Abstract
Using nonequilibrium molecular dynamics simulations and nonequilibrium Green's function method, we investigate the thermoelectric properties of a series of zigzag and chiral carbon nanotubes which exhibit interesting diameter and chirality dependence. Our calculated results indicate that these carbon nanotubes could have higher ZT values at appropriate carrier concentration and operating temperature. Moreover, their thermoelectric performance can be significantly enhanced via isotope substitution, isoelectronic impurities, and hydrogen adsorption. It is thus reasonable to expect that carbon nanotubes may be promising candidates for highperformance thermoelectric materials.
Introduction
As it can directly convert waste heat into electric power, thermoelectric material
is expected to be one of the promising candidates to meet the challenge of energy
crisis. The performance of a thermoelectric material is quantified by the dimensionless
figure of merit
As an interesting quasionedimensional nanostructure with many unusual properties, carbon nanotubes [CNTs] have attracted a lot of attention from the science community since their discovery [5]. However, few people believe that CNTs could be promising thermoelectric materials. This is probably due to the fact that although CNTs can have much higher electrical conductivities, their thermal conductivities are also found to be very high [611]. As a result, the ZT values of CNTs predicated from previous works [10,12] are rather small (approximately 0.0047). Prasheret al. [13] found the socalled 'CNT bed' structure could reduce the thermal conductivity of CNTs. However, the random network of the samples may weaken the electronic transport, and the room temperature ZT value is estimated to be 0.2. Jiang et al. [14] investigated the thermoelectric properties of singlewalled CNTs using a nonequilibrium Green's function approach [NEGF]. They found that CNTs exhibit very favorable electronic transport properties, but the maximum ZT value is only 0.2 at 300 K. The possible reason is the neglect of nonlinear effect [15] in the phonon transport, and the corresponding thermal conductivity was overestimated. If the thermal conductivity can be significantly reduced without many changes to their electronic transport, CNTs may have very favorable thermoelectric properties. In this work, we use a combination of nonequilibrium molecular dynamics simulations and NEGF method to study the thermoelectric properties of a serial of CNTs with different diameters and chiralities. They are the zigzag (7,0), (8,0), (10,0), (11,0), (13,0), (14,0) and the chiral (4,2),(5,1), (6,2), (6,4), (8,4), (10,5), and all are semiconductors in their pristine form. By cooperatively manipulating the electronic and phonon transports, we shall see that these CNTs could be optimized to exhibit much higher ZT values by isotope substitution, isoelectronic impurities, and hydrogen adsorption. It is thus reasonable to expect that CNTs may be promising candidates for highperformance thermoelectric materials.
Computational details
The phonon transport is studied using the nonequilibrium molecular dynamics [NEMD] simulations as implemented in the LAMMPS software package (Sandia National Laboratories, Livermore, CA, USA)[16]. The Tersoff potential [17] is adopted to solve Newtonian equations of motion according to the MüllerPlathe algorithm [18] with a fixed time step of 0.5 fs. We carry out a 300ps constant temperature simulation and a 200ps constant energy simulation to make sure that the system has reached a steady state. The nanotubes are then divided into 40 equal segments with periodic boundary condition, and the first and twentyfirst segments are defined as the hot and cold regions, respectively. The coldest atom in the hot region and the hottest one in the cold region swap their kinetic energies every hundreds of time steps, and then temperature gradient responses and thermal flux maintain via atom interactions in neighboring segments [19,20]. The electronic transport is calculated using the NEGF method as implemented in the AtomistixToolKit code (Quantum Wise A/S, Copenhagen, Denmark) [21,22]. The nanotube is modeled by a central part connected by the left and right semiinfinite one. We use the TroullierMartins nonlocal pseudopotentials [23] to describe the electronion interactions. The exchangecorrelation energy is in the form of PW91 [24], and the cutoff energy is set to be 150 Ry. We use a double ζ basis set plus polarization for the carbon atoms, and the Brillouin zone is sampled with 1 × 1 × 100 MonkhorstPack meshes. The mixing rate of the electronic Hamiltonian is set as 0.1, and the convergent criterion for the total energy is 4 × 10^{5}eV.
Results and discussions
We begin with the phonon transport of these CNTs using the NEMD simulations, where
the phononinduced thermal conductivity [κ_{p}] is calculated according to Fourier's law
Figure 1. Lattice thermal conductivity of tube (4,2) as a function of temperature. Results from the CallawayHolland model and the nonequilibrium molecular dynamics simulations are both shown.
For lowdimensional systems, one should pay special attention to the size effect when
discussing the thermal conductivity. Both the experiment measurements [27,28] and molecular dynamics simulations [29,30] indicate that the κ_{p }of CNTs depends on their length, which is different from that of bulk materials. Here,
we use a simple approach [20] by calculating the thermal conductivity at different tube lengths and then using
a linear fitting according to the formula
