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Theoretical study of heat conduction in carbon nanotube hetero-junctions
Physics Letters A 374 (2010) 1860–1865
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Physics Letters A www.elsevier.com/locate/pla
Theoretical study of heat conduction in carbon nanotube hetero-junctions Cuilan Ren a,b , Zijian Xu a , Wei Zhang a,∗ , Yong Li a,b , Zhiyuan Zhu a , Ping Huai a a b
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Graduate University of the Chinese Academy of Sciences, Beijing 100049, China
a r t i c l e
i n f o
Article history: Received 5 August 2009 Received in revised form 11 February 2010 Accepted 11 February 2010 Available online 13 February 2010 Communicated by R. Wu Keywords: Carbon nanotube hetero-junction Thermal conductivity Point defects Phonon spectrum
a b s t r a c t Thermal conductivity of single-wall carbon nanotubes with linear, “Y” and “X” hetero-junctions is studied using nonequilibrium molecular dynamics method. The three types of junctions with almost equal number of point defects show a similar decrease in thermal conductivity compared to that of the pristine carbon nanotube. Thermal conductivity of the nanotubes with different number of point defects is also calculated for a qualitative comparison. The thermal conductivity decreases with the defect number increasing. When the defects form an interface, they can cause a larger decrease in the thermal conductivity of CNT than when they are distributed dispersedly. The number of defects contained in the nanojunction determines the thermal conductivity to a great extent more than the geometrical forms of the nanojunctions. Phonon spectra of these junctions are also analyzed to explore the mechanism of degradation in thermal conductivity of these junctions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction With the development of micro/nanotechnology, carbon nanotubes (CNTs) has become one of the most promising candidates for building nanometer-scale electronic devices because of their noble electronic, mechanical, chemical and thermal properties [1–7]. Hetero-junctions with linear, T, Y and X shapes formed by different single-walled carbon nanotubes (SWCNTs) have been proposed recently [8–10]. For instance, quantum dot [11,12] and electronic rectification behavior [13,14] were observed in hetero-junction formed by two SWCNTs with different chiralities, while the multiterminal hetero-junctions with T, Y and X shapes could work as transistor devices [15–17]. Multiterminal junctions have been synthesized by mechanical manipulation, thermal annealing and ion beam irradiation method [8,9,18,19]. A vast amount of experimental and theoretical studies have been conducted to probe into their properties. For example, the mechanical and thermal stability of T and X junctions has been investigated using molecular dynamics (MD) simulation, and the results showed that the junctions have high thermal stability and mechanical strength which made CNT junctions have potentials to support the design of quality enhanced future nanoscale devices [20–22]. Andriotis et al. [23] have simulated the electronic current versus voltage (I-V) characteristics of Y junctions using an efficient
*
Corresponding author. Tel.: +86 21 39194793; fax: +86 21 39194793. E-mail address: [email protected] (W. Zhang).
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Green’s function embedding scheme and showed that rectifying behavior exists in the symmetric Y junctions. Via MD simulation, the thermal rectification behavior in linear CNT junctions has been investigated by Wu and Li [24], their results showed that the heat was conducted preferably in one direction along the axial of linear junction than in the opposite direction. There have been increasing studies in heat conduction in nanoscale materials both experimentally and theoretically [25–29]. The thermal conductivity of CNTs along cylindrical axis was suggested to be about 3000 W/mK by experimental measurements which are ensemble averages over different CNTs on the mat samples and insufficient to reveal the special thermal properties of the CNTs [25,26]. In addition, an even higher value of 6600 W/mK was obtained by an MD simulation at room temperature [30]. Several studies about the thermal conduction in linear, Y, X nanojunctions were carried out using nonequilibrium MD (NEMD) method and showed that there was a significant temperature gradient at junction area on temperature profile due to the topological structure of the junction [31–33]. It is also found that the thermal conductivity of CNTs has a dependence on the temperature, the tube length, defects and impurities [26,27,34–37]. The thermal conductivity λ becomes larger and has a relationship with tube length L as L β in a tube length of micrometers both theoretically and experimentally [38–40]. This remarkable dependence on tube length has made it difficult to directly compare the thermal conductivity. The vacancies, isotope impurities and other imperfections in CNTs could decrease thermal conductivity significantly [31,41–43]. It has shown that mere 20% of isotope can reduce the thermal
