Volume 5 Issue 8 - September 19, 2008
Electronic excitations and deexcitations in narrow-gap carbon nanotubes
Ming-Fa Lin* and Chih-Wei Chiu

Department of Physics, College of Sciences, National Cheng Kung University
*Email: mflin@mail.ncku.edu.tw

Nanotechnology 18, 435401

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The quasi-one-dimensional carbon nanotubes (CNs) have prompted numerous interesting studies because of their nanoscaled cylindrical structures. The geometric structure dominates over band structures and thus essential physical properties e.g., the low-frequency electronic excitations and the inelastic scatterings. From the tight-binding model and LDA calculations, a 1D CN may be (I) a gapless, (II) narrow-gap, or (III) moderate-gap system, which depends on radius (r) and chiral angle (θ). These calculated results can be directly verified by scanning tunneling spectroscopy. The main characteristics of type-II CNs would be directly reflected in low-frequency Coulomb excitations and deexcitations. In this work, we mainly study the Coulomb decay rates of the lowest conduction and valence bands in narrow-gap CNs. The dependence on radius, wave vector and deexcitation mechanisms would be investigated in detail.

The geometric structure of CNs is characterized by the parameters (m,n), e.g., (m,0) zigzag nanotubes. We use the nearest-neighbor tight-binding model with curvature effect to evaluate the band structures. Only three nearest-neighbor interactions, γ1, γ2 and γ3, are taken into account. They are identical and equal to γ0 (=3.033 eV) for a 2D monolayer graphene. When a graphite sheet is rolled up in the cylindrical form, these three interactions along the different directions might not be the same with one another, which is due to the curvature effect from the misorientation of orbitals. The electronic states are described by the angular momentum J along the transverse direction and the wave vector k along the nanotube axis. The (m,0) CNs with m=3I (I is an integer) are narrow-gap semiconductors. For type-II CNs, the transfer integrals become γ1=γ0 (1-π/8m2) and γ2=γ3=γ0. Each subband is denoted by the angular momentum or the subband index J=1,2...,2m. The lowest one is J=2m/3. The (12,0), (15,0), (18,0) and (21,0) CNs, as shown in Fig. 1(a), are chosen for a model study. Their lowest subbands are, respectively, J=8, 10, 12 ,and 14. The energy dispersions are given by
where b=14.2 nm is the C-C band length. k is confined within the first Brillouin zone. The occupied valence-band states (v) and the unoccupied conduction-band states (c) are symmetric with respect to the Fermi level EF=0. The energy dispersions are parabolic at small k and linear at others. The linear part is roughly described by Ec~3γ0b|k|/2 and hardly affected by r. The band-edge states are located at k=0. The curvature of the parabolic part increases as r grows. The energy gap (Eg) exhibits the opposite behavior, e.g., (52.0 meV, 33.3 meV, 23.1 meV and 17.0 meV) for ((12,0), (15,0), (18,0) and (21,0)) CNs. Such small Eg might be lower than the thermal energy at room temperature. Here Eg and parabolic energy bands completely originate from the curvature effect.

The longitudinal dielectric function (ε) of CNs is used in understanding the elementary excitations. A cylindrical CN exhibits rich and interesting excitation spectra, since the transferred momentum (q) and angular momentum (L) are conserved in the electron-electron (e-e) interactions. Each excitation spectrum is characterized by q and L. The L=0 mode, with the lowest excitation frequency, is sufficient to demonstrate the low-energy excitations; hence, it could be used to comprehend many-particle properties of free carriers near the Fermi level. The temperature (T)-dependent dielectric function is given by

nF is the Fermi-Dirac distribution function. kB is the Boltzmann constant. The chemical potential μ remains zero for any T because of the symmetry of the low energy bands about the Fermi level. ε0=2.4 is the background dielectric constant contributed from high-energy excitations. V(q,L=0)=4πe2I0(qr)K0(qr) is the bare Coulomb interaction. I0(qr) [K0(qr)] is the first [second] kind of modified Bessel function of zero order. The unit of q is 105/cm in the following results.
Fig.1 (a) The first conduction bands are shown for (12,0), (15,0), (18,0) and (21,0) CNs. The imaginary part (b) and the real part (c) of the L=0 dielectric function are shown at q=0.1, and different r's & T's.

