Kohn Anomalies in Graphene Nanoribbons
Ken-ichi Sasaki, Masayuki Yamamoto, Shuichi Murakami, Riichiro Saito, Mildred S. Dresselhaus, Kazuyuki Takai, Takanori Mori, Toshiaki Enoki, Katsunori Wakabayashi
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Kohn Anomalies in Graphene Nanoribbons
Ken-ichi Sasaki ∗ and Masayuki Yamamoto National Institute for Materials Science, Namiki, Tsukuba 305-0044, Japan
Shuichi Murakami
Department of Physics, Tokyo Institute of Technology,Ookayama, Meguro-ku, Tokyo 152-8551, Japan andPRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan
Riichiro Saito
Department of Physics, Tohoku University, Sendai 980-8578, Japan
Mildred S. Dresselhaus
Department of Physics, Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Cambridge, MA 02139-4307
Kazuyuki Takai, Takanori Mori, and Toshiaki Enoki
Department of Chemistry, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Katsunori Wakabayashi
International Center for Materials Nanoarchitectonics,National Institute for Materials Science, Namiki, Tsukuba 305-0044, Japan andPRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan (Dated: October 29, 2018)The quantum corrections to the energies of the Γ point optical phonon modes (Kohn anomalies) ingraphene nanoribbons are investigated. We show theoretically that the longitudinal optical modesundergo a Kohn anomaly effect, while the transverse optical modes do not. In relation to Ramanspectroscopy, we show that the longitudinal modes are not Raman active near the zigzag edge, whilethe transverse optical modes are not
Raman active near the armchair edge. These results are usefulfor identifying the orientation of the edge of graphene nanoribbons by G band Raman spectroscopy,as is demonstrated experimentally. The differences in the Kohn anomalies for nanoribbons and formetallic single wall nanotubes are pointed out, and our results are explained in terms of pseudospineffects.
I. INTRODUCTION
Graphene nanoribbons (NRs) are rectangular sheetsof graphene with lengths up to several micrometersand widths as small as nanometers.
NRs can beregarded as unrolled single wall nanotubes (SWNTs).Since SWNTs exhibit either metallic or semiconductingbehavior depending on the diameter and chirality of thehexagonal carbon lattice of the tube, it is expected thatthe electronic properties of NRs depend on the width and“chirality” of the edge. In fact it has been predictedthat their electronic properties near the zigzag edge arequite different from those near the armchair edge.
Thus the characterization of the NRs as well as SWNTsis a matter of prime importance.Raman spectroscopy has been widely used for thecharacterization of SWNTs andgraphenes.
Recently, it has been shownthat the frequencies and spectral widths of the Γ pointoptical phonons (called the G band in Raman spectra)depend on the position of the Fermi energy E F and thechirality of the metallic SWNT. The Fermi en-ergy dependence of the Raman spectra can be used to determine the position of the Fermi energy, and the chi-rality dependence of the Raman spectra provides detailedinformation on the electronic properties near the Fermienergy of metallic SWNTs. These dependences originatefrom the fact that the conduction electrons of a metalpartly screen the electronic field of the ionic lattice. Kohnpointed out that the ability of the electrons to screen theionic electric field depends strongly on the geometry ofthe Fermi surface, and this screening leads to a change inthe frequency of a specific phonon and an increase in itsdissipation (the Kohn anomalies). The Kohn anomalies(KAs) in graphene systems are unique in the sense thatthey can occur with respect to the Γ point phonons whilethe KA in a normal metal occurs with respect to phononswith 2 k F where k F is the Fermi wave vector. The unique-ness of graphene comes from the geometry of the Fermisurface given by the Dirac cone. Since the geometry of the Fermi surface of NRs, andthe energy band structure of NRs depend on the orien-tation of the edge, one may expect that the KAsof NRs depend on the “chirality” of the edge like theKAs of metallic SWNTs. In this paper, we study KAsin graphene NRs with zigzag and armchair edges. A NRwith a zigzag (armchair) edge is hereafter referred to asa Z-NR (an A-NR) for simplicity (see Fig. 1(a) for an N Z-NR with a width W = N ℓ where ℓ ≡ a cc / a cc (= 0 . et al . observed that the intensity of the Dband near the armchair edge of highly ordered pyrolyticgraphite (HOPG) is much stronger than that near thezigzag edge of HOPG. This is confirmed for single layergraphene by the experiments of You et al . who alsoshow that the D band has a very strong laser polarizationdependence. However, a strong D band intensity appearsonly ∼
20 nm from the edges, so that the observation ofthis effect requires a precise experimental technique. This paper is organized as follows. In Sec. II, we studyKAs in Z-NRs and A-NRs. In Sec. III, we point out thatthe LO modes in Z-NRs (TO modes in A-NRs) are notRaman active and show experimental results. In Sec. IV,the mechanism of the edge dependent KAs and Ramanintensities is explained in terms of the pseudospin. Adiscussion and summary are given in Sec. V and Sec. VI.
