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Low-frequency electronic excitations in doped carbon nanotubes
Physica B 292 (2000) 117}126
Low-frequency electronic excitations in doped carbon nanotubes M.F. Lin *, F.L. Shyu Department of Physics, National Cheng Kung University, Taiwan 70101, Taiwan, ROC Department of Electronics Engineering, Fortune Institute of Technology, Kaohsiung 842, Taiwan, ROC Received 20 September 1999; received in revised form 8 February 2000
Abstract We study the low-frequency electronic excitations of a doped carbon nanotube. The longitudinal dielectric function, the loss function, and the plasmon frequencies are calculated within the random phase approximation. They strongly depend on the transferred momentum (q), the transferred angular momentum (¸), the Fermi energy, and the nanotube geometry (the radius and the chiral angle). All the doped carbon nanotubes could exhibit the ¸"0 acoustic plasmon. There also exists the ¸"1 optical plasmon, when the Fermi energy is su$ciently high. There are several important di!erences between type-I nanotubes and type-II nanotubes. The local-"eld corrections on the loss spectra and the plasmon frequencies are discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Mf; 71.45.Gm; 73.20.Dx Keywords: Electronic excitations; Plasmons; Carbon nanotubes
1. Introduction Iijima [1] recently reported observations of carbon nanotubes, which are graphite sheets rolled up in the cylindrical form. Carbon nanotubes, with radii between 3.39 and 150 As , represent an interesting one-dimensional (1D) system. This system has generated many interesting studies, such as structural [2}10] and electronic properties [11}18]. When carbon nanotubes are packed together, they could form a 3D carbon nanotube bundle [2}7] or a 2D}3D carbon nanotube "lm [8}10]. Moreover, * Corresponding author. Tel.: #886-622757575-65272; fax: #11-886-62747995. E-mail address: m#[email protected] (M.F. Lin).
metallic atoms or molecules [19}26] could be intercalated into carbon nanotubes. Experimental measurements on electrical resistivity [19}21] and Raman spectrum [22] indicate that electrons are transferred from intercalants to carbon atoms, or vice versa. Charge transfer could also occur, when carbon nanotubes lie on some substrates, e.g., carbon nanotubes on gold [27]. In this work, we mainly study the low-frequency electronic excitations in a doped carbon nanotube. The dependence of the electronic excitations on the transferred momentum (q), the transferred angular momentum (¸), the nanotube geometry (the radius r and the chiral angle h), and the Fermi energy (E ) is investigated. $ The local-"eld e!ects on the low-frequency electronic excitations are also discussed.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 9 9 - 3
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The electronic structures of the undoped carbon nanotubes strongly depend on the nanotube geometry [11}18]. As has been discussed [28], a 1D carbon nanotube is a rolled-up 2D graphite sheet, the structure of which is fully speci"ed by a lattice vector R"ma #na . a and a are the primitive lattice vectors of a 2D graphite sheet. The parameters (m, n) uniquely de"ne a 1D carbon nanotube. The radius and the chiral angle of an (m, n) carbon nanotube are, respectively, r""R"/2p "(3(m#mn#n)/2p and h" tan\[!(3n/(2m#n)]. b"1.42 As is the C}C bond length. Type I (II) carbon nanotubes, with 2m#nO3i ("3i; i is an integer), are predicted to be semiconductors (metals) [11}18]. The theoretical predictions had been veri"ed from the STM measurements [27,29]. All the doped carbon nanotubes are metallic. The theoretical calculation [30] predicts that the low-energy n band is hardly a!ected by the intercalation, and the Fermi level rigidly shifts from zero to a "nite value. The rigidband model is suggested to be suitable for studying the low-frequency properties of the doped carbon nanotubes. Such a model had been used to study magnetic [28,31,32] and transport properties [33,34]. It will be further used to investigate the low-frequency electronic excitations. A carbon-based system has the n and p bands which are, respectively, formed by the 2p and X (2s, 2p , 2p ) orbitals. Experimental measurements V W from the electron-energy-loss spectroscopy (EELS) show that the n plasmon frequency is &5}7 eV [35}40] and the n#p plasmon frequency is &22}27 eV [35}39]. Moreover, the undoped typeII (metallic) carbon nanotubes are predicted to exhibit the low-frequency plasmon at &0.1}1 eV [41]. All the doped carbon nanotubes have free carriers in the conduction bands (the nH bands); therefore, they are expected to exhibit the lowfrequency plasmons. Such plasmons are signi"cantly a!ected by the Fermi energy. For a doped carbon nanotube, doping would change the lowfrequency electronic excitations, but not the n plasmon or the p plasmon. The high-frequency
electronic excitations are thus neglected in this study. The low-frequency electronic excitations in the doped (or metallic) nanotube systems had been qualitatively studied within the electron}gas model [42}54]. A 1D carbon nanotube [42}52] could exhibit the electronic excitations of discrete angular momenta (¸'s); that is, there are ¸-decoupled electronic excitations in a 1D cylindrical nanotube. In this work, the real n band obtained from the tightbinding model [13,14] is used to study the electronic excitations of the doped nanotube systems. The above-mentioned characteristics also exist in the present study. The low-frequency excitation properties are quite complicated since the dependence of the n band on the nanotube geometry, the longitudinal wave vector, and the transverse angular momentum is very strong. The calculated results could be tested by the re#ection-electronenergy-loss spectroscopy (REELS) with a high resolution of &10 meV [53]. The paper is organized as follows. In Section 2, the dielectric function of a doped carbon nanotube is calculated within the random phase approximation (RPA). The calculated results are discussed in Section 3. Concluding remarks are given in Section 4. 2. Electronic excitations of a doped carbon nanotube The electronic structures of carbon nanotubes strongly depend on the nanotube geometry, and so do the electronic excitations. The n band is calculated from the tight-binding model [13,14]. The energy dispersion relations of the (m, n) carbon nanotube are [28] E (J, k)"$c
;cos
1#4 cos
3b(k cos h#J sin h/r) 2
(3b(k sin h!J cos h/r) 2
#4 cos
(3b(k sin h!J cos h/r) 2
. (1)
The wave functions are For the details of the nanotube geometry and the n-band structure see Ref. [28].