Table 1. Summary of the NEMDcalculated room temperature κ_{p }of a series of zigzag and chiral nanotubes
We now move to the discussions of electronic transport using the NEGF approach. Figure 2 shows the calculated electronic transmission function [T(E)] for the abovementioned zigzag and chiral series. Within the rigidband picture, E > 0 corresponds to the ntype doping, while E < 0 corresponds to the ptype doping. Here, we focus on the electron ballistic transport and ignore the weak electronphonon scattering. We see that all the investigated CNTs exhibit quantized transmission which can be essentially derived from their energy band structures. The vanishing transmission function around the Fermi level is consistent with the fact that all of them are semiconductors. It is interesting to find that those CNTs with a larger diameter have a symmetrically distributed transmission function near the Fermi level. However, this is not the case for the smaller diameter CNTs such as(7,0), (8,0), and (4,2), where we see that the number of first conduction channel is two for the ptype doping and one for the ntype doping. By integrating [33] the calculated T(E), one can easily obtain the Seebeck coefficient (S), the electrical conductance [G], and the electronic thermal conductance [λ_{e}] within the linear response limit. Here, we choose the zigzag (10,0) and chiral (6,4) as two typical examples and plot in Figure 3 the corresponding transport coefficients at 300 K as a function of chemical potential [μ]. Note that the chemical potential indicates the doping level or carrier concentration of the system, and the ntype doping corresponds to μ > 0, while ptype corresponds to μ < 0. As can be seen in Figure 3a, b, both G and λ_{e }of these two CNTs vanish around the Fermi level (μ = 0) since this area corresponds to the band gap of the systems. When the chemical potential moves to the edge of the first conduction channels, there is a sharp increase of G and λ_{e }For both the (10,0) and (6,4) tubes, the S shown in Figure 3c is rather symmetric about the Fermi level, which can be attributed to the symmetrically distributed first conduction channels (see Figure 2). The absolute value of the Seebeck coefficient reaches the maximum value at μ ≈ ± κ_{B}T and then decreases until vanish near the edge of band gap.
Figure 2. Calculated electron transmission function for a series of (a) zigzag and (b) chiral tubes.
Figure 3. Calculated transport coefficients at 300 K as a function of chemical potential. These are for zigzag (10,0) and chiral (6,4). (a) Electric conductance, (b) electronic thermal conductance, and (c) Seebeck coefficient with (d) the corresponding ZT values.
It should be mentioned that we have used the term 'conductivity' for the phonon transport
but 'conductance' for the electronic transport. To avoid arbitrary definition of crosssectional
area in lowdimensional system such as CNTs, we rewrite the figure of merit as
Up to now, we are dealing with room temperature, and the corresponding ZT values are still not comparable to that of the best commercial materials. Moreover, a thermoelectric material may be needed to operate at different temperatures for different applications. We thus perform additional transport calculations where the temperature ranges from 250 to 1,000 K. Figure 4 plots the calculated ZT values as a function of temperature for the abovementioned zigzag and chiral series. At each temperature, two ZT values are shown which correspond to the optimized ptype and ntype doping in each tube. Except for the small (4,2) tube with a maximum ZT value at 300 K, we see in Figure 4 that the thermoelectric performance of other CNTs can be significantly enhanced at a relatively higher temperature. The maximum ZT values achieved are 3.5 for the zigzag (10,0) at 800 K and 4.5 for the chiral (6,4) at 900 K. These values are very competitive with that of conventional refrigerators or generators. It is interesting to note that among the investigated CNTs, both the (10,0) and (6,4) tubes have an intermediate diameter (0.7 to 0.8 nm), and those with larger or smaller diameters have a relatively less favorable thermoelectric performance. On the other hand, we see that almost all the zigzag tubes exhibit a peak ZT value at an intermediate temperature (700 to 800 K). In contrast, the peak for the chiral series moves roughly from 300 to 900 K as the tube diameter is increased. Our calculated results thus provide a simple map by which one can efficiently find the best CNT for the thermoelectric applications at different operating temperatures.
Figure 4. Optimized ZT values as a function of temperature. These are for a series of (a) zigzag and (b) chiral tubes. The results for the ptype and ntype doping are both shown.
To further improve the thermoelectric performance of these CNTs, we have considered
isotope substitution which is believed to reduce the phononinduced thermal conductance
without changing the electronic transport properties [3436]. Here, we choose (10,0) as an example since it has the highest ZT value among those in the zigzag series, and the zigzag tubes are usually easier to
be fabricated in or to be selected from the experiments than the chiral ones. In our
calculations, the ^{12}C atoms in the (10,0) tube are randomly substituted by^{13}C atoms at different concentrations. The corresponding lattice thermal conductance
as well as the ZT value at 800 K is shown in Figure 5 with respect to the pristine values. Due to the mass difference between ^{12}C and ^{13}C, we see that the calculated thermal conductance of the (10,0) tube decreases with