II. KOHN ANOMALIESA. Zigzag NRs
KAs are relevant to the electron-phonon (el-ph) matrixelement for electron-hole pair creation. The electron-holepair creation should be a vertical transition for the Γpoint optical phonon, and the KA effect, such as an in-crease in the dissipation of the phonon, is possible onlywhen the energy band gap of a NR is smaller than theenergy of the phonon (about 0.2eV ). First, we studythe energy band structures of Z-NRs whose lattice andphonon modes are shown in Figs. 1(a) and (c,d), respec-tively. Z-NRs have a metallic energy band structure re-gardless of their widths as shown in Fig. 2(a). The metal-licity of Z-NRs is due to the edge states forming a flatenergy band at E = 0. Similarly, armchair SWNTs havea metallic energy band regardless of their diameters asshown in Fig. 2(b). It is interesting to imagine that an N = 2 n − n, n ) armchairSWNT by cutting the circumferential C-C bonds alongthe tube axis as shown in Fig. 1(b). For example, froma (5 ,
5) armchair SWNT, we get an N = 9 Z-NR. Themetallicity of armchair SWNTs is preserved in Z-NRs bythe edge states. FIG. 1: (a) The lattice structure of a Z-NR. The solid(empty) circles denote the A (B) sublattice. We use integers I ∈ [0 , M ] and J ∈ [0 , N ] for the axes. The width (length) W ( L ) of a Z-NR is given by Nℓ ( aM where a ≡ √ a cc ). (b) An( n, n ) armchair SWNT is cut along its axis and flattened outto make an N = 2 n − δγ , = 0 and δγ , = − δγ , . The TOmode is characterized by δγ , = − δγ , and δγ , = δγ , ,and δγ , , δγ , , δγ , are defined by the inset.FIG. 2: The energy band structure of an N = 9 Z-NR (a) andthat of a (5 ,
5) armchair SWNT (b). (c) | V L /δγ | as a functionof ka . The matrix elements of the vertical transitions, denotedby the arrows (1,2) in (a) and (b), are shown as the solid anddashed curves, respectively. The phonon eigenvector of the LO (TO) mode is par-allel (perpendicular) to the zigzag edge as shown inFig. 1c(d). By the displacement of a C-atom, a bondlength increases or decreases depending on the positionof the bond. A change of bond length causes a changeof the three nearest-neighbor hopping integrals from − γ (= 2 . − γ + δγ ,a ( a = 1 , ,
3) (see the inset toFig. 1(c,d)). The electron eigen function in the presenceof δγ ,a is given by a linear combination of those in theabsence of δγ ,a . In other words, a shift δγ ,a works asa perturbation by which an electron in the valence bandmay be transferred to a state in the conduction band.This is an electron-hole pair creation process due to alattice deformation. We derive the el-ph matrix elementas follows.Let ψ IJ A ( ψ I ′ J B ) denote the wave function of an electronat an A-site (B-site) at site IJ ( I ′ J where I ′ ≡ I + 1 / P I ( ψ IJ A ) ∗ δγ , ψ I ′ J B is the amplitude for the pro-cess that an electron at the B-sites with I ′ J is trans-ferred by the perturbation δγ , into the A-sites with IJ (see Fig. 1(a)). Note that ( ψ IJ A ) ∗ indicates complex con-jugation of ψ IJ A and δγ , for the LO and TO modes isconstant. By introducing the Bloch function ( φ J A , φ J B )as ψ IJ A = ( e iI ( ka ) / √ M ) φ J A and ψ I ′ J B = ( e iI ′ ( ka ) / √ M ) φ J B where k is the wave vector along the zigzag edge, weobtain P I ( ψ IJ A ) ∗ δγ , ψ I ′ J B = ( φ J A ) ∗ δγ , e ika/ φ J B . By de-riving the matrix elements for δγ , and δγ , in a similarmanner, we obtain the el-ph matrix element for a verticalelectron-hole pair creation process such as V + U , where V = X J (cid:18) φ J A − φ J B (cid:19) † (cid:18) e − i ka δγ , + e i ka δγ , c . c . (cid:19) (cid:18) φ J A φ J B (cid:19) ,U = X J,J ′ (cid:18) φ J ′ A − φ J ′ B (cid:19) † (cid:18) δγ , δ J ′ ,J +1 δγ , δ J ′ ,J − (cid:19) (cid:18) φ J A φ J B (cid:19) . (1) V and U represent the el-ph interaction for the δγ , ( δγ , ) perturbation which acts for the same J and the δγ , perturbation which acts for the nearest neighborpair of J and J ′ , respectively. For V in Eq. (1), c . c . repre-sents the complex conjugation of e − i ka δγ , + e i ka δγ , .It is also noted that the minus signs in front of φ J B inEq. (1) come from the fact that the Bloch function withenergy E in the conduction band is given by ( φ J A , − φ J B )when the Bloch function with energy − E in the valenceband is ( φ J A , φ J B ). This is a property of the nearest-neighbor tight-binding Hamiltonian with two sublattices.The matrix elements V and U in Eq. (1) can be rewrittenas V = V T + V L (2)where V T = 2 i ( δγ , + δγ , ) cos (cid:18) ka (cid:19) X J Im (cid:2) φ J ∗ A φ J B (cid:3) ,V L = 2 i ( δγ , − δγ , ) sin (cid:18) ka (cid:19) X J Re (cid:2) φ J ∗ A φ J B (cid:3) , (3)and U = 2 iδγ , X J Im (cid:2) φ J +1 ∗ A φ J B (cid:3) . (4)By assuming that the perturbation δγ ,a is proportionalto a change of the bond length, we have δγ , = 0 and δγ , = − δγ , for the LO mode, while δγ , = − δγ , and δγ , = δγ , for the TO mode (see Fig. 1(c) and(d)).Thus, for the LO mode, both V T and U vanish becausethe LO mode satisfies δγ , + δγ , = 0 and δγ , = 0,respectively. The non-vanishing matrix element for theLO mode is given by V L only. By setting δγ , = − δγ , and introducing a shift δγ due to a bond stretching as δγ , ≡ δγ cos( π/
