sign of the exchange couplings between them. Of course, spin density at edge boundaries or vacancy defects and defects contribute to the magnetic susceptibility. And especially the consideration of the exchange coupling of spin density along an edge or between pairs of edges is of interest, e.g., for the subject of “spin-tronics”. But this deserves separate consideration. Next it is surmised that equations of the same form apply in some local sense regardless of translational symmetries. That is, it is surmised that there are nearly-non-bonding MOs localized in regions where local counts of different degree starred and unstarred vertices are unbalanced as described by equation (1). For the case of boundary with the misbalance of equation (3) all on a site of a given type, the resultant non-bonding MOs though localized transverse to the boundary are delocalized along the edge, and should manifest an (effective) exchange coupling of ferromagnetic sign – and indeed couple with long-range ferromagnetic order along the quasi-1dimensional boundary region. This then gives a simple means for controlling the crucial frontierorbital dispositions – as we imagine should aid in nano-technological design. Evidently the characteristics of the frontier orbitals for a range of graphene decorations have been elucidated to manifest novel features, which should be widely useful.

Supporting Information Supporting Information contains analytical derivation of the expression for the decay range and the details regarding the imposition of zero band gap in the calculation for the infinite graphene strip with zigzag edges.

Acknowledgement The authors acknowledge helpful discussions with Dr. Y. Ortiz and Dr. L. Bytautas. T. G. thanks Indo-US Science and Technology Forum (IUSSTF) and SERB for the fellowship grant (SERB Indo-US postdoctoral fellowship 2017).


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The degree of localization of non-bonding orbitals in defected or decorated graphenic materials appear to be localized around the defect with a power-law decay away from the defect. For a defect consisting of a translationally symmetric boundary, such orbitals appear to be exponentially localized at the boundary.

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Spin-Density Localization in Graphene at Boundaries and at Vacancy

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