The calculated effective masses 
1. Introduction
Singlewalled carbon nanotube (SWCNTs) can be thought of as graphene a single graphene sheet wrapped up to form a oneatomthick cylinders. CNTs were discovered by Iijima [1] in 1991, since then the excellent charge transport properties of CNTs have been of great interest, for its great potential applications in nanoelectronics, such as highspeed fieldeffect transistors (FETs) [2, 3], singleelectron memories [4], and chemical sensors [5]. The CNT has an atomic and electronic structure that gives it unique advantage as an FET channel. The band gap of the semiconducting SWCNT is inversely proportional to the tube diameter, which allows such tubes to be used in various different applications. CNTs display outstanding electrical properties such as ballistic transport or diffusive transport with long mean free path, which is of the order of a micrometer. Ballistic transport in CNTs has been experimentally demonstrated for lowbias conditions at low temperatures [6, 7]. Highperformance CNT transistors operating close to the ballistic limit have also been reported [810]. Besides, one of the most important advantages is the CNT’s excellent transport properties due to the high carrier mobility. The experimentally obtained carrier mobilities are of the orders 10^{4} cm^{2}/Vs [11, 12] so exceptional device characteristics can indeed be expected. Current transport in long metallic CNTs, however, is found to saturate at ~ 25 µA at high biases, and the saturation mechanism is attributed to phonon scattering [13]. On the other hand, for short length metallic tubes, the current is found not to saturate but to increase well beyond the above limit [14, 15].
Nevertheless, carrier transport in these shorter tubes is still influenced by phonon scattering, and warrants a detailed physical understating of the scattering mechanisms due to its implications on device characteristics for both metallic as well as semiconducting CNTs. And there have been many theoretical studies on the calculation of carrier scattering rates and mobilities in CNTs using semiclassical transport simulation based on the Boltzmann equation [1622]. Similarly, phonon mode calculations for CNTs are also performed with varying degrees of complexity: continuum and forceconstant models [2325] to firstprinciples based methods [2628]. The determination of electronphonon coupling strength is performed by using tight binding calculations [2931] as well as firstprinciples techniques [32]. Nonequilibrium Green’s function formalism also has been employed to treat the effects of phonon scattering in CNT [3335].
In this work, we will show our physical simulation on the carrier mobilities under acoustic phonon scattering process. This work is organized as follows. In section II, we start with the basic properties of CNTs. A brief summary of the electronphonon scattering is discussed in Section III. In this section, we will review the latest theoretical developments aimed at exploring the effect of electronphonon interactions on carrier mobility. In the last section, we will describe the simulation approach we use. In this section, we also present the simulation results to discuss the acoustic phonon scattering effect on the charge carrier mobility.
2. Electronic strcutures of CNTs
2.1. Structure of CNTs
The SWCNT is a hollow cylindershaped molecule with a diameter in the order of 1 nm. SWCNT can be viewed conceptually as graphene sheets rolled up into concentric cylinders. The atomic structure of a singlewalled CNT is conveniently explained in terms of two vectors
2.2. Electronic structure of CNTs
The electronic structure of a SWCNT is deduced from the energy dispersion of graphene. The band structure of the SWCNT is found by imposing periodic boundary conditions around the circumference of the tube,
where
Where
For the armchair SWCNTs, these nanotubes are truly metallic and have two bands crossing at the Fermi level (Figure 2a). The bands stem from the quantization lines drawn in Figure 2d in the reciprocal lattice. The corners of the hexagons are the Kpoints, where the conduction and the valence band of the graphene dispersion touch. One of the quantization lines (thick dashed line) passes through two Kpoints making the tube metallic.
Figure 2b shows the band structure for a (15, 0) zigzag tube which is metallic judging from the degenerate band crossing the Fermi level. The bands stem from the quantization lines drawn in Figure 2e. It is seen that the bands touching at the Fermi level are two times degenerate. However, the band structure is calculated from the dispersion graphene, while the CNT has a curvature around the circumference of the tube. The curvature slightly modifies the band structure by moving the Kpoints.For the zigzag tube with n ≠ 3*integer, such as (16, 0) zigzag tube (Figure 2c), in the reciprocal space the quantization lines do not cross the Kpoints. It has a band gap in the order of ~ 1eV.
So, theoretical studies have shown that a singlewalled CNT can be either metallic or semiconducting depending on its chirality and diameter. The armchair SWCNTs are a group of truly metallic conductors with two bands crossing the Fermi level. For nm=3*integer, the nanotubes would be quasisemiconducting with a small band gap proportional to 1/d^{2}. Typical band gaps are in the order of tens of meV. Finally, a group of zigzag and chiral SWCNTs is semiconducting (nm ≠ 3*integer) with bigger band gaps. The band gap of these tubes are in the order of ~1 eV and scales as
2.3. Electronic transport of CNTs
For the metallic armchair CNTs, the valence and conduction bands cross at the Fermi level just as in the case of graphene. The two crossing bands provide the tube with two conducting channels at and close to the Fermi level, where in each of these bands, two electrons of opposite spins can coexists. By the Landauer formula, the conductance is then:
where
Different band structures are obtained for a truly metallic, a quasimetallic and a semiconducting nanotube. The various band structures are illustrated in Figure 2 which displays a quasimetallic (15,0) zigzag, a semiconducting (16,0) zigzag, and an armchair (9,9) SWCNT band structure. In the case of the armchair tube, the two bands conduct the current while in the case of quasimetallic zigzag or chiral SWCNTs, a small energy gap of few meV exists due to the nanotube curvature. This gap is important at lowtemperatures and can suppress electron transport. However, at roomtemperature, the thermal energy is larger than the gap and the tubes show metallic behavior. Semiconducting tubes possess an energygap of ≈ 0.51 eV, where zigzag SWCNTs have their DOS singularities at the Γpoint.
CNT filed effect transistor (CNTFET) can distinguish between the two character types. In the case of metallic tubes, the conductance is V_{G} independent, with the crossing bands providing conducting electrons independently on the V_{G}, i.e., the gate potential does not change the number of conduction channels. On the other hand, the conductance in semiconducting tubes is strongly affected by the V_{G} and can change by orders of magnitude. The CNT has an atomic and electronic structure that gives it unique advantage as an FET channel.
3. Electronphonon scattering in CNTs
CNTs have been extensively explored for nanoelectronic applications due to their excellent electrical properties. Scattering plays an important role on carrier transport CNTs [13, 14]. It has demonstrated that at finite temperatures or high biases, electronphonon scattering becomes significant. It can be divided into the low and highenergy regimes, corresponding to acousticphonon scattering and optical or zoneboundaryphonon scattering. Because of the light mass and strong bonds, the opticalphonon energy is very high in CNTs