6) in Eq. (3), we have V L = 2 √ iδγ sin (cid:18) ka (cid:19) X J φ J A φ J B . (5)From Eq. (5), it is understood that the electron-hole pairsaround ka = 0 are not excited since | V L | is proportionalto sin ( ka/ we have P J φ J A φ J B ≈ | V L /δγ | as a function of ka by the solid curves for the two low-est energy electron-hole pair creation processes which aredenoted by the arrows in Fig. 2(a). In Fig. 2(c), wesee that the edge states appearing as a flat energy bandat 2 π/ < ka < π do not contribute to electron-holepair creation (solid line 1). In Fig. 2(c), we also plot | V L /δγ | for a (5 ,
5) armchair SWNT for comparison bythe dashed curves for the two lowest energy electron-holepair creation processes which are denoted by the arrowsin Fig. 2(b). As for the lowest energy electron-hole pairs(the dashed line 1), | V L | increases with increasing ka dueto sin( ka/
2) in Eq. (5). This indicates a constant value of P J φ J A φ J B for the case of armchair SWNTs. In Fig. 2(c),we see that the matrix element of the next lowest energyelectron-hole pairs vanishes at k satisfying ∂ǫ/∂k | k = 0(van Hove singularity) for both the Z-NR and armchairSWNT. The same behavior is observed for higher sub-bands, and a large density of states due to the van Hovesingularities of the sub-bands is not effective in producingthe electron-hole pair.For the TO mode, V L vanishes owing to δγ , − δγ , =0. Moreover, it can be shown thatIm (cid:2) φ J ∗ A φ J B (cid:3) = 0 , Im[ φ J +1 ∗ A φ J B ] = 0 , (6)since an analytic solution of ( φ J A , φ J B ) for Z-NRs is givenin Ref. 44 as φ J A = (cid:20) g sin Jφ + sin( J + 1) φ (cid:21) C ( g, φ ) ,φ J B = (cid:20) ǫ ( g, φ ) g sin( J + 1) φ (cid:21) C ( g, φ ) . (7)Here g ≡ ka/ C ( g, φ ) is a normalization con-stant, φ is the wave number which is determined by theboundary condition: φ N +1A = 0, and ǫ ( g, φ ) is the en-ergy eigenvalue in units of − γ . The energy dispersionrelation is given by ǫ ( g, φ ) = g + 1 + 2 g cos φ . Since C ∗ ( g, φ ) C ( g, φ ) is a real number, we get Eq. (6). As aconsequence, we have V T = 0 and U = 0 in Eqs. (3) and(4). Thus, the TO modes give both V = 0 and U = 0 forany vertical electron-hole pair creation process, and theTO modes decouple from the electron-hole pairs. Thisshows the absence of the KA for the TO modes in Z-NRs.A renormalized phonon energy is written as a sumof the unrenormalized energy ¯ hω and the self-energyΠ( ω ). Throughout this paper, we assume a constantvalue for ¯ hω as ¯ hω = 1600cm − (0.2eV) both for theLO and TO modes. The self-energy is given by time-dependent second-order perturbation theory asΠ( ω ) =2 X eh (cid:18) | V L | ¯ hω − E eh + i Γ / − | V L | ¯ hω + E eh + i Γ / (cid:19) × ( f h − f e ) , (8)where the factor 2 comes from spin degeneracy, f h , e =(1 + exp( β ( E h , e − E F )) − is the Fermi distribution func-tion, E e ( E h ) is the energy of an electron (a hole), and E eh ≡ E e − E h is the energy of an electron-hole pair. InEq. (8), the decay width Γ is determined self-consistentlyby Γ / − Im [Π( ω )]. The self-consistent calculation be-gins by putting Γ / γ into the right-hand side ofEq. (8). By summing the right-hand side, we have a newΓ / / − Im [Π( ω )] and we put it into the right-hand side again. This calculation is repeated until Π( ω )is converged. We use Eq. (5) with δγ ≡ g off u ( ω ) /ℓ for V L in Eq. (8). Here g off is the off-site electron-phonon matrixelement and u ( ω ) is the amplitude of the LO mode. Weadopt g off = 6 . A similar value is obtained by afirst-principles calculation with the local density approxi-mation. We use a harmonic oscillator model which gives u ( ω ) = p ¯ h/ M c N u ω where M c is the mass of a carbonatom and N u is the number of hexagonal unit cells.In Fig. 3(a), we plot the renormalized energy ¯ hω +Re [Π( ω )] as a function of E F for the LO and TO modesin a N = 9 Z-NR at room temperature (300K). Sincethe TO mode decouples from electron-hole pairs, theself-energy Π( ω ) vanishes and the frequency of the TOmode does not change from ω . On the other hand, theLO mode exhibits a KA effect. The error-bars extend-ing up to ± Γ / ≈ − when E F = 0eV. The value of Γdecreases quickly when E F > . E F > ¯ hω / For comparison, we show the renormalized energies ofthe LO and TO modes for a (5 ,
5) armchair SWNT asa function of E F by the black curves in Fig. 3(a). TheTO mode exhibits no broadening but a softening witha constant energy ∼ −
30 cm − . The absence of broad-ening is due to the fact that the Bloch function can betaken as a real number for lowest energy sub-bands, i.e., for vertical transition denoted by the dashed line 1 inFig. 2(b) even when a Z-NR is rolled to form an arm-chair SWNT. The details are given in Sec. IV. For theLO mode, the broadening of the Z-NR is smaller thanthat of the armchair SWNT because | V L | of a Z-NR issmaller than that of armchair SWNTs for the lower en-ergy bands (see Fig. 2(c)). In fact, the real part of theright-hand side of Eq. (8) is a negative (positive) valuewhen E eh > ¯ hω ( E eh < ¯ hω ). Thus, electron-hole pairswith higher (lower) energy contribute to the frequencysoftening (hardening). Since the energies of the edgestates for Z-NR are smaller than ¯ hω , the edge statesmay contribute a frequency hardening like the one shownaround E F = 0 for a (5 ,
5) SWNT. The absence of thehardening confirms that the edge states do not contributeto Π( ω ) because | V L | is negligible. FIG. 3: (color online) (a) The E F dependence of the renor-malized energies of the LO and TO modes in a N = 9 Z-NR(red) and of the LO and TO modes in a (5 ,
5) SWNT (black).(b) The E F dependence of the renormalized frequencies of theLO and TO modes in a N = 9 A-NR (red) and of the LO andTO modes in a (9 ,
0) SWNT (black).