the increasing concentration of ^{13}Catoms. Of course, if half or more ^{12}Catoms are substituted, the situation is reversed. The thermal conductance can be
well fitted by a double exponential function
Figure 5. Lattice thermal conductance and the ZT value. Calculated lattice thermal conductance (red) and optimized ZT value (blue) at 800 K for the (10,0) tube, where the ^{12}C atoms are substituted by ^{13}C atoms with different concentrations. Note that the results are given with respect to those of the pure ^{12}C tube.
Introducing isoelectronic impurities is another effective way to localize phonon and reduce lattice thermal conductance due to impurity scattering [37]. Here, we choose Si as an example and consider a very low concentration where one C atom in a (10,0) supercell containing three primitive cells is replaced by a Si atom. The resulting product has a nominal formula of C_{119}Si and is schematically shown in Figure 6a. As the mass difference between C and Si is even larger, we find that the phononderived thermal conductance of C_{119}Si is significantly reduced by 45% to 60% compared with that of the pristine(10,0) tube in the temperature range from 300 to 900 K. On the other hand, since C and Si atoms have the same electron configuration, one may expect that Si doping will not change much of the electronic transport properties. Indeed, our calculations only find a small weakening of the power factor (S^{2}G). As a result, we see in Figure 6c that there is an overall increase of the ZT value at the temperature range of 300 to 700 K. The Sidoped product has a maximum ZT = 4.0 at T = 600 K compared with the pristine value of 3.5 at T = 800 K. It is worth to mention that in a wide temperature range (450 to 800 K), the ZT values of the Sidoped product are all higher than 3.0 which is very beneficial for their thermoelectric applications.
Figure 6. Fully relaxed structures and optimized ZT values of CNTs. Top view of the fully relaxed structures for (a) Sidoped (10,0) tube with nominal formula of C_{119}Si and (b) the (10,0) tube chemisorbed with hydrogen atoms having nominal formula of C_{40}H_{2}. (c) Plots of the optimized ZT value as a function of temperature for theC_{119}Si and C_{40}H_{2 }products, and the results for the pristine (10,0) tube is also shown for comparison.
A similar improvement of the thermoelectric performance can be achieved by hydrogen adsorption on the (10,0) tube. As shown in Figure 6b, two hydrogen atoms are chemisorbed on top of a CC bond along the tube axis, and the product has a concentration of C_{40}H_{2}. Our calculated results indicate that such hydrogen adsorption causes deformation of the (10,0) tube and reduces both the phonon and electroninduced thermal conductance while keeping the S^{2}G less affected. For example, the calculated λ_{p }at 600 K is 0.072 nW/K, which is much lower than that found for the pristine (10,0) tube (0.21 nW/K). The calculated λ_{e }also decreases from 0.089 to 0.062 nW/K. At the same time, we find that the S^{2}G of the chemisorbed product (9.47 × 10^{13 }W/K^{2}) is slightly lower than that of the pristine (10,0) tube (1.28 × 10^{12 }W/K^{2}). As a result, the calculated ZT value at 600 K increases significantly from 2.6 to 4.2 which is even higher than the highest value of the pristine (10,0) tube. The chemisorptions of hydrogen also increase the ZT value at other temperatures, as indicated in Figure 6c. It is interesting to note that the temperaturedependent behavior almost coincides with that from Si doping, especially in the temperature region from 400 to 700 K.
Summary
In summary, our theoretical calculations indicate that by appropriate ntype and ptype doping, one can obtain much higher ZT values for both the zigzag and armchair CNTs, and those tubes with an intermediate diameter (0.7 to 0.8 nm) seems to have better thermoelectric properties than others. With the zigzag (10,0) as an example, we show that the phononinduced thermal conductance can be effectively reduced by isotope substitution, isoelectronic impurities, and hydrogen adsorption, while the electronic transport is less affected. As a result, the ZT value can be further enhanced and is very competitive with that of the best commercial materials. To experimentally realize this goal, one needs to fabricate CNTs with specific diameter and chirality, and the tube length should be at least 1 μm. This may be challenging but very possible, considering the fact that the (10,0) tube was successfully produced by many means, such as by direct laser vaporization [38], electric arc technique [39], and chemical vapor deposition [40], and can be selected from mixed or disordered samples using a DNAbased separation process [41].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
XJT carried out the NEGF and NEMD calculations. HJL participated in the design of the study and discussions of the theoretical results. YWW, HYL, and LP participated in the implementation of the LAMMPS and ATK codes. JS and XFT participated in the discussions of related experimental works. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the '973 Program' of China (grant no. 2007CB607501), the National Natural Science Foundation (grant no. 51172167), and the Program for New Century Excellent Talents in the University. We also acknowledge the financial support from the interdiscipline and postgraduate programs under the 'Fundamental Research Funds for the Central Universities'. All the calculations were performed in the PC Cluster from Sugon Company of China.
References