1 HH (J, k) W " ; (J, k)$ ; (J, k) . ( I (2 "H (J, k)"
(2)
M.F. Lin, F.L. Shyu / Physica B 292 (2000) 117}126
c ("3.033 eV) [13,14] is the resonance integral, and H is the nearest-neighbor Hamiltonian matrix element. ; and ; are the two tight binding functions of the 2p orbitals. The electronic X states are described by the angular momentum J along the azimuthal direction and the wave vector k along the nanotube axis. The subband nearest E"0 is denoted by J . The supercripts c and v, respectively, represent the nH and n bands. The n bands are symmetric to the nH bands about the zero energy. For the undoped carbon nanotubes, electrons only occupy the n band with E(J, k))0 at ¹"0. An (m, n) carbon nanotube with 2m#n"3i (O3i) is a metal (semiconductor). A metallic carbon nanotube (type II) has the linear J subbands intersecting at the Fermi level E "0. $ A semiconducting carbon nanotube (type I) has an energy gap E "bc /r inversely proportional to the nanotube radius. In general, a carbon nanotube has many parabolic subbands of di!erent J's except that the J subbands in a metallic nanotube are linear. As for a doped carbon nanotube, electrons could occupy the nH bands as well as the n bands. The characteristics of the electronic structures will be directly re#ected in the electronic excitations. The electronic excitations of a cylindrical carbon nanotube are characterized by the transferred momentum and angular momentum, because ¸ and q are conserved in the electron}electron (e}e) Coulomb interactions. The dielectric function of a carbon nanotube, which is calculated from the RPA [54], is given by [55] e(q, ¸, u)"e !<(q, ¸)s(q, ¸, u), (3) where e ("2.4) is the background dielectric con stant due to the high-frequency excitations. <(q, ¸)" 4peI (qr)K (qr) is the 1D bare Coulomb interac* * tion. I [K ] is the modi"ed Bessel function of the * * "rst (second) kind of the order ¸. The linear response function (or the bare polarizability) in Eq. (3) is 2 s(q, ¸,u)" (2p)
For the details of the n-electron response function see Ref. [55].
119
f (EFY(J#¸, k#q))!f (EF(J, k)) ; dk EFY(J#¸, k#q)!EF(J, k)!(u#iC) ( F FY ;"1J#¸, k#q, h "e *(Ye OW"J, k, h2",
(4a)
where "1J#¸, k#q, h "e *(Ye OW"J, k , h2" W 1 HH (J#¸, k#q)HH (J, k) " 1$ , (4b) 4 "HH (J#¸, k#q)HH (J, k)" and f in Eq. (4a) is the Fermi}Dirac distribution function. C is a phenomenological level broadening parameter. When an undoped carbon nanotube (E "0) is perturbed by an external potential at $ ¹"0, electrons are excited from the occupied nband states to the unoccupied nH-band states. That is to say, they only exhibit the inter-n-band excitations. But for a doped carbon nanotube, electrons are excited from the occupied nH-band states to the unoccupied nH-band states in addition to the intern-band excitations. That is to say, there are two kinds of electronic excitations, inter-n-band excitations and intra-n-band excitations. The low-frequency electronic excitations, with u)2E , $ mainly come from the intra-n-band excitations. However, the inter-n-band excitations would a!ect the real part of the dielectric function and thus the low-frequency plasmon frequency. Apparently, the electronic excitations of doped carbon nanotubes are more complicated than those of undoped carbon nanotubes. # and ! in Eq. (4b), respectively, correspond to the intra- and the inter-n-band excitations. The short-range electronic correlations might affect the electronic excitations. The above-mentioned RPA does not take into account the exchange}correlation e!ects. In general, there are two methods for improving the RPA [47,56]. They are the Hubbard approximation (HA) [56,57] and the local-"eld theory of Singwi et al. [56,58]. The latter includes the exchange}correlation e!ects. The linear response function in Eq. (4a) does not have an analytic form. Therefore, it needs too much calculation time to get the modi"ed plasmon frequency. The former, which includes only the exchange e!ects, is taken to see the local-"eld e!ects. Within the Hubbard approximation, the ladder
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vertex integral [56] for the irreducible polarizability is solved in a rough fashion. The dielectric function remains in a similar form as that of the RPA (Eq. (3)). However, the irreducible polarizability s&(q, ¸, u) is related to the bare polarizability s(q, ¸, u) by the formula [56,57] s(q, ¸, u) s&(q, ¸, u)" 1#<(q, ¸)G&(q, ¸)s(q, ¸, u)/e and <((q#k 0), ¸) $ G&(q, ¸)" 2<(q, ¸),
(5a)
(5b)
where k (0) is the Fermi momentum of the occu$ pied J subband. The static factor G&(q, ¸) is due to the local-"eld e!ects of the exchange holes. Eqs. (3)}(5) will be utilized to examine the local-"eld e!ects on the electronic excitations.
a doped carbon nanotube could exhibit the lowfrequency plasmons of di!erent ¸'s. We "rst see the electronic excitations of the doped (17, 0) nanotube with E "0.5 eV. Electrons $ occupy the nH bands of J "11 and 23 as well as all the n bands. The dielectric function of the ¸"0 excitations is shown in Fig. 1(a) at q"0.05 and C"0. The unit of q is As \, here and henceforth. The real [e (q, ¸, u)] and the imaginary [e (q, ¸, u)] parts of the dielectric function are shown by the solid and the dashed curves, respectively. For the low-frequency ¸"0 excitations, electrons are excited from the occupied nH-band state of (J , k) to the unoccupied nH-band state of (J , k#q). The intra-n-band excitation energy is u "E(J , k#q)!E(J , k). When the electronic excitation of the (J , k) state corresponds to a critical point (a minimum, a maximum, or a saddle
3. The calculated results We mainly use the RPA to calculate the dielectric function, the loss spectrum, and the plasmon frequency. In the following calculations, the type-I (17, 0) and the type-II (10, 10) nanotubes are chosen for a detailed study. They are, respectively, called zig-zag and armchair nanotubes. The former has r"6.65 As and h"03, and the latter has r"6.78 As and h"!303. Their radii are almost the same, while the di!erent chiral angles lead to important di!erences in the electronic structures, e.g., the number of subbands and the energy spacing between two subbands. The undoped (17, 0) nanotube is a semiconductor with E "0.647 eV, and the undoped (10, 10) nanotube is a metal with E "0. Charge carriers could be introduced into carbon nanotubes in a controlled manner by means of intercalation [19}26], as could be done for graphite [59}61]. The typical E of graphite intercalation $ compounds is about 0.5}1.5 eV. E "0.5 and $ 0.9 eV (E "0.5 and 1.1 eV) are taken for the doped $ (17, 0) ((10, 10)) nanotube. At lower (higher) Fermi energy, a doped carbon nanotube has one occupied subband (two occupied subbands) in the nH bands, if the state degeneracy is not taken into account. The Fermi energy would determine whether
Fig. 1. The dielectric function of the (17, 0) carbon nanotube is calculated at E "0.5 eV, q"0.05, and (a) ¸"0 and (b) ¸"1. $ The unit of q is As \, here and henceforth. The real part e and the imaginary part e are, respectively, shown by the solid and the dashed curves. The arrows point at the vanishing e .
