Nonlinear thermal transport and negative differential thermal conductance in graphene nanoribbons
Jiuning Hu, Yan Wang, Ajit Vallabhaneni, Xiulin Ruan, Yong P. Chen
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Nonlinear thermal transport and negative differential thermal conductance ingraphene nanoribbons
Jiuning Hu,
1, 2, a) Yan Wang, Ajit Vallabhaneni, Xiulin Ruan,
3, 2 and Yong P. Chen
4, 2, 1, b) School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907,USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907,USA School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907,USA Department of Physics, Purdue University, West Lafayette, Indiana 47907,USA
We employ classical molecular dynamics to study the nonlinear thermal transport in graphene nanoribbons(GNRs). For GNRs under large temperature biases beyond linear response regime, we have observed the onsetof negative differential thermal conductance (NDTC). NDTC is tunable by varying the manner of applyingthe temperature biases. NDTC is reduced and eventually disappears when the length of the GNR increases.We have also observed NDTC in triangular GNRs, where NDTC exists only when the heat current is fromthe narrower to the wider end. These effects may be useful in nanoscale thermal managements and thermalsignal processing utilizing GNRs.Graphene, an atomic monolayer of graphite, hasemerged as one of the most interesting materials incondensed matter physics and nanotechnology. Besidesits unusual electronic properties, graphene also hasunique thermal properties, e.g., high thermal conduc-tivities ( ∼ Graphene nanoribbons(GNRs) are promising in many applications, such as theirelectronic band-gap tunability and edge chirality depen-dent thermal transport. So far, little attention has beenpaid to nonlinear thermal transport in GNRs, thoughthese nonlinear effects have been explored in ideal atomicchains, molecular junctions and quantum dots. Here, we demonstrate negative differential thermal con-ductance (NDTC) in GNRs. Analogous to the electroniccounterpart, NDTC is a useful ingredient in developingGNR-based thermal management and signal manipula-tion devices, such as the thermal amplifiers and ther-mal logic gates. We study the thermal transport in GNRs using clas-sical molecular dynamics (MD) simulations. The many-body empirical Brenner potential is employed to de-scribe the carbon-carbon interactions. This method havebeen applied in many graphene-based systems. Thestructures of GNRs in this study are shown in the in-set (rectangular GNR) of Fig. 1 and the inset (triangu-lar GNR) of Fig. 3. The atoms denoted by squares arefixed in position, while those denoted by left- and right-pointing triangles are placed in two Nos´e-Hoover thermostats at different temperatures T L and T R , respec-tively. The equations of motion for atoms without posi-tion being fixed are: ddt p i = F i − γ i p i (1) a) [email protected] b) [email protected] where p i is the momentum of the i -th atom, F i is thetotal force acting on the i -th atom, and γ i is the Nos´e-Hoover dynamic parameter. For the atoms denoted bycircles, γ i ≡
0, and it recovers the NVE (constant numberof atoms, volume, and energy) ensemble. For the atomsin the left and right thermostats, γ i obeys the equation ddt γ i = h N L ( R ) k B P i ∈ L ( R ) p i m i − T L ( R ) τ T L ( R ) , (2)where τ is the thermostat relaxation time, N L ( R ) is thenumber of atoms in the thermostat, k B is the Boltzmannconstant and m is the mass of the carbon atom. Moredetails on our numerical calculation method can be foundelsewhere. First, we study the thermal transport in a rectangu-lar GNR with armchair top and bottom edges shown inthe inset of Fig. 1 (we have obtained qualitatively similarconclusions for GNRs with zigzag edges). Since the GNRis symmetrical, we only consider T L ≤ T R and define thetemperature difference ∆ T ≡ T R − T L . The temperature T R is kept as a constant. As we can see from both curvesin Fig. 1, for small temperature difference (e.g., ∆ T <
60K for T R = 300 K and ∆ T <