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conductivity as much as 50% [38,44]. However, the mechanism of degradation of thermal conductivity of nanojunction due to the defects has not been understood sufficiently. In this Letter, the thermal conductivity of SWCNTs with linear, “Y”, “X” hetero-junctions and different numbers of 5-7-7-5 defects is investigated using NEMD simulation. We show that the number of point defects affects the thermal conductivity of the nanojunctions more markedly than the geometrical forms of the junctions. With the increase of point defects, they can form an interface, which cause a larger decrease in the thermal conductivity of CNT than when they are distributed dispersedly. Phonon spectra of these junctions are also analyzed to explore the physical mechanism of degradation in thermal conductivity of these junctions. This Letter is organized as follows. In Section 2, we briefly describe the models of different junctions considered in this work and the theoretical method. In Section 3, we elaborate the calculation results and discussions. The conclusions are given in Section 4. 2. Models and methodology Although the pristine SWCNTs contain only hexagonal rings of carbon atoms, the formation of multiterminal junctions requires the presence of topological defects in the form of non-hexagonal rings. In this work, several pentagon–heptagon and octagon defects are introduced to connect the branches, and all carbon atoms maintain sp2 configurations in order to maximize stability. The three types of junctions as well as the pristine SWCNT are shown in Fig. 1: (a) the pristine (10, 10) SWCNT, (b) a linear junction obtained from a (10, 10) SWCNT connecting to a (20, 0) SWCNT with 10 pairs of 5-7 defects, (c) a Y junction obtained from a (10, 10) armchair trunk connecting to two (10, 0) zigzag branches with 8 pairs of 5-7 defects and 1 pair of octagons, (d) an X junction which is a joint of two (5, 5) armchair branches and two (10, 0) zigzag branches with 4 pairs of 5-7 defects and 5 pairs of heptagons. Branches of the junctions form an angle of 30◦ . All these junctions are about 24.5 nm long and contain about 4000 atoms. They are relaxed using the conjugate gradient (CG) method. Thermal conductivity of each system is calculated using NEMD method with empirical potentials [32,34]. The interaction between C atoms is modeled by Tersoff–Brenner bond-order potential with the cutoff distance of 2.0 Å [45]. This potential has been widely used for simulating carbon systems [32,33]. The long-range van der Waals (vdW) interaction modeled with Lennard-Jones (L-J) potential is also taken into account. The L-J parameters for carbon are ε = 0.0028 eV, σ = 3.4 Å [46]. During the simulations, classical equations of motion are integrated by velocity Verlet algorithm with a fixed time step of 0.5 fs [47,48]. The thermal conductivity λ of a pristine nanotube is defined by the Fourier’s law [35]
J=
1 dQ A dt
= −λ
dT dl
,
Fig. 1. The schematic simulation set-up for thermal conductivity calculation. (a) The pristine SWCNT, (b) the linear junction, (c) the “Y” junction, (d) the “X” junction. The black parts of each system are fixed rigidly during the simulation, the red and green parts are thermostated for relative high and cool slabs to get the temperature gradient, the gray parts between fixed and temperature controlled slabs are in an attempt to reduce reflecting of heat from the edge, the blue parts are free during simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
bon nanojunctions and SWCNTs with point defects, the temperature profile is a piecewise function, the thermal conductivity of the whole configuration is defined as follows
λ=
l R l + R int + R r
(2)
where R l = T l / J and R r = T r / J represent the thermal resistance of left and right segment respectively, R int = T int / J is the interface resistance [31]. T int is the temperature jump of the interface, and T l / T r is the temperature difference between the left/right side of the interface and the thermal bath. The T l , T int and T r are obtained after the linear fit of the corresponding segments of the temperature profile. In the simulation, velocity scaling method is used to control the temperature of the cold or hot slab
v i = v i
T def Tk
(3)
,
where T def represents T 1 or T 2 while T k is the instantaneous temperature of the kth slabs. The additional kinetic energy obtained from the hot slab or removed from the cold slab at each time step is described as
(1)
where J is the steady-state heat flux density which represents the energy Q transmitted across the area A in a time interval dt, while dT /dl is the temperature gradient along the tube axis of the system. Initially, the system is divided into 100 equal slabs. After evolving at the desired T 0 for 5 ps to reach a steady state, the two slabs at the ends of the tube are kept at the temperature of T 1 = T 0 − T and T 2 = T 0 + T with T = 20 K, respectively. Then, a run of 1.5 ns (3 × 106 time steps) is taken for the evolution, and the last 1.25 ns are used to get the heat flux density and the temperature gradient. For the pristine nanotube, the temperature gradient is estimated from the linear part of the temperature profile. Then the thermal conductivity can be got using formula (1). While for car-
,
E =
nk m 2 v i − v 2i , 2
(4)
i =1
where nk is the number of atoms in the kth slab. The heat flux density is defined by
J=
1 A
n steps j =1
| E 1 ( j ) + E 2 ( j )| , n steps t j =1
(5)
where A is the cross sectional area which is a ring of van der Waals thickness 3.4 Å. E 1 ( j ) and E 2 ( j ) correspond to the additional kinetic energy obtained from the hot slab or removed from the cold slab at each time step to make sure the thermostated slabs at specified temperatures. The temperature gradient is obtained by averaging instantaneous temperatures of each slab dur-
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Fig. 2. Thermal conductivity of pristine SWCNT at 200 K as a function of tube length.
Fig. 3. Thermal conductivity of the pristine SWCNT, the linear, “Y” and “X” junctions.