M.F. Lin, F.L. Shyu / Physica B 292 (2000) 117}126
point) in the energy-wave vector space, it would lead to a singular structure in the response function (Eq. (4a)) or the dielectric function. e (e ) exhibits the discontinuities (the logarithmic divergencies) and the square-root divergencies (the square-root divergencies). e only has a zero near the negative divergence; moreover, it corresponds to a vanishing e . It will induce a sharp plasmon peak in the excitation spectrum (Fig. 3(a)) at u"u , mainly owing to the absence of the e}h excitations. A doped (17, 0) carbon nanotube thus exhibits the ¸"0 low-frequency plasmon. It is also noticed that the ¸"0 plasmon (or the vanishing e and e ) does not exist in the undoped (17, 0) nanotube [41], since a type I nanotube is a semiconductor. e of the ¸"1 excitations is shown in Fig. 1(b) to see the ¸-dependence. The ¸"1 excitations have more excitation channels, as compared with the ¸"0 excitations. There are more singular structures in e as ¸ increases. e might have several zeros. However, e is not vanishing at e P0. Therefore, a prominent plasmon peak in the excitation spectrum (Fig. 3(a)) should be absent. Similar results are obtained for other ¸O0 excitations. Hence, the low-frequency plasmons do not have ¸*1 modes at E "0.5 eV. $ The dielectric function might drastically change as the Fermi energy varies. When the Fermi energy increases from E "0.5 to 0.9 eV, electrons in the $ (17, 0) carbon nanotube could occupy the second subbands J"12 and 22 in addition to the "rst subband J "11 and 23. The excitation channels increase with the occupied subbands, and so does the number of singular structures in e (Figs. 2(a) and 1(a)). e of ¸"0 vanishes at a higher frequency, since more free carriers contribute to the collective excitations. Moreover, e approaches zero at a slower rate. The oscillator strength of collective excitations is inversely proportional to the derivative of e versus u [50,51]. Therefore, the excitation spectrum would become strong in the increasing of E . $ The increase of E could make e of the ¸"1 $ excitations vanish at certain frequencies, as shown in Fig. 2(b). e could vanish at e P0 (pointed out by the arrow). This result means that the ¸"1 plasmon exists in the doped (17, 0) nanotube at E "0.9 eV, but not the doped (17,10) nanotube at $
121
Fig. 2. Same plots as Fig. 1, but shown at E "0.9 eV. The inset $ shows the details for the vanishing e .
E "0.5 eV. However, e does not vanish at e P0 $ for the ¸*2 excitations. The ¸*2 plasmons are absent at E "0.9 eV. The Fermi energy deter$ mines whether there are %*1 plasmons. A doped carbon nanotube could exhibit the ¸"1 (¸"2) plasmon, when the Fermi level is above the second (third) subband of the nH bands. For example, the doped (17, 0) nanotube could exhibit the ¸"1 and 2 plasmons at E 'E "0.62 eV and $ $ E 'E "1.37 eV, respectively. The critical Fermi $ $ energies, E and E , strongly depend on the $ $ nanotube radius and the chiral angle. The energy spacing between two subbands is larger for small carbon nanotubes, so the Fermi level needs to be relatively high to exhibit the ¸"1 and 2 plasmons. The loss function, de"ned as Im[!1/e], is calculated at C"10 meV for a closer study of the low-frequency plasmons. The results of di!erent ¸'s at q"0.05 are shown in Fig. 3(a) for the type-I (17, 0) nanotube. Each pronounced peak is
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crtical momenta. These di!erences illustrate that the dependence of the electronic excitations on the nanotube geometry is strong. The ¸-dependent electronic excitations are strongly a!ected by the transferred momentum. The ¸"0 excitation spectra at di!erent q's are shown in Fig. 4(a) for the (17, 0) nanotube at E "0.5 eV. The plasmon frequency clearly grows $ with q. However, the intensity of the loss function quickly declines in the increasing of q's. The collective excitations are completely damped by the e}h excitations at momenta beyond the critical momentum (q ). The prominent plasmon peak thus vanishes at q'q , e.g., the loss function of the (17, 0) nanotube at q"0.25 (the heavy solid curve). Compared with the ¸"0 plasmon peak in the (17, 0) nanotube, that of the (10, 10) nanotube could survive at larger momentum. For example, at
Fig. 3. The loss functions of di!erent ¸'s are calculated at q"0.05 (a) for the (17, 0) nanotube with E "0.5 or 0.9 eV, and $ (b) for the (10, 10) nanotube at E "0.5 or 1.1 eV. The insets $ show the loss functions which are not clear at the main "gures. Such results are calculated within the RPA. The result within the HA is also shown in (a) for the (17, 0) nanotube with E "0.5 eV $ (the dashed}circled curve).
identi"ed as the low-frequency plasmon. The ¸"0 plasmon exists at E "0.5 eV, while there are no $ ¸*1 plasmons (inset in Fig. 3(a)). The ¸"1 plasmon peak would appear in the loss function, when the Fermi energy is su$ciently high, e.g., the loss function of the (17, 0) nanotube at E "0.9 eV (the $ light-dashed curve). Similar results are obtained for the (10, 10) nanotube (Fig. 3(b)). That is to say, the (10, 10) nanotube exhibits the ¸"0 plasmon (the ¸"0 and 1 plasmons) at lower (higher) Fermi energy. On the other hand, the main di!erences between the type-I (17, 0) nanotube and the type-II (10, 10) nanotube include the intensity of excitation spectrum, the plasmon frequency, and the critical momentum (see below). The former exhibits weaker loss spectra, lower plasmon frequencies, and smaller
Fig. 4. (a) The ¸"0 loss functions of di!erent q's are calculated at E "0.5 eV for the (17, 0) nanotube. The inset shows $ the spectrum at E "0 and q"0.05. (b) The ¸"0 loss func$ tions of the (10, 0) nanotube are shown at di!erent E 's for $ q"0.5 or 0.3.
M.F. Lin, F.L. Shyu / Physica B 292 (2000) 117}126
q"0.3, the ¸"0 plasmon peak exists in the (10, 10) nanotube (Fig. 4(b)), but not the (17, 0) nanotube. The dependence of the loss function on the Fermi energy is further investigated. The type-II (10, 10) nanotube, as shown in Fig. 4(b), could exhibit the prominent ¸"0 plasmon peak in the loss function at any E . When the Fermi energy gradually in$ creases from zero, the plasmon frequency and the intensity of collective excitations remain almost identical at small q and E (E , e.g., the loss $ $ function at q"0.05 and E ) 0.9 eV. All type-II $ carbon nanotubes exhibit a similar result. This special result might be related to the J subband closest to the zero energy. The J subband of the (10, 10) nanotube is approximately linear at E(0.5 c [13,14]. The small-q excitations of ¸"0 mainly result from the J subband, and the highest e}h excitation energy is u +1.5bqc . e (u) generally exhibits a remarkably negative di vergence at u"u and then approaches zero at u . u "1.5c bq hardly varies with E at small q, $ and so do the plasmon frequency and the derivative of e versus u. On the other hand, the ¸"0 loss function of the (10, 10) nanotube is signi"cantly a!ected by the Fermi energy at E 'E (E " $ $ $ 0.94 eV) or large q. At E 'E (e.g., E "1.1 eV), $ $ $ the second occupied subbands contribute to electronic excitations in addition to the "rst occupied subbands. More free carriers take part in the collective excitations. Consequently, the plasmon frequency and the strength of collective excitations are greatly enhanced. At large q (e.g., q"0.3), electrons are excited from the occupied states with a linear dispersion relation to the unoccupied states with a nonlinear dipersion relation. The e}h excitation energy is not equal to 1.5bqc for the case of large q. The loss spectra apparently depend on E at large $ q. The plasmon frequency increases with the Fermi energy, while a simple relation between the intensity of collective excitations and the Fermi energy might be absent. The type-I (17, 0) nanotube contrasts sharply with the type-II (10, 10) nanotube in the E -de$ pendence of the ¸"0 loss functions. For the former, the ¸"0 plasmon does not exist at E "0 $ (inset in Fig. 4(a)) owing to the semiconducting characteristic. Moreover, the plasmon frequency
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Fig. 5. The momentum-dependent plasmon frequency is shown at various E 's for the (17, 0) nanotube within (a) the RPA and $ (b) the HA.
