1、Optical Properties and Raman Spectroscopyof Carbon NanotubesRiichiro Saito1and Hiromichi Kataura21Department of Electronic-Engineering, The University ofElectro-Communications1-5-1, Chofu-gaoka, Chofu, Tokyo 182-8585, Japanrsaitotube.ee.uec.ac.jp2Department of Physics, Tokyo Metropolitan University1
2、-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japankatauraphys.metro-u.ac.jpAbstract. The optical properties and the resonance Raman spectroscopy of singlewall carbon nanotubes are reviewed. Because of the unique van Hove singulari-ties in the electronic density of states, resonant Raman spectroscopy
3、has provideddiameter-selective observation of carbon nanotubes from a sample containing nano-tubes with different diameters. The electronic and phonon structure of single wallcarbon nanotubes are reviewed, based on both theoretical considerations and spec-troscopic measurements.The quantum propertie
4、s of Single-Wall Carbon Nanotubes (SWNTs) dependon the diameter and chirality, which is defined by the indices (n,m) 1,2.Chirality is a term used to specify a nanotube structure, which does not havemirror symmetry. The synthesis of a SWNT sample with a single chirality isan ultimate objective for ca
5、rbon nanotube physics and material science re-search, but this is still difficult to achieve with present synthesis techniques.On the other hand, the diameter of SWNTs can now be controlled signifi-cantly by changing the furnace growth temperature and catalysts 3,4,5,6.Thus, a mixture of SWNTs with
6、different chiralities, but with a small rangeof nanotube diameters is the best sample that can be presently obtained.Resonance Raman spectroscopy provides a powerful tool to investigate thegeometry of SWNTs for such samples and we show here that metallic andsemiconducting carbon nanotubes can be sep
7、arately observed in the resonantRaman signal.In this paper, we first review theoretical issues concerning the electronand phonon properties of a single-walled carbon nanotube. We then describethe electronic and phonon density of states of SWNTs. In order to discussresonant Raman experiments, we make
8、 a plot of the possible energies ofoptical transitions as a function of the diameter of SWNTs.Then we review experimental issues concerning the diameter-controlledsynthesis of SWNTs and Raman spectroscopy by many laser frequencies.The optical absorption measurements of SWNTs are in good agreement wi
9、ththe theoretical results.M. S. Dresselhaus, G. Dresselhaus, Ph. Avouris (Eds.): Carbon Nanotubes,Topics Appl. Phys. 80, 213247 (2001)c? Springer-Verlag Berlin Heidelberg 2001微信公共号：材料匠214Riichiro Saito and Hiromichi Kataura1Theoretical IssuesThe electronic structure of carbon nanotubes is unique in
10、solid-state physicsin the sense that carbon nanotubes can be either semiconducting or metallic,depending on their diameter and chirality 1,2. The phonon properties arealso remarkable, showing unique one-dimensional (1D) behavior and specialcharacteristics related to the cylindrical lattice geometry,
11、 such as the RadialBreathing Mode (RBM) properties and the special twist acoustic mode whichis unique among 1D phonon subbands.Using the simple tight-binding method and pair-wise atomic force con-stant models, we can derive the electronic and phonon structure, respectively.These models provide good
12、approximations for understanding the experimen-tal results for SWNTs.1.1Electronic Structure and Density of States of SWNTsWe now introduce the basic definitions of the carbon nanotube structureand of the calculated electronic and phonon energy bands with their specialDensity of States (DOS). The st
13、ructure of a SWNT is specified by the chiralvector ChCh= na1+ ma2 (n,m),(1)where a1and a2are unit vectors of the hexagonal lattice shown in Fig.1.The vector Chconnects two crystallographically equivalent sites O and A ona two-dimensional (2D) graphene sheet, where a carbon atom is located ateach ver
