Thermoelectric properties of graphene nanoribbons with surface roughness Huaping Xiao, Wei Cao, Tao Ouyang, Xiaoyan Xu, Yingchun Ding, and Jianxin Zhong

Citation: Appl. Phys. Lett. 112, 233107 (2018); doi: 10.1063/1.5031909 View online: https://doi.org/10.1063/1.5031909 View Table of Contents: http://aip.scitation.org/toc/apl/112/23 Published by the American Institute of Physics

APPLIED PHYSICS LETTERS 112, 233107 (2018)

Thermoelectric properties of graphene nanoribbons with surface roughness Huaping Xiao,1,a) Wei Cao,1 Tao Ouyang,1,b) Xiaoyan Xu,1 Yingchun Ding,2 and Jianxin Zhong1

1 Hunan Key Laboratory for Micro-Nano Energy Materials and Device and Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China 2 College of Optoelectronics Technology, Chengdu University of Information Technology, Chengdu 610225, People’s Republic of China

(Received 2 April 2018; accepted 22 May 2018; published online 6 June 2018) We theoretically investigate the ballistic thermoelectric performance of graphene nanoribbons with surface roughness using the nonequilibrium Green’s function method. The results show that the surface roughness could dramatically reduce the thermal conductance of graphene nanoribbons, and thus lead to the boosting of thermoelectric performance of graphene (the figure of merit can be as high as 3.7 at room temperature). Meanwhile, the electron transport properties of different edged rough graphene nanoribbons exhibit distinctive anisotropic behaviors, i.e., the thermal power of armchair edged nanoribbons significantly increases, while that of zigzag edged remains nearly unchanged, which is mainly attributed to the edge effect. The findings presented in this paper qualify surface roughness as an efficient approach to enhance the thermoelectric performance of graphene nanoribbons. Published by AIP Publishing. https://doi.org/10.1063/1.5031909

Thermoelectric materials, which can convert heat into electricity, and vice versa, have been attracting extensive attention recently. The efficiency of thermoelectric conversion is characterized by the dimensionless figure of merit ZT ¼ S2 Ge T=ðjph þ je Þ, where S is the Seebeck coefficient, Ge the electronic conductance, T the absolute temperature, and je (jph ) the thermal conductance contributed by electrons (phonons). Materials with ZT 1 are considered as good thermoelectrics, while their ZT values need to be at least 3 so that the transform efficiency could compete with conventional power generators and refrigerators.1 The usual method to increase ZT is suppressing the phonon thermal conductivity. However, it is difficult to greatly improve the thermoelectric performance due to the strong coupling between the electronic and phononic transport parameters.1,2 During the past decades, a series of breakthroughs3,4 have proven that low-dimensional or micro-nano systems tend to exhibit better thermoelectric performance than traditional bulk materials, which opens up new avenues in thermoelectric applications. Graphene, a typical 2D material, shows great potential for fabricating micro-nano devices because of its outstanding properties,5–8 e.g., high electron mobility,9 strong mechanical strength10 and high thermal conductivity.11,12 In addition to these superb properties, the thermoelectric performance of graphene has already been studied well. However, the results are unsatisfactory (ZT 0:0513) due to the ultra-high thermal conductance.14 Therefore, plenty of strategies have been put forward to reduce the thermal conductance and improve the thermoelectric efficiency of graphene, such as edge disorder,15 defect engineering,16 isotope engineering,17 and superlattice structures.18 Different from specific structural engineering, the surface roughness (SR) caused by an uneven substrate is a very common phenomenon during the fabrication. Such an uneven a)

E-mail: [email protected] E-mail: [email protected]

