





Electronic excitations and deexcitations in
narrowgap carbon nanotubes MingFa Lin^{*} and ChihWei
Chiu


  

The quasionedimensional 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 lowfrequency electronic excitations and the
inelastic scatterings. From the tightbinding model and LDA
calculations, a 1D CN may be (I) a gapless, (II) narrowgap, or
(III) moderategap 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 typeII CNs would be directly reflected in
lowfrequency Coulomb excitations and deexcitations. In this work,
we mainly study the Coulomb decay rates of the lowest conduction and
valence bands in narrowgap 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 nearestneighbor tightbinding model with curvature effect
to evaluate the band structures. Only three nearestneighbor
interactions, γ_{1}, γ_{2} and γ_{3}, are taken into account.
They are identical and equal to γ_{0} (=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 pπ 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 narrowgap semiconductors. For typeII
CNs, the transfer integrals become γ_{1}=γ_{0}
(1π/8m^{2}) and γ_{2}=γ_{3}=γ_{0}. 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 CC band
length. k is confined within the first Brillouin zone. The
occupied valenceband states (v) and the unoccupied
conductionband states (c) are symmetric with respect to the
Fermi level E_{F}=0. The energy
dispersions are parabolic at small k and linear at others.
The linear part is roughly described by
E^{c}~3γ_{0}bk/2 and
hardly affected by r. The bandedge states are located at
k=0. The curvature of the parabolic part increases as r
grows. The energy gap (E_{g}) 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 E_{g} might be lower than
the thermal energy at room temperature. Here E_{g} 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 electronelectron (ee) interactions. Each excitation
spectrum is characterized by q and L. The L=0
mode, with the lowest excitation frequency, is sufficient to
demonstrate the lowenergy excitations; hence, it could be used to
comprehend manyparticle properties of free carriers near the Fermi
level. The temperature (T)dependent dielectric function is
given by
n_{F} is the FermiDirac
distribution function. k_{B} 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. ε_{0}=2.4 is the background
dielectric constant contributed from highenergy excitations.
V(q,L=0)=4πe^{2}I_{0}(qr)K_{0}(qr) is the bare
Coulomb interaction. I_{0}(qr) [K_{0}(qr)] is the first
[second] kind of modified Bessel function of zero order. The unit of
q is 10^{5}/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
(ε_{2}) of
narrowgap 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
ε_{2} at
intraband singleparticle excitation energy ω_{ex}=E^{c}(k+q)E^{c}(k). At low
T, this peak with lower ω_{ex} mainly comes from the
excitations of the free carriers near the bandedge state
(k=0). As T grows, the finalstate number declines
quickly in the vicinity of k=0, and the free carrier
population on the linearpart bands increases simultaneously. The
excitations within the linearpart subbands are more and more
eminent. ω_{ex}
is gradually close to 3γ_{0}bq/2, and the
smaller CNs need higher temperature to lead ω_{ex}→3γ_{0}bq/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 linearpart energy
dispersions and the E_{g}'s changing with
nanotube radius. The real part of the dielectric function
(ε_{1}) exhibits
a special structure that falls rapidly from the maximum to the
minimum at ω_{ex}, and then approaches to
zero [Fig. 1(c)]. If ε_{1} is approaching to zero at
small ε_{2}, 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
(ω_{p}) and
strength are getting higher with the increase of T and
r. Such a strong plasmon mode does not contribute to the
lowenergy 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 eh excitations near k=0, place great importance on
the inelastic ee scatterings. They are profoundly affected
by T and r.
Electronelectron interactions
could affect heavily the lowenergy properties of the CNs. They
induce the rich excitation spectra and lead to the prominent decay
rates of the lowenergy states at finite T. Each k
state could be occupied by electrons and holes at nonzero T.
The decay rates associated with electrons and holes are worthy of a
discussion. Due to the band symmetry, the conduction and
valenceband electron decay rates are, respectively, equal to the
valence and conductionband hole decay rates. We only focus on the
electron decay rates in the lowest conduction and valence bands. The
decay energy is defined as ω_{de} =E^{h}(k,2m/3)E^{h'}(k+q,2m/3).
Besides the c→c deexcitations under positive ω_{de} (ω_{de}^{+}) at
T=0, there are the deexcitations under negative ω_{de} (ω_{de}^{}) 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 selfenergy 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 ee Coulomb interaction is
n_{B} is the Bose distribution
function. The first term Im[V^{eff}(ω_{de})] in Eq. (2) is
proportional to Im[1/ε(ω_{de})]. The second term
n_{B}(ω_{de})[1n_{F}(E^{h'}(k+q,2m/3))]
is associated with the number of deexcitation channels, where
n_{B}(ω_{de}) and n_{F}(E^{h'}(k+q,2m/3))
indicate the number of ω_{de} transfers and the
finalstate distribution function, respectively. The restrictions of
n_{B}(ω_{de}) accentuate the
importance of small(q,ω_{de}) transfers.
n_{B}(ω_{de}) declines rapidly as
ω_{de} grows,
especially for n_{B}(ω_{de}^{+}). 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 ω_{de}^{+} and
ω_{de}^{} intraband
deexcitations and ω_{de}^{+} interband
deexcitations. For example, the bandedge state is deexcited to the
conductionband states under the ω_{de}^{} intraband
transfers and the valenceband states under the ω_{de}^{+} interband
transfers, but not the ω_{de}^{+} intraband
transfers. The decay rate of this state is about 200 meV
(τ~20 fs) for the (12,0) CN. 1/τ with the
ω_{de}^{+} 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 ω_{de}^{} intraband
transfers except k near the bandedge 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 (ω_{de}≥E_{g}) 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
10^{2}.
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 smallerE_{g} CN has the stronger
loss spectrum. However, the E_{g} 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 valenceband states, as shown in Fig. 4, are
worth an indepth study. They also represent the hole decay rates of
the conductionband 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 valenceband states
[1/τ(k=0,v)=70 meV for (12,0) CN] are smaller
than those of the conductionband 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 finalstate distribution
functions [Figs. 4(a) and 4(b)]. As to the interband v→c
decay, the more finalstate number can enhance the decay rates, but
its contributions are covered by the small n_{B}(ω_{de}) due to
largeω_{de}
transfers [Fig. 4(c)]. That is to say, few c→v interband
excitations are available to be the effective decay
channels.
In conclusion, the temperatureinduced electronic
excitations and deexcitations in narrowgap 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 ee interactions exhibit the intraband and
interband eh 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
valenceband states have significant differences in the wave vector
dependence of the decay rates. The dependence on wave vector appears
weak on the conductionband states. For the valenceband states, the
decay rates decline fast when the initial states deviate from the
bandedge state. These calculated results could be verified by
experimental measurements from the femtosecond timeresolved
photoelectron, transmission, and fluorescence
spectroscopies.


  






