On the forbidden graphene's ZO (out-of-plane optic) phononic band-analog vibrational modes in fullerenes
Jesús N. Pedroza-Montero, Ignacio L. Garzón, Huziel E. Sauceda
OOn the forbidden graphene’s ZO (out-of-plane optic) phononic band-analogvibrational modes in fullerenes
Jes´us N. Pedroza-Montero, Ignacio L. Garz´on, and Huziel E. Sauceda
3, 4, ∗ Programa de Doctorado en Nanociencias y Nanotecnolog´ıas,CINVESTAV, Av. Instituto Polit´ecnico Nacional 2508, M´exico Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 20-364, 01000 CDMX, M´exico Machine Learning Group, Technische Universit¨at Berlin, 10587 Berlin, Germany BASLEARN, BASF-TU joint Lab, Technische Universit¨at Berlin, 10587 Berlin, Germany (Dated: January 12, 2021)The study of nanostructures’ vibrational properties is at the core of nanoscience research, they areknown to represent a fingerprint of the system as well as to hint the underlying nature of chemicalbonds. In this work we focus on addressing how does the vibrational density of states (VDOS) ofthe carbon fullerene family ( C n : n = 20 →
720 atoms) evolves from the molecular to the bulkmaterial (graphene) behavior using density functional theory. We found that the fullerene’s VDOSsmoothly converges to the graphene characteristic shape-line with the only noticeable discrepancyin the frequency range of the out-of-plane optic (ZO) phonon band in graphene. From a comparisonof both systems we obtain as main results that: 1)The pentagonal faces in the fullerenes impedethe existence of the analog of the high frequency graphene’s ZO phonons, 2)which in the context ofphonons this could be interpreted as a compression (by 43%) of the ZO phonon band by decreasingits maximum allowed radial-optic vibration frequency. 3)As a result, the deviation of fullerene’sVDOS relative to graphene should result on important thermodynamical implications. The obtainedinsights can be extrapolated to other structures containing pentagonal rings such as nanostructureor as pentagonal defects in graphene.
I. INTRODUCTION
Since the discovery of the C fullerene in 1985, [1] acascade of theoretical and experimental studies on thephysics and chemistry of novel carbon nanostructuresemerged. As a consequence of the very interestingproperties shown by carbon based nanomaterials, suchas mechanical, electronic, optical, and chemical ones, agreat number of fundamental investigations and tech-nological applications have been developed since.[2–7]Despite the plethora of research done on carbon nanoma-terials like fullerenes, nanotubes, nanoflakes, etc., [8–16]still some physicochemical properties have not been fullyinvestigated. These include their vibrational propertiesand how they are related to their size and morphology,being the fullerene family a particular case.During the last couple of decades, the vibrational prop-erties of metal nanostructures have gained a lot of atten-tion. [17–19] This is not only due to their relevance inthe design of nanodevices, [20] but also because of a no-torious development of vibrational spectroscopies, open-ing the opportunity to directly compare experimentalmeasurements and theoretical calculations. [21] For ex-ample, exploiting the symmetry breaking in supportedmetal nanoparticles (NPs), Carles’ group developed atechnique to directly measure the vibrational density ofstates (VDOS). [22, 23] Such a breakthrough [24] lead to ∗ [email protected]; Part of this work was done at Fritz-Haber-Institut der Max-Planck-Gesellschaft, 14195 Berlin, Germany the understanding of intricate experimental results us-ing well established theoretical results, [21, 25–29] whichlink the VDOS as a NP vibrational fingerprint that corre-lates to the morphology of the system. [18, 21] Contrast-ing these results, organic macro-molecules’ spectroscopyrelies only on Raman and IR spectroscopy to experimen-tally study them. Hence, given the nature of these exper-imental measurements, we have access to only a selectionof vibrational modes and not to their full VDOS.A fundamental question in nanoscience is how the tran-sition from molecular (physical and chemical) proper-ties evolve to bulk behavior as a function of the systemsize. In this regard, several results