ing the statistical process, and the temperature calculation is based on the theorem of equipartition of energy:
Tk =
m 3nk k B
nk
v 2i ,
(6)
i =1
where v 2i is the mean-square speed of carbon atoms in the kth slab and k B is the Boltzmann’s constant. As shown in Fig. 1, the periodic boundary condition is not applied to the systems due to the linear asymmetry, the ends of the branches of the junctions are kept fixed rigidly during the simulation to avoid from being displaced by the pseudo transfer of momentum. Several slabs between the fixed and the temperature controlled slabs are assigned to reduce the edge effects. In order to calculate the heat flux and the temperature gradient dT /dl for each junction in a consistent way, for the “Y”, “X” junctions at each time step, the energies adding to the hot slab or removing from the cold slab in the two branches are summed together, and the temperature of each slab is the average temperature of the corresponding slabs in the two branches. 3. Results and discussions The thermal conductivity of (10, 10) SWCNTs at 200 K with length within the range of 6–130 nm is tested. As shown in Fig. 2, the thermal conductivity increases with the tube length, and diverges with L as L 0.88 , which is in agreement with experimental results [40]. Herein, we focus on a series of models at the same length of about 24.5 nm, and the comparison between them makes sense and the length effects needn’t be involved with. The thermal conductivity of SWCNTs with linear, “Y” and “X” hetero-junctions at the temperatures ranging from 50 to 500 K is shown in Fig. 3, and they are compared with that of the pristine CNT which is defect-free. The thermal conductivity of linear, “Y”, “X” junctions increases with temperature. However, the values are close to each other and lower than that of the pristine CNT in the temperature range considered. As reported in Ref. [33], the thermal conductivity of SWCNTs with different X-shape junctions was lower than that of the corresponding pristine SWCNT and some of the values are similar to each other. The calculation reveals that the slope of the thermal conductivity of pristine SWCNT is about 16.5 W m−1 K−2 and the thermal conductivity of pristine SWCNT at 300 K is about 6000 W m−1 K−1 , which can be compared with
Fig. 4. Heat flux of the pristine SWCNT, the linear, “Y” and “X” junctions.
that of previous literatures [32,33]. However, it is different from that one in Ref. [38]. There may be several reasons for this. Firstly, we use a larger heat source of 40 atoms, which could lead to a higher value in heat flux [33]. Secondly, the temperature difference between the hot slab and the cool slab is 40 K in our work, while it is 20 K in Ref. [38]. The difference in temperature between the two ends can cause some difference in the temperature gradient. Thirdly, the boundary conditions are different. The ends of the models in our simulations are kept fixed rigidly during the simulation to avoid being displaced by the pseudo transfer of momentum, and the parts between fixed and temperature controlled slabs are in an attempt to reduce reflecting of heat from the edge, while it uses free boundary condition in Ref. [38]. We attempt to find the mechanism for the similar reduction of the thermal conductivity of these nanojunctions. The dependence of the heat flux on the temperature is similar to each other as shown in Fig. 4. Fig. 5 shows the temperature profiles of the pristine nanotube, linear, “Y” and “X” junctions at 300 K. The pristine SWCNT has a linear temperature profile while there exists a large discontinuity in the temperature profile at the junction area for
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Fig. 5. Temperature profiles at 300 K. (a) The pristine SWCNT, (b) the linear junction, (c) the “Y” junction, (d) the “X” junction.
each hetero-junction, and the temperature gradient changes with different junctions. As reported earlier, the discontinuity in temperature profiles has been observed in different imperfect structures such as an SWCNT with several vacancies, intermolecular junctions, SWCNT boundary as well as the system consisting of a SWCNT and water [31–33,41,42]. The mismatch of atom arrangements results in the sharp temperature gradient as well as the great local thermal resistance and thus reduces the thermal conductivity significantly. As is known, the interface in crystal could resist heat and electronic flow across the matter [27]. Similarly, when a junction formed by two SWCNTs with different chiralities, the topological defects at the junction area could be thought of as a certain type of interface because they induced the thermal boundary resistance against heat flow across the tube. In fact, carbon nanotubes have natural defects such as vacancies, holes, dangling bonds. Unlike these defects, Stone–Wales (5-7) defects could maintain all carbon atoms in sp2 configurations. To clarify the defect effect as well as the interface effect on the heat conduction properties of CNTs, the thermal conductivity of (10, 10) SWCNTs with 1, 2, 4, 5 and 10 pairs of 5-7-7-5 defects is also calculated and compared with that of nanojunctions. The defects are distributed uniformly in the central two slabs of the pristine nanotube. In another case, 10 pairs of 5-7-7-5 defects are distributed dispersedly on the entire surface. As shown in Fig. 6(a), in the temperature range of 50–500 K, the thermal conductivity decreases significantly and monotonically with defect concentration increasing. However, the thermal conductivity of the SWCNT with 10 pairs of 5-7-7-5 defects is almost the same as that of those junctions which have a similar number of defects by comparison of Figs. 6(a) and 3. An interface tends to form with the increasing of defects in the central slabs of a SWCNT. Fig. 6(b) shows that the thermal conductivity of the SWCNT with 10 pairs of 5-7-7-5 defects distributed dispersedly on the whole surface is higher than that of the SWCNT with 10 pairs of 5-7-7-5 defects limited in the central slabs. This is because the 10 pairs of 5-7-7-5 defects gathered in the central slabs can be envisaged to have formed an interface. The results
Fig. 6. (a) Thermal conductivity of SWCNTs with 1, 2, 4, 5, 10 pairs of 5-7-7-5 defects distributed uniformly in the central two slabs. (b) Thermal conductivity of SWCNTs with 10 pairs of 5-7-7-5 defects distributed uniformly in the central two slabs and distributed dispersedly on the entire surface. (c) Thermal conductivity of SWCNTs with 5 pairs of 5-7-7-5 defects, 4 pairs of 5-7-7-5 defects and one 5-8-5 defect, 10 pairs of 5-7-7-5 defects, 9 pairs of 5-7-7-5 defects and one 5-8-5 defect.
