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Two-phonon Raman bands of single-walled carbon nanotubes: a case study

Two-phonon Raman bands of single-walled carbon nanotubes: a case study Valentin N. Popov Faculty of Physics, University of Sofia, BG-1164 Sofia, Bulgaria (Dated: November 9, 2018) It has been long accepted that the second-order Raman bands in carbon nanotubes are enhanced through the double-resonance mechanism. Although separate aspects of this mechanism have been studied for a few second-order Raman bands, including the most intense defect-induced D band and the two-phonon 2D band, a complete computational approach to the second-order bands is still lacking. Here, we propose such an approach, entirely based on a symmetry-adapted non-orthogonal tight-binding model with ab-initio-derived parameters. As a case study, we consider nanotube (6, 5), for which we calculate the two-phonon spectrum. We investigate in detail the 2D band and identify three contributions to it: a non-dispersive one and two dispersive ones, which are found to depend on the electron and phonon dispersion of the nanotube, and on the laser excitation. We also predict two-phonon bands, which are not allowed in the parent structure graphene. The obtained two-phonon bands are in very good agreement with the available experimental data. The symmetry-adapted formalism makes feasible the calculation of the two-phonon Raman bands of any observable nanotube. I. INTRODUCTION for certain laser excitations, which has been explained by the the double-resonant (DR) scattering mechanism, initially proposed for graphene. In the early days of the In the last few decades, the layered carbon materi- modeling of these bands in graphene, simplifying assump- als like fullerenes, nanotubes, and few-layer graphene tions such as constant electronic linewidth, limiting the have been a subject of intense experimental and theoret- calculations only to high-symmetry directions in the Bril- ical study, because of their unique properties, originating louin zone, and exact DR conditions have been used (for a from their zero-, one-, two-, and three-dimensionality. In review, see, e.g., Ref. 10). Only recently, theoretical two- particular, significant progress has already been achieved phonon bands in graphene, which are in good agreement in the synthesis and the study of the properties and ap- 10,11 with the experimental data, have been reported. The 2,3 plication of carbon nanotubes The application of nan- success of the latter studies is based on the explicit calcu- otubes in nanoelectronics requires their precise structural lation of the couplings and the electronic linewidth, and characterization. For this purpose, the Raman scattering performing the integration of the quantum-mechanical by phonons is the experimental technique of choice, being expression for the Raman intensity over the entire Bril- a fast and nondestructive characterization method. louin zone for both electrons and phonons. The so-called single-walled nanotube (or briefly nan- The theoretical description of the two-phonon bands otube) can be viewed as obtained by rolling up a graphene of nanotubes is much more complex than for graphene, sheet into a seamless cylinder. It has a few intense first- mainly, because of the quantum confinement in one di- order Raman bands, arising from the radial-breathing mension and the large variety of nanotube types. Re- mode (RBM ) and the longitudinal and transverse tan- cently, a step towards the understanding of the defect- gential modes (G modes). The RBM frequency is found induced D mode in nanotubes vs diameter and energy has to be inversely-proportional to the nanotube radius and been made using the hexagonal symmetry of graphene is normally used for fast sample characterization. The 12,13 and geometrical considerations. This approach al- G modes also depend on the nanotube radius and can lows assigning the two-phonon bands to pairs of defi- be used to support the assignment of the Raman spec- nite phonons. However, the two-phonon band shape can tra to particular nanotubes but also allow for differenti- only be predicted, taking into account the couplings and ating between metallic and semiconducting nanotubes. the electronic linewidth, and carrying out full integra- Intense second-order Raman bands are also observed in tion over the Brillouin zone. In a recent publication, nanotubes. These bands arise from scattering processes, this has been done for a few narrow nanotubes using ab- which can involve two phonons with opposite momenta initio and tight-binding approaches, neglecting, however, (two-phonon bands) or a phonon and a defect (defect- the effects of the nanotube curvature. induced bands). The most intense two-phonon band (so- −1 called 2D band), usually observed around 2700 cm , is Here, we propose a computational approach to the due to electron/hole scattering by two transverse opti- calculation of the two-phonon Raman spectra of car- cal (TO) phonons with opposite momenta. Other two- bon nanotubes, which is entirely based on a symmetry- phonon bands have also been measured (for summary, adapted ab-initio-based non-orthogonal tight-binding see, e.g., Ref. 7). The two-phonon spectra contain valu- (NTB) model. This model has been used for more than able information about the phonon dispersion of the a decade for the successful prediction of the electronic nanotubes. band structure and phonon dispersion, and the first-order 15–19 The D and 2D band intensity is found to be enhanced Raman bands of several hundred nanotube types. arXiv:1804.03402v2 [cond-mat.mes-hall] 21 Jun 2018 2 The model describes the curvature effects on the phys- the phonons are labeled by the one-dimensional wavevec- ical properties of carbon nanotubes, which are essential tor q, q ∈ [0, 1) 2π/T , and the integer quantum number for nanotube diameters below about 1 nm. As a case λ, λ ∈ [0, N). The selection rules for scattering of elec- ′ ′ study, we consider the narrow nanotube (6, 5) and per- trons by phonons are k = k+q±2π/T and l = l+λ±L; form a complete calculation of the two-phonon bands L = (Nν + n)/n˜, where ν is an integer number such that at a number of laser excitations and discuss the con- L is an integer number. These selection rules can be de- tributions to the 2D band, as well as the appearance rived similarly to those in Ref.18 by including Umklapp of two-phonon bands, which are symmetry-forbidden for processes. Notice that the integer quantum number is the parent structure graphene. non-conserving. In quantum mechanics, the two-phonon process can The paper is organized as follows. The theoretical background is presented in Sec. II. The