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Effect of triangle vacancy on thermal transport in boron nitride nanoribbons
Solid State Communications 151 (2011) 460–464
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effect of triangle vacancy on thermal transport in boron nitride nanoribbons Kaike Yang, Yuanping Chen ∗ , Yuee Xie, X.L. Wei, Tao Ouyang, Jianxin Zhong Laboratory for Quantum Engineering and Micro-Nano Energy Technology and Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China
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info
Article history: Received 23 October 2010 Received in revised form 15 December 2010 Accepted 4 January 2011 by P. Sheng Available online 8 January 2011 Keywords: A. Hexagonal boron nitride nanoribbons C. Triangle vacancy D. Thermal transport
abstract Recently, triangle vacancy in hexagonal boron nitride is observed experimentally. Using nonequilibrium Green’s function method, we investigate thermal transport properties of boron nitride nanoribbons (BNNRs) with a triangle vacancy. The effect of triangle vacancy on the phonon transmission of zigzagedged BNNRs (Z-BNNRs) is different from that of armchair-edged BNNRs (A-BNNRs). The triangle vacancy induces antiresonant dips in the spectrum of Z-BNNRs. Moreover, the boron-terminated triangle vacancy causes antiresonant zero-transmission dip and the number of the zero-transmission dip increases with the geometrical size of triangle vacancy. For the A-BNNRs with triangle vacancy, except some antiresonant dips, a resonant peak is found in the transmission. The antiresonant and resonant phenomena are explained by analyzing local density of states and local thermal currents. Although the antiresonant dip and the resonant peak are both originated from quasibound states, their distributions of local thermal currents are distinct, which leads to the transport discrepancy. In addition, the thermal conductance of BNNRs decreases linearly with increasing the vacancy size. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The progress in materials growth and control techniques has fabricated the single-layer of hexagonal boron nitride (h-BN) successfully [1]. Based on the single-layer structure, boron nitride nanoribbons (BNNRs) with different edges and widths can be produced by cutting mechanically [2]. This novel nanomaterial has numerous intriguing physical properties, such as intrinsic halfmetallicity [3], strong mechanical properties, and ultraviolet laser characteristics [4,5]. Our previous study has shown that BNNRs also possess outstanding thermal transport properties [6]. The value of the thermal conductivity of BNNRs can approach 2000–3000 W/mK depending on the specific size and edge shape of the nanoribbons. The thermal conductance of zigzag-edged BNNRs (Z BNNRs) is found to be about 20% larger than that of armchair-edged BNNRs (A-BNNRs) at room temperature, indicating anisotropic thermal transport. The exceptional thermal properties establish BNNRs as a promising candidate for building nanodevices [7,8]. Recently, the triangle vacancies with different sizes are observed in the h-BN nanosheets by high-resolution transmission electron microscopy [9,10]. The edges of triangle vacancy are mostly zigzag-edged. Theoretical calculations show that the structural stabilities of triangle vacancy depend on the environmental condition of the system [11–13]. The electronic and magnetic properties of the vacancy in the h-BN have been investigated
∗
Corresponding author. Tel.: +86 0731 58292468; fax: +86 0731 58292468. E-mail addresses: [email protected] (Y. Chen), [email protected] (J. Zhong).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.01.002
[14,15]. The presence of triangle vacancy is found to be able to alter the distribution of charge density in the h-BN sheets [13]. As for the quasi-one-dimensional BNNRs, it is reported that the vacancy could induce an electron localized state and spontaneous magnetization [16,17]. This indicates the significance of the vacancy in determining the electronic and magnetic properties of the nanoribbons. Nevertheless, to the best of our knowledge, research on the vacancy on the thermal transport through BNNRs has not received deserved attention until now. Therefore, what happens when the triangle vacancy impacts on the thermal transport of BNNRs is interesting and also important in the design of thermal devices. In this paper, the thermal transport of BNNRs with triangle vacancy as shown in Fig. 1 is studied. We first investigate the effect of triangle vacancy on the thermal transport of Z -BNNR by comparing perfect and defective structures. It is found that triangle vacancy could induce antiresonant dips in the spectrum of Z BNNRs. Moreover, the boron-terminated triangle vacancy causes an antiresonant zero-transmission dip around ω = 400–450 cm−1 and the number of the zero-transmission dip increases with the geometrical size of triangle vacancy. Then we study the effect of triangle vacancy on the thermal transport of A-BNNRs. It is found that, except for some antiresonant dips, there is a resonant peak in the transmission when the vacancy size becomes large. The origination of the dip and the peak is analyzed by phonon local density of states (LDOS) and local thermal currents. The results show that, although the dip and the peak are both originated from quasibound states, their distributions of local thermal currents are distinct. Corresponding to the zero-transmission dip, loop thermal currents around the vacancy are formed in the defective Z -BNNR, whereas the currents display clear transport channel
K. Yang et al. / Solid State Communications 151 (2011) 460–464
461