substrate could be realized in experiments by utilizing photochemical etching and other methods. Many research efforts have paid attention to the distinctive electronic and chemical properties of rough graphene on different substrates.19–23 Moreover, the armchair edged graphene nanoribbons’ field effect transistors in the presence of a 2D SR structure could affect the device performance metrics strongly.24 These works reveal the unusual meaning of roughness structures on graphene. Considering previous works mentioned above, it is natural to ask whether such geometric irregularity could dramatically reduce the thermal transport and thereby enhance the thermoelectric performance of graphene. Consequently, in this paper, we study the thermoelectric properties of graphene nanoribbons with surface roughness (SRGNRs) which can be regarded as a one-dimensional random structure along the transport direction as schematically shown in Fig. 1. The images of structures of SRGNRs acquired from different sides are shown in Figs. 1(a)–1(c). To model such a one-dimensional random structure, we utilize a simplified method outlined by Garcia and Stoll25 to obtain a series of Zn values. Then, we take these Zn values to redefine the Z coordinates of atoms, and the SR distribution function is described by the relation between X and Z coordinates. The widths of pristine armchair and zigzag edged graphene nanoribbons are chosen as NA ¼ 8 and NZ ¼ 8. In this work, the variations taken into consideration are the correlation length Cl, the roughness height H and the length of the central part Nlength . The non-equilibrium Green’s function (NEGF) scheme26,27 is employed to obtain the ballistic phonon and electron transport properties. It should be noted here that the phonon (electron)-phonon interactions are neglected for the weak coupling of electron-phonon and large phonon mean free path in graphene.28 For electronic transport, the tightbinding Hamiltonian29 can be expressed as X X H¼e cþ eihij cþ (1) i ci t i ci ; hii

b)

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FIG. 1. Schematic diagram of armchair edged graphene nanoribbons at present of surface roughness: (a) top view, (b) side view, and (c) front view. The box of dotted line indicates a supercell.

where the on-site energy e is set to 0 and the bond length related hopping energy tl is given by tl ¼ t0 exp ð3:37ðl=a 1ÞÞ,30 where t0 is the pZ pZ hopping parameter of pristine graphene ˚ ) are the nearest neighbor C–C bond (2.7 eV) and l and a (1.42 A length in rough and pristine graphene. Due to the minor influence on ZT values, the on-site energy of vacancy has been ignored in this work (the result is not shown here). The electronic transmission function Te ðEÞ is calculated as Te ðEÞ ¼ TrðGr CL Ga CR Þ: The function LðmÞ is defined as ð 2 1 @f ðE; lÞ m ðmÞ L ¼ dE; Te ðEÞðE lÞ h 1 @E

(2)

(3)

where f ðE; lÞ ¼ 1=ðexp ½ðE lÞ=kB T þ 1Þ is the FermiDirac distribution function and l the chemical potential. Accordingly, the Seebeck coefficient (S ¼ Lð1Þ =ðeTLð0Þ Þ), the electronic conductance (r ¼ e2 Lð0Þ ) and the electron thermal conductance (je ¼ ½Lð2Þ ðLð1Þ Þ2 =Lð0Þ =T) can be computed. By using the second-generation reactive empirical bond order potential,31 we can calculate the phonon transmission function Tph ðxÞ,32 and the thermal conductance can be obtained from jph

2 h ¼ 2pkB T 2

ð1

x2

0

ehx=kB T ðehx=kB T 1Þ2

Tph ðxÞdx:

(4)

the transmission values decrease dramatically and the staircase-like transmission spectrum is completely destroyed. In other words, the available phonon transport channels have been broken and phonons will encounter strong scattering when passing through the SRGNRs, especially for high energy phonons (they are almost completely scattered). Such suppression in transmission leads to significant reduction in phononic thermal conductance of SRGNRs, as shown in Figs. 2(c) and 2(d). For instance, at room temperature, the thermal conductance of A-SRGNRs is 0.1 nW/K which is one order of magnitude lower than that of pristine A-GNRs (1.17 nW/K). Meanwhile, one can note that owing to the complete scattering of high energy phonons, the growth rate of the thermal conductance tends to be smooth and slow for SRGNRs as the temperature increases. In order to elucidate the reason for the SR effect on the thermal transport, the force constant difference (FCD) and SR distributions are depicted in Figs. 3(a) and 3(b). The FCD on every atom is the average value of the force constant matrix in xx, yy and zz directions. For simplicity, we treat the atoms with the same X coordinate as a single point and SRGNRs are completely a 1D structure now. It can be seen clearly that the FCD displays random oscillation like the distribution of SR. As the consequence of the inhomogeneity of the force constant caused by SR, the phonons will encounter scattering when transmitting through the roughness area, and

Finally, combining the above equations, the thermoelectric figure of merit ZT can be expressed as ZT ¼