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The phonon Umklapp process makes contribution to the thermal resistance in the CNTs [35]. At lower temperatures, fewer phonons are excited to participate in the phonon Umklapp process. However, with the temperature increasing, more and more phonons are excited, resulting in the rapid decrease of the mean phonon relaxation time [51], which leads to a drastic change of the thermal conductivity. The phonon mean free path of individual pristine SWCNT has been estimated to be an order of microns at room temperature [36]. The introduction of topological defects to CNTs and boundary scatterings can cause random Umklapp scatterings which play a main role in reducing the phonon mean free path. Che et al. [41] have reported the introduction of vacancies to a SWCNT can reduce the thermal conductivity significantly because of lacking channels for phonons to bypass the vacancy site. The change of the density of phonon modes, especially the reduction of the C–C bond-stretch characteristic peak which is the primary heat conduction mode, should result from the structure differences between the pristine CNT and the junctions, which also results in the discontinuity in the temperature profile and the decrease in the thermal conductivity. 4. Conclusions
Fig. 7. Total phonon spectra at 300 K. (a) The pristine SWCNT, (b) the linear junction, (c) the “Y” junction, (d) the “X” junction, (e) SWCNT with 10 pairs of 5-7-7-5 defects.
shown in Fig. 6(b) further testify the effect of defect concentration, and further confirm the effect of the interface on the heat conduction in SWCNTs. As shown in Fig. 6(c), thermal conductivity of SWCNTs with 5 (or 10) pairs of 5-7-7-5 defects is almost not affected when one of the 5-7-7-5 defect is substituted by a 5-8-5 defect. It means that SWCNTs with the same number of point defects have a similar thermal conductivity. Therefore, we can infer that the thermal conductivity of the three nanojunctions is similar to each other just because they have the similar numbers of point defects. As is well known, phonons play a dominant role in the heat conduction of carbon nanostructures as compared to electrons. The phonon contribution to the heat capacity is ∼ 102 times larger than that of electrons for CNTs at room temperature, while it is ∼ 104 times for graphite, which makes it reasonable to neglect the role of electron contribution [49]. There are 3N phonon modes in a CNT system of N atoms. The calculation of thermal conductivity of CNTs could be assumed as a steady-state process, and various phonon modes contribute to the thermal conductivity [50]. The normalized phonon spectra at 300 K calculated by Fourier transformation of the velocity autocorrelation functions of the simulated systems with correlation time length of 1 ps are analyzed to elucidate how the defects cause thermal resistance in the process of heat conduction. As shown in Fig. 7, the total phonon spectra for the pristine CNT, the three junctions as well as the SWCNT with 10 pairs of 5-7-7-5 defects distributed concentratedly show that the primary peak exists at around 50 THz which should mainly corresponds to the C–C bond stretching modes, and other minor peaks at around 20 THz. The magnitudes of the primary peaks of linear, Y, X junctions as well as the SWCNT with 10 pairs of 5-7-7-5 defects are about 8.64%, 18.35%, 30.88% and 20.0% smaller than that of the pristine CNT, respectively. The phonon spectra for junctions and imperfect CNTs seemingly have higher minor peaks at around 20 THz than that of the pristine SWCNT.
The thermal conductivity of SWCNTs with linear, “Y” and “X” hetero-junctions is studied employing NEMD methods with empirical potentials. The calculations reveal that the thermal conductivity of linear, “Y” and “X” junctions with almost equal number of defects are close to each other and all of them are lower than that of pristine nanotubes. There is a discontinuity in the temperature profile at the junction area, which results in a large temperature gradient and a great local thermal resistance, and thus reduces the thermal conductivity significantly compared to that of the pristine CNT. Thermal conductivity of SWCNTs with different number and different kind of point defects is also calculated for a qualitative comparison. The thermal conductivity decreases monotonically with the increasing of the number of topological defects in the centre of CNTs. The phonon spectra analyses demonstrated that the density of phonon modes of junctions is different from that of the pristine CNTs, especially the primary peak. The magnitudes of the primary peaks of the linear, “Y” and “X” junctions are all smaller than that of the pristine CNT. The number of defects influences the thermal conductivity of the nanojunctions more significantly than their geometrical forms, and the interface plays an important role in the heat conduction mechanism of CNTs. Acknowledgements This work is partly supported by the Key Project of the Knowledge Innovation Program (KJCX3-SYW-N10) of Chinese Academy of Sciences, the CAS Hundred Talents Program, the National Natural Science Foundation of China (Grant No. 10874197), the Scientific Research Starting Foundation of the Ministry of Human Resources and Social Security of China for Returned Overseas Chinese Scholars, and Shanghai Municipal Science and Technology Commission (09ZR1438300). We thank the Shanghai Supercomputer Centre for the use of the Dawning 5000A supercomputer. References [1] P.L. McEuen, Nature 393 (1998) 15. [2] Y. Huang, X. Duan, Y. Cui, L.J. Lauhon, K. Kim, C.M. Lieber, Science 294 (2001) 1313. [3] H.G. Craighead, Science 290 (2000) 1532. [4] L. Chico, V.H. Crespi, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 76 (1996) 971. [5] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Science 294 (2001) 1317.