obtained results be viewed as a sequence of virtual processes, namely, are discussed in Sec. III. The paper ends up with con- electron-hole creation, scattering of electrons or holes clusions, Sec. IV. by two phonons with opposite momenta, and finally electron-hole annihilation. The wavevector is conserved for each virtual process, while the energy is conserved only for the entire two-phonon process. The correspond- II. THEORETICAL BACKGROUND ing two-phonon Raman intensity can be described by an expression, derived in fourth-order quantum-mechanical A nanotube can be considered as obtained by cutting 3,4,10,11,22 perturbation theory, out a rectangle of graphene, defined by a pair of or- ~ ~ thogonal lattice vectors T and C, and rolling it along C into a seamless cylinder. This rolled-up nanotube can X X M M M M fc cb ba ai be characterized by the radius R = kCk/2π, translation I ∝ δ (E − E ) (1) i f ΔE ΔE ΔE ic ib ia period T ≡ kTk, as well as by the chiral angle θ, which f c,b,a is the angle between C and the nearest zigzag of carbon Here, ΔE = E − iγ and E = E − E ; E is atoms. All structural parameters of the rolled-up nan- iu iu iu i u i the energy of the initial state; E = E , where E otube can be expressed by means of the nearest-neighbor i L L is the incident photon energy (laser excitation); E , interatomic distance and the hexagonal indices (n, m) of u u = a, b, c, f, are the energies of the intermediate (a, b, c) ~ ~ C or (n˜, m ˜ ) of T. The former notation is traditionally and final (f) states of the system of photons, electrons, used to specify uniquely the nanotube and is accepted holes, and phonons. M are the matrix elements be- uv here as well. Normally, the total energy of the rolled-up tween initial, intermediate, and final states. M and ai nanotube is not minimal and the atomic structure of the M are momentum matrix elements. M and M are fc ba cb nanotube has to be subjected to structural relaxation, electron-phonon matrix elements. The electron-photon which is a necessary step before performing phonon cal- and electron-phonon matrix elements are calculated ex- culations. plicitly within the NTB model. γ = γ + γ , where c v The straightforward calculation of the electronic states γ and γ are the halfwidths of conduction (c) and va- c v and phonons for a large variety of nanotubes is accom- lence (v) states, respectively. The summation over the panied by insurmountable computational difficulties, be- phonon wavevectors q and the electron wavevector k is cause of the very large translational unit cells of most carried out over a mesh of 400 points in the Brillouin of the observed nanotubes. Fortunately, the nanotubes zone. The Raman intensity is enhanced for vanishing of have screw symmetry that allows reducing the compu- one, two, or three of the E ’s in Eq. 1, and therefore iu tational efforts by resorting to two-atom unit cells. The single, double, and triple resonances can occur. The in- latter approach has been used for calculation of the elec- 15 17 cident and scattered light are assumed polarized along tronic states and phonons of several hundred nan- the nanotube axis and only Stokes processes are consid- otubes within the NTB model. In this model, the Hamil- ered in this work. tonian and overlap matrix elements are derived as a function of the interatomic separation from an ab-initio study on carbon dimers and the Slater-Koster scheme III. RESULTS AND DISCUSSION is adopted for the angular dependence of the matrix ele- ments. The imposing of the translational periodicity and ro- We consider nanotube (6, 5) for exemplifying the be- tational boundary conditions on the solutions of the elec- havior of the two-phonon bands vs laser excitation. The tron and phonon eigenvalue problems results in labeling translational unit cell of the nanotube contains N = 182 of these solutions by a pair of indices. The electronic two-atom unit cells. The atomic structure of the nan- states are labeled by the one-dimensional wavevector k, otube is relaxed, retaining its circular cylindrical form. k ∈ [0, 1) 2π/T , and the integer quantum number l, In this case, as independent structural parameters, we l ∈ [0, N), where N is the number of two-atom unit cells choose the radius R, the translation period T , and the in the translational unit cell of the nanotube. Similarly, coordinates of the second atom relative to the first atom 3 in the zeroth two-atom unit cell, and obtain the relaxed in the parabolic bands, associated with the transition ˚ ˚ parameters R = 3.76 A and T = 40.67 A. E = 2.20 eV. (  H9 J 72#. ' DY:YFRHUH W S 7  (QH HUJ9 \ /2 /2 72=$#. 2=2. 7# /2# /$ /$ EXQLWV 5DPDQ,QWHQVLW\ DU /. 72$# 7$ 7$ 2= 2= 5DPD6KQ LWIFP $= $= 3KRQRQ)UHTXHQF\ FP FIG. 2. (a) Calculated two-phonon Raman spectrum of nan- otube (6, 5) at laser excitation E = 2.35 eV. The most in- tense two-phonon bands are labelled. The spectrum is slightly D:YHYHFRW U S 7 upshifted for clarity. FIG. 1. (a) Electronic band structure of nanotube (6, 5) close The phonon dispersion of nanotube (6, 5) consists of to the Fermi energy, chosen as zero. The optical transition 6N branches and their representation in the Brillouin E is denoted by vertical arrows. (b) Phonon dispersion of nanotube (6, 5) in the extended zone representation. The zone for the translational unit cell is not very informa- pairs of close vertical lines bracket the phonons, taking part in tive, because the branches would cover densely the graph. scattering between the extrema of the parabolic bands for the Alternatively, one can draw fewer phonon branches in the E transition. (c) Halfwidths of valence (γ ) and conduction 22 v extended zone representation, namely, 6ν, where ν is (γ ) states of nanotube (6, 5). The corresponding halfwidths the greatest common divisor of n and m. In the case for graphene (dotted lines) are provided for comparison. of nanotube (6, 5), ν = 1 and therefore there are only 6 branches in this representation. These branches have in- The electronic band structure of nanotube (6, 5) in the plane longitudinal optical (LO), transverse optical (TO), entire Brillouin zone and for energies up to ±1.5 eV rel- longitudinal acoustic (LA), and transverse acoustic (TA) ative to the Fermi energy E = 0 eV in shown in Fig. character, and out-of-plane optical (ZO) and acoustic 1a. The valence and the conduction bands in this energy (ZA) character. The unfolded phonon dispersion con- range are bands of graphene of mainly π and π charac- sists of parts of width 2π/T characterized by different ter, folded along the cutting lines of the Brillouin zone of values of the integer quantum number (Fig. 1b). The graphene. None of the cutting lines passes through the