It is noted that, for the renormalized energies of theLO and TO modes in NRs shown in Fig. 3, we have notincluded all the possible intermediate electron-hole pairstates created by a given phonon mode in evaluating P eh in Eq. (8). For example, vertical transition denoted bythe arrow 3 in Fig. 2(a) may be included as a possibleintermediate state in evaluating P eh in Eq. (8) althoughsuch intermediate state does not satisfy the momentumconservation. In this paper, we do not consider the con-tribution of momentum non-conserving electron-hole paircreation processes in evaluating P eh in Eq. (8). B. Armchair NRs
Next we study the KA in A-NRs. The zigzag SWNTsare cut along their axis and flattened out to make A-NRs.It is known that one third of zigzag SWNTs exhibit ametallic band structure. It is interesting to note thatif we cut the bonds along the axis of a metallic zigzagSWNT in order to make an A-NR, the obtained A-NRhas an energy gap. Namely, unrolling a metallic (3 i, N = 3 i − i + 1 ,
0) SWNTresults in a gap-less N = 3 i A-NR and unrolling a semi-conducting (3 i +2 ,
0) SWNT results in a N = 3 i +1 A-NRwith an energy gap. The one-third periodicity of metal-licity is maintained even if zigzag SWNTs are unrolled bycutting the bonds. Since metallicity is a necessary con-dition for the KA, we study the KA in N = 3 i metallicA-NRs here.In order to specify the lattice structure of an A-NR,we use integers I ∈ [0 , N ] and J ∈ [0 , M ] in Fig. 4(a). Ina box specified by IJ in Fig. 4(a), there are two A atomsand two B atoms. For convenience, we divide the twoA (B) atoms into up-A (up-B) and down-A (down-B),as shown in Fig. 4(a). The wave function then has fourcomponents: ( e i ( k ℓ ) J / √ M )( φ I uA , φ I uB , φ I dA , φ I dB ) t where k is the wave vector along the armchair edge. FIG. 4: (a) The lattice structure of an A-NR. (b,c) Theeigenvectors of the Γ point LO and TO phonon modes areillustrated. The LO mode satisfies δγ , = − δγ , and δγ , = δγ , . The TO mode is characterized by δγ , = 0and δγ , = − δγ , . In Fig. 4(b) and (c), we show phonon eigenvectorsof the Γ point LO and TO modes, respectively. Theeigenvector of the LO (TO) mode is parallel (perpen-dicular) to the armchair edge. The LO mode satisfies δγ , = − δγ , and δγ , = δγ , , while the TO modesatisfies δγ , = 0 and δγ , = − δγ , . Since δγ , and δγ , are perturbations that do not mix φ u and φ d , theelectron-hole pair creation matrix element can be dividedinto the following two parts: V u = X I,I ′ (cid:18) φ I ′ uA − φ I ′ uB (cid:19) † (cid:18) δγ , δ I ′ ,I + δγ , δ I ′ ,I +1 h . c . (cid:19) (cid:18) φ I uA φ I uB (cid:19) ,V d = X I,I ′ (cid:18) φ I ′ dA − φ I ′ dB (cid:19) † (cid:18) δγ , δ I ′ ,I + δγ , δ I ′ ,I − h . c . (cid:19) (cid:18) φ I dA φ I dB (cid:19) , (9)where the Hermite conjugate (h . c . ) of V u ( V d ) is defined as δγ , δ I ′ ,I + δγ , δ I ′ ,I − ( δγ , δ I ′ ,I + δγ , δ I ′ ,I +1 ). Wecan rewrite Eq. (9) as V u = 2 i X I (cid:0) δγ , Im (cid:2) φ I ∗ uA φ I uB (cid:3) + δγ , Im (cid:2) φ I +1 ∗ uA φ I uB (cid:3)(cid:1) ,V d = 2 i X I (cid:0) δγ , Im (cid:2) φ I ∗ dA φ I dB (cid:3) + δγ , Im (cid:2) φ I ∗ dA φ I +1dB (cid:3)(cid:1) . (10)The perturbation δγ , mixes φ I u and φ I d as U = X I φ I uA − φ I uB φ I dA − φ I dB † e ik ℓ δγ , δγ , δγ , e − ik ℓ δγ , φ I uA φ I uB φ I dA φ I dB , (11) so that U can be rewritten as U = i δγ , X I (cid:0) Im (cid:2) e ik ℓ φ I ∗ uA φ I dB (cid:3) − Im (cid:2) φ I ∗ uB φ I dA (cid:3)(cid:1) . (12)Now, it can be shown that each energy eigenstate sat-isfies the following equations (see Appendix A), X I φ I ∗ uA φ I uB = X I φ I ∗ dA φ I dB , X I φ I +1 ∗ uA φ I uB = X I φ I ∗ dA φ I +1dB . (13)Due to these conditions, the TO mode causes a specialcancellation between V u and V d as V u + V d = 0 since δγ , + δγ , = 0. In addition, we obtain U = 0 from δγ , = 0. Thus the TO modes in A-NRs decouple fromelectron-hole pairs and do not undergo a KA. For the LOmode, on the other hand, there is no cancellation between V u and V d , and the matrix element for the LO mode isgiven by V L ≡ U + V u + V d . By setting δγ , = δγ , , − δγ , = δγ and δγ , = δγ , we calculate V L and put itinto Eq. (8) to obtain Π( ω ).The solid curves in Fig. 3(b) give the E F dependenceof the renormalized frequencies ¯ hω + Re [Π( ω )] for theLO and TO modes in a N = 9 A-NR at room temper-ature. The frequency of the TO mode is given by ω showing that the TO mode decouples from electron-holepairs. The LO mode undergoes a KA. For comparison,we show the renormalized frequency of the LO and TOmodes in a (9 ,
