Abstract. 1. Introduction:

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1 Isotope Effect on the Thermal Conductvty of Graphene Hengj Zhang 1, Geunsk Lee 1, Alexandre F. Fonseca, Tamme L. Borders 3, Kyeongjae Cho 1,4, * 1 Department of Physcs, Unversty of Texas at Dallas, Rchardson, Texas 75080, USA Departamento de Físca, Insttuto de Cêncas Exatas (ICEx), Unversdade Federal Flumnense, Volta Redonda, RJ, , Brazl 3 Department of Chemstry, Unversty of North Texas, Denton, Texas Department of Matersals Scence and Engneerng, Unversty of Texas at Dallas, Rchardson, Texas 75080, USA *correspondng author: kjcho@utdallas.edu Abstract The thermal conductvty (TC) of solated graphene wth dfferent concentratons of sotopes (C 13 ) s studed wth equlbrum molecular dynamcs method at 300K. In the lmt of pure C 1 or C 13 graphene, TC of graphene n zgzag and armchar drectons are ~630 W/mK and ~1000W/mK, respectvely. We fnd that the TC of graphene can be maxmally reduced by ~80%, n both armchar and zgzag drectons, when a random dstrbuton of C 1 and C 13 s assumed at dfferent dopng concentratons. Therefore, our smulaton results suggest an effectve way to tune the TC of graphene wthout changng ts atomc and electronc structure, thus yeldng a promsng applcaton for nanoelectroncs and thermoelectrcty of graphene based nano-devce. 1. Introducton: Snce t was fabrcated n 004 [1], graphene, a monolayer of sp-bonded network of carbon atoms, has attracted much attenton for ts unque electronc propertes []. Meanwhle, both recent theory and experment studes [3, 4] have revealed that solated graphene has shown an unusual hgh thermal transport capablty, whch s of great mportance n thermal management of nanonelectroncs. Dfferent from conventonal metallc materals, thermal energy carrers for graphene are manly n the form of phonon vbratons [3-5], and phonon contrbuton to TC s approxmately 50 tmes larger than electron contrbuton at room temperature [3], whch suggests that electron thermal transport s neglgble n our case. Furthermore, the electronc contrbuton to TC would be ndependent of sotope effect as C 13 s electroncally dentcal to C 1. For these reasons, we wll study only the phonon contrbuton to TC n ths paper. In addton to utlzng ts hgh TC, another possble applcaton of graphene has been nvestgated for thermoelectrc energy converson [6], where low TC but hgh electrc conductvty for graphene s requred for obtanng hgh thermoelectrc effcency. Such effcency s expressed σs as ZT = T, where σ s the electrc conductvty, S s the Seebeck coeffcent, and κ s the κ thermal conductvty. To acheve hgh ZT for graphene, a general scheme s to mnmze κ whle keepng σ and S less changed. One practcal method s to dope the graphene wth the stable sotope C 13 snce electronc structure of graphene s unchanged n ths dopng. In a pure crystal, one wthout defects or dslocatons, phonon scatterng n the presence of dfferent sotopes has been strongly correlated wth changes n thermal conductvty. Smlar to the observaton for