selection rules for the wavevector and the integer quan- K and K points of the Brillouin zone of graphene and tum number impose restrictions on the allowed values therefore there are no linear bands that cross the Fermi of the phonon wavevectors ±q for scattering of electrons energy in the Brillouin zone of the nanotube. Thus, un- between the parabolic bands for transition E to the like graphene, nanotube (6, 5) is a semiconductor. Due to regions between the pairs of close vertical lines in Fig. the presence of electronic bands with extrema (so-called 1b. parabolic bands), the electronic density of states for such The Raman intensity, Eq. 1, depends crucially on the bands has singularities at the energies of the extrema, electronic linewidth. The calculated linewidth for nan- so-called van Hove singularities (vHs) (not shown). The otube (6, 5) exhibits sharp spikes, arising from the sin- absorption of electromagnetic radiation is enhanced for gularities of the electronic density of states, while that photon energies, matching mirror pairs of vHs with re- for graphene is a smooth function of energy (Fig. 1c). spect to the Fermi energy. In the mentioned energy The singularities of the linewidth effectively decrease the range, the NTB model predicts only two optical transi- Raman intensity close to the transition energies of the tions for nanotube (6, 5), which take place between the nanotube. first and the second pair of mirror vHs and are denoted The derived electronic band structure and phonon as E and E , respectively. Here, we will be interested dispersion, halfwidths, electron-photon and electron- 11 22 4 cally zero for energies below 2.35 eV but increases up to E −1 a few tens of cm at 2.60 eV. The integrated Raman in- tensity exhibits three peaks with separation between the adjacent peaks of about h ¯ω, where ω is a characteristic ( K Z TO phonon frequency (Fig. 3b). We have not attempted fitting the 2D band with two Lorentzians and deriving the Raman excitation profile for each of them, because of the strong overlap of the two peaks of this band. ( K Z ( ( ( ( F F F F U([FLWDWLRQ H9 /DVH T Q,QWHQVLW\ DUEXQLWV 5DPD T (QHUJ\ H9 (QHUJ\ H9 ( ( ( ( Y Y Y DY:HRYHUFW   S 7 DY:HRYHUFW  S 7 5DPD6KQ LWIFP ( ( F F FIG. 3. (a) Calculated Raman 2D band of nanotube (6, 5) vs Z T laser excitation E . The 2D band shape evolves from a single- L T to two-peaked structure (peaks 1 and 2) with increasing E from 2.15 to 2.60 eV with a step of 0.01 eV (see right panel Z T axis). (b) Integrated Raman intensity of the 2D band (in arb. (QUJ\ H9 Z T units) vs E . Three peaks, centered at E = E , E + ¯hω, L L 22 22 Y Y and E + 2¯hω, where ω is the TO phonon frequency, are RQ)UHTXHQF\ FP 22  3KRQ clearly seen. The line is a guide to the eye. DY:HRYHUFW  S 7 DY:HRYHUFW  S 7 FIG. 4. Part of the electronic band structure of nanotube (6, 5) in the extended zone representation with characteristic phonon matrix elements are used in Eq. 1 for calculating diagrams for ee and eh two-phonon processes with largest (a) the two-phonon Raman spectrum of nanotube (6, 5) at single resonance, (b) double resonance, and (c) triple reso- E = 2.35 eV (see Fig. 2). The most intense band nance contribution to the 2D band. The diagrams are closed −1 in this spectrum is positioned at ≈ 2630 cm . This polygons of arrows, denoting the virtual processes. (d) Part of band arises from TO phonons in the vicinity of the K the TO branch of nanotube (6, 5) in the extended zone rep- point of the hexagonal graphene Brillouin zone and is resentation, including phonons, relevant to the two-phonon usually denoted as 2TO@K or simply as 2D band. The processes in (a), (b), and (c). 2D band, along with two less intense bands - TOLA@K at −1 −1 ≈ 2450 cm and 2LO@Γ at ≈ 3200 cm - are observed The two-peaked structure of the Raman spectrum, Fig. in the Raman spectra of the parent structure graphene. 3a, and the three-peaked structure of the integrated in- −1 In nanotubes, bands around 1950 cm have been ob- tensity, Fig. 3b, can be explained with the vHs of the served and assigned to scattering processes TOLA@Γ or electronic density of states and the small denominator LOLA@Γ (for discussion, see, e.g., Ref. 7). Here, we find in Eq. 1 for small E ’s. In the case of mirror pairs of iu that such combination modes have negligible intensity conduction parabolic bands and mirror pairs of valence and are unlikely to be connected to the observed Raman parabolic bands with respect to k = π/T , which is char- bands. On the basis of our results, we assign such exper- acteristic for nanotubes (Fig. 1a), there are two types of imental bands to the predicted here combination bands scattering processes that give rise to the 2D band (see TOZA@K and TOZO@K. Such combination modes are also Ref. 10): not observed in graphene, because scattering of electrons (a) ee (or hh) processes, with twice scattering of an by ZA and ZO phonons is not allowed. However, these electron (or a hole) by phonons with opposite momenta, scattering processes become allowed in nanotubes due to for which E = E −E +E , E = E −E +E −¯hω (or ia L c v ib L c their curved surface. ‘ E = E −E +E −¯hω), and E = E −E +E −2h ¯ω, ib L v ic L c v We focus on the behavior of the 2D band vs laser ex- (b) eh processes with scattering of an electron and a citation E in the range from 2.15 up to 2.60 eV (Fig. hole by two phonons with opposite momenta, for which 3a). It is clearly seen, that the 2D band undergoes an E = E − E + E , E = E − E + E − ¯hω (or ia L c v ib L c ‘ ‘ ‘ evolution from a single, Lorentz-like shape at small E E = E −E +E −¯hω), and E = E −E +E −2h ¯ω. L ib L v ic L c c v to a two-peaked structure at large E . The separation First of all, single resonances are present for pro- between the two components of the 2D band is practi- cesses, for which one of the E ’s vanishes. However, iu 5 this does not yield a significant increase of the Raman perimental data. intensity unless the process involves initial and final elec- The evolution of the 2D band of any nanotube with tron wavevectors at a vHs. Such electron wavevectors are laser excitation is expected to show similar behavior as connected by phonon wavevectors, close to the wavevec- that for nanotube (6, 5). Namely, the 2D band will split tor q between the extrema of mirror pairs of conduc- into a two-peaked structure with increasing E . The red 0 L tion bands and mirror pairs of valence bands (Fig. 4a). shift and blue shift of the constituent peaks will depend Thus, the scattering phonons will have frequency, close on the TO branch and the optical transition of the nan- to ω ≡ ω(q ) (Fig. 4d) and the Raman shift of the 2D otube. For small enough slope of the TO