The imaginary part of the dielectric functions (ε2) of narrow-gap CNs is presented in Fig. 1(b) at L=0, q=0.1, and different T's. Temperature can induce some free electrons to occupy the conduction band. Holes also exist in the valence bands. Such two kinds of free carriers are excited to the other states of the same subband. They cause a prominent peak in ε2 at intraband single-particle excitation energy ωex=Ec(k+q)-Ec(k). At low T, this peak with lower ωex mainly comes from the excitations of the free carriers near the band-edge state (k=0). As T grows, the final-state number declines quickly in the vicinity of k=0, and the free carrier population on the linear-part bands increases simultaneously. The excitations within the linear-part subbands are more and more eminent. ωex is gradually close to 3γ0bq/2, and the smaller CNs need higher temperature to lead ωex→3γ0bq/2. Moreover, the peak strength becomes stronger with T or r increasing. These results directly reflect the properties of the band structures, that is, the consistency of the linear-part energy dispersions and the Eg's changing with nanotube radius. The real part of the dielectric function (ε1) exhibits a special structure that falls rapidly from the maximum to the minimum at ωex, and then approaches to zero [Fig. 1(c)]. If ε1 is approaching to zero at small ε2, the plasmon peak exists in the loss spectrum (Fig. 2).

The L=0 loss function,
Fig.2 The (q=0.1, L=0) loss functions for different r's and T's
defined as Im[-1/ε(L=0)], is shown in Fig. 2(a) at q= 0.1 and different T's. Each pronounced peak is identified as the intraband plasmon. Both plasmon frequency (ωp) and strength are getting higher with the increase of T and r. Such a strong plasmon mode does not contribute to the low-energy decay channels, since its frequency is much higher than the decay energy of the excited state. However, at lower frequency (the inset of Fig. 2(a) and Fig. 2(b)), the weak loss spectra, due to the e-h excitations near k=0, place great importance on the inelastic e-e scatterings. They are profoundly affected by T and r.

Electron-electron interactions could affect heavily the low-energy properties of the CNs. They induce the rich excitation spectra and lead to the prominent decay rates of the low-energy states at finite T. Each k state could be occupied by electrons and holes at non-zero T. The decay rates associated with electrons and holes are worthy of a discussion. Due to the band symmetry, the conduction- and valence-band electron decay rates are, respectively, equal to the valence- and conduction-band hole decay rates. We only focus on the electron decay rates in the lowest conduction and valence bands. The decay energy is defined as ωde =Eh(k,2m/3)-Eh'(k+q,2m/3). Besides the c→c deexcitations under positive ωde (ωde+) at T=0, there are the deexcitations under negative ωde (ωde-) and more decay channels (v→v, c→v and v→c) at finite T. All decay transitions must be satisfied by the conservation of energy and momentum. The Coulomb decay rates could be obtained from the electron self-energy or the Golden Rule by Fermi. The inelastic scattering rate of the (k,2m/3,h) state at T is given by
where the square of the effective e-e Coulomb interaction is

nB is the Bose distribution function. The first term Im[-Veff(ωde)] in Eq. (2) is proportional to Im[-1/ε(ωde)]. The second term nB(-ωde)[1-nF(Eh'(k+q,2m/3))] is associated with the number of deexcitation channels, where nB(-ωde) and nF(Eh'(k+q,2m/3)) indicate the number of -ωde transfers and the final-state distribution function, respectively. The restrictions of nB(-ωde) accentuate the importance of small-(q,ωde) transfers. nB(-ωde) declines rapidly as ωde grows, especially for nB(ωde+). That is, the probability of the transfer with absorbing energy is smaller than that with emitting energy.