150 K for T R = 600 K),the thermal current increases approximately linearly as∆ T increases, as expected from Fourier’s law. Interest-ingly, for some range of higher ∆ T , the thermal currentdecreases as ∆ T increases (the dashed boxes in Fig. 1),indicating the onset of NDTC. It is a reasonable approxi-mation to consider thermal current as proportional to theproduct of thermal conductivity κ of the GNR and ∆ T .Our previous study has shown that κ increases with theaverage temperature T ≡ ( T L + T R ) / T R − ∆ T / T (labeled at the right vertical axis andindicated by right-pointing arrows for Fig. 1-3 and in thesubplot of Fig. 4(b)) as a function of ∆ T in all figures.Since T decreases with ∆ T , κ decreases with increasing∆ T . The resulting trend of the thermal current as a func- ∆ T = T R − T L (K) T h e r m a l c u rr e n t( µ W ) T RS =300 KT RS =600 K 200250300350400450500550600 T = ( T R + T L ) / ( K ) T RS =300 KT RS =600 K n m FIG. 1. Thermal current (left vertical axis) and average tem-perature (right vertical axis) vs. temperature difference ∆ T .The dashed boxes highlight NDTC. The inset shows the struc-ture of the GNR ( ∼ × (cid:4) denotes fixed bound-ary atoms. ◭ ( ◮ ) denotes atoms in the left (right) thermostat. denotes the remain atoms in the bulk. ∆ T = T R − T L (K) T h e r m a l c u rr e n t( µ W ) T = ( T R + T L ) / ( K ) FIG. 2. Thermal current (left vertical axis) and average tem-perature (right vertical axis) vs. temperature difference ∆ T in GNRs with the similar structure as the GNR in the in-set of Fig. 1, except for different lengths. In all these plots, T R = 300 K and T L is varied from 300 K to 30 K. tion of ∆ T is thus a competition between decreasing κ and increasing ∆ T . In the ∆ T range displaying NDTC,the decrease of κ with ∆ T dominates. We have foundthat there is no NDTC (shown in Fig. 4) if T L is largerthan the constant T R , i.e., if T increases with ∆ T (thuswithout the above competition). Note that for large ∆ T beyond linear response, strictly speaking thermal con-ductivity is not well defined. Thus, in the above expla-nation, κ is considered to be an effective, average ther-mal conductivity. Similar arguments have been appliedin analysing thermal transport in 1D atomic chains. Second, we study the length dependence of NDTC inGNRs. For all three GNRs of different lengths in Fig. 2, ∆ T = | T R − T L | (K) T h e r m a l c u rr e n t( µ W ) RS =300 KT LS =300 K 180200220240260280300 T = ( T R + T L ) / ( K ) n m T L T R FIG. 3. Thermal current (left vertical axis) and average tem-perature (right vertical axis) vs. temperature difference ∆ T in triangular GNR shown in the inset. The labels for theGNR structure have the same meaning as that in the inset inFig. 1. The dashed box highlights NDTC. T R = 300 K while T L is varied from T R to 30 K. As theGNR length is increased, the ∆ T value for the onset ofNDTC increases and the ∆ T range where NDTC existsshrinks. We thus suggest that NDTC will eventually dis-appear if the length of GNR exceeds some critical value.We have verified this using LAMMPS package and ve-locity scaling MD, and found no NDTC in a 50 nm longGNR with similar structure as that studied in Fig. 1.Besides these nonlinear effects in symmetrical GNRs,we also explore the possibility of NDTC in an asymmet-rical triangular GNR, shown in the inset of Fig. 3. Ourprevious study has pointed out that thermal rectificationexists in this asymmetrical GNR. As we see from Fig. 3,here the nonlinear thermal transport is also direction-dependent. NDTC appears when the temperature of thenarrower end is held at T L = 300 K and the tempera-ture T R of the wider end is varied from 300 K to 30 K(solid line in Fig. 3). However, there is no NDTC whenthe values of T L and T R are interchanged (dashed linein Fig. 3). This provides another possibility to controlthe nonlinear thermal transport and NDTC in GNRs byengineering the shape of GNRs.In general, the way to tune the thermal current in thetwo-terminal thermal devices is very different from thatin any two-terminal electronic devices. In the latter case,only the voltage difference matters. However, in thermaldevices, the average temperature T is as important asthe temperature difference ∆ T in controlling the ther-mal current. For example, consider T = α ∆ T + T withconstants α and T , and we have T L = ( α − )∆ T + T and T R = ( α + )∆ T + T . The thermal currents andaverage temperature T as a function of ∆ T are plottedin Fig. 4 for the rectangular GNR shown in the inset inFig. 1, where T = 300 K and α is tuned from -0.5 to 0.5(indicated by the dashed curved arrow in Fig. 4). The T h e r m a l c u rr e n t( µ W ) α =0.5 α =0.25 α =0 α =−0.25 α =−0.50 50 100 150 200 250 300 350200250300350400 T = ( T R + T L ) / ( K ) ∆ T = | T R − T L | (K) (a)(b) α =−0.5 α =−0.5 α =0.5 α =0.5 FIG. 4. Thermal current (a) and average temperature (b) vs.temperature difference ∆ T for different values of α for theGNR shown in the inset of Fig. 1. Note that α = 0 . T L ( R ) fixed at 300 K while T R ( L ) is varied. solid curve in Fig. 1 corresponds to α = − .