S2 rT : je þ jph

(5)

To reduce the fluctuation of our results and obtain good statistics, for each roughness magnitude, up to twenty different geometric configurations are used to get the sample average (over twenty different geometric configurations are enough to obtain good results in our test work). It should be mentioned that the theory of nonequilibrium vertex correction (NVC)33,34 is another appropriate and efficient way to deal with disordered systems. First, the phononic transmission spectra of SRGNRs with armchair (A-SRGNRs) and zigzag (Z-SRGNRs) edges are given in Figs. 2(a) and 2(b), respectively. For comparison, the phonon transmissions of pristine GNRs are also presented. It can be seen clearly that flat GNRs display staircase-like transmission spectra and the transmission values correspond to the number of available transport channels. As the geometric irregularity is applied,

FIG. 2. The phonon transmission Tph ðxÞ as a function of frequency for (a) armchair-edged and (b) zigzag-edged SRGNRs compared with pristine GNRs. The phonon thermal conductance jph vs. temperature for (c) armchair-edged and (d) zigzag-edged SRGNRs compared with pristine GNRs.

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FIG. 3. The distribution of roughness height and force constant difference along the X-axis for (a) armchair and (b) zigzag edged structures.

thus give rise to the decrease in the thermal conductance of SRGNRs as shown in Figs. 2(c) and 2(d). To analyze the electron transport and thermoelectric properties, we present the electron transmission and thermal power S2 r of SRGNRs at room temperature in Figs. 4(a)–4(d), respectively. For A-SRGNRs, the electron transmission is completely collapsed and a new band gap arises

FIG. 4. The electron transmission Te ðEÞ for (a) armchair-edged and (b) zigzag-edged structures. The thermal power S2 r at room temperature for (c) armchair-edged and (d) zigzag-edged structures.

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compared with the pristine ones in Fig. 4(a). It is known that the energy gaps in armchair edged graphene nanoribbons are sensitive to the quantum confinement and edge shapes.35 That is to say, the SR greatly changes the edge shapes of A-GNRs and opens a band gap around Fermi level. Owing to the linear relation between the Seebeck coefficient and the energy gap of semiconductors,36 there is a sharp rise in thermal power in Fig. 4(c), which ensures the outstanding thermoelectric performance of A-SRGNRs. Moreover, the chemical potential corresponding to the maximum value has been moved closer to the Fermi level, which can be more easily effected for doping. In the case of Z-SRGNRs, the electron transmission still keeps the staircase-like structures above the Fermi level as shown in Fig. 4(b). With increasing energy, such staircase-like structures become even more unclear. Another important fact is that there is no new band gap around Fermi level. Consequently, as shown in Fig. 4(d), the thermal power of Z-SRGNRs is weakly affected compared to that of flat Z-GNRs. These discrepancies between A-SRGNRs and Z-SRGNRs also agree with the results about ballistic transport in edge-disordered GNRs.37 Finally, the averaged maximum ZT values and the corresponding standard deviations are plotted as a function of different geometric variations in Figs. 5(a)–5(c). Interestingly, the different edged SRGNRs present a discrepancy in growth trends with the increase in those variations. For A-SRGNRs, the growth trends of different variations are evident due to the significantly reduced thermal conductance and enhanced thermal power. As shown in Figs. 5(a) and 5(b), ZT values are sensitive to the variations H and Cl which control the roughness magnitude of SR. As H increases or Cl decreases, the SR turns out to be rougher and a higher ZT value is found. Then, we fix the roughness level of SR and investigate the influence of the central length in Fig. 5(c). It is obvious that longer A-SRGNRs tend to show better thermoelectric performance and ZT values exhibit a clear linear relationship with Nlength .

FIG. 5. The averaged figure of merit ZT for armchair and zigzag edged SRGNRs at room temperature as a function of (a) H, (b) Cl and (c) Nlength , where the banded line corresponds to the standard deviation. The dashed line at 0.05 represents the ZT value of pristine GNRs. (d) The averaged figure of merit ZT for A-SRGNRs and pristine A-GNRs at room temperature as a function of width.