Slack GA: New materials and performance limits for thermoelectric cooling. In CRC Handbook of Thermoelectrics. Edited by Rowe DM. Boca Raton: CRC Press; 1995:407.

Bejan A, Allan AD: Heat Transfer Handbook. New York: Wiley; 2003.

Hicks LD, Dresselhaus MS: Effect of quantumwell structures on the thermoelectric figure of merit.
Phys Rev B 1993, 47:1272712731. Publisher Full Text

Hicks LD, Dresselhaus MS: Thermoelectric figure of merit of a onedimensional conductor.
Phys Rev B 1993, 47:1663116634. Publisher Full Text

Iijima S: Helical microtubules of graphitic carbon.
Nature 1991, 354:5658. Publisher Full Text

Hone J, Whitney M, Piskoti C, Zettl A: Thermal conductivity of singlewalled carbon nanotubes.
Phys Rev B 1999, 59:R2514R2516. Publisher Full Text

Hone J, Llaguno MC, Nemes NM, Johnson AT, Fischer JE, Walters DA, Casavant MJ, Schmidt J, Smalley RE: Electrical and thermal transport properties of magnetically aligned single wall carbon nanotube films.
Appl Phys Lett 2000, 77:666668. Publisher Full Text

Berber S, Kwon YK, Tomanek D: Unusually high thermal conductivity of carbon nanotubes.
Phys Rev Lett 2000, 84:46134616. PubMed Abstract  Publisher Full Text

Kim P, Shi L, Majumdar A, McEuen PL: Thermal transport measurements of individual multiwalled nanotubes.
Phys Rev Lett 2001, 87:215502. PubMed Abstract  Publisher Full Text

Yu C, Shi L, Yao Z, Li D, Majumdar A: Thermal conductance and thermopower of an individual singlewall carbon nanotube.
Nano Lett 2005, 5:18421846. PubMed Abstract  Publisher Full Text

Pop E, Mann D, Cao J, Wang Q, Goodson K, Dai H: Negative differential conductance and hot phonons in suspended nanotube molecular wires.
Phys Rev Lett 2005, 95:155505. PubMed Abstract  Publisher Full Text

Purewal MS, Hong BH, Ravi A, Chandra B, Hone J, Kim P: Scaling of resistance and electron mean free path of singlewalled carbon nanotubes.
Phys Rev Lett 2007, 98:186808. PubMed Abstract  Publisher Full Text

Prasher RS, Hu XJ, Chalopin Y, Mingo N, Lofgreen K, Volz S, Cleri F, Keblinski P: Turning carbon nanotubes from exceptional heat conductors into insulators.
Phys Rev Lett 2009, 102:105901. PubMed Abstract  Publisher Full Text

Jiang JW, Wang JS, Li BW: A nonequilibrium green's function study of thermoelectric properties in singlewalled carbon nanotubes.
J Appl Phys 2011, 109:014326. Publisher Full Text

Wang JS, Wang J, Zeng N: Nonequilibrium Green's function approach to mesoscopic thermal transport.

Plimpton S: Fast parallel algorithms for shortrange molecular dynamics.
J Comput Phys 1995, 117:119. Publisher Full Text

Tersoff J: Modeling solidstate chemistry: interatomic potentials for multicomponent systems.
Phys Rev B 1989, 39:55665568. Publisher Full Text

MüllerPlathe F: A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity.
J Chem Phys 1997, 106:6082. Publisher Full Text

Osman MA, Srivastava D: Temperature dependence of the thermal conductivity of singlewall carbon nanotubes.
Nanotechnology 2001, 12:2124. Publisher Full Text

Schelling PK, Phillpot SR, Keblinski P: Comparison of atomiclevel simulation methods for computing thermal conductivity.