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Physics Letters A www.elsevier.com/locate/pla
Theoretical study of heat conduction in carbon nanotube hetero-junctions Cuilan Ren a,b , Zijian Xu a , Wei Zhang a,∗ , Yong Li a,b , Zhiyuan Zhu a , Ping Huai a a b
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Graduate University of the Chinese Academy of Sciences, Beijing 100049, China
a r t i c l e
i n f o
Article history: Received 5 August 2009 Received in revised form 11 February 2010 Accepted 11 February 2010 Available online 13 February 2010 Communicated by R. Wu Keywords: Carbon nanotube hetero-junction Thermal conductivity Point defects Phonon spectrum
a b s t r a c t Thermal conductivity of single-wall carbon nanotubes with linear, “Y” and “X” hetero-junctions is studied using nonequilibrium molecular dynamics method. The three types of junctions with almost equal number of point defects show a similar decrease in thermal conductivity compared to that of the pristine carbon nanotube. Thermal conductivity of the nanotubes with different number of point defects is also calculated for a qualitative comparison. The thermal conductivity decreases with the defect number increasing. When the defects form an interface, they can cause a larger decrease in the thermal conductivity of CNT than when they are distributed dispersedly. The number of defects contained in the nanojunction determines the thermal conductivity to a great extent more than the geometrical forms of the nanojunctions. Phonon spectra of these junctions are also analyzed to explore the mechanism of degradation in thermal conductivity of these junctions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction With the development of micro/nanotechnology, carbon nanotubes (CNTs) has become one of the most promising candidates for building nanometer-scale electronic devices because of their noble electronic, mechanical, chemical and thermal properties [1–7]. Hetero-junctions with linear, T, Y and X shapes formed by different single-walled carbon nanotubes (SWCNTs) have been proposed recently [8–10]. For instance, quantum dot [11,12] and electronic rectification behavior [13,14] were observed in hetero-junction formed by two SWCNTs with different chiralities, while the multiterminal hetero-junctions with T, Y and X shapes could work as transistor devices [15–17]. Multiterminal junctions have been synthesized by mechanical manipulation, thermal annealing and ion beam irradiation method [8,9,18,19]. A vast amount of experimental and theoretical studies have been conducted to probe into their properties. For example, the mechanical and thermal stability of T and X junctions has been investigated using molecular dynamics (MD) simulation, and the results showed that the junctions have high thermal stability and mechanical strength which made CNT junctions have potentials to support the design of quality enhanced future nanoscale devices [20–22]. Andriotis et al. [23] have simulated the electronic current versus voltage (I-V) characteristics of Y junctions using an efficient
*
Corresponding author. Tel.: +86 21 39194793; fax: +86 21 39194793. E-mail address: [email protected] (W. Zhang).
0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.02.028
Green’s function embedding scheme and showed that rectifying behavior exists in the symmetric Y junctions. Via MD simulation, the thermal rectification behavior in linear CNT junctions has been investigated by Wu and Li [24], their results showed that the heat was conducted preferably in one direction along the axial of linear junction than in the opposite direction. There have been increasing studies in heat conduction in nanoscale materials both experimentally and theoretically [25–29]. The thermal conductivity of CNTs along cylindrical axis was suggested to be about 3000 W/mK by experimental measurements which are ensemble averages over different CNTs on the mat samples and insufficient to reveal the special thermal properties of the CNTs [25,26]. In addition, an even higher value of 6600 W/mK was obtained by an MD simulation at room temperature [30]. Several studies about the thermal conduction in linear, Y, X nanojunctions were carried out using nonequilibrium MD (NEMD) method and showed that there was a significant temperature gradient at junction area on temperature profile due to the topological structure of the junction [31–33]. It is also found that the thermal conductivity of CNTs has a dependence on the temperature, the tube length, defects and impurities [26,27,34–37]. The thermal conductivity λ becomes larger and has a relationship with tube length L as L β in a tube length of micrometers both theoretically and experimentally [38–40]. This remarkable dependence on tube length has made it difficult to directly compare the thermal conductivity. The vacancies, isotope impurities and other imperfections in CNTs could decrease thermal conductivity significantly [31,41–43]. It has shown that mere 20% of isotope can reduce the thermal