0) zigzag SWNT as the black curves inFig. 3(b). It is found that | Π( ω ) | for the LO mode ofa N = 9 A-NR is smaller than | Π( ω ) | for the LO modeof a (9 ,
0) zigzag SWNT. It is because there are two lin-ear energy bands near the K and K’ points in metal-lic zigzag SWNTs, while there is only one linear energyband in metallic A-NRs, and the KAs in A-NRs are sup-pressed slightly as compared to those in metallic zigzagSWNTs. We note that the broadening in A-NRs is stilllarger than that in Z-NRs because of the absence of theedge states near the armchair edges. The TO mode ofa (9 ,
0) zigzag SWNT is down shifted. However, thereis no dependence on E F , which indicates that only highenergy electron-hole pairs contribute to the self-energy.We will explore the KA effect for the TO mode in zigzagSWNTs in Sec. IV.We have considered NRs with a long length (10 µm )in calculating the self-energies Π( ω ) shown in Fig. 3(a)and (b). For NRs with short lengths, the effect of thelevel spacing on Π( ω ) is not negligible. For example, thelevel spacing in armchair SWNTs becomes 0 . L = 30nm, which is comparable to ¯ hω /
2. Thus the levelspacing affects the resonant decay. The effect of the levelspacing on the KAs in NRs will be reported elsewhere.
III. RAMAN INTENSITYA. Raman Activity
In the Raman process, an incident photon excites anelectron in the valence energy band into a state in theconduction energy band. Then the photo-excited elec-tron emits or absorbs a phonon. The matrix elementfor the emission or absorption of a phonon is given bythe el-ph matrix element for the scattering between anelectron state in the conduction energy band and a statein the conduction energy band, which is in contrast tothat for the el-ph matrix element for electron-hole paircreation which is relevant to the matrix element from astate in the valence energy band to a state in the conduc-tion energy band. This matrix element for the emissionor absorption of a phonon is given by removing the mi-nus sign from − φ J B (or − φ I uB , dB ) of the final state in theelectron-hole pair creation matrix elements in Eqs. (1),(9), and (11). This operation is equivalent to replacingIm (Re) with Re (Im) in Eqs. (3), (4), (10) and (12).As a result, the Raman intensity of the TO (LO) modesin Z-NRs is proportional to Re (cid:2) φ J ∗ A φ J B (cid:3) (Im (cid:2) φ J ∗ A φ J B (cid:3) ).Thus the TO modes are Raman active, while the LOmodes are not. Because the TO modes in Z-NRs are freefrom the KA, the G band Raman spectra exhibit theoriginal frequency of the TO modes, ¯ hω . For A-NRs,on the other hand, the cancellation between V u and V d occurs for the TO modes, and the TO modes are thennot Raman active, while the LO modes are Raman ac-tive. Since the LO modes in A-NRs undergo KAs, therenormalized frequencies, ¯ hω + Π( ω ), will appear be-low the original frequencies of the LO modes by about20 cm − (see Fig. 3). Thus, the G band spectra in A-NRs can appear below those in Z-NRs, which is usefulin identifying the orientation of the edge of NRs by Gband Raman spectroscopy. Our results are summarizedin TABLE I combined with the results for armchair andzigzag SWNTs.For the Raman intensity of armchair SWNTs, we ob-tain the same conclusion as that of Z-NRs, that is, theTO modes are Raman active, while the LO modes arenot. For metallic zigzag SWNTs, on the other hand, the cancellation between V u and V d which occurs for the TOmodes in A-NRs is not valid. Then the TO modes, inaddition to the LO modes, can be Raman active. How-ever, as we will show in Sec. IV, since the matrix elementfor the emission or absorption of the TO modes vanishesat the van Hove singularities of the electronic sub-bands,then we can conclude that the TO modes are hardly ex-cited in resonant Raman spectroscopy. It is interesting tocompare these results with another theoretical results forthe Raman intensities of SWNTs. In Ref. 48, it is shownusing bond polarization theory that the Raman inten-sity is chirality dependent. In particular, for an armchair(zigzag) SWNT, the A g TO (LO) mode is a Raman ac-tive mode, while the A g LO (TO) mode is not Ramanactive. These results for SWNTs are consistent with ourresults. We think that it is natural that the Raman in-tensity does not change by unrolling the SWNT since theRaman intensity is proportional to the number of carbonatoms in the unit cell and is not sensitive to the smallfraction of carbon atoms at the boundary.