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The maximum ZT value for A-SRGNRs can reach 3.7 with H ¼ 2, Cl ¼ 0:5 and Nlength ¼ 35. In contrast, the thermoelectric performance of Z-SRGNRs is not satisfactory enough. Despite the fact that the ZT values for Z-SRGNRs have slightly increased compared to pristine Z-GNRs, the influence of roughness variations H and Cl seems to be general compared with A-SRGNRs. As shown in Fig. 5(c), the ZT values for Z-SRGNRs with different lengths almost remain the same and the stable value is about 0.35. The reason can be explained by decreased thermal conductance and unchanged thermal power mentioned before. Before concluding, the width influence on the thermoelectric performance of A-SRGNRs is also explored. In Fig. 5(d), we present the ZT values of A-SRGNRs as a function of width compared with pristine A-GNRs. It is well known that A-GNRs are semimetallic (gapless) if NA equals 3n þ 2 and semiconductors if NA equals 3n or 3n þ 1.35,38 As a result of higher Seebeck coefficient of semiconductors, A-GNRs with NA ¼ 3n or 3n þ 1 exhibit better thermoelectric performance. Similar width behavior could also be found in the A-SRGNRs. In summary, the thermoelectric performance of graphene nanoribbons with surface roughness has been investigated using NEGF approach. Our results demonstrate that the surface roughness structures can significantly improve the thermoelectric efficiency of graphene nanoribbons, especially for armchair edged ones. This mainly originated from the greatly reduced thermal conductance as a result of the irregular force constant difference. In addition, the electronic properties of ASRGNRs have been significantly improved, while those for ZSRGNRs are basically unchanged. This discrepancy could be attributed to the effect of the edge shape on graphene band gaps. Using this strategy, the maximum ZT higher than 1 can be easily achieved in A-GNRs. Our findings will be of great interest in the designing and fabrication of thermoelectric devices based on graphene in practice. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11304264, 11304262, and 11574014), Scientific Research Fund of Hunan Provincial Education Department (No. 17B252), the Program for Changjiang Scholars and Innovative Research Team in University (IRT13093), the Hunan Provincial Natural Science Foundation of China (No. 2018JJ2380), and the Foundation Department of Education of Sichuan Province (No. 2017Z031). 1

T. M. Tritt, Annu. Rev. Mater. Res. 41(1), 433 (2011). C. B. Vining, Nat. Mater. 8, 83 (2009). 3 M. S. Dresselhaus, G. Chen, M. Y. Tang, R. G. Yang, H. Lee, D. Z. Wang, Z. F. Ren, J.-P. Fleurial, and P. Gogna, Adv. Mater. 19(8), 1043 (2007). 4 G. Jeffrey Snyder and E. S. Toberer, Nat. Mater. 7, 105 (2008). 2