investigate the thermal transport properties, we employ the stressfree boundary condition [19], and the tight-binding Hamiltonian is given by H =
−
Hα + (uL )T V LC uC + (uC )T V CR uR ,
(1)
α=L,C ,R
where Hα = 12 (˙uα )T u˙ α + 21 (uα )T K α uα represents harmonic oscillators (α = L, C , R denote the left lead, central region, and right lead, respectively). uα is a column vector consisting of all the out-of-plane displacement variables in the α region, and u˙ α is the corresponding conjugate momenta. K α is the spring constant matrix [20]. V LC = (V CL )T or V RC = (V CR )T is the coupling matrix of the left or right leads to the central region. Based on the Hamiltonian, the phonon transmission of out-ofplane mode can be calculated by using nonequilibrium Green’s function (NEGF) method [21–25]. According to the NEGF scheme, the retarded Green’s function for the BNNRs is expressed as
Gr (ω) =
(ω + i0+ )2 I − K C −
r − L
Fig. 1. (Color online) Schematic view of (a) a Z -BNNR and (b) an A-BNNR with a triangle vacancy. The widths of Z -BNNR and A-BNNR are labeled by NZ and NA , respectively. N describes the side length of triangle vacancy. NU denotes the width between the triangle vacancy and the upper-boundary of Z -BNNRs, while NL represents the width between the center of triangle vacancy and the upperboundary of A-BNNRs.
corresponding to the resonant peak in the defective A-BNNR. Finally, we discuss the influence of triangle vacancy on the thermal conductance of BNNRs. 2. Model and method The geometries of Z -BNNR and A-BNNR with a triangle vacancy are respectively shown in Fig. 1(a) and (b). The different colors represent two different types of atom. It is noted that when the blue (dark color) stands for a nitrogen atom, the inner edges of triangle vacancy are boron-terminated. Conversely, if the orange (light color) stands for a nitrogen atom, the triangle vacancy is nitrogen-terminated. Previous studies demonstrate that the boron-terminated triangle vacancy would be stable [11]. Recently, some investigations show that the nitrogen-terminated triangle vacancy is more stable than the boron-terminated vacancy [13]. Here we study the thermal transport of BNNR with boron- or nitrogen-terminated triangle vacancy. The devices consist of three parts: left lead, central region and right lead. The semi-infinite left and right leads are assumed to be in thermal equilibrium and their widths are labeled by NZ and NA for Z -BNNR and A-BNNR, respectively. The side length of triangle vacancy is represented by N as illustrated in the lower-right of Fig. 1. To describe the position of the triangle vacancy, NU is used to denote the width between the vacancy and the upper-boundary of Z -BNNRs. While NL represents the width between the center of the triangle vacancy and the upper-boundary of A-BNNRs. For phonons in the BNNRs, there are three types of vibrational modes [18]: two in-plane (xyplane) modes and one out-of-plane mode which is perpendicular to the xy-plane (z-direction). Since there is no coupling between out-of-plane mode and in-plane modes, the Hamiltonian of outof-plane and in-plane modes can be completely decomposed. Here we consider the thermal transport of the out-of-plane mode. To
−
−1 r −
,
(2)
R
where ω∑ is the vibrational frequency of phonons, I is an identity r Cβ r βC matrix. denotes the retarded self-energy for β = V gβ V thermal leads (β = L, R), and gβr is the surface Green’s function of leads which can be computed by recursive iteration technique [26,27]. After the retarded Green’s function is obtained, one can calculate the transmission coefficient T [ω] and the DOS [28]: T [ω] = Tr[Gr (ω)ΓL Ga (ω)ΓR ],
(3)
DOS(ω) = iωTr(G − G )/π , r
a
where Ga (ω) = [Gr (ω)]Ď and Γβ = i
(4)
∑
r
β
−
∑a β
describes
the interaction between the left or right leads and the central region. Meanwhile, the LDOS at site j:ρj (ω) = iω(Grjj − Gajj )/π and the local thermal currents between two sites j and k [29,30]: Jj→k (ω) = Im[Kjkα Gnkj (ω)] can also be computed, where Gn (ω) = Gr (ω)[ΓL f (TL ) + ΓR f (TR )]Ga (ω) and Kjkα (Grjj or Gnkj ) is the corresponding matrix element. Utilizing the transmission coefficient, the thermal conductance σ (T ) can be obtained by
σ (T ) =
h¯ 2π
∞
∫ 0
∂ f (ω) T [ω]ω dω, ∂T
(5)
here f = {exp[h¯ ω/(kB T )] − 1}−1 is the Bose–Einstein distribution function for heat carriers. 3. Results and discussion Fig. 2(a) shows the transmission coefficient as a function of phonon frequency for Z -BNNR with smallest triangle vacancy, i.e., one atomic defect. The solid and dotted lines separately represent the transmission of Z -BNNR with boron-terminated and nitrogen-terminated vacancy. As a comparison, the transmission coefficient for perfect Z -BNNR is also plotted (dashed line). One can find that in the low frequency region (0 < ω < 100 cm−1 ), the transmission of defective Z -BNNR is almost coincided with that of perfect nanoribbon and exhibits a clear stepwise structure. This is because long wavelength phonons are nearly not scattered by the defect. However, since short wavelength phonons will be scattered in the presence of vacancy defect, the quantum plateaus in the high frequency are destroyed. The transmission curve of defective Z BNNR is below than that of perfect nanoribbon and the phonon transmission is suppressed. Moreover some dips at specific frequencies are shown in the transmission spectrum for both boronterminated and nitrogen-terminated triangular vacancy Z -BNNR. Particularly, it is found that at ω = 422 cm−1 the transmission
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Fig. 3. (Color online) Transmission coefficients T [ω] (solid line) and DOS (dashed line) as a function of phonon frequency ω for Z -BNNRs with triangle vacancy of different size: (a) N = 3 and (b) N = 13. Other parameters are the same as in Fig. 2.