and the oscillator strength of collective excitations increase with the Fermi energy at any momentum (not shown). The dispersion relation of plasmon frequency with q is important in understanding the characteristics of the low-frequency plasmons. u (q) of the type-I (17, 0) nanotube is shown in Fig. 5(a) at di!erent E 's. u of the ¸"0 plasmon rapidly $ grows with q. The strong q-dependence directly re#ects the n-band characteristic, the strong wavevector dependence. At small q, the plasmon frequency of ¸"0 is well described by u J q"ln(qr)" . The q-dependence of u is similar to that in a 1D quantum wire [62,63]. u approaches zero at qP0, so the ¸"0 plasmon belongs to an acoustic plasmon. This plasmon is a quantum mode of the electronic collective oscillations along the nanotube axis. The longitudinal plasma oscillations would behave as a propagating wave. The
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¸"0 plasmon is completely damped by the e}h excitations at q'q . As for the ¸"1 plasmon, it could exist only at E 'E , e.g., E "0.9 eV $ $ $ for the (17, 0) nanotube. The ¸"1 plasmon is an optical plasmon because of a "nite frequency at qP0. It corresponds to the coupled plasma oscillations along the axial and azimuthal directions. The transverse plasma oscillations are in the dipole form, i.e., they behave as a standing wave. The ¸"1 and 0 plasmons could survive at a larger q for a doped nanotube with a higher Fermi en ergy. The important di!erences between the ¸"1 plasmon and the ¸"0 plasmon include the collective plasma oscillations, the dependence of the Fermi energy, and the plasmon frequency. The critical momentum is larger for the ¸"0 plasmon, for which, thus, the experimental measurements are comparatively easy. The ¸"0 plasmon has a larger Ru /Rq; therefore, its plasma wave propagates relatively rapidly.
Fig. 6. The momentum-dependent plasmon frequency is shown for (a) the (10, 10) carbon nanotube at various E 's, and (b) $ various carbon nanotubes at E "0.5 eV. $
The above-mentioned characteristics of the low-frequency plasmons in type-I nanotubes could also be found in type-II nanotubes, e.g., the (10, 10) nanotube (Fig. 6(a)). They include the strong q-dependence of the ¸"0 plasmon, the strong dependence of q on E , the existence of the ¸"1 $ plasmon for E'E , the acoustic plasmon of $ ¸"0, and the optical plasmon of ¸"1. On the other hand, there are several important di!erences between type-I and -II nanotubes (Fig. 5(a)). The former do not have the ¸"0 plasmon at E "0. The ¸"0 plasmon frequency of type-I $ nanotubes is signi"cantly a!ected by the Fermi energy. But for type-II nanotubes, it hardly depends on E at small q and E (E . The oscil$ $ $ lator strength of the ¸"0 plasmon is weaker for the former. The ¸"1 plasmon frequency has a strong q-dependent dispersion relation for type-II nanotubes, while it is almost dispersionless for type-I nanotubes. Moreover, the critical momentum of the low-frequency plasmons is smaller for type-I nanotubes. The e!ects due to the nanotube geometry deserve a closer investgation. The ¸"0 plasmon frequency is shown in Fig. 6(b) for the doped carbon nanotubes with di!erent chiral angles. Type-I and -II nanotubes, respectively, have the nearly same plasmon frequency at small q. Moreover, the zigzag nanotubes with h"03 have lower plasmon frequencies and larger critical momenta, as compared with other nanotubes. The low-frequency plasmons depend on the detailed geometric structure at relatively large q. The low-frequency plasmons are also investigated within the HA. The bare polarizability s(q, ¸, u) in Eq. (5a) is reduced by the short-range electronic correlations. The dielectric response within the HA is smaller than that within the RPA. The real part of the dielectric function would vanish at a lower frequency, and the imaginary part of the dielectric function becomes small. Therefore, within the HA, the plasmon frequency is lower (Figs. 5(a) and (b)), and the oscillator strength of collective excitations is stronger (Fig. 4(a)). The local-"eld e!ects do not change the basic features of the low-frequency plasmons. However, they might modify the plasmon frequencies and the loss spectra.
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4. Concluding remarks In this work, we have studied the low-frequency electronic excitations in the doped carbon nanotubes within the RPA. The dielectric function, the loss function, and the plasmon frequency are investigated. The real (imaginary) part of the dielectric function exhibits discontinuities and square-root divergences (logarithmic divergences and square-root divergences). These special structures come from the e}h excitations. The low-frequency plasmons are identifed from the most prominent peak in the loss function. They strongly depend on the Fermi energy, the nanotube geometry, the transferred momentum, and the transferred angular momentum. The predicted excitation spectra and plasmon frequencies could be veri"ed by the high-resolution REELS. A doped 1D carbon nanotube could exhibit the ¸"0 acoustic plasmon. The plasmon frequency clearly grows with q, and the oscillator strength of collective excitations rapidly declines in the increasing of q's. The ¸"1 optical plasmon also exists in a doped carbon nanotube, when the Fermi energy is above the second subband (E 'E ). The criti$ $ cal Fermi energy E is principally determined by $ the nanotube geometry. It is smaller for a larger carbon nanotube. The main di!erences between the ¸"1 and 0 plasmon include the collective plasma oscillations, the dependence of the Fermi energy, and the q-dependence of plasmon frequency. There are several important di!erences between type-II and -I nanotubes in the ¸"0 and 1 plasmons. The ¸"0 plasmon exists in type-II carbon nanotubes even at E "0, but not type-I carbon nanotubes. $ For the former, the ¸"0 plasmon frequency is almost independent of the Fermi energy at small q and E (E . However, that of the latter is $ $ greatly enhanced by the increase of E . Type-II $ carbon nanotubes have higher plasmon frequencies, larger critical momenta, and stronger loss function. Hence, their ¸"0 plasmons are relatively easily observed in the measurements of loss functions. Moreover, the q-dependence of the ¸"1 plasmon frequency is weaker for type-I carbon nanotubes. The local-"eld e!ects would modify the loss spectra and the plasmon frequencies, while the
125
basic features of the low-frequency plasmons remain unchanged.
Acknowledgements This work was supported in part by the National Science Council of Taiwan, the Republic of China under Grant No. NSC 89-2112-M-006-011.