branch, splitting 0 0 band will be equal to 2ω (Fig. 3a). For the mentioned of the 2D band may not be observed at all. phonon wavevectors, E , E , and E turn to zero at ia ib ic The characteristic three-peak shape of the integrated E = E , E = E + h ¯ω , or E = E + 2h ¯ω , re- intensity vs laser excitation is quite different from that L 22 L 22 0 L 22 0 spectively. This corresponds to the derived three-peaked in graphene because the former is intrinsically connected structure of the integrated intensity of the 2D band (Fig. to the characteristic vHs in nanotubes, which are not 3b). present in graphene. Also, unlike graphene, it is difficult Double resonances are possible only for type ee/hh pro- to define dispersion rate of the 2D band, because of the cesses but the largest contribution comes from processes complex shape of this band. However, dispersion rate with initial or final electron wavevector at a vHs. There can be associated with each of the peaks of the 2D band are two phonon wavevectors for such processes, q and 1 and can be deduced from the slope of the part of the TO q (Fig. 4b). These phonons generally have different fre- 2 branch, which is relevant to the 2D band. quencies ω ≡ ω(q ) and ω ≡ ω(q ) (Fig. 4d) and give 1 1 2 2 The behavior of the remaining two-phonon bands is rise to two peaks of the 2D band with Raman shifts of similar to that of the 2D band. Namely, the Ra- 2ω and 2ω (Fig. 2a). In this case, E and E simul- 1 2 ia ib man bands have significant intensity for laser excitation taneously turn to zero at E = E + 2h ¯ω or E = L 22 1 L roughly in the range between the optical transition E ii E + 2h ¯ω . These processes will contribute to a double 22 2 and E + 2h ¯ω. For E close to E , single resonance ii L ii peak of the integrated intensity at E = E + 2h ¯ω and L 22 1 processes give rise to a non-dispersive two-phonon band. E = E + 2h ¯ω . However, because ω − ω is much L 22 2 2 1 With increasing E , double and triple resonance pro- smaller than the electronic linewidth, the two peaks are cesses become dominant and the two-phonon band splits likely to be observed as a single one at E = E + 2h ¯ω L 22 into two dispersive components. where ω ∈ [ω , ω ]. 1 2 Finally, triple resonances are possible only for type eh processes. In this case, the intensity is largest for initial IV. CONCLUSIONS or final electronic states at a vHs. Similarly to the case of double resonance, there are again two phonon wavevec- tors, q and q , satisfying this condition (Fig. 4c). The 1 2 We presented a computational approach to the calcula- corresponding phonon frequencies ω and ω are gener- tion of the two-phonon bands of carbon nanotubes, based 1 2 ally different (Fig. 4d) and give rise to two peaks of the on a symmetry-adapted non-orthogonal tight-binding 2D band with Raman shifts of 2ω and 2ω (Fig. 3a). 1 2 model. As a case study, we considered the narrow nan- All three factors E in the denominator of Eq. 1 vanish iu otube (6, 5) and analyzed the evolution of the 2D Raman simultaneously at E = E + 2h ¯ω or E = E + 2h ¯ω . L 22 1 L 22 2 band vs laser excitation. We found that this band splits Triple resonance processes will then contribute to the into two dispersive peaks with increasing laser excitation peak of the integrated intensity at E = E +2h ¯ω, where L 22 energy. Such behavior is expected for any two-phonon ω ∈ [ω , ω ]. 1 2 band of any nanotube with dispersion rate of the dis- The provided description of the contributions of the persive peaks, depending on the electronic and phonon various scattering processes to the 2D band allows ana- dispersion, and laser excitation. The adopted symmetry- lyzing the complex shape of this band. With increasing adapted approach significantly reduces the computa- E from below E , the 2D band is initially a single- L 22 tional efforts in comparison with the approach, based peak one, mainly due to single resonance processes. At only on the translational symmetry of the nanotubes, higher E , the 2D band shape evolves into a two-peak L and allows one to derive the two-phonon Raman bands one, due to the dominant contributions of double and of any observable carbon nanotube. triple resonance processes; the two peaks of the 2D band are red-shifted and blue-shifted, depending on the behav- ior of the TO phonon branch. We note that for such E ACKNOWLEDGMENTS there will also be a contribution to the 2D band from single resonances due to TO phonons with frequency ω , but it is much smaller than those from double and triple V.N.P. acknowledges financial support by the Na- resonances and cannot to be resolved. The predicted tional Science Fund of Bulgaria under grant DN18/9- two-peaked 2D band corresponds well to the recent ex- 11.12.2017. 6 1 11 M. S. Dresselhaus, A. Jorio, M. Hofmann, G. Dresselhaus, V. N. Popov and P. Lambin, Eur. Phys. J. B 85, 418 and R. Saito, Nano Lett. 10, 751 (2010). (2012). 2 12 S. Reich, C. Thomsen, and J. Maultzsch, Carbon Nan- F. Herziger, A. Vierck, J. Laudenbach, and J. Maultzsch, otubes: Basic Concepts and Physical Properties (Wiley- Phys. Rev. B 92, 235409 (2015). VCH, New York, 2004). A. Vierck, F. Gannott, M. Schweiger, J. Zaumseil, and A. Jorio, G. Dresselhaus, and M. S. Dresselhaus, eds., J. Maultzsch, Carbon 117, 360 (2017). “Carbon nanotubes: Advanced topics in the synthesis, L. G. Moura, M. V. O. Moutinho, P. Venezuela, F. Mauri, structure, properties, and applications,” in Topics Appl. A. Righi, M. S. Strano, C. Fantini, and M. A. Pimenta, Physics, Vol. 111 (Springer, Berlin, 2008). Carbon 117, 41 (2017). 4 15 C. Thomsen and S. Reich, “Raman scattering in carbon V. N. Popov and L. Henrard, Phys. Rev. B 70, 115407 nanotubes,” in Light Scattering in Solid IX, edited by (2004). M. Cardona and R. Merlin (Springer Berlin Heidelberg, V. N. Popov, L. Henrard, and P. Lambin, Phys. Rev. B Berlin, Heidelberg, 2007) p. 115. 72, 035436 (2005). 5 17 P. T. Araujo, I. O. Maciel, P. B. C. Pesce, M. A. Pimenta, V. N. Popov and P. Lambin, Phys. Rev. B 73, 085407 S. K. Doorn, H. Qian, A. Hartschuh, M. Steiner, L. Grigo- (2006). rian, K. Hata, and A. Jorio, Phys. Rev. B 77, 241403(R) V. N. Popov and P. Lambin, Phys. Rev. B 74, 075415 (2008). (2006). 6 19 A. Jorio, A. G. SouzaFilho, G. Dresselhaus, M. S. Dres- V. N. Popov and P. Lambin, Nano Res. 3, 822 (2010). selhaus, A. K. Swan, M. S. Unlu¨, B. B. Goldberg, M. A. D. Porezag, T. Frauenheim, T. K¨ohler, G. Seifert, and Pimenta, J. H. Hafner, C. M. Lieber, and R. Saito, Phys. R. Kaschner, Phys. Rev. B 51, 12 947 (1995). Rev. B 65, 155412 (2002). T. Vukovi´c, I. 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Two-phonon Raman bands of single-walled carbon nanotubes: a case study

Popov, Valentin N.