Fig. 2. (Color online) (a) Transmission coefficient T [ω] as a function of phonon frequency ω for Z -BNNRs. The solid, dotted and dashed lines separately represent the transmission of Z -BNNR with smallest boron-terminated triangle vacancy, smallest nitrogen-terminated triangle vacancy and perfect Z -BNNR. (b) DOS for Z BNNR with smallest boron-terminated triangle vacancy. The inset in (a) shows LDOS (circles) and local thermal currents (arrows) corresponding to the antiresonant zero-transmission dip at ω = 422 cm−1 . Other parameters of the Z -BNNRs are chosen to be NZ = 14 and NU = 5.
reduces to zero in Z -BNNR with boron-terminated triangle vacancy (see the solid line). This zero-transmission dip indicates that the phonon transmission is totally blocked due to the triangle vacancy. To further show the influence of defect on the phonon transport properties, in Fig. 2(b) we depict the DOS profile for Z BNNR with smallest boron-terminated triangle vacancy. It is found many peaks induced by Van Hove singularity corresponding to the positions of transmission steps of perfect nanoribbon. Except these peaks, there is a sharp DOS peak corresponding to the zerotransmission dip. In the inset of Fig. 2(a) we plot the phonon LDOS corresponding to this dip. One can see that the state is tightly localized around the triangle vacancy of the nanoribbon, i.e., it is a quasibound state in which phonons are mostly confined in a limited region. This indicates that the antiresonant zero-transmission dip is induced by the quasibound state. The local thermal currents at ω = 422 cm−1 are also calculated (see the inset in Fig. 2(a)). It is shown that the thermal currents have a large magnitude, encircle the vacancy and form loop thermal current distributions. There is no thermal current flowing through the ribbon. Consequently, the antiresonant zero-transmission dip appears in the spectrum of defective Z -BNNR. In order to investigate the dependence of phononic antiresonant transmission on the size of triangle vacancy, in Fig. 3 we show the transmission coefficients (solid line) and the DOS profiles (dashed line) for Z -BNNRs with different boron-terminated triangle vacancy size N. In contrast to the case of Z -BNNR with a boron-terminated atomic vacancy, one can see from Fig. 3 that the number of the antiresonant zero-transmission dip increases with the geometrical size of the triangle vacancy. As the vacancy size N = 3, there are three antiresonant dips in the frequency region 410 cm−1 < ω < 450 cm−1 (see Fig. 3(a)). Moreover, each dip corresponds to a DOS peak. The calculations of LDOS indicate that the states are mostly localized in the structure (not shown). This suggests that the increase of the number of the antiresonant dip is due to the increase of quasibound state. When the vacancy becomes large, in Fig. 3(b) it is found that there are five antiresonant dips
Fig. 4. (Color online) (a) Transmission coefficient T [ω] as a function of phonon frequency ω for A-BNNRs. The solid, dotted and dashed lines represent the transmission of A-BNNR with smallest boron-terminated triangle vacancy, smallest nitrogen-terminated triangle vacancy and perfect A-BNNR, respectively. (b) Transmission coefficient and (c) DOS for A-BNNR with larger boron-terminated triangle vacancy (N = 5). The inset in (b) shows LDOS (circles) and local thermal currents (arrows) corresponding to the resonant peak at ω = 396 cm−1 . Other parameters of the A-BNNRs are NA = 19 and NL = 10.