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Low-frequency electronic excitations in doped carbon nanotubes M.F. Lin *, F.L. Shyu Department of Physics, National Cheng Kung University, Taiwan 70101, Taiwan, ROC Department of Electronics Engineering, Fortune Institute of Technology, Kaohsiung 842, Taiwan, ROC Received 20 September 1999; received in revised form 8 February 2000
Abstract We study the low-frequency electronic excitations of a doped carbon nanotube. The longitudinal dielectric function, the loss function, and the plasmon frequencies are calculated within the random phase approximation. They strongly depend on the transferred momentum (q), the transferred angular momentum (¸), the Fermi energy, and the nanotube geometry (the radius and the chiral angle). All the doped carbon nanotubes could exhibit the ¸"0 acoustic plasmon. There also exists the ¸"1 optical plasmon, when the Fermi energy is su$ciently high. There are several important di!erences between type-I nanotubes and type-II nanotubes. The local-"eld corrections on the loss spectra and the plasmon frequencies are discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Mf; 71.45.Gm; 73.20.Dx Keywords: Electronic excitations; Plasmons; Carbon nanotubes
1. Introduction Iijima [1] recently reported observations of carbon nanotubes, which are graphite sheets rolled up in the cylindrical form. Carbon nanotubes, with radii between 3.39 and 150 As , represent an interesting one-dimensional (1D) system. This system has generated many interesting studies, such as structural [2}10] and electronic properties [11}18]. When carbon nanotubes are packed together, they could form a 3D carbon nanotube bundle [2}7] or a 2D}3D carbon nanotube "lm [8}10]. Moreover, * Corresponding author. Tel.: #886-622757575-65272; fax: #11-886-62747995. E-mail address: m#[email protected] (M.F. Lin).
metallic atoms or molecules [19}26] could be intercalated into carbon nanotubes. Experimental measurements on electrical resistivity [19}21] and Raman spectrum [22] indicate that electrons are transferred from intercalants to carbon atoms, or vice versa. Charge transfer could also occur, when carbon nanotubes lie on some substrates, e.g., carbon nanotubes on gold [27]. In this work, we mainly study the low-frequency electronic excitations in a doped carbon nanotube. The dependence of the electronic excitations on the transferred momentum (q), the transferred angular momentum (¸), the nanotube geometry (the radius r and the chiral angle h), and the Fermi energy (E ) is investigated. $ The local-"eld e!ects on the low-frequency electronic excitations are also discussed.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 9 9 - 3
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The electronic structures of the undoped carbon nanotubes strongly depend on the nanotube geometry [11}18]. As has been discussed [28], a 1D carbon nanotube is a rolled-up 2D graphite sheet, the structure of which is fully speci"ed by a lattice vector R"ma #na . a and a are the primitive lattice vectors of a 2D graphite sheet. The parameters (m, n) uniquely de"ne a 1D carbon nanotube. The radius and the chiral angle of an (m, n) carbon nanotube are, respectively, r""R"/2p "(3(m#mn#n)/2p and h" tan\[!(3n/(2m#n)]. b"1.42 As is the C}C bond length. Type I (II) carbon nanotubes, with 2m#nO3i ("3i; i is an integer), are predicted to be semiconductors (metals) [11}18]. The theoretical predictions had been veri"ed from the STM measurements [27,29]. All the doped carbon nanotubes are metallic. The theoretical calculation [30] predicts that the low-energy n band is hardly a!ected by the intercalation, and the Fermi level rigidly shifts from zero to a "nite value. The rigidband model is suggested to be suitable for studying the low-frequency properties of the doped carbon nanotubes. Such a model had been used to study magnetic [28,31,32] and transport properties [33,34]. It will be further used to investigate the low-frequency electronic excitations. A carbon-based system has the n and p bands which are, respectively, formed by the 2p and X (2s, 2p , 2p ) orbitals. Experimental measurements V W from the electron-energy-loss spectroscopy (EELS) show that the n plasmon frequency is &5}7 eV [35}40] and the n#p plasmon frequency is &22}27 eV [35}39]. Moreover, the undoped typeII (metallic) carbon nanotubes are predicted to exhibit the low-frequency plasmon at &0.1}1 eV [41]. All the doped carbon nanotubes have free carriers in the conduction bands (the nH bands); therefore, they are expected to exhibit the lowfrequency plasmons. Such plasmons are signi"cantly a!ected by the Fermi energy. For a doped carbon nanotube, doping would change the lowfrequency electronic excitations, but not the n plasmon or the p plasmon. The high-frequency
electronic excitations are thus neglected in this study. The low-frequency electronic excitations in the doped (or metallic) nanotube systems had been qualitatively studied within the electron}gas model [42}54]. A 1D carbon nanotube [42}52] could exhibit the electronic excitations of discrete angular momenta (¸'s); that is, there are ¸-decoupled electronic excitations in a 1D cylindrical nanotube. In this work, the real n band obtained from the tightbinding model [13,14] is used to study the electronic excitations of the doped nanotube systems. The above-mentioned characteristics also exist in the present study. The low-frequency excitation properties are quite complicated since the dependence of the n band on the nanotube geometry, the longitudinal wave vector, and the transverse angular momentum is very strong. The calculated results could be tested by the re#ection-electronenergy-loss spectroscopy (REELS) with a high resolution of &10 meV [53]. The paper is organized as follows. In Section 2, the dielectric function of a doped carbon nanotube is calculated within the random phase approximation (RPA). The calculated results are discussed in Section 3. Concluding remarks are given in Section 4. 2. Electronic excitations of a doped carbon nanotube The electronic structures of carbon nanotubes strongly depend on the nanotube geometry, and so do the electronic excitations. The n band is calculated from the tight-binding model [13,14]. The energy dispersion relations of the (m, n) carbon nanotube are [28] E (J, k)"$c
;cos
1#4 cos
3b(k cos h#J sin h/r) 2
(3b(k sin h!J cos h/r) 2
#4 cos
(3b(k sin h!J cos h/r) 2
. (1)
The wave functions are For the details of the nanotube geometry and the n-band structure see Ref. [28].