2 sotope effects on TC of Ge, damond, and boron ntrde nanotubes [7, 8], t s nterestng to check how effectvely sotope dopng method wll reduce TC of Graphene. Modelng of thermal transport can be acheved by usng Bolzmann transport equaton (BTE) [9, 10], or molecular dynamcs (MD) smulatons [11]. In BTE method, the sngle mode relaxaton tme (SMRT) approxmaton s a commonly used technque nvolvng the assgnment of the relaxaton tme to dfferent phonon scatterng mechansms. The relaxaton tme can be ether ftted to the expermental TC value [1] or determned from MD smulatons [10]. In MD smulaton approach, TC can be predcted from ether nonequlbrum MD [13], where a temperature s appled across the smulaton cell, or equlbrum MD [14], where the so-called Green-Kubo method s used to compute TC from heat current fluctuatons. One advantage for MD method s that there s no assumpton needed for phonon nteractons. As long as the phonon dsperson and anharmoncty of the potental are accurate, MD method provdes a robust way to accurately compute thermal transport. In Secton, we ntroduce Green-Kubo method and descrbe the smulaton procedures to compute TC. In Secton 3, we show the sotope effects on thermal transport of graphene, and dscuss the reason for TC reducton and ts possble applcaton for thermoelectrc applcaton. Secton 4 presents a summary and conclusons.. Green-Kubo MD smulaton Method: The Green-Kubo formula [14] derved from lnear response theory can express thermal conductvty tensor n terms of equlbrum heat current-current autocorrelaton n the form, κ αβ 1 τ m = Jα ( τ ) J β (0) dτ Ωk B T 0 Where Ω s the system volume defned as the area of graphene multpled wth van der Waals thckness (3.35 Angstrom), k B s the Boltzmann constant, T s the system temperature, and τ m s the tme requred to be longer than the tme for current-current correlatons to decay to zero [11]. J α,β s denoted as the heat flux n α or β drecton, ts expresson s commonly defned as, d J = r ( t) E ( t) () dt E 1 = mv + φ (3) Where r, v and E are the poston, velocty and ste energy of atom respectvely. φ s the potental energy at ste. Then, we use Hardy s defnton [15], J = E v + rj ( Fj v )] j (1) [, (4) where r j =r -r j, F j s the force exerted on atom by atom j. One mert of Hardy s defnton s that t s ndependent of par-wse or many-body potental formulas. Based on the above equatons, heat current at each step s recorded on dsk as a quantty defned by atom postons, veloctes, and

3 nter-atomc forces, whch can be extracted from atomstc smulatons. The last step s to compute TC tensor wth the known heat current n equaton (1). Fg.1 Structure of graphene unt cell wth 11 carbon atoms. In the equlbrum MD smulatons, we used the second generaton REBO carbon potental for ts accuracy n descrbng bond strength and anharmoncty of carbon materals [16]. A unt cell of graphene, wth a perodc boundary condton as shown n Fg.1, has been thermalzed to 300K for 00 ps wth Berendson thermostats. Afterwards, the heat current of graphene s recorded every fs n nne mcrocanoncal ensemble smulatons wth uncorrelated ntal condtons. Each mcrocanoncal smulaton needs to run up to 10 ns to obtan converged TC value. Compared wth non-equlbrum molecular dynamcs (NEMD), the Green-Kubo method s ndeed less computatonally effcent; however, t s free of troubles such as fnte sze and boundary effects, whch are commonly unavodable n NEMD. 3. Results and Dscusson: We computed the TC of mass defect free graphene as 630 W/mk and 1000 W/mk n armchar and zgzag drecton, respectvely. Ths s lower than the reported expermental data [4, 5]. The reason has to do wth the dscrepances on phonon dsperson between experment measurement and theoretcal data predcted wth orgnal REBO potental [17]. Although the absolute TC of graphene s overall underestmated n our smulaton, the relatve dfference of TC caused by sotope mass defect s stll physcally meanngful, and the normalzed TC reducton seen n Fg. 4 properly reflects the sotope effect on TC of graphene. One prmary test done before studyng sotope effects on thermal transport s the convergence test for graphene wth dfferent unt cell perodc boundary lengths. As shown n Fg., we have tested three cases wth the boundary length of 1.6, 3.4 and 5.1 nm respectvely, and the converged result suggests that the majorty contrbuton from dfferent phonon vbraton modes are well ncluded n our smulatons. Then, to be effcent, we choose the 1.6 nm unt cell as the structure for the followng smulatons.

4 Fg. Graphene TC convergence tests wth dfferent unt cell boundary lengths. κ s the average of TC n zgzag and armchar drectons. The frst thng we learned from the smulatons s that TC of pure C 1 or C 13 graphene n zgzag drecton s ~58% greater than that n armchar drecton. Ths can be conceptually explaned from the acoustc phonon dsperson and Grünesen parameter of graphene calculated wth REBO potental as shown n Fg. 3. In the SMRT approxmaton, the BTE method can express TC as a summaton of each phonon mode contrbuton, κ = κ q ( q), and ph κ = C ( q) v ( q) τ ( q) where and q represent phonon modes and wave momentum, C (q) s the specfc heat of the phonons, whch s constant n the classcal case. v s the group velocty for mode, τ ph s the relaxaton tme and T s the temperature. From Fg.3 (a), we notce that both out of plane acoustc (ZA) and transverse acoustc (TA) modes n zgzag drecton have apparently larger group velocty than that n armchar drecton, longtudnal acoustc (LA) modes group velocty n both drectons s qute smlar. Then, from the Fg.3 (b), the Grünesen parameter, whch s plotted as a functon of phonon mode and momentum q, shows a smaller dfference n zgzag and armchar drectons for each vbraton mode. As the Grünesen parameters for LA, TA and ZA are smlar, t s reasonable to assume that the relaxaton tmes for phonon vbraton n zgzag and armchar drecton are the same. Based on the above analyss, we conclude that the dfference n the TC between zgzag and armchar drectons manly comes from ther dfferent group veloctes.