corresponding to five DOS peaks which imply that there would be five quasibound states in the defective Z -BNNR. From these results, one can conclude that the phononic antiresonant transmission in the Z -BNNRs is very sensitive to the structural size of the triangle vacancy. In Fig. 4(a) we show the transmission coefficient versus phonon frequency for A-BNNR with smallest triangle vacancy. The solid and dotted lines stand for the transmission curves of A-BNNR with boron- and nitrogen-terminated vacancy, respectively. The transmission coefficient of perfect nanoribbon is also plotted for the convenience of comparison (dashed line). One can see that the influence of the vacancy on the phonon transmission of A-BNNR
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thermal transport. However, in the presence of triangle vacancy, the transport channels will be destroyed by the defect. Phonons in the destructive channels are blocked and can’t transmit through the ribbons. As the vacancy size N = 2k − 1 (k is a positive integer), it is obvious that there are k channels destroyed for the Z -BNNRs. With respect to the A-BNNRs, because of the effective width (i.e., the lead width subtract the vacancy size) decreasing to NA − (2k − 1), there are also k channels destroyed. Therefore, the thermal conductance shows linear attenuation due to the gradually reducing phonon transport channels. These results imply that the thermal conductance of BNNRs can be significantly modulated by manipulating the geometrical size of triangle vacancy. 4. Conclusions Fig. 5. (Color online) Thermal conductance σ versus triangle vacancy size N for (a) Z -BNNRs and (b) A-BNNRs with various widths at room temperature (300 K), the solid and hollow symbols represent the thermal conductance of BNNRs with boron- and nitrogen-terminated triangle vacancy, respectively. Other parameters are NU = 5 in the Z -BNNRs and NL = 10 in the A-BNNRs.
is analogous to that of Z -BNNR by comparing Figs. 4(a) and 2(a). In the low frequency region, quantum plateaus are found. At high frequency the transmission shows some oscillations. Nevertheless, it is noticed that there is a narrow phonon band-gap located around ω = 400–430 cm−1 in the transmission of A-BNNRs with boronor nitrogen-terminated vacancy because of the armchair edges. In Fig. 4(b) we depict the transmission coefficient for A-BNNR including larger boron-terminated triangle vacancy. One can see that the transmission is sharply suppressed and some dips induced by antiresonance at specific frequencies are shown, such as ω = 443 and 451 cm−1 . Except these dips, it is interesting to find that there is a peak at ω = 396 cm−1 in the transmission of defective A-BNNR, which can also be found in the transmission of A-BNNR with larger nitrogen-terminated triangle vacancy (not shown). This suggests that there exists phononic resonant transport in the structure. Similar to the case of antiresonant dip, this resonant peak also corresponds to a high DOS peak as shown in Fig. 4(c). The inset of Fig. 4(b) depicts the phonon LDOS corresponding to the resonant peak. It can be seen that the state is strongly confined around the vacancy defect. This implies that the resonant peak in the transmission of defective A-BNNR is also originated from a quasibound state. In the inset of Fig. 4(b), the local thermal currents at ω = 396 cm−1 are calculated. It is found that the thermal currents in the center of the nanoribbon are much bigger and form a clear channel along the transport direction, which is distinct from the current distributions of antiresonant zero-transmission dip in the defective Z -BNNR. In this case the thermal currents could flow through the ribbon, and thus the resonant peak is observed. Moreover, our calculations indicate that the number of the resonant peak increases with the triangle vacancy size (not shown). The effect of triangle vacancy on the thermal conductance of BNNRs is discussed in Fig. 5. Fig. 5(a) and (b) respectively show the thermal conductance as a function of triangle vacancy size N for Z -BNNRs and A-BNNRs. The solid and hollow symbols represent the results of BNNRs with boron-terminated and nitrogen-terminated vacancy. It can be seen that the all thermal conductance curves decrease linearly with the increase of N, and the thermal conductance of wide BNNRs is larger than that of narrow nanoribbons for a fixed N. These phenomena can be qualitatively explained by the variation of transport channel. The number of phonon channels in the perfect Z -BNNRs equals the ribbon width NZ . While the number of phonon channels in the perfect A-BNNRs equals (NA − 1)/2 (NA is odd). Thus, the wider the ribbons, the more phonon channels that contribute to the