have been publishedsuch as the birth of the localized surface plasmon reso-nance in Au nanoclusters [30], the evolution of thermody-namical properties on metallic nanoparticles [25, 31–33],the formation of surface plasmons in sodium nanoclus-ters [34], as well as the convergence of molecular vibra-tional spectra to a bulk-like phonon density of states inmetal NPs. [35] In the particular case of metal NPs, asmooth transition of the VDOS from the nanoscale tothe bulk has been reported, study that is missing in thecase of carbon nanostructures.In this work, we present a theoretical investigation ofthe vibrational properties of fullerenes and its depen-dence with size (20 →
720 atoms) at the density functionaltheory (DFT) theory level. In particular, we analyse theevolution of the fullerene VDOS toward the bulk ma-terial (graphene) regime. Analysis that leads to an in-teresting and counterintuitive difference between the vi-brational modes in fullerenes and the phonon bands ingraphene: a severe restriction on the out-of-plane (ra-dial) optical band (ZO-modes) in fullerenes in the neigh- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n bourhood of the Γ symmetric point. These results showthat the presence of the pentagonal faces in the fullerenefamily hinders the smooth convergence of their VDOS tothe bulk graphene one. Furthermore, theses findings hintthat pentagonal impurities in graphene can have far moreimportant implications on its vibrational properties thanexpected. effective pagefull page with figure x 902 pts x 114.5 mm trajectory: Text in fig. and its caption < 18 pt However: regular text 18 pt
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460 cm -1
670 cm -1
450 cm -1
630 cm -1
750 cm -1 Missing peak in fullerenes
ZO peakZA peak
Upper limit for Z modes
FIG. 1. VDOS (black curve) of fullerenes. The red and bluecurves represent the radial and tangential contributions tothe total VDOS, respectively. The VDOS was constructedusing a Gaussian broadening with a width of 20 cm − . Bulkgraphene VDOS (bottom panel) was taken from Ref. [36] II. RESULTS AND DISCUSSIONA. Vibrational density of states
1. The C case Figure 1 shows the VDOS of fullerenes from 20 to 720atoms (diameters from 0.4 to 2.6 nm) with I h symme-try. As a consequence of the finite size of the fullerenes,the frequency spectrum is discrete and it has a finite acoustic gap ν AG (lowest frequency value). The vibra-tional spectrum extends from ν AG up to ∼ − ,although in the case for C , it is notorious a smaller fre-quency distribution range which goes from ∼
480 cm − to 1340 cm − . It is well known that the C fullereneis a controversial molecule, in which the reported cal-culated structure strongly depends on the utilized levelof theory due to its high electron correlation and multi-configuration character. [37–43] Often, different method-ologies differ in the molecular point group of the cageground state of C , which is reflected on the values ofthe interatomic forces and consequently in its vibrationalspectrum. In particular, the main differences are locatednear the lowest frequency region. In a previous theo-retical work, Saito and Miyamoto [37] reported a fre-quency distribution range that goes from ∼
115 to ∼ − using density functional theory (DFT) with thehybrid functional B3PW91. Another study by Sch¨utt el al. recently reported a vibrational spectrum rang-ing from ∼
100 to ∼ − using a machine learningmethod based on deep neural networks, SchNet, [42, 43]trained on DFT(PBE+vdW TS ) [44] level of theory. Sim-ilar results can be obtained with other machine learn-ing approaches. [45–47] In these two works, the ν AG arelower than in our case because the lower symmetry groupconsidered as a global minimum, D h instead of I h . Inthe case of the highest frequency, also known as cut-offfrequency ν COF , the lower value displayed in C com-pared to other fullerenes is attributed to the nature of thecarbon–carbon bond in C coming from the fact that thegeometrical structure is only formed by pentagonal faces.The rest of the studied fullerenes are more electronicallystable and different level of theories give consistent re-sults, showing a smoother transition between differentsizes.