TABLE I: Dependences of the KAs and Raman intensitieson the Γ point optical phonon modes in NRs and metallicSWNTs. (cid:13) and × represent ‘occurrence’ and ‘absence’, re-spectively. △ means that the KA is possible, but the broad-ening effect weakens due to the presence of the edge states. ▽ means that the KA is possible, but the E F dependence issuppressed by the decoupling from the metallic energy bandcrossing at the Dirac point.mode Kohn anomaly Raman active
Z-NRs LO △ × TO × (cid:13) A-NRs LO (cid:13) (cid:13) TO × × Armchair SWNTs LO (cid:13) × (rolled Z-NRs) TO ▽ (cid:13)
Zigzag SWNTs LO (cid:13) (cid:13) (rolled A-NRs) TO ▽ ×
B. Comparison With Experiment
We prepare graphene samples by means of the cleav-age method to observe the frequency of the G bandRaman spectra. In many cases, graphene samples ob-tained by the cleavage method show that the angles be-tween the edges have an average value equal to multiplesof 30 ◦ . This is consistent with the results by You etal . Figure 5(a) shows an optical image of the exfoliatedgraphene with the edges crossing each other with an an-gle of ∼ ◦ . This angle can be considered as evidence ofthe presence of edges composed predominantly of zigzagor armchair edges. Note that, the obtained sample ischaracterized as a multi-layer graphene. We estimatedthe number of layers to be about five based on the be-havior of the G’ band. The sample is placed on a SiO (100) surface 300 nm in thickness.The Raman study was performed using a Jobin-YvonT64000 Raman system. The laser energy is 2.41eV(514.5nm), the laser power is below 1mW and the laserspot is about 1 µm in diameter. Figure 5(b) shows spec-tra for finding the position dependence of the G band.The results show that the G band frequency dependson the position of laser spot. When the spot is focusednear the upper edge (A) or far from the edge, the posi-tion of the G band is almost similar to that of graphite(1582cm − ). However, a softening of the G band isclearly seen when the laser spot moves to the vicinityof the lower edge (B).Based on our theoretical studies, we find that the Gband observed near the upper edge consists only of theTO mode, while the G band observed near the lower edgecomes from the LO mode, because the G band near thelower edge shows a down shift which is considered to bedue to the KA effect of the LO mode. Then, we speculatethat the upper (lower) edge is dominated by the zigzag(armchair) edge.It should be noted that our experiment does not provethat the observed down shift of the G band near thelower edge is due to the KA effect, since we do not ex-amine the E F dependence. There is a possibility thatthe observed downshift of the G band is related to me-chanical effects. Mohiuddin et al . observed that theG band splits into two peaks due to uniaxial strain andboth peaks exhibit redshifts with increasing strain. Theedge in this work somehow has half of it suspended andthis may decrease the vibration energy. This effect mayexplain why for the upper edge in Fig. 5 the frequen-cies are slightly downshifted compared with the spectrataken at the center of the graphene sample. Moreover, itis probable that the physical edge in this work is a mix-ture of armchair and zigzag edges. Note also that wemeasured the D band in order to confirm that the iden-tification of the orientation of the edge is consistent withthe fact that the D band intensity is stronger near thearmchair edge than the zigzag edge.
However, wecould not resolve the difference of the intensity near theupper and lower edges clearly. Zhou et al. observed bymeans of high-resolution angle-resolved photo-emissionspectroscopy experiment for epitaxial graphene that theD band gives rise to a kink structure in the electron self-energy and pointed out that an interplay between theel-ph and electron-electron interactions plays an impor-tant role in the physics relating to the D band. We willconsider these issues further in the future. IV. PSEUDOSPIN
In the preceding sections we have shown for Z-NRs thatthe LO modes are not Raman active and that the TOmodes do not undergo KAs. These results originate fromthe fact that the Bloch function is a real number. Besides,
FIG. 5: (color online) (a) An optical image of a graphenesample. The sample is characterized as a multi-layer graphene( ∼ the TO modes in A-NRs are not Raman active and donot undergo KAs. This is due to the cancellation between V u and V d for the TO modes. The LO modes in A-NRs undergo KAs since the Bloch function is a complexnumber. The absence or presence of a relative phasebetween the φ A and φ B of the Bloch function is essentialin deriving our results. In this section, we explain thephase of the Bloch function in terms of the pseudospin, and clarify the effect of the zigzag and armchair edges onthe phase of the Bloch function. A. Absence of a Pseudospin Phase in Z-NRs
Here we use the effective-mass model in order to un-derstand the reason why the Bloch function in Z-NRs isa real number. In the effective-mass model, the Blochfunctions in the conduction energy band near the K andK’ points are given by ϕ K ( k x , k y ) = 1 √ e iθ ! ,ϕ K ′ ( k x , k y ) = 1 √ − e − iθ ′ ! , (14)where k x and k y ( k ′ x and k ′ y ) are measured from the K(K’) point and the angle θ ( θ ′ ) is defined by k x + ik y ≡| k | e iθ ( k ′ x + ik ′ y ≡ | k ′ | e iθ ′ ). It is noted that k x ( k y ) istaken as parallel to the zigzag (armchair) edge. Then k y is reflected into − k y at the zigzag edge, and the scatteredstate is given by ϕ K ( k x , − k y ) = 1 √ e − iθ ! , (15)as shown in Fig. 6. FIG. 6: In k -space, we consider a state near the K point (solidcircle) and states which are scattered by the zigzag and arm-chair edges (empty circles). k x ( k y ) is taken as parallel tothe zigzag (armchair) edge. The arrows in the insets indi-cate the direction of the pseudospins. The h σ y i component ofthe pseudospin is reversed at the zigzag