In conclusion, we have studied the thermal transport properties of BNNRs with a triangle vacancy by using the NEGF approach. It is shown that the effect of triangle vacancy on the phonon transmission of Z -BNNRs is different from that of A-BNNRs. The triangle vacancy induces antiresonant dips in the spectrum of Z BNNRs. Moreover, the boron-terminated triangle vacancy causes antiresonant zero-transmission dip and the number of the zerotransmission dip increases with the geometrical size of the triangle vacancy. As for the A-BNNRs, except some antiresonant dips, an interesting resonant peak is found in the transmission due to the triangle vacancy. The antiresonant and resonant phenomena are explained by analyzing phonon LDOS and local thermal currents. Although the antiresonant dip and the resonant peak are both originated from quasibound states, their distributions of local thermal currents are distinct, which leads to the transport discrepancy. In addition, the thermal conductance of BNNRs decreases linearly with increasing the vacancy size. These results indicate that the thermal transport of BNNRs can be effectively tuned through manipulating the triangle vacancy. Our findings could be useful for building BNNR-based thermoelectric devices. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 51006086, 11074213 and 10774127), the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708068), the Open Fund based on innovation platform of Hunan colleges and universities (No. 09K034), and the Hunan Provincial Innovation Foundation for Postgraduate (No. CX2010B253). References [1] K.S. Novoselov, D. Jiang, F. Schedin, T.J. Booth, V.V. Khotkevich, S.V. Morozov, A.K. Geim, Proc. Natl. Acad. Sci. 102 (2005) 10451. [2] M.Y. Han, B. Özyilmaz, Y. Zhang, P. Kim, Phys. Rev. Lett. 98 (2007) 206805. [3] F. Zheng, G. Zhou, Z. Liu, J. Wu, W. Duan, B.L. Gu, S.B. Zhang, Phys. Rev. B 78 (2008) 205415. [4] Z.G. Chen, J. Zou, G. Liu, F. Li, Y. Wang, L. Wang, X.L. Yuan, T. Sekiguchi, H.M. Cheng, G.Q. Lu, ACS Nano 2 (2008) 2183. [5] K. Watanabe, T. Taniguchi, H. Kanda, Nat. Mater. 3 (2004) 404. [6] T. Ouyang, Y.P. Chen, Y.E. Xie, K.K. Yang, Z.G. Bao, J.X. Zhong, Nanotechnology 21 (2010) 245701. [7] B. Li, L. Wang, G. Casati, Phys. Rev. Lett. 93 (2004) 184301. [8] L. Wang, B. Li, Phys. Rev. Lett. 99 (2007) 177208. [9] J.C. Meyer, A. Chuvilin, G.A. Siller, J. Biskupek, U. Kaiser, Nano Lett. 9 (2009) 2683. [10] C. Jin, F. Lin, K. Suenaga, S. Iijima, Phys. Rev. Lett. 102 (2009) 195505. [11] S. Azevedo, J.R. Kaschny, C.M.C. de Castilho, F. de Brito Mota, Nanotechnology 18 (2007) 495707. [12] S. Okada, Phys. Rev. B 80 (2009) 161404. [13] L.C. Yin, H.M. Cheng, R. Saito, Phys. Rev. B 81 (2010) 153407. [14] M.S. Si, D.S. Xue, Phys. Rev. B 75 (2007) 193409. [15] A. Du, Y. Chen, Z. Zhu, R. Amal, G.Q. Lu, S.C. Smith, J. Am. Chem. Soc. 131 (2009) 17354. [16] S. Tang, Z. Cao, Comput. Mater. Sci. 48 (2010) 648. [17] M. Topsakal, E. Aktürk, S. Ciraci, Phys. Rev. B 79 (2009) 115442.
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[18] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotube, Imperial College, London, 1998. [19] K.F. Graff, Wave Motion in Elastic Solids, Dover, New York, 1991. [20] Y. Xiao, X.H. Yan, J.X. Cao, J.W. Ding, Y.L. Mao, J. Xiang, Phys. Rev. B 69 (2004) 205415. [21] J.S. Wang, J. Wang, N. Zeng, Phys. Rev. B 74 (2006) 033408. [22] N. Mingo, Phys. Rev. B 74 (2006) 125402. [23] T. Yamamoto, K. Watanabe, Phys. Rev. Lett. 96 (2006) 255503. [24] T. Ouyang, Y.P. Chen, K.K. Yang, J.X. Zhong, Europhys. Lett. 88 (2009) 28002.