1 HH (J, k) W " ; (J, k)$ ; (J, k) . ( I (2 "H (J, k)"
(2)
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c ("3.033 eV) [13,14] is the resonance integral, and H is the nearest-neighbor Hamiltonian matrix element. ; and ; are the two tight binding functions of the 2p orbitals. The electronic X states are described by the angular momentum J along the azimuthal direction and the wave vector k along the nanotube axis. The subband nearest E"0 is denoted by J . The supercripts c and v, respectively, represent the nH and n bands. The n bands are symmetric to the nH bands about the zero energy. For the undoped carbon nanotubes, electrons only occupy the n band with E(J, k))0 at ¹"0. An (m, n) carbon nanotube with 2m#n"3i (O3i) is a metal (semiconductor). A metallic carbon nanotube (type II) has the linear J subbands intersecting at the Fermi level E "0. $ A semiconducting carbon nanotube (type I) has an energy gap E "bc /r inversely proportional to the nanotube radius. In general, a carbon nanotube has many parabolic subbands of di!erent J's except that the J subbands in a metallic nanotube are linear. As for a doped carbon nanotube, electrons could occupy the nH bands as well as the n bands. The characteristics of the electronic structures will be directly re#ected in the electronic excitations. The electronic excitations of a cylindrical carbon nanotube are characterized by the transferred momentum and angular momentum, because ¸ and q are conserved in the electron}electron (e}e) Coulomb interactions. The dielectric function of a carbon nanotube, which is calculated from the RPA [54], is given by [55] e(q, ¸, u)"e !<(q, ¸)s(q, ¸, u), (3) where e ("2.4) is the background dielectric con stant due to the high-frequency excitations. <(q, ¸)" 4peI (qr)K (qr) is the 1D bare Coulomb interac* * tion. I [K ] is the modi"ed Bessel function of the * * "rst (second) kind of the order ¸. The linear response function (or the bare polarizability) in Eq. (3) is 2 s(q, ¸,u)" (2p)
For the details of the n-electron response function see Ref. [55].
119
f (EFY(J#¸, k#q))!f (EF(J, k)) ; dk EFY(J#¸, k#q)!EF(J, k)!(u#iC) ( F FY ;"1J#¸, k#q, h "e *(Ye OW"J, k, h2",
(4a)
where "1J#¸, k#q, h "e *(Ye OW"J, k , h2" W 1 HH (J#¸, k#q)HH (J, k) " 1$ , (4b) 4 "HH (J#¸, k#q)HH (J, k)" and f in Eq. (4a) is the Fermi}Dirac distribution function. C is a phenomenological level broadening parameter. When an undoped carbon nanotube (E "0) is perturbed by an external potential at $ ¹"0, electrons are excited from the occupied nband states to the unoccupied nH-band states. That is to say, they only exhibit the inter-n-band excitations. But for a doped carbon nanotube, electrons are excited from the occupied nH-band states to the unoccupied nH-band states in addition to the intern-band excitations. That is to say, there are two kinds of electronic excitations, inter-n-band excitations and intra-n-band excitations. The low-frequency electronic excitations, with u)2E , $ mainly come from the intra-n-band excitations. However, the inter-n-band excitations would a!ect the real part of the dielectric function and thus the low-frequency plasmon frequency. Apparently, the electronic excitations of doped carbon nanotubes are more complicated than those of undoped carbon nanotubes. # and ! in Eq. (4b), respectively, correspond to the intra- and the inter-n-band excitations. The short-range electronic correlations might affect the electronic excitations. The above-mentioned RPA does not take into account the exchange}correlation e!ects. In general, there are two methods for improving the RPA [47,56]. They are the Hubbard approximation (HA) [56,57] and the local-"eld theory of Singwi et al. [56,58]. The latter includes the exchange}correlation e!ects. The linear response function in Eq. (4a) does not have an analytic form. Therefore, it needs too much calculation time to get the modi"ed plasmon frequency. The former, which includes only the exchange e!ects, is taken to see the local-"eld e!ects. Within the Hubbard approximation, the ladder
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vertex integral [56] for the irreducible polarizability is solved in a rough fashion. The dielectric function remains in a similar form as that of the RPA (Eq. (3)). However, the irreducible polarizability s&(q, ¸, u) is related to the bare polarizability s(q, ¸, u) by the formula [56,57] s(q, ¸, u) s&(q, ¸, u)" 1#<(q, ¸)G&(q, ¸)s(q, ¸, u)/e and <((q#k 0), ¸) $ G&(q, ¸)" 2<(q, ¸),
(5a)
(5b)
where k (0) is the Fermi momentum of the occu$ pied J subband. The static factor G&(q, ¸) is due to the local-"eld e!ects of the exchange holes. Eqs. (3)}(5) will be utilized to examine the local-"eld e!ects on the electronic excitations.
a doped carbon nanotube could exhibit the lowfrequency plasmons of di!erent ¸'s. We "rst see the electronic excitations of the doped (17, 0) nanotube with E "0.5 eV. Electrons $ occupy the nH bands of J "11 and 23 as well as all the n bands. The dielectric function of the ¸"0 excitations is shown in Fig. 1(a) at q"0.05 and C"0. The unit of q is As \, here and henceforth. The real [e (q, ¸, u)] and the imaginary [e (q, ¸, u)] parts of the dielectric function are shown by the solid and the dashed curves, respectively. For the low-frequency ¸"0 excitations, electrons are excited from the occupied nH-band state of (J , k) to the unoccupied nH-band state of (J , k#q). The intra-n-band excitation energy is u "E(J , k#q)!E(J , k). When the electronic excitation of the (J , k) state corresponds to a critical point (a minimum, a maximum, or a saddle
3. The calculated results We mainly use the RPA to calculate the dielectric function, the loss spectrum, and the plasmon frequency. In the following calculations, the type-I (17, 0) and the type-II (10, 10) nanotubes are chosen for a detailed study. They are, respectively, called zig-zag and armchair nanotubes. The former has r"6.65 As and h"03, and the latter has r"6.78 As and h"!303. Their radii are almost the same, while the di!erent chiral angles lead to important di!erences in the electronic structures, e.g., the number of subbands and the energy spacing between two subbands. The undoped (17, 0) nanotube is a semiconductor with E "0.647 eV, and the undoped (10, 10) nanotube is a metal with E "0. Charge carriers could be introduced into carbon nanotubes in a controlled manner by means of intercalation [19}26], as could be done for graphite [59}61]. The typical E of graphite intercalation $ compounds is about 0.5}1.5 eV. E "0.5 and $ 0.9 eV (E "0.5 and 1.1 eV) are taken for the doped $ (17, 0) ((10, 10)) nanotube. At lower (higher) Fermi energy, a doped carbon nanotube has one occupied subband (two occupied subbands) in the nH bands, if the state degeneracy is not taken into account. The Fermi energy would determine whether
Fig. 1. The dielectric function of the (17, 0) carbon nanotube is calculated at E "0.5 eV, q"0.05, and (a) ¸"0 and (b) ¸"1. $ The unit of q is As \, here and henceforth. The real part e and the imaginary part e are, respectively, shown by the solid and the dashed curves. The arrows point at the vanishing e .