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V. Barone, O. Hod, and G. E. Scuseria, Nano Lett. 6(12), 2748 (2006). € M. Y. Han, B. Ozyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett. 98(20), 206805 (2007). 7 L. Yang, C.-H. Park, Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 99(18), 186801 (2007). 8 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81(1), 109 (2009). 9 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(5696), 666 (2004). 10 C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321(5887), 385 (2008). 11 H.-S. Zhang, Z.-X. Guo, X.-G. Gong, and J.-X. Cao, J. Appl. Phys. 112(12), 123508 (2012). 12 D. L. Nika, E. P. Pokatilov, A. S. Askerov, and A. A. Balandin, Phys. Rev. B 79(15), 155413 (2009). 13 Y. Chen, T. Jayasekera, A. Calzolari, K. W. Kim, and M. Buongiorno Nardelli, J. Phys.: Condens. Matter 22(37), 372202 (2010). 14 A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, Nano Lett. 8(3), 902 (2008). 15 H. Sevinc¸li and G. Cuniberti, Phys. Rev. B 81(11), 113401 (2010). 16 J. Haskins, A. Kınacı, C. Sevik, H. Sevinc¸li, G. Cuniberti, and T. C¸a gın, ACS Nano 5(5), 3779 (2011). 17 V.-T. Tran, J. Saint-Martin, P. Dollfus, and S. Volz, Sci. Rep. 7(1), 2313 (2017). 18 H. Karamitaheri, M. Pourfath, R. Faez, and H. Kosina, J. Appl. Phys. 110(5), 054506 (2011). 19 R. Decker, Y. Wang, V. W. Brar, W. Regan, H.-Z. Tsai, Q. Wu, W. Gannett, A. Zettl, and M. F. Crommie, Nano Lett. 11(6), 2291 (2011). 20 M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams, Nano Lett. 7(6), 1643 (2007). 21 C. H. Lui, L. Liu, K. F. Mak, G. W. Flynn, and T. F. Heinz, Nature 462, 339 (2009). 22 J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, Nature 446, 60 (2007). 23 C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, Nat. Nanotechnol. 5, 722 (2010). 24 M. Sanaeepur, A. Y. Goharrizi, and M. J. Sharifi, IEEE Trans. Electron Devices 61(4), 1193 (2014). 25 N. Garcia and E. Stoll, Phys. Rev. Lett. 52(20), 1798 (1984). 26 J. S. Wang, J. Wang, and J. T. L€ u, Eur. Phys. J. B 62(4), 381 (2008). 27 T. Yamamoto and K. Watanabe, Phys. Rev. Lett. 96(25), 255503 (2006). 28 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(15), 151911 (2008). 29 T. C. Li and S.-P. Lu, Phys. Rev. B 77(8), 085408 (2008). 30 V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, Phys. Rev. B 80(4), 045401 (2009). 31 W. Brenner Donald, A. Shenderova Olga, A. Harrison Judith, J. Stuart Steven, N. Boris, and B. Sinnott Susan, J. Phys.: Condens. Matter 14(4), 783 (2002). 32 J.-W. Jiang, J.-S. Wang, and B. Li, J. Appl. Phys. 109(1), 014326 (2011). 33 Y. Ke, K. Xia, and H. Guo, Phys. Rev. Lett. 100(16), 166805 (2008). 34 Y. Ke, F. Zahid, V. Timoshevskii, K. Xia, D. Gall, and H. Guo, Phys. Rev. B 79(15), 155406 (2009). 35 Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97(21), 216803 (2006). 36 H. J. Goldsmid and J. W. Sharp, J. Electron. Mater. 28(7), 869 (1999). 37 D. A. Areshkin, D. Gunlycke, and C. T. White, Nano Lett. 7(1), 204 (2007). 38 K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54(24), 17954 (1996). 6

Citation: Appl. Phys. Lett. 112, 233107 (2018); doi: 10.1063/1.5031909 View online: https://doi.org/10.1063/1.5031909 View Table of Contents: http://aip.scitation.org/toc/apl/112/23 Published by the American Institute of Physics

APPLIED PHYSICS LETTERS 112, 233107 (2018)

Thermoelectric properties of graphene nanoribbons with surface roughness Huaping Xiao,1,a) Wei Cao,1 Tao Ouyang,1,b) Xiaoyan Xu,1 Yingchun Ding,2 and Jianxin Zhong1

1 Hunan Key Laboratory for Micro-Nano Energy Materials and Device and Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China 2 College of Optoelectronics Technology, Chengdu University of Information Technology, Chengdu 610225, People’s Republic of China

(Received 2 April 2018; accepted 22 May 2018; published online 6 June 2018) We theoretically investigate the ballistic thermoelectric performance of graphene nanoribbons with surface roughness using the nonequilibrium Green’s function method. The results show that the surface roughness could dramatically reduce the thermal conductance of graphene nanoribbons, and thus lead to the boosting of thermoelectric performance of graphene (the figure of merit can be as high as 3.7 at room temperature). Meanwhile, the electron transport properties of different edged rough graphene nanoribbons exhibit distinctive anisotropic behaviors, i.e., the thermal power of armchair edged nanoribbons significantly increases, while that of zigzag edged remains nearly unchanged, which is mainly attributed to the edge effect. The findings presented in this paper qualify surface roughness as an efficient approach to enhance the thermoelectric performance of graphene nanoribbons. Published by AIP Publishing. https://doi.org/10.1063/1.5031909