The coupling is through the strain dependence of the band gap. Depending on the tube, the dominant coupling can either be through the stretching or the twisting of the tube. The linear temperature dependence comes from the thermal occupation of the (smallmomentum transfer) acoustic phonon responsible for backscattering.In addition to the lowenergy acoustic phonons, electron (or hole) scattering by the radial breathing mode (RBM) is important in the low bias regime. The RBM phonon energy is inversely proportional to the tube diameter8, and its energy is comparable to the thermal energy at room temperature for tubes in the diameter range of d_{CNT} = 1.5–2.0 nm, which are of interest for electronic applications. As the acoustic mean free path is very longof the order of a micrometre at room temperatureelectrons can be accelerated up to the RBM energy not only thermally, but also by an applied bias of a few Vcm^{−1}
Unlike acoustic phonon scattering, optical phonon scattering is very strong in CNTs; optical phonons contract and elongate the C–C bond length and lead to a strong modulation of the electronic structure. However, for electrons to emit an optical phonon, their energies must be larger than the optical phonon energy. This can only be achieved under high bias conditions. Such scattering processes were first observed in metallic tubes [13, 14, 15] and later in semiconducting tubes [39]. At large sourcedrain biases, the electrons in the tube can accelerate to energies well above the Fermi energy, and these hot electrons can scatter very efficiently by emitting optical and zoneboundary phonons. The scattering rate for this process is
Where
In summary, the inelastic scattering rates determining transport properties of CNTs vary by four orders of magnitude depending on the energy of the electrons and their angular momentum (subband index) as shown in Figure 4 [40]. The weakest is the acoustic (primarily RBM) phonon scattering, which has linear temperature dependence. The optical phonon scattering rate, which is two orders of magnitude stronger, is nearly temperature independent. Finally, another two orders of magnitude stronger than the optical phonon scattering is impact excitation.
The different conduction band edges are labelled as _{i} and the resulting electronic excitations are denoted as E_{iij}. Subscripts bs and fs stand for the back and forward scattering. b), Calculated phonon scattering rate for a (25,0) nanotube showing weak acoustic phonon scattering and strong optical phonon scattering. c), Calculated inelastic scattering rate for a (19,0) nanotube over a wide carrier energy range. Different colours correspond to the scattering rates of electrons in bands with different circumferential angular momentum. The vertical lines show the bottoms of the conduction bands 2 (blue), 3 (cyan) and 4 (green) with respect to the fundamental band edge _{1}. Some of the characteristic peaks in the scattering, due to the longtitudinal (LA) acoustic phonons (APh), radial breathing mode (RBM), longitudinal (LO) and transverse (TO) optical phonons (OPh) and impact electronic excitation (IExc), are labelled. In b and c the electron scattering rate is shown as a function of the excess energy of the electron above the first conductionband minimum. (Figure 2 in ref. 40)
4. Acoustic phonons scattering effect on carrier mobility of semiconducting SWCNTs
A number of groups have reported modeling and simulation studies of the carrier transport in CNTs [4148]. Our intent in this section is not to review these works. Instead, we briefly describe the techniques we currently use to study the intrinsic carrier mobility of semicoducting SWCNTs. The semiconducting zigzag SWCNTs have large intrinsic carrier mobility due to the weak acoustic phonon scattering. Although recently much experimental progress has been achieved on improving the charge carrier mobility of semiconducting CNTs [8, 11, 49, 50], there are a lot of works on the theoretical understanding of the carrier mobility in the semiconducting zigzag SWCNTs [16, 18, 22, 51]. The carrier mobility of the semiconducting zigzag SWCNT can reach 7.9×10^{4} cm^{2}/Vs at room temperature experimentally [8]. Even the higher mobility up to 1.2×10^{5} cm^{2}/Vs for a 4.6 nm diameter semiconducting zigzag SWCNT at room temperature has been predicted by the zonefolding method approximation [16]. Perebeinos
4.1. Acoustic phonons scattering based on the deformationpotential theory
The spedific conducitivity of a threedimensional solid can be written as:
where
Bardeen and Shockley [52] derived an analytical expression for the intrinsic carrier mobility (μ) by assuming that the change of the energy of the electron scattered by an acoustic phonon is proportional to the deformation:
Here
where
The deformation potential theory proposed by Bardeen and Schockly is based on the face that when the change of lattice is very small, the variations of the top of valance band and the bottom of conduction band are linearly related to the variation of lattice constant, therefore the energy of the top of valence band (
Where
The matrix element obtained form perturbational potential is
Upon considering the crystal as continuous medium, timeaveraging and summing for the whole crystal, one can get the total average kinetic energy for the whole crystal :
When the temperatures is higher than Debye temperature, based on classical law of equipartition energy, we have
Where
For the very small energy of phonon the scattering may be considered as elastic, and by summing up the probabilities of phonon absorption and emission. From quantum theory of solid, the reciprocal of relaxation time is:
With the effective mass approximation, we can get:
In semiconductor physics the mobility is defined by
By using Boltzmann distribution function, we can get the charge mobility in one dimensional crystal:
where
where
4.2. Calculation method and results
To calculate the carrier mobility of the semiconducting zigzag SWCNTs, there are three parameters to be determined as shown in the above formula, namely,
Electronically, SWCNTs can behave as either metallic or semiconducting depending on the chirality of their atomic arrangements and diameter. The band structure calculations have predicted that the armchair SWCNTs with (
The stretching modulus was evaluated by compressing and elongating the semiconducting zigzag SWCNTs along the longitudinal direction. For the evaluation of the elastic properties all atomic positions were fully relaxed. Typically the unstrained configurations were calculated first and then the strain was applied in steps of 0.25% in units of strain percentage for strains less than 1%. The results of these simulations are presented in Figure 5. It clearly demonstrates the parabolic form of the strain energy as a function of the strain, reminiscent to the parabolic potential energy derived from Hook’s law for the macroscopic springs. It is interesting to note that the same strain can lead to the increasingly high deformation energies in SWCNT with larger
From the shape of the band structure, we have calculated the effective masses of the electrons and holes of the semiconducting zigzag SWCNTs. We can fit two curves the energy
The deformationpotential constant
The calculated effective masses, the stretching modulus, and the deformationpotential constants of the semiconducting zigzag SWCNTs are summarized in Table 1. The electron and hole mobilities at room temperature can be calculated by the Eq (23) from these three parameters are also displayed in Table 1. We plotted the mobilities of electrons and holes of the semiconducting zigzag SWCNTs calculated as a function of the diameter in Figure 9.