Condensed Matter , Volume 2018 (1804) – Apr 10, 2018
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2469-9950
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ARCH-3331
DOI
10.1103/PhysRevB.98.085413
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Abstract

Valentin N. Popov Faculty of Physics, University of Sofia, BG-1164 Sofia, Bulgaria (Dated: November 9, 2018) It has been long accepted that the second-order Raman bands in carbon nanotubes are enhanced through the double-resonance mechanism. Although separate aspects of this mechanism have been studied for a few second-order Raman bands, including the most intense defect-induced D band and the two-phonon 2D band, a complete computational approach to the second-order bands is still lacking. Here, we propose such an approach, entirely based on a symmetry-adapted non-orthogonal tight-binding model with ab-initio-derived parameters. As a case study, we consider nanotube (6, 5), for which we calculate the two-phonon spectrum. We investigate in detail the 2D band and identify three contributions to it: a non-dispersive one and two dispersive ones, which are found to depend on the electron and phonon dispersion of the nanotube, and on the laser excitation. We also predict two-phonon bands, which are not allowed in the parent structure graphene. The obtained two-phonon bands are in very good agreement with the available experimental data. The symmetry-adapted formalism makes feasible the calculation of the two-phonon Raman bands of any observable nanotube. I. INTRODUCTION for certain laser excitations, which has been explained by the the double-resonant (DR) scattering mechanism, initially proposed for graphene. In the early days of the In the last few decades, the layered carbon materi- modeling of these bands in graphene, simplifying assump- als like fullerenes, nanotubes, and few-layer graphene tions such as constant electronic linewidth, limiting the have been a subject of intense experimental and theoret- calculations only to high-symmetry directions in the Bril- ical study, because of their unique properties, originating louin zone, and exact DR conditions have been used (for a from their zero-, one-, two-, and three-dimensionality. In review, see, e.g., Ref. 10). Only recently, theoretical two- particular, significant progress has already been achieved phonon bands in graphene, which are in good agreement in the synthesis and the study of the properties and ap- 10,11 with the experimental data, have been reported. The 2,3 plication of carbon nanotubes The application of nan- success of the latter studies is based on the explicit calcu- otubes in nanoelectronics requires their precise structural lation of the couplings and the electronic linewidth, and characterization. For this purpose, the Raman scattering performing the integration of the quantum-mechanical by phonons is the experimental technique of choice, being expression for the Raman intensity over the entire Bril- a fast and nondestructive characterization method. louin zone for both electrons and phonons. The so-called single-walled nanotube (or briefly nan- The theoretical description of the two-phonon bands otube) can be viewed as obtained by rolling up a graphene of nanotubes is much more complex than for graphene, sheet into a seamless cylinder. It has a few intense first- mainly, because of the quantum confinement in one di- order Raman bands, arising from the radial-breathing mension and the large variety of nanotube types. Re- mode (RBM ) and the longitudinal and transverse tan- cently, a step towards the understanding of the defect- gential modes (G modes). The RBM frequency is found induced D mode in nanotubes vs diameter and energy has to be inversely-proportional to the nanotube radius and been made using the hexagonal symmetry of graphene is normally used for fast sample characterization. The 12,13 and geometrical considerations. This approach al- G modes also depend on the nanotube radius and can lows assigning the two-phonon bands to pairs of defi- be used to support the assignment of the Raman spec- nite phonons. However, the two-phonon band shape can tra to particular nanotubes but also allow for differenti- only be predicted, taking into account the couplings and ating between metallic and semiconducting nanotubes. the electronic linewidth, and carrying out full integra- Intense second-order Raman bands are also observed in tion over the Brillouin zone. In a recent publication, nanotubes. These bands arise from scattering processes, this has been done for a few narrow nanotubes using ab- which can involve two phonons with opposite momenta initio and tight-binding approaches, neglecting, however, (two-phonon bands) or a phonon and a defect (defect- the effects of the nanotube curvature. induced bands). The most intense two-phonon band (so- −1 called 2D band), usually observed around 2700 cm , is Here, we propose a computational approach to the due to electron/hole scattering by two transverse opti- calculation of the two-phonon Raman spectra of car- cal (TO) phonons with opposite momenta. Other two- bon nanotubes, which is entirely based on a symmetry- phonon bands have also been measured (for summary, adapted ab-initio-based non-orthogonal tight-binding see, e.g., Ref. 7). The two-phonon spectra contain valu- (NTB) model. This model has been used for more than able information about the phonon dispersion of the a decade for the successful prediction of the electronic nanotubes. band structure and phonon dispersion, and the first-order 15–19 The D and 2D band intensity is found to be enhanced Raman bands of several hundred nanotube types. arXiv:1804.03402v2 [cond-mat.mes-hall] 21 Jun 2018 2 The model describes the curvature effects on the phys- the phonons are labeled by the one-dimensional wavevec- ical properties of carbon nanotubes, which are essential tor q, q ∈ [0, 1) 2π/T , and the integer quantum number for nanotube diameters below about 1 nm. As a case λ, λ ∈ [0, N). The selection rules for scattering of elec- ′ ′ study, we consider the narrow nanotube (6, 5) and per- trons by phonons are k = k+q±2π/T and l = l+λ±L; form a complete calculation of the two-phonon bands L = (Nν + n)/n˜, where ν is an integer number such that at a number of laser excitations and discuss the con- L is an integer number. These selection rules can be de- tributions to the 2D band, as well as the appearance rived similarly to those in Ref.18 by including Umklapp