Interface thermal resistance and thermal rectification

APPLIED PHYSICS LETTERS 99, 051917 (2011)

Interface thermal resistance and thermal rectification in hybrid graphene-graphane nanoribbons: A nonequilibrium molecular dynamics study A. Rajabpour,1 S. M. Vaez Allaei,2,a) and F. Kowsary1 1

Department of Mechanical Engineering, University of Tehran, Tehran 14395-515, Iran Department of Physics, University of Tehran, Tehran 14395-547, Iran

2

(Received 27 April 2011; accepted 14 July 2011; published online 5 August 2011) The thermal conductivity of hybrid graphene-graphane nanoribbons (GGNRs) have been investigated using nonequilibrium molecular dynamics simulations. The interface between graphene and graphane leads to a Kapitza resistance with strongly dependence on the imposed heat flux direction. We introduce GGNRs as promising thermal rectifiers at room temperature. By C 2011 American Institute calculating phonon spectra, underlying mechanisms were investigated. V of Physics. [doi:10.1063/1.3622480]

Graphene,1 a mono layer sheet of graphite or twodimensional honeycomb lattice of sp2 bonded carbon, has attracted many attentions in recent decade due to its exceptional and unique electronic2–4 and chemical properties.5 Moreover, the thermal properties of graphene is promising for thermal transport and heat management in nanoelectronic devices.6 Recent measurements and theoretical studies revealed that graphene has an ultrahigh thermal conductivity j even more than that of carbon nanotubes.6–8 This suggests graphene as a good candidate for heat management in nanoelectronic devices.9 Graphane was also theoretically proposed10 and experimentally synthesized11,12 by selectively hydrogenating graphene nanoribbons and transforming the bond type between carbon atoms from sp2 to sp3. Hydrogenation can be utilized to cover graphene ribbon totally (graphane—with special electrical, mechanical, and thermal properties)10,13,14 or partially to construct structures with controllable properties. One of these structures is graphene-graphane nanoribbon (GGNR) or patterned hydrogen functionalized graphene, which has recently been proposed as a stable structure with interesting electronic and magnetic properties.15–18 These properties are strongly dependent on the interface between graphene and graphane nanoribbons. These structures with intrinsic thermal asymmetries lead us to investigate the possibility of introducing a promising thermal rectifier. These systems can be used in nanostructured devices with wide range of applicability, similar to recently proposed thermal rectifiers, i.e., graphene nanoribbons and carbon nanotubes.19–24 In this Letter, we investigate the thermal transport in GGNRs using non-equilibrium molecular dynamic (NEMD) simulations and find that the interface between the two parts of the system shows significant thermal resistance and also makes the system an appropriate thermal rectifier. Since at room temperature the contribution of electrons to the thermal conductivity of graphene structure is less than that of phonons, classical NEMD simulation is an appropriate tool for exploring the thermal properties of graphene a)

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structure.6,21–24 By imposing 35 K temperature gradient on our system, it remains close to the room temperature. All simulations are carried out by LAMMPS package, using adaptive intermolecular reactive bond order (AIREBO) potential for C-C and C-H bonding interactions.25–27 We note that the estimated thermal conductivity of the system can depend on chosen interatomic potentials; thus, we just explore the variations of j in our investigated systems. Equations of motions are integrated with velocity Verlet algorithm with a time step of Dt ¼ 1 fs. A periodic boundary condition is applied to y direction, and a fixed boundary condition is imposed on right and left edges of the system (schematically shown in Fig. 1(a)). Nose-Hoover thermostat is applied about 300 ps to equilibrate and relax the system. By rescaling atomic velocities at each time step, Dt, specific amount of kinetic energy D is added into and subtracted from the hot and cold regions, respectively.28 When the system reaches to steady state, the heat flux can be calculated by j ¼ D=ADt where A is the cross-section area. In all simulations the local temperature T is also averaged over half million time steps. All simulations are averaged over at least five samples with different initial conditions to lower statistical uncertainties less than 6%. In order to be able to compare our results with other reports, we use 0.144 nm for the thickness of the ribbons in all simulations.7,14 The value of temperature gradient is used to obtain j from Fourier’s law: j ¼ jrT. Our calculations indicate that the thermal conductivity of 20 nm 5 nm graphene and graphane nanoribbons (as schematically shown in Fig. 1(a)) are equal to 292.8 W/mK and 133.8 W/mK, respectively, in agreement to other reports.7,14,29 The mean temperature of the two systems were 300 K. As it is seen, the estimated j of graphane nanoribbon is considerably lower than that of graphene, which is due to the conversion of sp2 bondings in graphene to sp3 ones in graphane. We then simulate GGNR that it consists of graphene (left side) and graphane (right side) with equal lengths as schematically shown in Fig. 1(b). The total length of the system is set to 20 nm. In this structure, when the heat flux is applied, a local temperature drop is observed across the interface (Kapitza