Fig.3 (a) The Coulomb decay rates for the first conduction bands. (b) and (c) are those due to the decay channels with positive energy and the interband decay channels, respectively.
The wave vector dependence of the decay rates is shown in Fig. 3(a) at room temperature for the lowest conduction band. There is no simple relation between 1/τ and wave vector. The decay rates exhibit a weak oscillation at small k and almost remain unchanged at others. They are contributed unequally from the ωde+ and ωde- intraband deexcitations and ωde+ interband deexcitations. For example, the band-edge state is deexcited to the conduction-band states under the ωde- intraband transfers and the valence-band states under the ωde+ interband transfers, but not the ωde+ intraband transfers. The decay rate of this state is about 200 meV (τ~20 fs) for the (12,0) CN. 1/τ with the ωde+ intraband transfers quickly becomes larger from zero as k grows from k=0 [Fig. 3(b)], and it has the same magnitude as 1/τ with the ωde- intraband transfers except k near the band-edge state. As regards interband deexcitations, 1/τ decreases rapidly as k grows. The initial (k,c) states are principally deexcited to the final (k,v) states in the vicinity of k=0; therefore, the decay energy will generally become large as initial k grows. The interband decay occurs at the sufficiently high decay energy (ωdeEg) to such an extent that it is much slower than the intraband decay [Fig. 3(c)]. The ratio of 1/τ's between the former and the latter is about 102.

The radius dependence of the decay rates are greatly affected by T and decay mechanisms. 1/τ under the intraband transfers exhibits a monotonous change with r at T=300 K [Figs. 3(a) and 3(b)]. The larger the radius is, the smaller the decay rate is. But, the decay rates reversely vary with r when T<30 K. These features reflect the dependence of the loss spectrum on temperature and radius [Fig. 2(b)]. As to the interband decay, 1/τ increases as r grows at small k because the smaller-Eg CN has the stronger loss spectrum. However, the Eg effects are getting weaker and weaker with the increase of k. The r dependence of the decay rates behaves the opposite at large k.

Fig.4 (a) The Coulomb decay rates for the first valence bands. (b) and (c) are those due to the intraband decay with negative energy and interband decay, respectively.
The electron decay rates of the valence-band states, as shown in Fig. 4, are worth an in-depth study. They also represent the hole decay rates of the conduction-band states. The (k,v) and (k,c) states exhibit different decay behaviors about the magnitude and the wave vector dependence. The decay rates of the valence-band states [1/τ(k=0,v)=70 meV for (12,0) CN] are smaller than those of the conduction-band states, and they decline with k increasing [Fig. 4(a)]. Such features are based on different reasons for the intraband deexcitations and interband deexcitations. The former is limited by the final-state distribution functions [Figs. 4(a) and 4(b)]. As to the interband v→c decay, the more final-state number can enhance the decay rates, but its contributions are covered by the small nB(-ωde) due to large-ωde transfers [Fig. 4(c)]. That is to say, few c→v interband excitations are available to be the effective decay channels.

In conclusion, the temperature-induced electronic excitations and deexcitations in narrow-gap CNs have been investigated in this work. The band structures predominate over the excitation spectra and the electron decay rates. The Coulomb scattering rates strongly depend on decay mechanisms, radius, and wave vector. The e-e interactions exhibit the intraband and interband e-h excitations at finite temperature. Such two kinds of excitations, especially the former, can be effective decay channels. The decay rates with interband deexcitations increase as radius grows at room temperature. The electron conduction- and valence-band states have significant differences in the wave vector dependence of the decay rates. The dependence on wave vector appears weak on the conduction-band states. For the valence-band states, the decay rates decline fast when the initial states deviate from the band-edge state. These calculated results could be verified by experimental measurements from the femtosecond time-resolved photoelectron, transmission, and fluorescence spectroscopies.
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