14、tex of the honeycomb structure 7. When we join the line AB?tothe parallel line OB in Fig.1, we get a seamlessly joined SWNT classified bythe integers (n,m), since the parallel lines AB?and OB cross the honeycomblattice at equivalent points. There are only two kinds of SWNTs which havemirror symmetry
15、: zigzag nanotubes (n,0), and armchair nanotubes (n,n).The other nanotubes are called chiral nanotubes, and they have axial chiralsymmetry. The general chiral nanotube has chemical bonds that are notFig.1. The unrolled honeycomb latticeof a nanotube. When we connect sites Oand A, and sites B and B?,
16、 a nanotubecan be constructed. OA and OB definethe chiral vector Chand the transla-tional vector T of the nanotube, respec-tively. The rectangle OAB?B defines theunit cell for the nanotube. The figure isconstructed for an (n,m) = (4,2) nano-tube 2微信公共号：材料匠Optical Properties and Raman Spectroscopy of
17、 Carbon Nanotubes215parallel to the nanotube axis, denoted by the chiral angle in Fig.1. Herethe direction of the nanotube axis corresponds to OB in Fig.1. The zigzag,armchair and chiral nanotubes correspond, respectively, to = 0, = 30,and 0 | 30. In a zigzag or an armchair nanotube, respectively, o
18、ne ofthree chemical bonds from a carbon atom is parallel or perpendicular to thenanotube axis.The diameter of a (n,m) nanotube dtis given bydt= Ch/ =3aCC(m2+ mn + n2)1/2/(2)where aCCis the nearest-neighbor CC distance (1.42A in graphite), and Chis the length of the chiral vector Ch. The chiral angle
19、 is given by = tan13m/(m + 2n).(3)The 1D electronic DOS is given by the energy dispersion of carbon nano-tubes which is obtained by zone folding of the 2D energy dispersion relationsof graphite. Hereafter we only consider the valence and the conduction energy bands of graphite and nanotubes. The 2D
20、energy dispersion relationsof graphite are calculated 2 by solving the eigenvalue problem for a (2 2)Hamiltonian H and a (2 2) overlap integral matrix S, associated with thetwo inequivalent carbon atoms in 2D graphite,H =?2p0f(k)0f(k)?2p?and S =1sf(k)sf(k)1(4)where ?2pis the site energy of the 2p at
21、omic orbital andf(k) = eikxa/3+ 2eikxa/23coskya2(5)where a = |a1| = |a2| =3aCC. Solution of the secular equation det(H ES) = 0 implied by (4) leads to the eigenvaluesEg2D(k) =?2p 0w(k)1 sw(k)(6)for the C-C transfer energy 0 0, where s denotes the overlap of the elec-tronic wave function on adjacent
22、sites, and E+and Ecorrespond to theand the energy bands, respectively. Here we conventionally use 0as apositive value. The function w(k) in (6) is given byw(k) =?|f(k)|2=?1 + 4cos3kxa2coskya2+ 4cos2kya2.(7)In Fig.2 we plot the electronic energy dispersion relations for 2D graphite asa function of th
23、e two-dimensional wave vector k in the hexagonal Brillouinzone in which we adopt the parameters 0= 3.013eV, s = 0.129 and ?2p= 0微信公共号：材料匠216Riichiro Saito and Hiromichi KatauraFig.2. The energy dispersion relations for 2D graphite with 0= 3.013eV, s =0.129 and ?2p= 0 in (6) are shown throughout the
24、whole region of the Brillouinzone. The inset shows the energy dispersion along the high symmetry lines betweenthe , M, and K points. The valence band (lower part) and the conduction band (upper part) are degenerate at the K points in the hexagonal Brillouin zonewhich corresponds to the Fermi energy
25、2so as to fit both the first principles calculation of the energy bands of 2Dturbostratic graphite 8,9 and experimental data 2,10. The correspondingenergy contour plot of the 2D energy bands of graphite with s = 0 and ?2p= 0is shown in Fig.3. The Fermi energy corresponds to E = 0 at the K points.Nea