2. The hypothesis
Based on the fact that fullerenes are single layer carboncage structures, it is natural to assume that in the limitof large fullerene diameters, the VDOS should convergeto the graphene’s VDOS given its single layer character.To explore this hypothesis, in Fig. 1 we show the VDOSevolution with the size of the fullerenes and its compar-ison to bulk graphene (Fig. 1 bottom panel). Here wecan think of the graphene case as the case of fullerene
FIG. 2. VDOS comparison between fullerene C (purple)and graphene (black). Bulk graphene VDOS was taken fromRef.[36] The suppressed phonon ZO branch in fullerenes ishighlighted in red. with diameter → ∞ , where the twelve pentagonal facesare completely diluted. From intermediate sizes, such asthe C , we can see that the lineshape of the VDOSalready has the main features of the biggest fullerene inour study (i.e. C ). Furthermore, the fullerenes displaythe higher intensity (main) peak in the VDOS around 670cm − , a characteristic feature that emerges from the 60atom structure.
3. Types of vibrational modes
When comparing the VDOSs of the fullerene fam-ily and graphene, it is observed that the C fullerenepresents most of the features displayed by bulk graphene,with the obvious difference of the finite acoustic gap (Seelowest two panels in Fig. 1). However, it is worth tohighlight the mismatch located in the range of ∼ − , where the graphene peak around ∼
850 cm − is missing in the fullerene family (Fig. 1-bottom andFig. 2). In order to understand this difference, first we an-alyze the VDOS of graphene. Graphene is a 2D materialwhich presents vibrations/phonons that can be classifiedin two main groups, in-plane (XY) and out-of-plane (Z)vibrations (See Fig. 3). This means that in XY phononsthe atomic displacement will be contained in the sameplane of the graphene, whereas in Z phonons the dis-placements will be perpendicular to the graphene plane.Such decomposition of the VDOS for graphene has beendiscussed by Paulatto et al., [48] indicating that the Zvibrational modes can be divided in acoustic (ZA) andoptic (ZO) vibrations, and that their respective frequencyintervals are [0, ∼ ∼ ∼ − . [48–50]These two bands are highlighted in Fig. 3, ZA in redand ZO in green, as well as their own density of states,VDOS ZA and VDOS ZO . It is worth to stress that all Zvibrations/phonons are confined to frequencies below − .By using this idea in the case of fullerenes, we can sep-arate their VDOS contributions from normal modes withradial atomic displacements or out-of-shell (Z, red linesin Fig. 1) from those containing displacements in-shell (XY, blue lines in Fig. 1). In the case of the XY-VDOSfor fullerenes (blue line), we observe a smooth line-shapespreading along the whole frequency spectrum, since itcontains all the in-shell vibrational modes: TA, LA, TOand LO. On the other hand, for the Z modes (red line)shown in Fig. 1 from top to bottom, we can see thatthe Z-VDOS converge to a smooth flat curve above 800cm − , displaying two well defined peaks at 460 and 670cm − , consistent with those observed in graphene (seeFig. 1). B. The forbidden wave numbers in ZO branch infullerenes
As stated in the previous section, the Z modes infullerenes have a cut-off frequency of ∼
750 cm − , whilethe corresponding value for graphene is ∼
910 cm − (seeFig. 3). Fig. 2 shows the overlap between the VDOS ofC and graphene, clearly showing the fullerene’s miss-ing region in red. By analyzing the phonon bands ofgraphene, we hypothesize that the missing peak is dueto the lack of modes that would come from the highestfrequency region of the ZO branch (centered in the Γsymmetry point and marked by a red square in Fig. 3-A). The origin of this is the fact that some symmetriesof the wave vectors (i.e. vibrational eigenvectors) are notallowed in fullerenes. To illustrate the kind of vibrationalmodes in the ZO (out-of-plane optical) branch, in Fig. 4-awe show the highest frequency mode in the branch whichcorresponds to the high-symmetry Γ point. This consistsin anti-phase