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conductivity as much as 50% [38,44]. However, the mechanism of degradation of thermal conductivity of nanojunction due to the defects has not been understood sufficiently. In this Letter, the thermal conductivity of SWCNTs with linear, “Y”, “X” hetero-junctions and different numbers of 5-7-7-5 defects is investigated using NEMD simulation. We show that the number of point defects affects the thermal conductivity of the nanojunctions more markedly than the geometrical forms of the junctions. With the increase of point defects, they can form an interface, which cause a larger decrease in the thermal conductivity of CNT than when they are distributed dispersedly. Phonon spectra of these junctions are also analyzed to explore the physical mechanism of degradation in thermal conductivity of these junctions. This Letter is organized as follows. In Section 2, we briefly describe the models of different junctions considered in this work and the theoretical method. In Section 3, we elaborate the calculation results and discussions. The conclusions are given in Section 4. 2. Models and methodology Although the pristine SWCNTs contain only hexagonal rings of carbon atoms, the formation of multiterminal junctions requires the presence of topological defects in the form of non-hexagonal rings. In this work, several pentagon–heptagon and octagon defects are introduced to connect the branches, and all carbon atoms maintain sp2 configurations in order to maximize stability. The three types of junctions as well as the pristine SWCNT are shown in Fig. 1: (a) the pristine (10, 10) SWCNT, (b) a linear junction obtained from a (10, 10) SWCNT connecting to a (20, 0) SWCNT with 10 pairs of 5-7 defects, (c) a Y junction obtained from a (10, 10) armchair trunk connecting to two (10, 0) zigzag branches with 8 pairs of 5-7 defects and 1 pair of octagons, (d) an X junction which is a joint of two (5, 5) armchair branches and two (10, 0) zigzag branches with 4 pairs of 5-7 defects and 5 pairs of heptagons. Branches of the junctions form an angle of 30◦ . All these junctions are about 24.5 nm long and contain about 4000 atoms. They are relaxed using the conjugate gradient (CG) method. Thermal conductivity of each system is calculated using NEMD method with empirical potentials [32,34]. The interaction between C atoms is modeled by Tersoff–Brenner bond-order potential with the cutoff distance of 2.0 Å [45]. This potential has been widely used for simulating carbon systems [32,33]. The long-range van der Waals (vdW) interaction modeled with Lennard-Jones (L-J) potential is also taken into account. The L-J parameters for carbon are ε = 0.0028 eV, σ = 3.4 Å [46]. During the simulations, classical equations of motion are integrated by velocity Verlet algorithm with a fixed time step of 0.5 fs [47,48]. The thermal conductivity λ of a pristine nanotube is defined by the Fourier’s law [35]
J=
1 dQ A dt
= −λ
dT dl
,
Fig. 1. The schematic simulation set-up for thermal conductivity calculation. (a) The pristine SWCNT, (b) the linear junction, (c) the “Y” junction, (d) the “X” junction. The black parts of each system are fixed rigidly during the simulation, the red and green parts are thermostated for relative high and cool slabs to get the temperature gradient, the gray parts between fixed and temperature controlled slabs are in an attempt to reduce reflecting of heat from the edge, the blue parts are free during simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
bon nanojunctions and SWCNTs with point defects, the temperature profile is a piecewise function, the thermal conductivity of the whole configuration is defined as follows
λ=
l R l + R int + R r
(2)
where R l = T l / J and R r = T r / J represent the thermal resistance of left and right segment respectively, R int = T int / J is the interface resistance [31]. T int is the temperature jump of the interface, and T l / T r is the temperature difference between the left/right side of the interface and the thermal bath. The T l , T int and T r are obtained after the linear fit of the corresponding segments of the temperature profile. In the simulation, velocity scaling method is used to control the temperature of the cold or hot slab
v i = v i
T def Tk
(3)
,
where T def represents T 1 or T 2 while T k is the instantaneous temperature of the kth slabs. The additional kinetic energy obtained from the hot slab or removed from the cold slab at each time step is described as
(1)
where J is the steady-state heat flux density which represents the energy Q transmitted across the area A in a time interval dt, while dT /dl is the temperature gradient along the tube axis of the system. Initially, the system is divided into 100 equal slabs. After evolving at the desired T 0 for 5 ps to reach a steady state, the two slabs at the ends of the tube are kept at the temperature of T 1 = T 0 − T and T 2 = T 0 + T with T = 20 K, respectively. Then, a run of 1.5 ns (3 × 106 time steps) is taken for the evolution, and the last 1.25 ns are used to get the heat flux density and the temperature gradient. For the pristine nanotube, the temperature gradient is estimated from the linear part of the temperature profile. Then the thermal conductivity can be got using formula (1). While for car-
,
E =
nk m 2 v i − v 2i , 2
(4)
i =1
where nk is the number of atoms in the kth slab. The heat flux density is defined by
J=
1 A
n steps j =1
| E 1 ( j ) + E 2 ( j )| , n steps t j =1
(5)
where A is the cross sectional area which is a ring of van der Waals thickness 3.4 Å. E 1 ( j ) and E 2 ( j ) correspond to the additional kinetic energy obtained from the hot slab or removed from the cold slab at each time step to make sure the thermostated slabs at specified temperatures. The temperature gradient is obtained by averaging instantaneous temperatures of each slab dur-
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Fig. 2. Thermal conductivity of pristine SWCNT at 200 K as a function of tube length.
Fig. 3. Thermal conductivity of the pristine SWCNT, the linear, “Y” and “X” junctions.