26、r the K-point at the corner of the hexagonal Brillouin zone of graphite,w(k) has a linear dependence on k |k| measured from the K point asw(k) =32ka + .,for ka ? 1.(8)Thus, the expansion of (6) for small k yieldsEg2D(k) = ?2p (0 s?2p)w(k) + . ,(9)so that in this approximation, the valence and conduc
27、tion bands are symmet-ric near the K point, independent of the value of s. When we adopt ?2p= 0and take s = 0 for (6), and assume a linear k approximation for w(k), we getthe linear dispersion relations for graphite given by 12,13E(k) = 320ka = 320kaCC.(10)If the physical phenomena under considerati
28、on only involve small k vectors,it is convenient to use (10) for interpreting experimental results relevant tosuch phenomena.The 1D energy dispersion relations of a SWNT are given byE(k) = Eg2D?kK2|K2|+ K1?,?T k T, and = 1,N?,(11)微信公共号：材料匠Optical Properties and Raman Spectroscopy of Carbon Nanotubes
29、217Fig.3. Contour plot of the 2D electronic energy of graphite with s = 0 and ?2p= 0in (6). The equi-energy lines are circles near the K point and near the center of thehexagonal Brillouin zone, but are straight lines which connect nearest M points.Adjacent lines correspond to changes in height (ene
30、rgy) of 0.10and the energyvalue for the K, M and points are 0, 0and 30, respectively. It is useful to notethe coordinates of high symmetry points: K = (0,4/3a), M = (2/3a,0) and = (0,0), where a is the lattice constant of the 2D sheet of graphite 11where T is the magnitude of the translational vecto
31、r T, k is a 1D wavevector along the nanotube axis, and N denotes the number of hexagons ofthe graphite honeycomb lattice that lie within the nanotube unit cell (seeFig.1). T and N are given, respectively, byT =3ChdR=3dtdR,andN =2(n2+ m2+ nm)dR.(12)Here dRis the greatest common divisor of (2n + m) an
32、d (2m + n) for a(n,m) nanotube 2,14. Further K1and K2denote, respectively, a discreteunit wave vector along the circumferential direction, and a reciprocal latticevector along the nanotube axis direction, which for a (n,m) nanotube aregiven byK1= (2n + m)b1+ (2m + n)b2/NdRandK2= (mb1 nb2)/N,(13)wher
33、e b1and b2are the reciprocal lattice vectors of 2D graphite and aregiven in x,y coordinates byb1=?13,1?2a,b2=?13,1?2a.(14)The periodic boundary condition for a carbon nanotube (n,m) gives Ndiscrete k values in the circumferential direction. The N pairs of energydispersion curves given by (11) corres
34、pond to the cross sections of the two-dimensional energy dispersion surface shown in Fig.2, where cuts are made on微信公共号：材料匠218Riichiro Saito and Hiromichi Kataurathe lines of kK2/|K2|+K1. In Fig.4 several cutting lines near one of the Kpoints are shown. The separation between two adjacent lines and
35、the lengthof the cutting lines are given by the K1and K2vectors of (13), respectively,whose lengths are given by|K1| =2dt,and|K2| =2T=2dR3dt.(15)If, for a particular (n,m) nanotube, the cutting line passes through a K pointof the 2D Brillouin zone (Fig.4a), where the and energy bands of 2Dgraphite a
36、re degenerate (Fig.2) by symmetry, then the 1D energy bands havea zero energy gap. Since the degenerate point corresponds to the Fermi energy,and the density of states are finite as shown below, SWNTs with a zero bandgap are metallic. When the K point is located between two cutting lines,the K point
37、 is always located in a position one-third of the distance betweentwo adjacent K1lines (Fig.4b) 14 and thus a semiconducting nanotubewith a finite energy gap appears. The rule for being either a metallic or asemiconducting carbon nanotube is, respectively, that nm = 3q or nm ?=3q, where q is an inte
38、ger 2,8,15,16,17.Fig.4. The wave vector k for one-dimensionalcarbonnanotubesisshowninthetwo-dimensionalBrillouinzoneofgraphite(hexagon) as bold lines for (a) metallic and(b) semiconducting carbon nanotubes. Inthe direction of K1, discrete k values areobtained by periodic boundary conditions forthe c