of two-phonon bands, which are symmetry-forbidden for processes. Notice that the integer quantum number is the parent structure graphene. non-conserving. In quantum mechanics, the two-phonon process can The paper is organized as follows. The theoretical background is presented in Sec. II. The obtained results be viewed as a sequence of virtual processes, namely, are discussed in Sec. III. The paper ends up with con- electron-hole creation, scattering of electrons or holes clusions, Sec. IV. by two phonons with opposite momenta, and finally electron-hole annihilation. The wavevector is conserved for each virtual process, while the energy is conserved only for the entire two-phonon process. The correspond- II. THEORETICAL BACKGROUND ing two-phonon Raman intensity can be described by an expression, derived in fourth-order quantum-mechanical A nanotube can be considered as obtained by cutting 3,4,10,11,22 perturbation theory, out a rectangle of graphene, defined by a pair of or- ~ ~ thogonal lattice vectors T and C, and rolling it along C into a seamless cylinder. This rolled-up nanotube can X X M M M M fc cb ba ai be characterized by the radius R = kCk/2π, translation I ∝ δ (E − E ) (1) i f ΔE ΔE ΔE ic ib ia period T ≡ kTk, as well as by the chiral angle θ, which f c,b,a is the angle between C and the nearest zigzag of carbon Here, ΔE = E − iγ and E = E − E ; E is atoms. All structural parameters of the rolled-up nan- iu iu iu i u i the energy of the initial state; E = E , where E otube can be expressed by means of the nearest-neighbor i L L is the incident photon energy (laser excitation); E , interatomic distance and the hexagonal indices (n, m) of u u = a, b, c, f, are the energies of the intermediate (a, b, c) ~ ~ C or (n˜, m ˜ ) of T. The former notation is traditionally and final (f) states of the system of photons, electrons, used to specify uniquely the nanotube and is accepted holes, and phonons. M are the matrix elements be- uv here as well. Normally, the total energy of the rolled-up tween initial, intermediate, and final states. M and ai nanotube is not minimal and the atomic structure of the M are momentum matrix elements. M and M are fc ba cb nanotube has to be subjected to structural relaxation, electron-phonon matrix elements. The electron-photon which is a necessary step before performing phonon cal- and electron-phonon matrix elements are calculated ex- culations. plicitly within the NTB model. γ = γ + γ , where c v The straightforward calculation of the electronic states γ and γ are the halfwidths of conduction (c) and va- c v and phonons for a large variety of nanotubes is accom- lence (v) states, respectively. The summation over the panied by insurmountable computational difficulties, be- phonon wavevectors q and the electron wavevector k is cause of the very large translational unit cells of most carried out over a mesh of 400 points in the Brillouin of the observed nanotubes. Fortunately, the nanotubes zone. The Raman intensity is enhanced for vanishing of have screw symmetry that allows reducing the compu- one, two, or three of the E ’s in Eq. 1, and therefore iu tational efforts by resorting to two-atom unit cells. The single, double, and triple resonances can occur. The in- latter approach has been used for calculation of the elec- 15 17 cident and scattered light are assumed polarized along tronic states and phonons of several hundred nan- the nanotube axis and only Stokes processes are consid- otubes within the NTB model. In this model, the Hamil- ered in this work. tonian and overlap matrix elements are derived as a function of the interatomic separation from an ab-initio study on carbon dimers and the Slater-Koster scheme III. RESULTS AND DISCUSSION is adopted for the angular dependence of the matrix ele- ments. The imposing of the translational periodicity and ro- We consider nanotube (6, 5) for exemplifying the be- tational boundary conditions on the solutions of the elec- havior of the two-phonon bands vs laser excitation. The tron and phonon eigenvalue problems results in labeling translational unit cell of the nanotube contains N = 182 of these solutions by a pair of indices. The electronic two-atom unit cells. The atomic structure of the nan- states are labeled by the one-dimensional wavevector k, otube is relaxed, retaining its circular cylindrical form. k ∈ [0, 1) 2π/T , and the integer quantum number l, In this case, as independent structural parameters, we l ∈ [0, N), where N is the number of two-atom unit cells choose the radius R, the translation period T , and the in the translational unit cell of the nanotube. Similarly, coordinates of the second atom relative to the first atom 3 in the zeroth two-atom unit cell, and obtain the relaxed in the parabolic bands, associated with the transition ˚ ˚ parameters R = 3.76 A and T = 40.67 A. E = 2.20 eV. (  H9 J 72#. ' DY:YFRHUH W S 7  (QH HUJ9 \ /2 /2 72=$#. 2=2. 7# /2# /$ /$ EXQLWV 5DPDQ,QWHQVLW\ DU /. 72$# 7$ 7$ 2= 2= 5DPD6KQ LWIFP $= $= 3KRQRQ)UHTXHQF\ FP FIG. 2. (a) Calculated two-phonon Raman spectrum of nan- otube (6, 5) at laser excitation E = 2.35 eV. The most in- tense two-phonon bands are labelled. The spectrum is slightly D:YHYHFRW U S 7 upshifted for clarity. FIG. 1. (a) Electronic band structure of nanotube (6, 5) close The phonon dispersion of nanotube (6, 5) consists of to the Fermi energy, chosen as zero. The optical transition 6N branches and their representation in the Brillouin E is denoted by vertical arrows. (b) Phonon dispersion of nanotube (6, 5) in the extended zone representation. The zone for the translational unit cell is not very informa- pairs of close vertical lines bracket the phonons, taking part in tive, because the branches would cover densely the graph. scattering between the extrema of the parabolic bands for the Alternatively, one can draw fewer phonon branches in the E transition. (c) Halfwidths of valence (γ ) and conduction 22 v extended zone representation, namely, 6ν, where ν is (γ ) states of nanotube (6, 5). The corresponding halfwidths the greatest common divisor of n and m. In the case for graphene (dotted lines) are provided for comparison. of nanotube (6, 5), ν = 1 and therefore there are only 6 branches in this representation. These branches have in- The electronic band structure of nanotube (6, 5) in the plane longitudinal optical (LO), transverse optical (TO), entire Brillouin zone and for energies up to ±1.5 eV rel- longitudinal acoustic (LA), and transverse