5 Fg. 3 a) Phonon dspersons for C 1 graphene and b) Grünesen parameters of graphene as a functon of q momentum for each vbraton mode. ГM and ГK represent armchar and zgzag drectons of graphene. Phonon dsperson for graphene s calculated wth PHON code [18] for REBO carbon potental. In the sotope effect study, we generated a wde range of graphene samples wth dfferent C 13 concentratons randomly dstrbuted, snce t s a more realstc possble confguraton after the synthess of graphene. Recently, Mngo et al. have proposed a possble method to generate sotope clusters n graphene, and theoretcally demonstrated TC reducton usng non-equlbrum Green s functon method [19]. Fg.4 shows the normalzed graphene TC values as a functon of C 13 concentraton n armchar and zgzag drectons, and the maxmum TC reducton for graphene can be made between 5% and 75% of the C 13 atomc concentratons. Fg. 4 Normalzed Graphene TC as a functon of C 13 concentraton a) TC n armchar drecton b) TC n zgzag drecton. The nset shows TC as a functon of C 13 for low concentraton. The normalzng factors κ x and κ y are the values for pure graphene, 630 W/mK and 1000 W/mK, respectvely. The explanaton of the almost parabolc shape of TC n Fgure 4 can be found n an earler mportant relaton derved by Klemens [0]. As t s commonly accepted, the longer the phonon

6 mean free path s, the larger TC would be. Klemens found that, for sotope scatterng, the mean free path s proportonal to g 1 4 T, where g = M C ( ( C ) M, and T s the C M ) temperature. C and M represent the concentraton and the mass of the consttuent sotope atoms, respectvely. Thus, the mean free path drectly has to do wth the g factor, whch s the mass varaton of sotope atoms. In our smulaton, g factor reaches ts maxmum for around 50% of C 13, so there we have the mnmum phonon mean free path and the mnmum TC. As C 13 atomc concentraton approaches to zero or 100%, g becomes smaller, phonon free path s larger, and TC ncreases. It won t get to nfnty because of the Umklapp and other phonon scatterng mechansms. Among varous methods that modulate the TC of graphene, the sotope dopng method provdes an effcent way to mprove thermoelectrc effcency, snce sotope atoms do not change the electronc structure of graphene, the electrc conductvty and the Seebeck coeffcent reman the same after the C 13 dopng. As our smulatons suggest, κ n armchar and zgzag drectons can be reduced by up to 80%, whch means an mprovement of the ZT by a factor of 5 snce ZT s nversely proportonal to TC. Recently, two mportant smulaton studes were carred out on ZT of armchar [1] and zgzag [] graphene nanorbbons (GNRs). In both works, the authors looked for defect and dsordered structures that mnmze TC whle keep electrc conductvty less changed, thus maxmzng the ZT. Compared wth ths, the sotope dopng method appears to be another attractve method that would promote ZT even further for GNRs when t s combned wth those methods [1, ]. We beleve our results can motvate new experments to check the effcency of sotope dopng method for thermoelectrc applcaton. 4. Concluson: In summary, we have studed sotope effect on TC of solated graphene wth equlbrum MD methods. Our smulaton results suggest that TC of graphene can be effectvely reduced by up to 80% n armchar and zgzag drectons for sotope concentratons as low as 5%. The phenomenon that mass defect can reduce TC s explaned wth the relaton between phonon mean free path and mass varaton of the sotope mxtures [0]. Fnally, our study shows that dopng graphene wth carbon sotope C 13 could be a practcal way to reduce TC wthout changng ts electrc property, thus promotng thermoelectrc coeffcent. Acknowledgement: The authors are grateful to support from Lockheed Martn. G.L. acknowledges support from SWAN, and A.F.F. acknowledges partal support from the Brazlan agency CNPq. We also apprecate dscussons wth Davde Donado and Gula Gall.

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