Thermoelectric materials, which can convert heat into electricity, and vice versa, have been attracting extensive attention recently. The efficiency of thermoelectric conversion is characterized by the dimensionless figure of merit ZT ¼ S2 Ge T=ðjph þ je Þ, where S is the Seebeck coefficient, Ge the electronic conductance, T the absolute temperature, and je (jph ) the thermal conductance contributed by electrons (phonons). Materials with ZT 1 are considered as good thermoelectrics, while their ZT values need to be at least 3 so that the transform efficiency could compete with conventional power generators and refrigerators.1 The usual method to increase ZT is suppressing the phonon thermal conductivity. However, it is difficult to greatly improve the thermoelectric performance due to the strong coupling between the electronic and phononic transport parameters.1,2 During the past decades, a series of breakthroughs3,4 have proven that low-dimensional or micro-nano systems tend to exhibit better thermoelectric performance than traditional bulk materials, which opens up new avenues in thermoelectric applications. Graphene, a typical 2D material, shows great potential for fabricating micro-nano devices because of its outstanding properties,5–8 e.g., high electron mobility,9 strong mechanical strength10 and high thermal conductivity.11,12 In addition to these superb properties, the thermoelectric performance of graphene has already been studied well. However, the results are unsatisfactory (ZT 0:0513) due to the ultra-high thermal conductance.14 Therefore, plenty of strategies have been put forward to reduce the thermal conductance and improve the thermoelectric efficiency of graphene, such as edge disorder,15 defect engineering,16 isotope engineering,17 and superlattice structures.18 Different from specific structural engineering, the surface roughness (SR) caused by an uneven substrate is a very common phenomenon during the fabrication. Such an uneven a)

E-mail: [email protected] E-mail: [email protected]

substrate could be realized in experiments by utilizing photochemical etching and other methods. Many research efforts have paid attention to the distinctive electronic and chemical properties of rough graphene on different substrates.19–23 Moreover, the armchair edged graphene nanoribbons’ field effect transistors in the presence of a 2D SR structure could affect the device performance metrics strongly.24 These works reveal the unusual meaning of roughness structures on graphene. Considering previous works mentioned above, it is natural to ask whether such geometric irregularity could dramatically reduce the thermal transport and thereby enhance the thermoelectric performance of graphene. Consequently, in this paper, we study the thermoelectric properties of graphene nanoribbons with surface roughness (SRGNRs) which can be regarded as a one-dimensional random structure along the transport direction as schematically shown in Fig. 1. The images of structures of SRGNRs acquired from different sides are shown in Figs. 1(a)–1(c). To model such a one-dimensional random structure, we utilize a simplified method outlined by Garcia and Stoll25 to obtain a series of Zn values. Then, we take these Zn values to redefine the Z coordinates of atoms, and the SR distribution function is described by the relation between X and Z coordinates. The widths of pristine armchair and zigzag edged graphene nanoribbons are chosen as NA ¼ 8 and NZ ¼ 8. In this work, the variations taken into consideration are the correlation length Cl, the roughness height H and the length of the central part Nlength . The non-equilibrium Green’s function (NEGF) scheme26,27 is employed to obtain the ballistic phonon and electron transport properties. It should be noted here that the phonon (electron)-phonon interactions are neglected for the weak coupling of electron-phonon and large phonon mean free path in graphene.28 For electronic transport, the tightbinding Hamiltonian29 can be expressed as X X H¼e cþ eihij cþ (1) i ci t i ci ; hii

b)

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112, 233107-1

hi;ji

Published by AIP Publishing.

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FIG. 1. Schematic diagram of armchair edged graphene nanoribbons at present of surface roughness: (a) top view, (b) side view, and (c) front view. The box of dotted line indicates a supercell.

where the on-site energy e is set to 0 and the bond length related hopping energy tl is given by tl ¼ t0 exp ð3:37ðl=a 1ÞÞ,30 where t0 is the pZ pZ hopping parameter of pristine graphene ˚ ) are the nearest neighbor C–C bond (2.7 eV) and l and a (1.42 A length in rough and pristine graphene. Due to the minor influence on ZT values, the on-site energy of vacancy has been ignored in this work (the result is not shown here). The electronic transmission function Te ðEÞ is calculated as Te ðEÞ ¼ TrðGr CL Ga CR Þ: The function LðmÞ is defined as ð 2 1 @f ðE; lÞ m ðmÞ L ¼ dE; Te ðEÞðE lÞ h 1 @E

(2)