5. For smalltemperature difference in the linear response regime, theslope of thermal current vs. ∆ T is independent of α .In the nonlinear response regime (large ∆ T ), the systemtransitions from a regime with NDTC to a regime with-out NDTC when α is tuned from negative to positivevalues. We can see a strong correlation between the thetrend of the thermal current and that of the average tem-perature for different values of α in the range of ∆ T from100 K to 250 K where NDTC occurs for negative α . Fornegative α , since T decreases with ∆ T , the effective κ decreases with ∆ T , and the occurrence of NDTC can besimilarly explained as that for Fig. 1.There are two independent parameters to controlthe thermal transport in two-terminal devices, either( T L , T R ) or (∆ T, T ). Two-terminal thermal devices areactually analogous to three-terminal electronic devices.In the language of electronic transport of field effect tran-sistors (FETs), ∆ T plays the role of the drain-source volt-age difference in FETs, while α plays the role of the gatevoltage. Fig. 4 shows the ability to realize the FET-likebehaviour in GNRs.In summary, we have studied the nonlinear thermaltransport in rectangular and triangular GNRs underlarge temperature biases. We find that in short ( ∼ A. K. Geim and K. S. Novoselov, Nature Mater. , 183 (2007). A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, andA. K. Geim, Rev. Mod. Phys , 109 (2009). A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan,F. Miao, and C. N. Lau, Nano Lett. , 902 (2008). W. Cai, A. L. Moore, Y. Zhu, X. Li, S. Chen, L. Shi, and R. S.Ruoff, Nano Lett. , 1645 (2010). C. Faugeras, B. Faugeras, M. Orlita, M. Potemski, R. R. Nair,and A. K. Geim, ACS Nano , 1889 (2010). L. A. Jaureguia, Y. Yue, A. N. Sidorov, J. Hu, Q. Yu, G. Lopez,R. Jalilian, D. K. Benjamin, D. A. Delk, W. Wu, Z. Liu, X. Wang,Z. Jiang, X. Ruan, J. Bao, S. S. Pei, and Y. P. Chen, ECS Trans-actions , 73 (2010). J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T.Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R. S.Ruoff, and L. Shi, Science , 213 (2010). M. Y. Han, B. Ozyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett. , 206805 (2007). J. Hu, X. Ruan, and Y. P. Chen, Nano Lett. , 2730 (2009). B. Li, L. Wang, and G. Casati, Appl. Phys. Lett. , 143501(2006). W.-R. Zhong, P. Yang, B.-Q. Ai, Z.-G. Shao, and B. Hu, Phys.Rev. E , 050103 (May 2009). D. He, S. Buyukdagli, and B. Hu, Phys. Rev. B , 104302 (Sep2009). E. Pereira, Phys. Rev. E , 040101 (Oct 2010). D. He, B.-Q. Ai, H.-K. Chan, and B. Hu, Phys. Rev. E , 041131(Apr 2010). D. Segal, Phys. Rev. B , 205415 (May 2006). D. M.-T. Kuo and Y.-C. Chang, Jpn. J. Appl. Phys. , 064301(2010). L. Esaki, Phys. Rev. , 603 (Jan 1958). L. Wang and B. Li, Phys. Rev. Lett. , 177208 (Oct 2007). D. W. Brenner, Phys. Rev. B , 9458 (1990). C. Y. Wang, K. Mylvaganam, and L. C. Zhang, Phys. Rev. B , 155445 (2009). Z.-Y. Ong and E. Pop, Phys. Rev. B , 155408 (2010). J. Hu, S. Schiffli, A. Vallabhaneni, X. Ruan, and Y. P. Chen,Appl. Phys. Lett. , 133107 (2010). S. Nos´e, J. Chem. Phys. , 511 (1984). W. G. Hoover, Phys. Rev. A , 1695 (1985). J. Hu, X. Ruan, Z. Jiang, and Y. P. Chen, AIP Conf. Proc. ,135 (2009). S. Plimpton, J. Comput. Phys. , 1 (1995). Z. Huang and Z. Tang, Physica B373