edge, while the h σ x i component is preserved at the armchair edge. The relative phase of the wave function at the A andB sublattices can be characterized by the direction ofthe pseudospin. The pseudospin is defined by the ex-pectation value of the Pauli matrices σ x,y,z with re-spect to the Bloch function. For ϕ K ( k x , k y ), we have h σ x i = cos θ , h σ y i = sin θ , h σ z i = 0. For ϕ K ( k x , − k y ),we have h σ x i = cos θ , h σ y i = − sin θ , h σ z i = 0. Then h σ y i flips at the zigzag edge as shown in Fig. 6. Thus, due to an interference between the incoming and reflectedwaves, we have h σ y i = 0 for the Bloch function near thezigzag edge. The condition h σ y i = 0 means that theBloch function becomes a real number. In fact, since h σ y i = 2 P J Im (cid:2) φ J ∗ A φ J B (cid:3) and h σ x i = 2 P J Re (cid:2) φ J ∗ A φ J B (cid:3) for the Bloch function ( φ J A , φ J B ) in Eq. (7), the el-ph ma-trix elements V T and V L (Eq. (3)) are proportional to h σ y i and h σ x i , respectively. In Appendix B, we showthe relationship between the Bloch function ϕ K in theeffective-mass model (Eq. (14)) and the Bloch function( φ J A , φ J B ) in the tight-binding model (Eq. (7)).We point out that the condition h σ y i = 0 is not satis-fied in the case of armchair SWNTs (rolled Z-NRs) ex-cept for the lowest energy sub-bands of k y = 0. This isthe reason why we see in Fig. 3(a) that the TO modeexhibits no broadening but a softening with a constantenergy ∼ −
30 cm − in a (5 ,
5) SWNT. It is interestingto note that the Aharanov-Bohm flux applied along thetube axis shifts the cutting lines and can make k y for the lowest energy sub-bands nonzero. Thus theAharanov-Bohm flux makes that h σ y i = sin θ = 0 evenfor the lowest energy sub-bands, for which the TO modecan exhibit a broadening. Since the effective-mass model describes the physicswell in the long wave length limit, an advantage in theabove discussion of using the effective-mass model is thatit is not necessary for the edge of graphene NRs to be welldefined on an atomic scale in order that we have a can-cellation of h σ y i . This may be a reason why we observe asoftening of the G band near the edge of an armchair-richsample experimentally as shown in Sec. III B. B. Coherence of the Pseudospin in A-NRs
On the other hand, a state near the K point with( k x , k y ) is reflected by the armchair edge into a state nearthe K’ point with ( k ′ x , k ′ y ) = ( − k x , k y ), and the scatteredstate is given by ϕ K ′ ( − k x , k y ). In this case, by putting θ ′ = π − θ into ϕ K ′ ( k x , k y ) in Eq. (14), we obtain ϕ K ′ ( − k x , k y ) = 1 √ e iθ ! (16)which is identical to the initial Bloch function, ϕ K ( k x , k y ). Thus the relative phase between the A andB Bloch functions is conserved thorough the reflectionby the armchair edge so that the Bloch function can notbe reduced to a real number. Namely, the reflections bythe armchair edge preserve the pseudospin as shown inFig. 6. It is expected that the relative phase makes itpossible that KAs occur not only for the LO mode butalso for the TO mode near the armchair edge. However,as we have shown in Eq. (13), the armchair edge givesrise to the cancellation between V u and V d , so that theTO modes in A-NRs do not undergo KAs. We considerwhether Eq. (13) is satisfied in the case of zigzag SWNTsor not, in order to see if the KA effect is present in zigzagSWNTs or not. By applying the Bloch theorem to zigzagSWNTs, we have φ I uA = ( e iIka / √ N ) ϕ A ,φ I uB = ( e i ( I +1 / ka / √ N ) ϕ B ,φ I dA = ( e i ( I +1 / ka / √ N ) ϕ A ,φ I dB = ( e iIka / √ N ) ϕ B , (17)where we set ϕ φ = t ( ϕ A , ϕ B ). Using these equations, weobtain X I φ I ∗ uA φ I uB = 14 e i ka ϕ ∗ A ϕ B , X I φ I ∗ dA φ I dB = 14 e − i ka ϕ ∗ A ϕ B . (18)Thus the first equation in Eq. (13) is not satisfied forzigzag SWNTs. Similarly, we have X I φ I +1 ∗ uA φ I uB = 14 e − i ka ϕ ∗ A ϕ B , X I φ I ∗ dA φ I +1dB = 14 e i ka ϕ ∗ A ϕ B , (19)which shows that the second equation in Eq. (13) is notfulfilled, either. Thus the TO modes can undergo KAsbecause the cancellation between V u and V d is not possi-ble for zigzag SWNTs. In fact, by putting Eqs. (18) and(19) into Eq. (10), we get V u + V d = i ( δγ , + δγ , ) sin θ cos ka i ( δγ , − δγ , ) cos θ sin ka , (20)where we set ϕ ∗ A ϕ B = e iθ . Because the TO modes satisfy δγ , + δγ , = 0, V u + V d can take a nonzero value for V u + V d = i δγ , sin (cid:18) ka (cid:19) cos θ. (21)It is noted that Eq. (21) vanishes when θ = ± π/
2. Thiscondition θ = ± π/ ,
0) zigzag SWNTas shown in Fig. 3. In “metallic” zigzag SWNTs, thecurvature effect shifts the position of the cutting line of the metallic energy band from the Dirac point andproduces a small energy gap. In this case, thelow energy electron-hole pairs satisfy cos θ = 0 in Eq. (21)and they contribute to a frequency hardening of the TOmodes in metallic zigzag SWNTs. The curvature effectgives rise to a change of the Fermi surface and results inKAs for the TO modes. Similarly, the matrix element for the emission or ab-sorption of the TO modes in zigzag SWNTs is given by2 δγ , sin (cid:18) ka (cid:19) sin θ, (22)which does not vanish in general. This shows that the TOmodes in zigzag SWNTs can be Raman active. However,since the van Hove singularities of sub-bands in zigzagSWNTs are located on the k x axis (and satisfy θ = 0),the sin θ factor tells us that the TO modes are hardlyexcited in resonant Raman spectroscopy. V. DISCUSSION
The theoretical analysis performed in this paper isbased on the use of a simple tight-binding method whichincludes only the first nearest-neighbor hopping inte-gral and its variation due to the atomic displacements.The approximation used is partly justified because thedeformation-potential and the el-ph matrix element withrespect to the second nearest-neighbors is about one or-der of magnitude smaller than that of the first nearestneighbors.