99, 051917-1

C 2011 American Institute of Physics V

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FIG. 1. (Color online) (a) The schematic model of graphene with periodic boundary condition in y-direction and fixed boundary condition in x-direction. Effective thermal resistance R is defined between the distance L. (b) The schematic model of GGNR. H atoms are green-colored and bonded to C atoms on both sides in an alternating manner.

effect) as shown in Fig. 2. Considerable phonon scatterings at the interface of graphene and graphane nanoribbons lead to a local temperature drop along the heat flux direction. The temperature drop at the interface, DTK, can be related to the heat flux according to RK ¼ DTK/j, where RK is known as interface thermal resistance or Kapitza resistance.30,31 In order to investigate the thermal rectification of the system, we impose a hot and cold (cold and hot) heat baths to the left and right (right and left) sides of the system indicated by superscript þ (). We label the calculated values depending on the heat flux direction by þ and , corre spondingly. The values of Rþ K and RK are given in Table I. As shown, Kapitza resistance can significantly change by reversing the heat flux direction, which provides a good base for making a thermal rectifier. In order to measure the thermal rectification of the ribbons, we need to calculate the effective thermal resistance of the structure, R, between two edges of the ribbon according to R ¼ DTL/j, where L is corresponding length (Fig. 1(a)). The corresponding values Rþ and R in Table I indicate that the effective thermal resistance also depends on the heat flux direction as well. We define thermal rectification as Rþ R 100%: TR ¼ R

To make sure that obtained results do not depend on the method of simulation, instead of applying a specific heat flux, we have calculated TR by applying two hot and cold heat baths with determined temperatures22 as well. In this method, TR can be measured by (j jþ)/jþ, where j is the heat flux of the system. We also calculated the rest of param eters as well, i.e., Rþ, R, Rþ K , and RK , and they were in agreement with corresponding results presented in Table I. We utilized this method to study the effect of finite size of the ribbon as well (Fig. 3). We examined the validity of results for larger system sizes and varied the length and the width of the ribbon up to 100 and 10 nm, respectively. As shown, TR does not depend on the finite length of the system up to 100 nm. In order to understand the underlying mechanism of the phenomena, we calculated phonon spectra of two groups of atoms, corresponding to the two sides of the interface. The phonon spectrum have been extracted from their atomic velocities v as20 PðxÞ ¼

1 ð

eixt hvðtÞ:vðtÞidt;

(2)

0

(1)

Table I contains TR for effective (R) and Kapiza (RK) resistances and reveals a significant dependence on heat flux direction. As shown, R in GGNR is larger than that of single graphene or graphane nanoribbons, due to the large Kapitza thermal resistance particularly when the heat flux is from the left side to the right side direction (þ). The considerable changes in the magnitudes are specially important for nanoscale devices. The obtained TR in this structure is comparable to good thermal rectifiers recently proposed.21,22

where x denoted to the angular frequency. As it can be seen in Fig. 4, there are considerable mismatches between the two spectra which indicate remarkable phonon scattering at GGNR interface. There is not considerable overlap between the major peak of P(x) in graphene (blue) and graphane (green), which shows how phonons should be scattered when they go through the interface. This behaviour can clearly explain the considerable Kapitza resistance of the system. To study the underlying mechanism of the thermal rectification, usually the overlap of two spectra are compared20 TABLE 1. Kapitza resistances RK, effective thermal resistances R for GGNR, graphene and graphane nanoribbons for the cases of (þ) and (). TR is also presented for GGNR. Structure name

FIG. 2. Temperature profiles for heat flux in (a) (þ) direction and (b) () direction.