M.F. Lin, F.L. Shyu / Physica B 292 (2000) 117}126
point) in the energy-wave vector space, it would lead to a singular structure in the response function (Eq. (4a)) or the dielectric function. e (e ) exhibits the discontinuities (the logarithmic divergencies) and the square-root divergencies (the square-root divergencies). e only has a zero near the negative divergence; moreover, it corresponds to a vanishing e . It will induce a sharp plasmon peak in the excitation spectrum (Fig. 3(a)) at u"u , mainly owing to the absence of the e}h excitations. A doped (17, 0) carbon nanotube thus exhibits the ¸"0 low-frequency plasmon. It is also noticed that the ¸"0 plasmon (or the vanishing e and e ) does not exist in the undoped (17, 0) nanotube [41], since a type I nanotube is a semiconductor. e of the ¸"1 excitations is shown in Fig. 1(b) to see the ¸-dependence. The ¸"1 excitations have more excitation channels, as compared with the ¸"0 excitations. There are more singular structures in e as ¸ increases. e might have several zeros. However, e is not vanishing at e P0. Therefore, a prominent plasmon peak in the excitation spectrum (Fig. 3(a)) should be absent. Similar results are obtained for other ¸O0 excitations. Hence, the low-frequency plasmons do not have ¸*1 modes at E "0.5 eV. $ The dielectric function might drastically change as the Fermi energy varies. When the Fermi energy increases from E "0.5 to 0.9 eV, electrons in the $ (17, 0) carbon nanotube could occupy the second subbands J"12 and 22 in addition to the "rst subband J "11 and 23. The excitation channels increase with the occupied subbands, and so does the number of singular structures in e (Figs. 2(a) and 1(a)). e of ¸"0 vanishes at a higher frequency, since more free carriers contribute to the collective excitations. Moreover, e approaches zero at a slower rate. The oscillator strength of collective excitations is inversely proportional to the derivative of e versus u [50,51]. Therefore, the excitation spectrum would become strong in the increasing of E . $ The increase of E could make e of the ¸"1 $ excitations vanish at certain frequencies, as shown in Fig. 2(b). e could vanish at e P0 (pointed out by the arrow). This result means that the ¸"1 plasmon exists in the doped (17, 0) nanotube at E "0.9 eV, but not the doped (17,10) nanotube at $
121
Fig. 2. Same plots as Fig. 1, but shown at E "0.9 eV. The inset $ shows the details for the vanishing e .
E "0.5 eV. However, e does not vanish at e P0 $ for the ¸*2 excitations. The ¸*2 plasmons are absent at E "0.9 eV. The Fermi energy deter$ mines whether there are %*1 plasmons. A doped carbon nanotube could exhibit the ¸"1 (¸"2) plasmon, when the Fermi level is above the second (third) subband of the nH bands. For example, the doped (17, 0) nanotube could exhibit the ¸"1 and 2 plasmons at E 'E "0.62 eV and $ $ E 'E "1.37 eV, respectively. The critical Fermi $ $ energies, E and E , strongly depend on the $ $ nanotube radius and the chiral angle. The energy spacing between two subbands is larger for small carbon nanotubes, so the Fermi level needs to be relatively high to exhibit the ¸"1 and 2 plasmons. The loss function, de"ned as Im[!1/e], is calculated at C"10 meV for a closer study of the low-frequency plasmons. The results of di!erent ¸'s at q"0.05 are shown in Fig. 3(a) for the type-I (17, 0) nanotube. Each pronounced peak is
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M.F. Lin, F.L. Shyu / Physica B 292 (2000) 117}126
crtical momenta. These di!erences illustrate that the dependence of the electronic excitations on the nanotube geometry is strong. The ¸-dependent electronic excitations are strongly a!ected by the transferred momentum. The ¸"0 excitation spectra at di!erent q's are shown in Fig. 4(a) for the (17, 0) nanotube at E "0.5 eV. The plasmon frequency clearly grows $ with q. However, the intensity of the loss function quickly declines in the increasing of q's. The collective excitations are completely damped by the e}h excitations at momenta beyond the critical momentum (q ). The prominent plasmon peak thus vanishes at q'q , e.g., the loss function of the (17, 0) nanotube at q"0.25 (the heavy solid curve). Compared with the ¸"0 plasmon peak in the (17, 0) nanotube, that of the (10, 10) nanotube could survive at larger momentum. For example, at
Fig. 3. The loss functions of di!erent ¸'s are calculated at q"0.05 (a) for the (17, 0) nanotube with E "0.5 or 0.9 eV, and $ (b) for the (10, 10) nanotube at E "0.5 or 1.1 eV. The insets $ show the loss functions which are not clear at the main "gures. Such results are calculated within the RPA. The result within the HA is also shown in (a) for the (17, 0) nanotube with E "0.5 eV $ (the dashed}circled curve).
identi"ed as the low-frequency plasmon. The ¸"0 plasmon exists at E "0.5 eV, while there are no $ ¸*1 plasmons (inset in Fig. 3(a)). The ¸"1 plasmon peak would appear in the loss function, when the Fermi energy is su$ciently high, e.g., the loss function of the (17, 0) nanotube at E "0.9 eV (the $ light-dashed curve). Similar results are obtained for the (10, 10) nanotube (Fig. 3(b)). That is to say, the (10, 10) nanotube exhibits the ¸"0 plasmon (the ¸"0 and 1 plasmons) at lower (higher) Fermi energy. On the other hand, the main di!erences between the type-I (17, 0) nanotube and the type-II (10, 10) nanotube include the intensity of excitation spectrum, the plasmon frequency, and the critical momentum (see below). The former exhibits weaker loss spectra, lower plasmon frequencies, and smaller
Fig. 4. (a) The ¸"0 loss functions of di!erent q's are calculated at E "0.5 eV for the (17, 0) nanotube. The inset shows $ the spectrum at E "0 and q"0.05. (b) The ¸"0 loss func$ tions of the (10, 0) nanotube are shown at di!erent E 's for $ q"0.5 or 0.3.
M.F. Lin, F.L. Shyu / Physica B 292 (2000) 117}126
q"0.3, the ¸"0 plasmon peak exists in the (10, 10) nanotube (Fig. 4(b)), but not the (17, 0) nanotube. The dependence of the loss function on the Fermi energy is further investigated. The type-II (10, 10) nanotube, as shown in Fig. 4(b), could exhibit the prominent ¸"0 plasmon peak in the loss function at any E . When the Fermi energy gradually in$ creases from zero, the plasmon frequency and the intensity of collective excitations remain almost identical at small q and E (E , e.g., the loss $ $ function at q"0.05 and E ) 0.9 eV. All type-II $ carbon nanotubes exhibit a similar result. This special result might be related to the J subband closest to the zero energy. The J subband of the (10, 10) nanotube is approximately linear at E(0.5 c [13,14]. The small-q excitations of ¸"0 mainly result from the J subband, and the highest e}h excitation energy is u +1.5bqc . e (u) generally exhibits a remarkably negative di vergence at u"u and then approaches zero at u . u "1.5c bq hardly varies with E at small q, $ and so do the plasmon frequency and the derivative of e versus u. On the other hand, the ¸"0 loss function of the (10, 10) nanotube is signi"cantly a!ected by the Fermi energy at E 'E (E " $ $ $ 0.94 eV) or large q. At E 'E (e.g., E "1.1 eV), $ $ $ the second occupied subbands contribute to electronic excitations in addition to the "rst occupied subbands. More free carriers take part in the collective excitations. Consequently, the plasmon frequency and the strength of collective excitations are greatly enhanced. At large q (e.g., q"0.3), electrons are excited from the occupied states with a linear dispersion relation to the unoccupied states with a nonlinear dipersion relation. The e}h excitation energy is not equal to 1.5bqc for the case of large q. The loss spectra apparently depend on E at large $ q. The plasmon frequency increases with the Fermi energy, while a simple relation between the intensity of collective excitations and the Fermi energy might be absent. The type-I (17, 0) nanotube contrasts sharply with the type-II (10, 10) nanotube in the E -de$ pendence of the ¸"0 loss functions. For the former, the ¸"0 plasmon does not exist at E "0 $ (inset in Fig. 4(a)) owing to the semiconducting characteristic. Moreover, the plasmon frequency
123
Fig. 5. The momentum-dependent plasmon frequency is shown at various E 's for the (17, 0) nanotube within (a) the RPA and $ (b) the HA.