acoustic (TA) ative to the Fermi energy E = 0 eV in shown in Fig. character, and out-of-plane optical (ZO) and acoustic 1a. The valence and the conduction bands in this energy (ZA) character. The unfolded phonon dispersion con- range are bands of graphene of mainly π and π charac- sists of parts of width 2π/T characterized by different ter, folded along the cutting lines of the Brillouin zone of values of the integer quantum number (Fig. 1b). The graphene. None of the cutting lines passes through the selection rules for the wavevector and the integer quan- K and K points of the Brillouin zone of graphene and tum number impose restrictions on the allowed values therefore there are no linear bands that cross the Fermi of the phonon wavevectors ±q for scattering of electrons energy in the Brillouin zone of the nanotube. Thus, un- between the parabolic bands for transition E to the like graphene, nanotube (6, 5) is a semiconductor. Due to regions between the pairs of close vertical lines in Fig. the presence of electronic bands with extrema (so-called 1b. parabolic bands), the electronic density of states for such The Raman intensity, Eq. 1, depends crucially on the bands has singularities at the energies of the extrema, electronic linewidth. The calculated linewidth for nan- so-called van Hove singularities (vHs) (not shown). The otube (6, 5) exhibits sharp spikes, arising from the sin- absorption of electromagnetic radiation is enhanced for gularities of the electronic density of states, while that photon energies, matching mirror pairs of vHs with re- for graphene is a smooth function of energy (Fig. 1c). spect to the Fermi energy. In the mentioned energy The singularities of the linewidth effectively decrease the range, the NTB model predicts only two optical transi- Raman intensity close to the transition energies of the tions for nanotube (6, 5), which take place between the nanotube. first and the second pair of mirror vHs and are denoted The derived electronic band structure and phonon as E and E , respectively. Here, we will be interested dispersion, halfwidths, electron-photon and electron- 11 22 4 cally zero for energies below 2.35 eV but increases up to E −1 a few tens of cm at 2.60 eV. The integrated Raman in- tensity exhibits three peaks with separation between the adjacent peaks of about h ¯ω, where ω is a characteristic ( K Z TO phonon frequency (Fig. 3b). We have not attempted fitting the 2D band with two Lorentzians and deriving the Raman excitation profile for each of them, because of the strong overlap of the two peaks of this band. ( K Z ( ( ( ( F F F F U([FLWDWLRQ H9 /DVH T Q,QWHQVLW\ DUEXQLWV 5DPD T (QHUJ\ H9 (QHUJ\ H9 ( ( ( ( Y Y Y DY:HRYHUFW   S 7 DY:HRYHUFW  S 7 5DPD6KQ LWIFP ( ( F F FIG. 3. (a) Calculated Raman 2D band of nanotube (6, 5) vs Z T laser excitation E . The 2D band shape evolves from a single- L T to two-peaked structure (peaks 1 and 2) with increasing E from 2.15 to 2.60 eV with a step of 0.01 eV (see right panel Z T axis). (b) Integrated Raman intensity of the 2D band (in arb. (QUJ\ H9 Z T units) vs E . Three peaks, centered at E = E , E + ¯hω, L L 22 22 Y Y and E + 2¯hω, where ω is the TO phonon frequency, are RQ)UHTXHQF\ FP 22  3KRQ clearly seen. The line is a guide to the eye. DY:HRYHUFW  S 7 DY:HRYHUFW  S 7 FIG. 4. Part of the electronic band structure of nanotube (6, 5) in the extended zone representation with characteristic phonon matrix elements are used in Eq. 1 for calculating diagrams for ee and eh two-phonon processes with largest (a) the two-phonon Raman spectrum of nanotube (6, 5) at single resonance, (b) double resonance, and (c) triple reso- E = 2.35 eV (see Fig. 2). The most intense band nance contribution to the 2D band. The diagrams are closed −1 in this spectrum is positioned at ≈ 2630 cm . This polygons of arrows, denoting the virtual processes. (d) Part of band arises from TO phonons in the vicinity of the K the TO branch of nanotube (6, 5) in the extended zone rep- point of the hexagonal graphene Brillouin zone and is resentation, including phonons, relevant to the two-phonon usually denoted as 2TO@K or simply as 2D band. The processes in (a), (b), and (c). 2D band, along with two less intense bands - TOLA@K at −1 −1 ≈ 2450 cm and 2LO@Γ at ≈ 3200 cm - are observed The two-peaked structure of the Raman spectrum, Fig. in the Raman spectra of the parent structure graphene. 3a, and the three-peaked structure of the integrated in- −1 In nanotubes, bands around 1950 cm have been ob- tensity, Fig. 3b, can be explained with the vHs of the served and assigned to scattering processes TOLA@Γ or electronic density of states and the small denominator LOLA@Γ (for discussion, see, e.g., Ref. 7). Here, we find in Eq. 1 for small E ’s. In the case of mirror pairs of iu that such combination modes have negligible intensity conduction parabolic bands and mirror pairs of valence and are unlikely to be connected to the observed Raman parabolic bands with respect to k = π/T , which is char- bands. On the basis of our results, we assign such exper- acteristic for nanotubes (Fig. 1a), there are two types of imental bands to the predicted here combination bands scattering processes that give rise to the 2D band (see TOZA@K and TOZO@K. Such combination modes are also Ref. 10): not observed in graphene, because scattering of electrons (a) ee (or hh) processes, with twice scattering of an by ZA and ZO phonons is not allowed. However, these electron (or a hole) by phonons with opposite momenta, scattering processes become allowed in nanotubes due to for which E = E −E +E , E = E −E +E −¯hω (or ia L c v ib L c their curved surface. ‘ E = E −E +E −¯hω), and E = E −E +E −2h ¯ω, ib L v ic L c v We focus on the behavior of the 2D band vs laser ex- (b) eh processes with scattering of an electron and a citation E in the range from 2.15 up to 2.60 eV (Fig. hole by two phonons with opposite momenta, for which 3a). It is clearly seen, that the 2D band undergoes an E = E − E + E , E = E − E + E − ¯hω (or ia L c v ib L c ‘ ‘ ‘ evolution from a single, Lorentz-like shape at small E E = E −E +E −¯hω), and E = E −E +E −2h ¯ω. L ib L v ic L c c v to a two-peaked structure at large E . The separation First of all, single resonances are present for pro- between the two components of the 2D band is practi- cesses, for which one of the E ’s vanishes. However, iu 5 this does not yield a significant increase of the Raman perimental data. intensity unless the process involves