[25] K.K. Yang, Y.P. Chen, Y.E. Xie, T. Ouyang, J.X. Zhong, Europhys. Lett. 91 (2010) 46006. [26] M.P.L. Sancho, J.M.L. Sancho, J. Rubio, J. Phys. F: Met. Phys. 15 (1985) 851. [27] J.S. Wang, J. Wang, J.T. Lü, Eur. J. Phys. B 62 (2008) 381. [28] W. Zhang, T.S. Fisher, N. Mingo, Numer. Heat Transfer B 51 (2007) 333. [29] Y. Zhang, J.P. Hu, B.A. Bernevig, X.R. Wang, X.C. Xie, W.M. Liu, Phys. Rev. B 78 (2008) 155413. [30] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University, Cambridge, 1995.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effect of triangle vacancy on thermal transport in boron nitride nanoribbons Kaike Yang, Yuanping Chen ∗ , Yuee Xie, X.L. Wei, Tao Ouyang, Jianxin Zhong Laboratory for Quantum Engineering and Micro-Nano Energy Technology and Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China
article
info
Article history: Received 23 October 2010 Received in revised form 15 December 2010 Accepted 4 January 2011 by P. Sheng Available online 8 January 2011 Keywords: A. Hexagonal boron nitride nanoribbons C. Triangle vacancy D. Thermal transport
abstract Recently, triangle vacancy in hexagonal boron nitride is observed experimentally. Using nonequilibrium Green’s function method, we investigate thermal transport properties of boron nitride nanoribbons (BNNRs) with a triangle vacancy. The effect of triangle vacancy on the phonon transmission of zigzagedged BNNRs (Z-BNNRs) is different from that of armchair-edged BNNRs (A-BNNRs). The triangle vacancy induces antiresonant dips in the spectrum of Z-BNNRs. Moreover, the boron-terminated triangle vacancy causes antiresonant zero-transmission dip and the number of the zero-transmission dip increases with the geometrical size of triangle vacancy. For the A-BNNRs with triangle vacancy, except some antiresonant dips, a resonant peak is found in the transmission. The antiresonant and resonant phenomena are explained by analyzing local density of states and local thermal currents. Although the antiresonant dip and the resonant peak are both originated from quasibound states, their distributions of local thermal currents are distinct, which leads to the transport discrepancy. In addition, the thermal conductance of BNNRs decreases linearly with increasing the vacancy size. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The progress in materials growth and control techniques has fabricated the single-layer of hexagonal boron nitride (h-BN) successfully [1]. Based on the single-layer structure, boron nitride nanoribbons (BNNRs) with different edges and widths can be produced by cutting mechanically [2]. This novel nanomaterial has numerous intriguing physical properties, such as intrinsic halfmetallicity [3], strong mechanical properties, and ultraviolet laser characteristics [4,5]. Our previous study has shown that BNNRs also possess outstanding thermal transport properties [6]. The value of the thermal conductivity of BNNRs can approach 2000–3000 W/mK depending on the specific size and edge shape of the nanoribbons. The thermal conductance of zigzag-edged BNNRs (Z BNNRs) is found to be about 20% larger than that of armchair-edged BNNRs (A-BNNRs) at room temperature, indicating anisotropic thermal transport. The exceptional thermal properties establish BNNRs as a promising candidate for building nanodevices [7,8]. Recently, the triangle vacancies with different sizes are observed in the h-BN nanosheets by high-resolution transmission electron microscopy [9,10]. The edges of triangle vacancy are mostly zigzag-edged. Theoretical calculations show that the structural stabilities of triangle vacancy depend on the environmental condition of the system [11–13]. The electronic and magnetic properties of the vacancy in the h-BN have been investigated
∗
Corresponding author. Tel.: +86 0731 58292468; fax: +86 0731 58292468. E-mail addresses: [email protected] (Y. Chen), [email protected] (J. Zhong).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.01.002
[14,15]. The presence of triangle vacancy is found to be able to alter the distribution of charge density in the h-BN sheets [13]. As for the quasi-one-dimensional BNNRs, it is reported that the vacancy could induce an electron localized state and spontaneous magnetization [16,17]. This indicates the significance of the vacancy in determining the electronic and magnetic properties of the nanoribbons. Nevertheless, to the best of our knowledge, research on the vacancy on the thermal transport through BNNRs has not received deserved attention until now. Therefore, what happens when the triangle vacancy impacts on the thermal transport of BNNRs is interesting and also important in the design of thermal devices. In this paper, the thermal transport of BNNRs with triangle vacancy as shown in Fig. 1 is studied. We first investigate the effect of triangle vacancy on the thermal transport of Z -BNNR by comparing perfect and defective structures. It is found that triangle vacancy could induce antiresonant dips in the spectrum of Z BNNRs. Moreover, the boron-terminated triangle vacancy causes an antiresonant zero-transmission dip around ω = 400–450 cm−1 and the number of the zero-transmission dip increases with the geometrical size of triangle vacancy. Then we study the effect of triangle vacancy on the thermal transport of A-BNNRs. It is found that, except for some antiresonant dips, there is a resonant peak in the transmission when the vacancy size becomes large. The origination of the dip and the peak is analyzed by phonon local density of states (LDOS) and local thermal currents. The results show that, although the dip and the peak are both originated from quasibound states, their distributions of local thermal currents are distinct. Corresponding to the zero-transmission dip, loop thermal currents around the vacancy are formed in the defective Z -BNNR, whereas the currents display clear transport channel
K. Yang et al. / Solid State Communications 151 (2011) 460–464
461
investigate the thermal transport properties, we employ the stressfree boundary condition [19], and the tight-binding Hamiltonian is given by H =
−
Hα + (uL )T V LC uC + (uC )T V CR uR ,
(1)
α=L,C ,R
where Hα = 12 (˙uα )T u˙ α + 21 (uα )T K α uα represents harmonic oscillators (α = L, C , R denote the left lead, central region, and right lead, respectively). uα is a column vector consisting of all the out-of-plane displacement variables in the α region, and u˙ α is the corresponding conjugate momenta. K α is the spring constant matrix [20]. V LC = (V CL )T or V RC = (V CR )T is the coupling matrix of the left or right leads to the central region. Based on the Hamiltonian, the phonon transmission of out-ofplane mode can be calculated by using nonequilibrium Green’s function (NEGF) method [21–25]. According to the NEGF scheme, the retarded Green’s function for the BNNRs is expressed as
Gr (ω) =
(ω + i0+ )2 I − K C −
r − L
Fig. 1. (Color online) Schematic view of (a) a Z -BNNR and (b) an A-BNNR with a triangle vacancy. The widths of Z -BNNR and A-BNNR are labeled by NZ and NA , respectively. N describes the side length of triangle vacancy. NU denotes the width between the triangle vacancy and the upper-boundary of Z -BNNRs, while NL represents the width between the center of triangle vacancy and the upperboundary of A-BNNRs.