and the oscillator strength of collective excitations increase with the Fermi energy at any momentum (not shown). The dispersion relation of plasmon frequency with q is important in understanding the characteristics of the low-frequency plasmons. u (q) of the type-I (17, 0) nanotube is shown in Fig. 5(a) at di!erent E 's. u of the ¸"0 plasmon rapidly $ grows with q. The strong q-dependence directly re#ects the n-band characteristic, the strong wavevector dependence. At small q, the plasmon frequency of ¸"0 is well described by u J q"ln(qr)" . The q-dependence of u is similar to that in a 1D quantum wire [62,63]. u approaches zero at qP0, so the ¸"0 plasmon belongs to an acoustic plasmon. This plasmon is a quantum mode of the electronic collective oscillations along the nanotube axis. The longitudinal plasma oscillations would behave as a propagating wave. The
124
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¸"0 plasmon is completely damped by the e}h excitations at q'q . As for the ¸"1 plasmon, it could exist only at E 'E , e.g., E "0.9 eV $ $ $ for the (17, 0) nanotube. The ¸"1 plasmon is an optical plasmon because of a "nite frequency at qP0. It corresponds to the coupled plasma oscillations along the axial and azimuthal directions. The transverse plasma oscillations are in the dipole form, i.e., they behave as a standing wave. The ¸"1 and 0 plasmons could survive at a larger q for a doped nanotube with a higher Fermi en ergy. The important di!erences between the ¸"1 plasmon and the ¸"0 plasmon include the collective plasma oscillations, the dependence of the Fermi energy, and the plasmon frequency. The critical momentum is larger for the ¸"0 plasmon, for which, thus, the experimental measurements are comparatively easy. The ¸"0 plasmon has a larger Ru /Rq; therefore, its plasma wave propagates relatively rapidly.
Fig. 6. The momentum-dependent plasmon frequency is shown for (a) the (10, 10) carbon nanotube at various E 's, and (b) $ various carbon nanotubes at E "0.5 eV. $
The above-mentioned characteristics of the low-frequency plasmons in type-I nanotubes could also be found in type-II nanotubes, e.g., the (10, 10) nanotube (Fig. 6(a)). They include the strong q-dependence of the ¸"0 plasmon, the strong dependence of q on E , the existence of the ¸"1 $ plasmon for E'E , the acoustic plasmon of $ ¸"0, and the optical plasmon of ¸"1. On the other hand, there are several important di!erences between type-I and -II nanotubes (Fig. 5(a)). The former do not have the ¸"0 plasmon at E "0. The ¸"0 plasmon frequency of type-I $ nanotubes is signi"cantly a!ected by the Fermi energy. But for type-II nanotubes, it hardly depends on E at small q and E (E . The oscil$ $ $ lator strength of the ¸"0 plasmon is weaker for the former. The ¸"1 plasmon frequency has a strong q-dependent dispersion relation for type-II nanotubes, while it is almost dispersionless for type-I nanotubes. Moreover, the critical momentum of the low-frequency plasmons is smaller for type-I nanotubes. The e!ects due to the nanotube geometry deserve a closer investgation. The ¸"0 plasmon frequency is shown in Fig. 6(b) for the doped carbon nanotubes with di!erent chiral angles. Type-I and -II nanotubes, respectively, have the nearly same plasmon frequency at small q. Moreover, the zigzag nanotubes with h"03 have lower plasmon frequencies and larger critical momenta, as compared with other nanotubes. The low-frequency plasmons depend on the detailed geometric structure at relatively large q. The low-frequency plasmons are also investigated within the HA. The bare polarizability s(q, ¸, u) in Eq. (5a) is reduced by the short-range electronic correlations. The dielectric response within the HA is smaller than that within the RPA. The real part of the dielectric function would vanish at a lower frequency, and the imaginary part of the dielectric function becomes small. Therefore, within the HA, the plasmon frequency is lower (Figs. 5(a) and (b)), and the oscillator strength of collective excitations is stronger (Fig. 4(a)). The local-"eld e!ects do not change the basic features of the low-frequency plasmons. However, they might modify the plasmon frequencies and the loss spectra.
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4. Concluding remarks In this work, we have studied the low-frequency electronic excitations in the doped carbon nanotubes within the RPA. The dielectric function, the loss function, and the plasmon frequency are investigated. The real (imaginary) part of the dielectric function exhibits discontinuities and square-root divergences (logarithmic divergences and square-root divergences). These special structures come from the e}h excitations. The low-frequency plasmons are identifed from the most prominent peak in the loss function. They strongly depend on the Fermi energy, the nanotube geometry, the transferred momentum, and the transferred angular momentum. The predicted excitation spectra and plasmon frequencies could be veri"ed by the high-resolution REELS. A doped 1D carbon nanotube could exhibit the ¸"0 acoustic plasmon. The plasmon frequency clearly grows with q, and the oscillator strength of collective excitations rapidly declines in the increasing of q's. The ¸"1 optical plasmon also exists in a doped carbon nanotube, when the Fermi energy is above the second subband (E 'E ). The criti$ $ cal Fermi energy E is principally determined by $ the nanotube geometry. It is smaller for a larger carbon nanotube. The main di!erences between the ¸"1 and 0 plasmon include the collective plasma oscillations, the dependence of the Fermi energy, and the q-dependence of plasmon frequency. There are several important di!erences between type-II and -I nanotubes in the ¸"0 and 1 plasmons. The ¸"0 plasmon exists in type-II carbon nanotubes even at E "0, but not type-I carbon nanotubes. $ For the former, the ¸"0 plasmon frequency is almost independent of the Fermi energy at small q and E (E . However, that of the latter is $ $ greatly enhanced by the increase of E . Type-II $ carbon nanotubes have higher plasmon frequencies, larger critical momenta, and stronger loss function. Hence, their ¸"0 plasmons are relatively easily observed in the measurements of loss functions. Moreover, the q-dependence of the ¸"1 plasmon frequency is weaker for type-I carbon nanotubes. The local-"eld e!ects would modify the loss spectra and the plasmon frequencies, while the
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basic features of the low-frequency plasmons remain unchanged.
Acknowledgements This work was supported in part by the National Science Council of Taiwan, the Republic of China under Grant No. NSC 89-2112-M-006-011.
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