ing the statistical process, and the temperature calculation is based on the theorem of equipartition of energy:
Tk =
m 3nk k B
nk
v 2i ,
(6)
i =1
where v 2i is the mean-square speed of carbon atoms in the kth slab and k B is the Boltzmann’s constant. As shown in Fig. 1, the periodic boundary condition is not applied to the systems due to the linear asymmetry, the ends of the branches of the junctions are kept fixed rigidly during the simulation to avoid from being displaced by the pseudo transfer of momentum. Several slabs between the fixed and the temperature controlled slabs are assigned to reduce the edge effects. In order to calculate the heat flux and the temperature gradient dT /dl for each junction in a consistent way, for the “Y”, “X” junctions at each time step, the energies adding to the hot slab or removing from the cold slab in the two branches are summed together, and the temperature of each slab is the average temperature of the corresponding slabs in the two branches. 3. Results and discussions The thermal conductivity of (10, 10) SWCNTs at 200 K with length within the range of 6–130 nm is tested. As shown in Fig. 2, the thermal conductivity increases with the tube length, and diverges with L as L 0.88 , which is in agreement with experimental results [40]. Herein, we focus on a series of models at the same length of about 24.5 nm, and the comparison between them makes sense and the length effects needn’t be involved with. The thermal conductivity of SWCNTs with linear, “Y” and “X” hetero-junctions at the temperatures ranging from 50 to 500 K is shown in Fig. 3, and they are compared with that of the pristine CNT which is defect-free. The thermal conductivity of linear, “Y”, “X” junctions increases with temperature. However, the values are close to each other and lower than that of the pristine CNT in the temperature range considered. As reported in Ref. [33], the thermal conductivity of SWCNTs with different X-shape junctions was lower than that of the corresponding pristine SWCNT and some of the values are similar to each other. The calculation reveals that the slope of the thermal conductivity of pristine SWCNT is about 16.5 W m−1 K−2 and the thermal conductivity of pristine SWCNT at 300 K is about 6000 W m−1 K−1 , which can be compared with
Fig. 4. Heat flux of the pristine SWCNT, the linear, “Y” and “X” junctions.
that of previous literatures [32,33]. However, it is different from that one in Ref. [38]. There may be several reasons for this. Firstly, we use a larger heat source of 40 atoms, which could lead to a higher value in heat flux [33]. Secondly, the temperature difference between the hot slab and the cool slab is 40 K in our work, while it is 20 K in Ref. [38]. The difference in temperature between the two ends can cause some difference in the temperature gradient. Thirdly, the boundary conditions are different. The ends of the models in our simulations are kept fixed rigidly during the simulation to avoid being displaced by the pseudo transfer of momentum, and the parts between fixed and temperature controlled slabs are in an attempt to reduce reflecting of heat from the edge, while it uses free boundary condition in Ref. [38]. We attempt to find the mechanism for the similar reduction of the thermal conductivity of these nanojunctions. The dependence of the heat flux on the temperature is similar to each other as shown in Fig. 4. Fig. 5 shows the temperature profiles of the pristine nanotube, linear, “Y” and “X” junctions at 300 K. The pristine SWCNT has a linear temperature profile while there exists a large discontinuity in the temperature profile at the junction area for
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Fig. 5. Temperature profiles at 300 K. (a) The pristine SWCNT, (b) the linear junction, (c) the “Y” junction, (d) the “X” junction.
each hetero-junction, and the temperature gradient changes with different junctions. As reported earlier, the discontinuity in temperature profiles has been observed in different imperfect structures such as an SWCNT with several vacancies, intermolecular junctions, SWCNT boundary as well as the system consisting of a SWCNT and water [31–33,41,42]. The mismatch of atom arrangements results in the sharp temperature gradient as well as the great local thermal resistance and thus reduces the thermal conductivity significantly. As is known, the interface in crystal could resist heat and electronic flow across the matter [27]. Similarly, when a junction formed by two SWCNTs with different chiralities, the topological defects at the junction area could be thought of as a certain type of interface because they induced the thermal boundary resistance against heat flow across the tube. In fact, carbon nanotubes have natural defects such as vacancies, holes, dangling bonds. Unlike these defects, Stone–Wales (5-7) defects could maintain all carbon atoms in sp2 configurations. To clarify the defect effect as well as the interface effect on the heat conduction properties of CNTs, the thermal conductivity of (10, 10) SWCNTs with 1, 2, 4, 5 and 10 pairs of 5-7-7-5 defects is also calculated and compared with that of nanojunctions. The defects are distributed uniformly in the central two slabs of the pristine nanotube. In another case, 10 pairs of 5-7-7-5 defects are distributed dispersedly on the entire surface. As shown in Fig. 6(a), in the temperature range of 50–500 K, the thermal conductivity decreases significantly and monotonically with defect concentration increasing. However, the thermal conductivity of the SWCNT with 10 pairs of 5-7-7-5 defects is almost the same as that of those junctions which have a similar number of defects by comparison of Figs. 6(a) and 3. An interface tends to form with the increasing of defects in the central slabs of a SWCNT. Fig. 6(b) shows that the thermal conductivity of the SWCNT with 10 pairs of 5-7-7-5 defects distributed dispersedly on the whole surface is higher than that of the SWCNT with 10 pairs of 5-7-7-5 defects limited in the central slabs. This is because the 10 pairs of 5-7-7-5 defects gathered in the central slabs can be envisaged to have formed an interface. The results
Fig. 6. (a) Thermal conductivity of SWCNTs with 1, 2, 4, 5, 10 pairs of 5-7-7-5 defects distributed uniformly in the central two slabs. (b) Thermal conductivity of SWCNTs with 10 pairs of 5-7-7-5 defects distributed uniformly in the central two slabs and distributed dispersedly on the entire surface. (c) Thermal conductivity of SWCNTs with 5 pairs of 5-7-7-5 defects, 4 pairs of 5-7-7-5 defects and one 5-8-5 defect, 10 pairs of 5-7-7-5 defects, 9 pairs of 5-7-7-5 defects and one 5-8-5 defect.