initial and final elec- The evolution of the 2D band of any nanotube with tron wavevectors at a vHs. Such electron wavevectors are laser excitation is expected to show similar behavior as connected by phonon wavevectors, close to the wavevec- that for nanotube (6, 5). Namely, the 2D band will split tor q between the extrema of mirror pairs of conduc- into a two-peaked structure with increasing E . The red 0 L tion bands and mirror pairs of valence bands (Fig. 4a). shift and blue shift of the constituent peaks will depend Thus, the scattering phonons will have frequency, close on the TO branch and the optical transition of the nan- to ω ≡ ω(q ) (Fig. 4d) and the Raman shift of the 2D otube. For small enough slope of the TO branch, splitting 0 0 band will be equal to 2ω (Fig. 3a). For the mentioned of the 2D band may not be observed at all. phonon wavevectors, E , E , and E turn to zero at ia ib ic The characteristic three-peak shape of the integrated E = E , E = E + h ¯ω , or E = E + 2h ¯ω , re- intensity vs laser excitation is quite different from that L 22 L 22 0 L 22 0 spectively. This corresponds to the derived three-peaked in graphene because the former is intrinsically connected structure of the integrated intensity of the 2D band (Fig. to the characteristic vHs in nanotubes, which are not 3b). present in graphene. Also, unlike graphene, it is difficult Double resonances are possible only for type ee/hh pro- to define dispersion rate of the 2D band, because of the cesses but the largest contribution comes from processes complex shape of this band. However, dispersion rate with initial or final electron wavevector at a vHs. There can be associated with each of the peaks of the 2D band are two phonon wavevectors for such processes, q and 1 and can be deduced from the slope of the part of the TO q (Fig. 4b). These phonons generally have different fre- 2 branch, which is relevant to the 2D band. quencies ω ≡ ω(q ) and ω ≡ ω(q ) (Fig. 4d) and give 1 1 2 2 The behavior of the remaining two-phonon bands is rise to two peaks of the 2D band with Raman shifts of similar to that of the 2D band. Namely, the Ra- 2ω and 2ω (Fig. 2a). In this case, E and E simul- 1 2 ia ib man bands have significant intensity for laser excitation taneously turn to zero at E = E + 2h ¯ω or E = L 22 1 L roughly in the range between the optical transition E ii E + 2h ¯ω . These processes will contribute to a double 22 2 and E + 2h ¯ω. For E close to E , single resonance ii L ii peak of the integrated intensity at E = E + 2h ¯ω and L 22 1 processes give rise to a non-dispersive two-phonon band. E = E + 2h ¯ω . However, because ω − ω is much L 22 2 2 1 With increasing E , double and triple resonance pro- smaller than the electronic linewidth, the two peaks are cesses become dominant and the two-phonon band splits likely to be observed as a single one at E = E + 2h ¯ω L 22 into two dispersive components. where ω ∈ [ω , ω ]. 1 2 Finally, triple resonances are possible only for type eh processes. In this case, the intensity is largest for initial IV. CONCLUSIONS or final electronic states at a vHs. Similarly to the case of double resonance, there are again two phonon wavevec- tors, q and q , satisfying this condition (Fig. 4c). The 1 2 We presented a computational approach to the calcula- corresponding phonon frequencies ω and ω are gener- tion of the two-phonon bands of carbon nanotubes, based 1 2 ally different (Fig. 4d) and give rise to two peaks of the on a symmetry-adapted non-orthogonal tight-binding 2D band with Raman shifts of 2ω and 2ω (Fig. 3a). 1 2 model. As a case study, we considered the narrow nan- All three factors E in the denominator of Eq. 1 vanish iu otube (6, 5) and analyzed the evolution of the 2D Raman simultaneously at E = E + 2h ¯ω or E = E + 2h ¯ω . L 22 1 L 22 2 band vs laser excitation. We found that this band splits Triple resonance processes will then contribute to the into two dispersive peaks with increasing laser excitation peak of the integrated intensity at E = E +2h ¯ω, where L 22 energy. Such behavior is expected for any two-phonon ω ∈ [ω , ω ]. 1 2 band of any nanotube with dispersion rate of the dis- The provided description of the contributions of the persive peaks, depending on the electronic and phonon various scattering processes to the 2D band allows ana- dispersion, and laser excitation. The adopted symmetry- lyzing the complex shape of this band. With increasing adapted approach significantly reduces the computa- E from below E , the 2D band is initially a single- L 22 tional efforts in comparison with the approach, based peak one, mainly due to single resonance processes. At only on the translational symmetry of the nanotubes, higher E , the 2D band shape evolves into a two-peak L and allows one to derive the two-phonon Raman bands one, due to the dominant contributions of double and of any observable carbon nanotube. triple resonance processes; the two peaks of the 2D band are red-shifted and blue-shifted, depending on the behav- ior of the TO phonon branch. We note that for such E ACKNOWLEDGMENTS there will also be a contribution to the 2D band from single resonances due to TO phonons with frequency ω , but it is much smaller than those from double and triple V.N.P. acknowledges financial support by the Na- resonances and cannot to be resolved. The predicted tional Science Fund of Bulgaria under grant DN18/9- two-peaked 2D band corresponds well to the recent ex- 11.12.2017. 6 1 11 M. S. Dresselhaus, A. Jorio, M. Hofmann, G. Dresselhaus, V. N. Popov and P. Lambin, Eur. Phys. J. B 85, 418 and R. Saito, Nano Lett. 10, 751 (2010). (2012). 2 12 S. Reich, C. Thomsen, and J. Maultzsch, Carbon Nan- F. Herziger, A. Vierck, J. Laudenbach, and J. Maultzsch, otubes: Basic Concepts and Physical Properties (Wiley- Phys. Rev. B 92, 235409 (2015). VCH, New York, 2004). A. Vierck, F. Gannott, M. Schweiger, J. Zaumseil, and A. Jorio, G. Dresselhaus, and M. S. Dresselhaus, eds., J. Maultzsch, Carbon 117, 360 (2017). “Carbon nanotubes: Advanced topics in the synthesis, L. G. Moura, M. V. O. Moutinho, P. Venezuela, F. Mauri, structure, properties, and applications,” in Topics Appl. A. Righi, M. S. Strano, C. Fantini, and M. A. Pimenta, Physics, Vol. 111 (Springer, Berlin, 2008). Carbon 117, 41 (2017). 4 15 C. Thomsen and S. 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Condensed Matter – arXiv (Cornell University)

Published: Apr 10, 2018

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