out-of-plane atomic displacements betweenfirst neighbors. Such vibrational mode is only allowedbecause of the hexagonal lattice of graphene (i.e. it hasan even number of atoms in the ring), while in the caseof fullerene family the presence of the pentagons pro-hibit the existence of this kind of vibrational mode. Anschematic example is constructed in Fig. 4-b, where themode in Fig. 4-a is being constructed from top to bottomin a lattice with a pentagonal ring, which then inevitablyends up in two neighbouring atoms moving in phase.Therefore, the incapability of lattices with pentagons tohost certain types of ZO modes limits the value of thecut-off frequency of Z-modes ν maxZ in fullerenes. In fact,the frequency value ν maxZ of the vibrational mode and itsshape (i.e. its eigenvector) not only depend on the sizebut also on the molecular symmetry point group (see Fig.SI-1). In general, the 12 pentagonal faces in fullerenesseverely restrict high-frequency Z optical modes, but al-lows the creation of slower optical modes (hence the am-plification of the ZO peak in Fig. 1 and 3-B).This should also be true for bulk material with hexag-onal lattice with pentagonal defects, since the odd num-ber of atoms in the pentagonal ring hinders such highlysymmetric vibrational motion, originating a lower fre-quency in-phase motion between first neighbors (Fig. 4-b,in green).Figure 3 highlights the contributions of the ZO (green) FIG. 3. Comparison of Z-phonon-bands between bulk graphene and fullerenes. A) The out of plane (Z) phonons are highlightedin red for acoustic (ZA) and green for optical (ZO) ones. B-1) ZO and ZA contribution to the graphene total VDOS. B-2)Radial (Z)-VDOS in C fullerene. Graphene’s VDOS was taken from Ref. [48]FIG. 4. a) Highest frequency ZO vibrational mode [49] ingraphene. b) Construction of a ZO vibrational mode in anhexagonal lattice with a pentagonal defect. In b) we cansee that the presence of pentagons will produce an in-phasemotion between first neighbors (marked in green), and con-sequently, the highest frequency ZO vibrational mode forfullerenes will be lower than the one in graphene. and ZA (red) phonon bands to the total VDOS, andtheir direct comparison to the fullerences’ Z-VDOS ispresented. From this figure we can clearly see the re-gion of the ZO phonon branch that would correspond tothe missing vibrational modes in the fullerene family.
C. Visualization of highest Z modes
In order to visually understand what was explainedabove, it is interesting to analyze how the ZO modes looklike in fullerenes, in particular the one corresponding tothe maximum allowed frequency, ν maxZ . The family ofthese modes is displayed in Fig. 5. From here, we cansee that starting from C to C , the shape of the as-sociated eigenvector v maxZ follows similar patterns. Theshape of the vibrational modes can be contrasted to thehighest frequency ZO phonon (Γ point) in Fig. 4-a. Inter-estingly, the ν maxZ value converges to ∼
710 cm − for allfullerenes in our study, with the exception of C giventhe fundamental differences in the structure and chemicalbonds. The inset plot in Fig. 5 shows the range of fre-quencies in which the ZO branch is defined in graphene(highlighted in green in Fig. 3) and where the ν maxZ val-ues lay relative to this reference interval. It is expectedthat as the diameter of the fullerene grows, the ν maxZ willslowly start increasing towards the graphene ZO branchmaximum value given that the contribution of the areaof the 12 pentagons relative to the rest of the fullerenesurface will start diluting. III. DISCUSSION
The present analysis on the evolution of fullerenes’VDOS with the diameter and its similarities and devi-ations from the bulk graphene VDOS reveals an inter-esting geometrical constraint on the type of vibrational effective pagefull page with figure x 902 pts x 114.5 mm trajectory: Text in fig. and its caption 22pt
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685 cm -1
678 cm -1 C
713 cm -1 C
700 cm -1 C
678 cm -1 C
696 cm -1 C
710 cm -1 C