(3)

where f ðE; lÞ ¼ 1=ðexp ½ðE lÞ=kB T þ 1Þ is the FermiDirac distribution function and l the chemical potential. Accordingly, the Seebeck coefficient (S ¼ Lð1Þ =ðeTLð0Þ Þ), the electronic conductance (r ¼ e2 Lð0Þ ) and the electron thermal conductance (je ¼ ½Lð2Þ ðLð1Þ Þ2 =Lð0Þ =T) can be computed. By using the second-generation reactive empirical bond order potential,31 we can calculate the phonon transmission function Tph ðxÞ,32 and the thermal conductance can be obtained from jph

2 h ¼ 2pkB T 2

ð1

x2

0

ehx=kB T ðehx=kB T 1Þ2

Tph ðxÞdx:

(4)

the transmission values decrease dramatically and the staircase-like transmission spectrum is completely destroyed. In other words, the available phonon transport channels have been broken and phonons will encounter strong scattering when passing through the SRGNRs, especially for high energy phonons (they are almost completely scattered). Such suppression in transmission leads to significant reduction in phononic thermal conductance of SRGNRs, as shown in Figs. 2(c) and 2(d). For instance, at room temperature, the thermal conductance of A-SRGNRs is 0.1 nW/K which is one order of magnitude lower than that of pristine A-GNRs (1.17 nW/K). Meanwhile, one can note that owing to the complete scattering of high energy phonons, the growth rate of the thermal conductance tends to be smooth and slow for SRGNRs as the temperature increases. In order to elucidate the reason for the SR effect on the thermal transport, the force constant difference (FCD) and SR distributions are depicted in Figs. 3(a) and 3(b). The FCD on every atom is the average value of the force constant matrix in xx, yy and zz directions. For simplicity, we treat the atoms with the same X coordinate as a single point and SRGNRs are completely a 1D structure now. It can be seen clearly that the FCD displays random oscillation like the distribution of SR. As the consequence of the inhomogeneity of the force constant caused by SR, the phonons will encounter scattering when transmitting through the roughness area, and

Finally, combining the above equations, the thermoelectric figure of merit ZT can be expressed as ZT ¼

S2 rT : je þ jph

(5)

To reduce the fluctuation of our results and obtain good statistics, for each roughness magnitude, up to twenty different geometric configurations are used to get the sample average (over twenty different geometric configurations are enough to obtain good results in our test work). It should be mentioned that the theory of nonequilibrium vertex correction (NVC)33,34 is another appropriate and efficient way to deal with disordered systems. First, the phononic transmission spectra of SRGNRs with armchair (A-SRGNRs) and zigzag (Z-SRGNRs) edges are given in Figs. 2(a) and 2(b), respectively. For comparison, the phonon transmissions of pristine GNRs are also presented. It can be seen clearly that flat GNRs display staircase-like transmission spectra and the transmission values correspond to the number of available transport channels. As the geometric irregularity is applied,

FIG. 2. The phonon transmission Tph ðxÞ as a function of frequency for (a) armchair-edged and (b) zigzag-edged SRGNRs compared with pristine GNRs. The phonon thermal conductance jph vs. temperature for (c) armchair-edged and (d) zigzag-edged SRGNRs compared with pristine GNRs.

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FIG. 3. The distribution of roughness height and force constant difference along the X-axis for (a) armchair and (b) zigzag edged structures.

thus give rise to the decrease in the thermal conductance of SRGNRs as shown in Figs. 2(c) and 2(d). To analyze the electron transport and thermoelectric properties, we present the electron transmission and thermal power S2 r of SRGNRs at room temperature in Figs. 4(a)–4(d), respectively. For A-SRGNRs, the electron transmission is completely collapsed and a new band gap arises

FIG. 4. The electron transmission Te ðEÞ for (a) armchair-edged and (b) zigzag-edged structures. The thermal power S2 r at room temperature for (c) armchair-edged and (d) zigzag-edged structures.