0.45  0.48  0.29  0.60  0.25  0.45  0.24  0.36  0.22  0.31 

0.27  0.81  0.27  0.54  0.26  0.41  0.25  0.34  0.23  0.28 

0.97  1.13  1.41  1.60  1.80  2.04  2.37  2.64  3.06  3.48 

4.52  5.35  2.15  14.3  1.32  13.9  1.12  13.7  1.06  13.5 

14.8  0.31  14.2  0.37  13.8  0.41  13.5  0.43  13.2  0.44 

1.31  0.92  15.5  0.14  65.6  0.28  130  0.52  209  0.88 

0.25  128  0.41  224  0.56  367  0.89  571  1.24  945 
(
It is found that the intrinsic electron mobility can reach 2×10^{5} cm^{2}/Vs at room temperature for
To understand the alternating behavior of DP constant, we examine the frontier molecular orbitals at the
It is thus expected that for
5. Conclusion
CNTFETs are important devices with potentially important applications in nanoelectronics. In this work, we have summarized the electronphonon scattering on the carrier transport. The carrier mobility of the semiconducting zigzag SWCNT scattered from the acoustic phonons is investigated by using firstprinciples calculations. We considered only the longitudinal acoustic phonon scattering process by using the deformationpotential theory. We found that the intrinsic carrier mobility can reach 10^{6} cm^{2}/Vs at room temperature for n=20, and the intriguing alternating behaviors of the carrier mobilities of the semiconducting zigzag SWCNTs are due to the curvature effects of the CNT. We believe that the detailed investigation of acoustic phonon scattering in CNTs [62] will also help us to study the carrier mobilities in other organic or inorganic materials by using the similar technique.
Acknowledgments
This work is supported by the Fundamental Research Funds for the Central Universities, a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Bo Xu thanks the support by the China Postdoctoral Science Foundation funded project (20100481119) and Jiangsu Planned Projects for Postdoctoral Research Funds (1002007B).
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