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The phonon Umklapp process makes contribution to the thermal resistance in the CNTs [35]. At lower temperatures, fewer phonons are excited to participate in the phonon Umklapp process. However, with the temperature increasing, more and more phonons are excited, resulting in the rapid decrease of the mean phonon relaxation time [51], which leads to a drastic change of the thermal conductivity. The phonon mean free path of individual pristine SWCNT has been estimated to be an order of microns at room temperature [36]. The introduction of topological defects to CNTs and boundary scatterings can cause random Umklapp scatterings which play a main role in reducing the phonon mean free path. Che et al. [41] have reported the introduction of vacancies to a SWCNT can reduce the thermal conductivity significantly because of lacking channels for phonons to bypass the vacancy site. The change of the density of phonon modes, especially the reduction of the C–C bond-stretch characteristic peak which is the primary heat conduction mode, should result from the structure differences between the pristine CNT and the junctions, which also results in the discontinuity in the temperature profile and the decrease in the thermal conductivity. 4. Conclusions
Fig. 7. Total phonon spectra at 300 K. (a) The pristine SWCNT, (b) the linear junction, (c) the “Y” junction, (d) the “X” junction, (e) SWCNT with 10 pairs of 5-7-7-5 defects.
shown in Fig. 6(b) further testify the effect of defect concentration, and further confirm the effect of the interface on the heat conduction in SWCNTs. As shown in Fig. 6(c), thermal conductivity of SWCNTs with 5 (or 10) pairs of 5-7-7-5 defects is almost not affected when one of the 5-7-7-5 defect is substituted by a 5-8-5 defect. It means that SWCNTs with the same number of point defects have a similar thermal conductivity. Therefore, we can infer that the thermal conductivity of the three nanojunctions is similar to each other just because they have the similar numbers of point defects. As is well known, phonons play a dominant role in the heat conduction of carbon nanostructures as compared to electrons. The phonon contribution to the heat capacity is ∼ 102 times larger than that of electrons for CNTs at room temperature, while it is ∼ 104 times for graphite, which makes it reasonable to neglect the role of electron contribution [49]. There are 3N phonon modes in a CNT system of N atoms. The calculation of thermal conductivity of CNTs could be assumed as a steady-state process, and various phonon modes contribute to the thermal conductivity [50]. The normalized phonon spectra at 300 K calculated by Fourier transformation of the velocity autocorrelation functions of the simulated systems with correlation time length of 1 ps are analyzed to elucidate how the defects cause thermal resistance in the process of heat conduction. As shown in Fig. 7, the total phonon spectra for the pristine CNT, the three junctions as well as the SWCNT with 10 pairs of 5-7-7-5 defects distributed concentratedly show that the primary peak exists at around 50 THz which should mainly corresponds to the C–C bond stretching modes, and other minor peaks at around 20 THz. The magnitudes of the primary peaks of linear, Y, X junctions as well as the SWCNT with 10 pairs of 5-7-7-5 defects are about 8.64%, 18.35%, 30.88% and 20.0% smaller than that of the pristine CNT, respectively. The phonon spectra for junctions and imperfect CNTs seemingly have higher minor peaks at around 20 THz than that of the pristine SWCNT.
The thermal conductivity of SWCNTs with linear, “Y” and “X” hetero-junctions is studied employing NEMD methods with empirical potentials. The calculations reveal that the thermal conductivity of linear, “Y” and “X” junctions with almost equal number of defects are close to each other and all of them are lower than that of pristine nanotubes. There is a discontinuity in the temperature profile at the junction area, which results in a large temperature gradient and a great local thermal resistance, and thus reduces the thermal conductivity significantly compared to that of the pristine CNT. Thermal conductivity of SWCNTs with different number and different kind of point defects is also calculated for a qualitative comparison. The thermal conductivity decreases monotonically with the increasing of the number of topological defects in the centre of CNTs. The phonon spectra analyses demonstrated that the density of phonon modes of junctions is different from that of the pristine CNTs, especially the primary peak. The magnitudes of the primary peaks of the linear, “Y” and “X” junctions are all smaller than that of the pristine CNT. The number of defects influences the thermal conductivity of the nanojunctions more significantly than their geometrical forms, and the interface plays an important role in the heat conduction mechanism of CNTs. Acknowledgements This work is partly supported by the Key Project of the Knowledge Innovation Program (KJCX3-SYW-N10) of Chinese Academy of Sciences, the CAS Hundred Talents Program, the National Natural Science Foundation of China (Grant No. 10874197), the Scientific Research Starting Foundation of the Ministry of Human Resources and Social Security of China for Returned Overseas Chinese Scholars, and Shanghai Municipal Science and Technology Commission (09ZR1438300). We thank the Shanghai Supercomputer Centre for the use of the Dawning 5000A supercomputer. References [1] P.L. McEuen, Nature 393 (1998) 15. [2] Y. Huang, X. Duan, Y. Cui, L.J. Lauhon, K. Kim, C.M. Lieber, Science 294 (2001) 1313. [3] H.G. Craighead, Science 290 (2000) 1532. [4] L. Chico, V.H. Crespi, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 76 (1996) 971. [5] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Science 294 (2001) 1317.
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