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compared with the pristine ones in Fig. 4(a). It is known that the energy gaps in armchair edged graphene nanoribbons are sensitive to the quantum confinement and edge shapes.35 That is to say, the SR greatly changes the edge shapes of A-GNRs and opens a band gap around Fermi level. Owing to the linear relation between the Seebeck coefficient and the energy gap of semiconductors,36 there is a sharp rise in thermal power in Fig. 4(c), which ensures the outstanding thermoelectric performance of A-SRGNRs. Moreover, the chemical potential corresponding to the maximum value has been moved closer to the Fermi level, which can be more easily effected for doping. In the case of Z-SRGNRs, the electron transmission still keeps the staircase-like structures above the Fermi level as shown in Fig. 4(b). With increasing energy, such staircase-like structures become even more unclear. Another important fact is that there is no new band gap around Fermi level. Consequently, as shown in Fig. 4(d), the thermal power of Z-SRGNRs is weakly affected compared to that of flat Z-GNRs. These discrepancies between A-SRGNRs and Z-SRGNRs also agree with the results about ballistic transport in edge-disordered GNRs.37 Finally, the averaged maximum ZT values and the corresponding standard deviations are plotted as a function of different geometric variations in Figs. 5(a)–5(c). Interestingly, the different edged SRGNRs present a discrepancy in growth trends with the increase in those variations. For A-SRGNRs, the growth trends of different variations are evident due to the significantly reduced thermal conductance and enhanced thermal power. As shown in Figs. 5(a) and 5(b), ZT values are sensitive to the variations H and Cl which control the roughness magnitude of SR. As H increases or Cl decreases, the SR turns out to be rougher and a higher ZT value is found. Then, we fix the roughness level of SR and investigate the influence of the central length in Fig. 5(c). It is obvious that longer A-SRGNRs tend to show better thermoelectric performance and ZT values exhibit a clear linear relationship with Nlength .

FIG. 5. The averaged figure of merit ZT for armchair and zigzag edged SRGNRs at room temperature as a function of (a) H, (b) Cl and (c) Nlength , where the banded line corresponds to the standard deviation. The dashed line at 0.05 represents the ZT value of pristine GNRs. (d) The averaged figure of merit ZT for A-SRGNRs and pristine A-GNRs at room temperature as a function of width.

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The maximum ZT value for A-SRGNRs can reach 3.7 with H ¼ 2, Cl ¼ 0:5 and Nlength ¼ 35. In contrast, the thermoelectric performance of Z-SRGNRs is not satisfactory enough. Despite the fact that the ZT values for Z-SRGNRs have slightly increased compared to pristine Z-GNRs, the influence of roughness variations H and Cl seems to be general compared with A-SRGNRs. As shown in Fig. 5(c), the ZT values for Z-SRGNRs with different lengths almost remain the same and the stable value is about 0.35. The reason can be explained by decreased thermal conductance and unchanged thermal power mentioned before. Before concluding, the width influence on the thermoelectric performance of A-SRGNRs is also explored. In Fig. 5(d), we present the ZT values of A-SRGNRs as a function of width compared with pristine A-GNRs. It is well known that A-GNRs are semimetallic (gapless) if NA equals 3n þ 2 and semiconductors if NA equals 3n or 3n þ 1.35,38 As a result of higher Seebeck coefficient of semiconductors, A-GNRs with NA ¼ 3n or 3n þ 1 exhibit better thermoelectric performance. Similar width behavior could also be found in the A-SRGNRs. In summary, the thermoelectric performance of graphene nanoribbons with surface roughness has been investigated using NEGF approach. Our results demonstrate that the surface roughness structures can significantly improve the thermoelectric efficiency of graphene nanoribbons, especially for armchair edged ones. This mainly originated from the greatly reduced thermal conductance as a result of the irregular force constant difference. In addition, the electronic properties of ASRGNRs have been significantly improved, while those for ZSRGNRs are basically unchanged. This discrepancy could be attributed to the effect of the edge shape on graphene band gaps. Using this strategy, the maximum ZT higher than 1 can be easily achieved in A-GNRs. Our findings will be of great interest in the designing and fabrication of thermoelectric devices based on graphene in practice. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11304264, 11304262, and 11574014), Scientific Research Fund of Hunan Provincial Education Department (No. 17B252), the Program for Changjiang Scholars and Innovative Research Team in University (IRT13093), the Hunan Provincial Natural Science Foundation of China (No. 2018JJ2380), and the Foundation Department of Education of Sichuan Province (No. 2017Z031). 1

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