corresponding to the resonant peak in the defective A-BNNR. Finally, we discuss the influence of triangle vacancy on the thermal conductance of BNNRs. 2. Model and method The geometries of Z -BNNR and A-BNNR with a triangle vacancy are respectively shown in Fig. 1(a) and (b). The different colors represent two different types of atom. It is noted that when the blue (dark color) stands for a nitrogen atom, the inner edges of triangle vacancy are boron-terminated. Conversely, if the orange (light color) stands for a nitrogen atom, the triangle vacancy is nitrogen-terminated. Previous studies demonstrate that the boron-terminated triangle vacancy would be stable [11]. Recently, some investigations show that the nitrogen-terminated triangle vacancy is more stable than the boron-terminated vacancy [13]. Here we study the thermal transport of BNNR with boron- or nitrogen-terminated triangle vacancy. The devices consist of three parts: left lead, central region and right lead. The semi-infinite left and right leads are assumed to be in thermal equilibrium and their widths are labeled by NZ and NA for Z -BNNR and A-BNNR, respectively. The side length of triangle vacancy is represented by N as illustrated in the lower-right of Fig. 1. To describe the position of the triangle vacancy, NU is used to denote the width between the vacancy and the upper-boundary of Z -BNNRs. While NL represents the width between the center of the triangle vacancy and the upper-boundary of A-BNNRs. For phonons in the BNNRs, there are three types of vibrational modes [18]: two in-plane (xyplane) modes and one out-of-plane mode which is perpendicular to the xy-plane (z-direction). Since there is no coupling between out-of-plane mode and in-plane modes, the Hamiltonian of outof-plane and in-plane modes can be completely decomposed. Here we consider the thermal transport of the out-of-plane mode. To
−
−1 r −
,
(2)
R
where ω∑ is the vibrational frequency of phonons, I is an identity r Cβ r βC matrix. denotes the retarded self-energy for β = V gβ V thermal leads (β = L, R), and gβr is the surface Green’s function of leads which can be computed by recursive iteration technique [26,27]. After the retarded Green’s function is obtained, one can calculate the transmission coefficient T [ω] and the DOS [28]: T [ω] = Tr[Gr (ω)ΓL Ga (ω)ΓR ],
(3)
DOS(ω) = iωTr(G − G )/π , r
a
where Ga (ω) = [Gr (ω)]Ď and Γβ = i
(4)
∑
r
β
−
∑a β
describes
the interaction between the left or right leads and the central region. Meanwhile, the LDOS at site j:ρj (ω) = iω(Grjj − Gajj )/π and the local thermal currents between two sites j and k [29,30]: Jj→k (ω) = Im[Kjkα Gnkj (ω)] can also be computed, where Gn (ω) = Gr (ω)[ΓL f (TL ) + ΓR f (TR )]Ga (ω) and Kjkα (Grjj or Gnkj ) is the corresponding matrix element. Utilizing the transmission coefficient, the thermal conductance σ (T ) can be obtained by
σ (T ) =
h¯ 2π
∞
∫ 0
∂ f (ω) T [ω]ω dω, ∂T
(5)
here f = {exp[h¯ ω/(kB T )] − 1}−1 is the Bose–Einstein distribution function for heat carriers. 3. Results and discussion Fig. 2(a) shows the transmission coefficient as a function of phonon frequency for Z -BNNR with smallest triangle vacancy, i.e., one atomic defect. The solid and dotted lines separately represent the transmission of Z -BNNR with boron-terminated and nitrogen-terminated vacancy. As a comparison, the transmission coefficient for perfect Z -BNNR is also plotted (dashed line). One can find that in the low frequency region (0 < ω < 100 cm−1 ), the transmission of defective Z -BNNR is almost coincided with that of perfect nanoribbon and exhibits a clear stepwise structure. This is because long wavelength phonons are nearly not scattered by the defect. However, since short wavelength phonons will be scattered in the presence of vacancy defect, the quantum plateaus in the high frequency are destroyed. The transmission curve of defective Z BNNR is below than that of perfect nanoribbon and the phonon transmission is suppressed. Moreover some dips at specific frequencies are shown in the transmission spectrum for both boronterminated and nitrogen-terminated triangular vacancy Z -BNNR. Particularly, it is found that at ω = 422 cm−1 the transmission
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Fig. 3. (Color online) Transmission coefficients T [ω] (solid line) and DOS (dashed line) as a function of phonon frequency ω for Z -BNNRs with triangle vacancy of different size: (a) N = 3 and (b) N = 13. Other parameters are the same as in Fig. 2.