Brandbyge M, Mozos JL, Ordejόn P, Taylor J, Stokbro K: Densityfunctional method for nonequilibrium electron transport.

Soler JM, Artacho E, Gale JD, García A, Junquera J, Ordejόn P, SánchezPortal D: The SIESTA method for ab initio orderN materials simulation.
J Phys: Conders Matter 2002, 14:27452779. Publisher Full Text

Troullier N, Martins JL: Efficient pseudopotentials for planewave calculations. II. Operators for fast iterative diagonalization.
Phys Rev B 1991, 43:88618869. Publisher Full Text

Perdew JP, Wang Y: Accurate and simple analytic representation of the electrongas correlation energy.
Phys Rev B 1992, 45:1324413249. Publisher Full Text

Cao JX, Yan XH, Xiao Y, Ding JW: Thermal conductivity of zigzag singlewalled carbon nanotubes: role of the umklapp process.

Gu Y, Chen Y: Thermal conductivities of singlewalled carbon nanotubes calculated from the complete phonon dispersion relations.

Wang ZL, Tang DW, Li XB, Zheng XH, Zhang WG, Zheng LX, Zhu YT, Jin AZ, Yang HF, Gu CZ: Lengthdependent thermal conductivity of an individual singlewall carbon nanotubes.
Appl Phys Lett 2007, 91:123119. Publisher Full Text

Chang CW, Okawa D, Garcia H, Majumdar A, Zettl A: Breakdown of Fourier's law in nanotube thermal conductors.
Phys Rev Lett 2008, 101:075903. PubMed Abstract  Publisher Full Text

Che J, Çağın T, Goddard WA III: Thermal conductivity of carbon nanotubes.
Nanotechnology 2000, 11:6569. Publisher Full Text

Padgett CW, Brenner DW: Influence of chemisorption on the thermal conductivity of singlewall carbon nanotubes.
Nano Lett 2004, 4:10511053. Publisher Full Text

Maiti A, Mahan GD, Pantelides ST: Dynamical simulations of nonequilibrium processes  heat flow and the Kapitza resistance across grain boundaries.
Solid State Commun 1997, 102:517521. Publisher Full Text

Cao JX, Yan XH, Xiao Y, Ding JW: Exact study of lattice dynamics of singlewalled carbon nanotubes.

Esfarjani K, Zebarjadi M, Kawazoe Y: Thermoelectric properties of a nanocontact made of twocapped singlewall carbon nanotubes calculated within the tightbinding approximation.

Zhang G, Li B: Thermal conductivity of nanotubes revisited: effects of Chirality, Isotope impurity, tube length, and temperature.
J Chem Phys 2005, 123:114714. PubMed Abstract  Publisher Full Text

Chang CW, Fennimore AM, Afanasiev A, Okawa D, Ikuno T, Garcia H, Li D, Majumdar A, Zettl A: Isotope effect on the thermal conductivity of boron nitride nanotubes.
Phys Rev Lett 2006, 97:085901. PubMed Abstract  Publisher Full Text

Balasubramanian G, Puri IK, Böhm MC, Leroy F: Thermal conductivity reduction through isotope substitution in nanomaterials: predictions from an analytical classical model and nonequilibrium molecular dynamics simulations.
Nanoscale 2011, 3:37143720. PubMed Abstract  Publisher Full Text

Chen J, Zhang G, Li B: Tunable thermal conductivity of Si_{1x}Ge_{x }nanowires.
Appl Phys Lett 2009, 95:073117. Publisher Full Text

Guo T, Nikolaev P, Thess A, Colbert DT, Smalley RE: Catalytic growth of singlewalled nanotubes by laser vaporization.
Chem Phys Lett 1995, 243:4954. Publisher Full Text

Journet C, Maser WK, Bernier P, Loiseau A, Chapelle ML, Lefrant S, Deniard P, Leek R, Fischer JE: Largescale production of singlewalled carbon nanotubes by the electricarc technique.
Nature 1997, 388:756758. Publisher Full Text

Kong J, Soh HT, Cassell AM, Quate CF, Dai H: Synthesis of individual singlewalled carbon nanotubes on patterned silicon wafers.
Nature 1998, 395:878881. Publisher Full Text

Zheng M, Jagota A, Semke ED, Diner BA, Mclean RS, Lustig SR, Richardson RE, Tassi NG: DNAassisted dispersion and separation of carbon nanotubes.
Nat Mater 2003, 2:338342. PubMed Abstract  Publisher Full Text