39、ircumferential direction of the carbonnanotubes, while in the direction of the K2vector, continuous k vectors are shown inthe one-dimensional Brillouin zone. (a) Formetallic nanotubes, the bold line intersectsa K point (corner of the hexagon) at theFermi energy of graphite. (b) For the semi-conducto
40、r nanotubes, the K point alwaysappears one-third of the distance betweentwo bold lines. It is noted that only a fewof the Nbold lines are shown near theindicated K point. For each bold line, thereis an energy minimum (or maximum) in thevalence and conduction energy subbands,giving rise to the energy
41、 differences Epp(dt)微信公共号：材料匠Optical Properties and Raman Spectroscopy of Carbon Nanotubes219The 1D density of states (DOS) in units of states/C-atom/eV is calculatedbyD(E) =T2N?N?=1?1?dE(k)dk?(E(k) E)dE,(16)where the summation is taken for the N conduction (+) and valence ()1D bands. Since the ener
42、gy dispersion near the Fermi energy (10) is linear,the density of states of metallic nanotubes is constant at the Fermi energy:D(EF) = a/(220dt), and is inversely proportional to the diameter of thenanotube. It is noted that we always have two cutting lines (1D energy bands)at the two equivalent sym
43、metry points K and K?in the 2D Brillouin zone inFig.3. The integrated value of D(E) for the energy region of E(k) is 2 forany (n,m) nanotube, which includes the plus and minus signs of Eg2Dandthe spin degeneracy.It is clear from (16) that the density of states becomes large when theenergy dispersion
44、 relation becomes flat as a function of k. One-dimensionalvan Hove singularities (vHs) in the DOS, which are known to be propor-tional to (E2 E20)1/2at both the energy minima and maxima (E0) ofthe dispersion relations for carbon nanotubes, are important for determin-ing many solid state properties o
45、f carbon nanotubes, such as the spectraobserved by scanning tunneling spectroscopy (STS), 18,19,20,21,22, opticalabsorption 4,23,24, and resonant Raman spectroscopy 25,26,27,28,29.The one-dimensional vHs of SWNTs near the Fermi energy come from theenergy dispersion along the bold lines in Fig.4 near
46、 the K point of the Bril-louin zone of 2D graphite. Within the linear k approximation for the energydispersion relations of graphite given by (10), the energy contour as shownin Fig.3 around the K point is circular and thus the energy minima of the1D energy dispersion relations are located at the cl
47、osest positions to the Kpoint. Using the small k approximation of (10), the energy differences EM11(dt)and ES11(dt) for metallic and semiconducting nanotubes between the highest-lying valence band singularity and the lowest-lying conduction band singular-ity in the 1D electronic density of states cu
48、rves are expressed by substitutingfor k the values of |K1| of (15) for metallic nanotubes and of |K1/3| and|2K1/3| for semiconducting nanotubes, respectively 30,31, as follows:EM11(dt) = 6aCC0/dtandES11(dt) = 2aCC0/dt.(17)When we use the number p (p = 1,2,.) to denote the order of the valence and co
49、nduction energy bands symmetrically located with respect tothe Fermi energy, optical transitions Epp?from the p-th valence band tothe p?-th conduction band occur in accordance with the selection rules ofp = 0 and p = 1 for parallel and perpendicular polarizations of theelectric field with respect to
50、 the nanotube axis, respectively 23. However,in the case of perpendicular polarization, the optical transition is suppressed微信公共号：材料匠220Riichiro Saito and Hiromichi Katauraby the depolarization effect 23, and thus hereafter we only consider theoptical absorption of p = 0. For mixed samples containin