However, we have not considered the ef-fect of the overlap integral. The overlap integral breaksthe particle-hole symmetry and may invalidate our re-sults. Besides, we have neglected the contribution of mo-mentum non-conserving electron-hole pair creation pro-cesses in evaluating P eh in Eq. (8). Although this is anapproximation which works well for thin NRs, the inclu-sion of the momentum non-conserving electron-hole paircreation processes may invalidate our results. We willelaborate on this idea in the future. VI. SUMMARY
In summary, the LO modes undergo KAs in grapheneNRs while the TO modes do not. This conclusion doesnot depend on the orientation of the edge. In Z-NRs,the Raman intensities of the LO modes are strongly sup-pressed because the wave function is a real number, andonly the TO modes are Raman active. As a result, theKA for the LO mode in Z-NRs would be difficult to ob-serve in Raman spectroscopy. In A-NRs, only the LOmodes are Raman active owing to the cancellation be-tween V u and V d . The “chirality” dependent Raman in-tensity derived for NRs is the same as the chirality depen-dent Raman intensity for SWNTs calculated in Ref. 48.The strong down shift of the LO mode makes it possi-ble to identify the orientation of edges of graphene bythe G band Raman spectroscopy due to the “chirality”dependent Raman intensity.0 Acknowledgment
K.S would like to thank Hootan Farhat and Prof.Jing Kong for discussions on the KAs in SWNTs.S.M acknowledges MEXT Grants (No. 21000004 andNo. 19740177). R.S acknowledges a MEXT Grant(No. 20241023). M.S.D acknowledges grant NSF/DMR07-04197. This work is supported by a Grant-in-Aidfor Specially Promoted Research (No. 20001006) fromMEXT.
APPENDIX A: DERIVATION OF EQ. (13)
In this Appendix, we derive Eq. (13) by using mir-ror and time-reversal symmetries. Let us introduce themirror-reflection operator M by M φ
In,k = φ I uA φ I uB φ I dA φ I dB ( I = 0 , . . . , N ) , (A1)where k is the wave vector along the armchair edge and n is the band index. By applying M to the energyeigen-equation H k φ n,k = E n,k φ n,k , we get M H k φ n,k = E n,k M φ n,k . Since the Hamiltonian satisfies
M H k M − = H − k , we obtain M φ n,k = e iφ φ n, − k where φ is a phasefactor.On the other hand, due to the time-reversal symmetry,we have φ ∗ n,k = e iφ ′ φ n, − k . Thus, by combing the time-reversal symmetry ( φ ∗ n,k = e iφ ′ φ n, − k ) with the mirrorsymmetry ( M φ n,k = e iφ φ n, − k ), we get M φ n,k = e iφ ′′ φ ∗ n,k , (A2)that is, φ I dB φ I dA φ I uB φ I uA = e iφ ′′ φ I ∗ uA φ I ∗ uB φ I ∗ dA φ I ∗ dB . (A3)Using this condition, one sees that Eq. (13) is satisfied. APPENDIX B: RELATIONSHIP BETWEENEQ. (7) AND EQ. (14)
Here we will show for Z-NRs that the Bloch functionderived using the tight-binding lattice model (Eq. (7)) isa superposition of incoming and reflected Bloch functionsderived using the effective-mass model (Eq. (14)). By rewriting the Bloch function of Z-NRs ( φ J A , φ J B ) inEq. (7) as φ J A = C ( g, φ )2 i (cid:18) g + e iφ (cid:19) e iJφ + c . c .,φ J B = C ( g, φ )2 i ǫ ( g, φ ) e iφ g e iJφ + c . c ., (B1)where c . c . denotes the complex conjugation of the firstterm, one sees that ( φ J A , φ J B ) is a real number as a resultof the cancellation of the imaginary part between the firstand second terms. By introducing a new Bloch function ϕ φ as ϕ φ ≡ C ( g, φ )2 ig e iφ g + e − iφ ǫ ( g, φ ) ! , (B2)the Bloch function ( φ J A , φ J B ) is expressed by φ J A φ J B ! = ϕ φ e iJφ + ϕ ∗ φ e − iJφ . (B3)Because of the different signs in the exponents of e iJφ and e − iJφ in Eq. 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