866 cm -1 f r e q . m ax . Z O [ c m - ] diameter [nm] Graphene ZO branch limit C
713 cm -1 C
700 cm -1 C
696 cm -1 C
710 cm -1 C
866 cm -1 f r e q . m ax . Z O [ c m - ] diameter [nm] Graphene ZO branch max. freq.Graphene ZO branch min. freq. C
678 cm -1 C , Symmetry: I h C , Symmetry: I FIG. 5. Graphical representation of the highest frequency vibrational mode corresponding to purely radial motions, which thecorresponds to the phonons in the ZO branch in graphene. The red and blue colors indicate phase and anti-phase radial atomicdisplacements, respectively. The inset plot shows the corresponding vibrational frequencies versus the fullerenes diameter. Thebottom and top doted lines indicate the frequency limits of the ZO branch in graphene. modes that fullerenes can host. Unlike the size evolu-tion towards the bulk behavior obtained for the VDOSof metal nanoparticles, [21] in the present case there willnot be a full convergency to the graphene VDOS sincethe pentagonal facets are required to close the cage struc-tures of fullerenes. In particular, in terms of the phononicbands in graphene, the ZO-equivalent normal mode os-cillations are greatly affected by the presence of pentag-onal faces, which sets a lower frequency ( ∼
750 cm − )and dynamically different ZO cut-off eigenstate (compar-ison between Fig. 4-a and Fig. 5), as well as a narrowerZO band ( ∼ d →∞ ν maxZ ( d ) = ν maxZO,Graphene , where d is the diameter of the fullerene.Such convergence is very slow as can be appreciated inFig. 5-inset. Nevertheless, we have to remember thatnormal modes are collective atomic oscillations, meaningthat either the interatomic interaction in the pentagonalfaces strengthen with the size (thereby increasing its fre-quency) or the pentagonal faces slowly become nodes inthe normal mode oscillation i.e. in the v maxZ eigenvec-tor, allowing then higher frequency oscillations localizedin the hexagons. By analysing the atomic amplitudes ofthe normal modes in Fig. 5, we found evidence of thelatter option. Such result would imply that, if the vibra-tional mode corresponding to the Γ symmetric point inthe ZO phonon band exists in a graphene material with pentagonal defects, those defects will have null ampli-tudes. A thorough analysis in this topic requires furtherevidence through numerical calculations and goes beyondthe scope of this article, then will be left as future work.To conclude, we have presented here a thorough anal-ysis of the peculiarities in the vibrational properties offullerences relative to the bulk graphene, and their po-tential thermodynamical differences due to the presenceof a compressed ZO-band in fullerenes. This work shouldopen question regarding the implications and dynamicsof pentagonal or other type of ring defects on graphene,but also it fills up a knowledge-gap in our understandingof the physical properties of carbon nanostructures.
IV. METHODOLOGY
All computations were done at the DFT level withinthe generalized gradient approximation (GGA), using thePerdew-Burke-Ernzerhof (PBE) parameterization. [51] Asplit valence basis set (def2-SVP) and the pseudopoten-tial for carbon with 4 valence electrons corresponding toECP2SDF were used, [8, 52] as implemented in the TUR-BOMOLE code. [53] To guarantee accurate frequenciescalculation, the structures were optimized with a forcetolerance of 10 − Hartree Bohr − . The computed har-monic frequencies were validated with reported calcula-tions for carbon dimer’s bond length, binding energy, andvibrational frequency. [54–60] Additionally, the whole vi-brational spectrum of C fullerene was calculated andcompared with experimental results [61] for validation.The VDOS was constructed using a Gaussian broaden-ing with a width of 20 cm − of the 3 N − ν ) = (cid:80) N − i =1 (cid:104) x | v i (cid:105) δ ( ν − ν i ), where x is a radial vector and { v i } is the set ofeigenvectors. Then, the in-plane (or more precisely forfullerenes, in-shell) contribution is computed via XY-VDOS=VDOS – Z-VDOS. V. ACKNOWLEDGEMENTS
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