GGNRðRþ KÞ GGNRðR KÞ GGNR(Rþ) GGNR(R) Graphene Graphane

Thermal resistance 1011(m2/W K)

TR

8.95 2.97 32.8 26.8 5.91 12.95

201.3%


23.8% — —

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Appl. Phys. Lett. 99, 051917 (2011)

S. Scandolo, K. Gordiz, S. Volz, and M. F. Miri for useful discussions. The partial support of high performance computing center of IKIU is also acknowledged. 1

FIG. 3. The variation of thermal rectification TR vs the GGNR length.

for þ and heat flux directions, and the difference between the two overlaps is corresponding to the rectification phenomenon. Our estimations show there is not considerably difference between the two overlaps in the range of their error bars. The same results or weak differences between overlaps have been recently reported as well.32,33 This feature could be originated from the small temperature gradient in the system. In summary, we have investigated the thermal rectification in GGNRs wherein the Kapitza thermal resistances play important roles. GGNRs exhibit a remarkable thermal rectification in addition to their interesting electronic and magnetic properties.15–18 Moreover, their fabrications are easier than other nanoscale thermal rectifiers, which could made them applicable in wide range of applications.18,34 The work was supported in part by the Research Council of the University of Tehran. We thank F. Shahbazi,

FIG. 4. (Color online) Phonon spectra of two groups of atoms near to the interface in graphene (blue line) and graphane (green line) nanoribbons for heat flux in (þ) and () directions.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 2 A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007). 3 A. K. Geim, Science 324, 1530 (2009). 4 C. Xu, H. Li, and K. Banerjee, IEEE Trans. Electron Devices 55, 3264 (2008). 5 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov, Nat. Mater. 6, 652 (2007). 6 A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, Nano Lett. 8, 902 (2008). 7 Z. X. Guo, D. Zhang, and X. G. Gong, Appl. Phys. Lett. 95, 163103 (2009). 8 S. Ghosh, I. Calizo, D. Teweldebrhan, E. P. Pokatilov, D. L. Nika, A. A. Balandin, W. Bao, F. Miao, and C. N. Lau, Appl. Phys. Lett. 92, 151911 (2008). 9 A. Vassighi and M. Sachdev, Thermal and Power Management of Integrated Circuits (Springer, New York, 2006). 10 J. Sofo, A. S. Chaudhari, G. D. Barber, Phys. Rev. B 75, 153401 (2007). 11 D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P. Blake, M. P. Halsall, A. C. Ferrari, D. W. Boukhvalov, M. I. Katsnelson, A. K. Geim, and K. S. Novoselov, Science 323, 610 (2009). 12 A. K. Singh and B. I. Yakobson, Nano Lett. 9, 1540 (2009). 13 Q. X. Pei, Y. W. Zhang, and V. B. Shenoy, Carbon 48, 898 (2010). 14 S. Chien, Y. Yang, and C. Chena, Appl. Phys. Lett. 98, 033107 (2011). 15 Y. H. Lu and Y. P. Feng, J. Phys. Chem. C 113, 20841 (2009). 16 Z. M. Ao, A. D. Hernndez-Nieves, F. M. Peeters, and S. Li, Appl. Phys. Lett. 97, 233109 (2010). 17 A. D. Hernndez-Nieves, B. Partoens, and F. M. Peeters, Phys. Rev. B 82, 165412 (2010). 18 P. Sessi, J. R. Guest, M. Bode, and N. P. Guisinger, Nano Lett. 9, 4343 (2009). 19 C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121 (2006). 20 B. Li, J. Lan, and L. Wang, Phys. Rev. Lett. 95, 104302 (2005). 21 J. Hu, X. Ruan, and Y. P. Chen, Nano Lett. 9, 2730 (2009). 22 N. Yang, G. Zhang, and B. Li, Appl. Phys. Lett. 95, 033107 (2009); N. Yang, G. Zhang, and B. Li, ibid. 93, 243111 (2008). 23 G. Wu and B. Li, Phys. Rev. B 76, 085424 (2007). 24 K. Takahashi, M. Inoue, and Y. Ito, Jap. J. Appl. Phys. 49, 02BD12(2010). 25 S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472 (2000). 26 See http://lammps.sandia.gov for more information about LAMMPS. 27 S. Plimpton, J. Comput. Phys. 117, 1 (1995). 28 P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B 65, 14 (2002). 29 J. A. Thomas, R. M. Iutzi, and A. J. H. McGaughey, Phys. Rev. B 82, 045413 (2010). 30 E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989). 31 A. Rajabpour and S. Volz, J. Appl. Phys. 108, 094324 (2010). 32 B. Hu, L. Yang, and Y. Zhang, Phys. Rev. Lett. 97, 124302 (2006). 33 S. Wang and X. Liang, Int. J. Therm. Sci. 50, 680 (2011). 34 D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phys. 93, 793 (2003).