Fig. 2. (Color online) (a) Transmission coefficient T [ω] as a function of phonon frequency ω for Z -BNNRs. The solid, dotted and dashed lines separately represent the transmission of Z -BNNR with smallest boron-terminated triangle vacancy, smallest nitrogen-terminated triangle vacancy and perfect Z -BNNR. (b) DOS for Z BNNR with smallest boron-terminated triangle vacancy. The inset in (a) shows LDOS (circles) and local thermal currents (arrows) corresponding to the antiresonant zero-transmission dip at ω = 422 cm−1 . Other parameters of the Z -BNNRs are chosen to be NZ = 14 and NU = 5.
reduces to zero in Z -BNNR with boron-terminated triangle vacancy (see the solid line). This zero-transmission dip indicates that the phonon transmission is totally blocked due to the triangle vacancy. To further show the influence of defect on the phonon transport properties, in Fig. 2(b) we depict the DOS profile for Z BNNR with smallest boron-terminated triangle vacancy. It is found many peaks induced by Van Hove singularity corresponding to the positions of transmission steps of perfect nanoribbon. Except these peaks, there is a sharp DOS peak corresponding to the zerotransmission dip. In the inset of Fig. 2(a) we plot the phonon LDOS corresponding to this dip. One can see that the state is tightly localized around the triangle vacancy of the nanoribbon, i.e., it is a quasibound state in which phonons are mostly confined in a limited region. This indicates that the antiresonant zero-transmission dip is induced by the quasibound state. The local thermal currents at ω = 422 cm−1 are also calculated (see the inset in Fig. 2(a)). It is shown that the thermal currents have a large magnitude, encircle the vacancy and form loop thermal current distributions. There is no thermal current flowing through the ribbon. Consequently, the antiresonant zero-transmission dip appears in the spectrum of defective Z -BNNR. In order to investigate the dependence of phononic antiresonant transmission on the size of triangle vacancy, in Fig. 3 we show the transmission coefficients (solid line) and the DOS profiles (dashed line) for Z -BNNRs with different boron-terminated triangle vacancy size N. In contrast to the case of Z -BNNR with a boron-terminated atomic vacancy, one can see from Fig. 3 that the number of the antiresonant zero-transmission dip increases with the geometrical size of the triangle vacancy. As the vacancy size N = 3, there are three antiresonant dips in the frequency region 410 cm−1 < ω < 450 cm−1 (see Fig. 3(a)). Moreover, each dip corresponds to a DOS peak. The calculations of LDOS indicate that the states are mostly localized in the structure (not shown). This suggests that the increase of the number of the antiresonant dip is due to the increase of quasibound state. When the vacancy becomes large, in Fig. 3(b) it is found that there are five antiresonant dips
Fig. 4. (Color online) (a) Transmission coefficient T [ω] as a function of phonon frequency ω for A-BNNRs. The solid, dotted and dashed lines represent the transmission of A-BNNR with smallest boron-terminated triangle vacancy, smallest nitrogen-terminated triangle vacancy and perfect A-BNNR, respectively. (b) Transmission coefficient and (c) DOS for A-BNNR with larger boron-terminated triangle vacancy (N = 5). The inset in (b) shows LDOS (circles) and local thermal currents (arrows) corresponding to the resonant peak at ω = 396 cm−1 . Other parameters of the A-BNNRs are NA = 19 and NL = 10.
corresponding to five DOS peaks which imply that there would be five quasibound states in the defective Z -BNNR. From these results, one can conclude that the phononic antiresonant transmission in the Z -BNNRs is very sensitive to the structural size of the triangle vacancy. In Fig. 4(a) we show the transmission coefficient versus phonon frequency for A-BNNR with smallest triangle vacancy. The solid and dotted lines stand for the transmission curves of A-BNNR with boron- and nitrogen-terminated vacancy, respectively. The transmission coefficient of perfect nanoribbon is also plotted for the convenience of comparison (dashed line). One can see that the influence of the vacancy on the phonon transmission of A-BNNR
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thermal transport. However, in the presence of triangle vacancy, the transport channels will be destroyed by the defect. Phonons in the destructive channels are blocked and can’t transmit through the ribbons. As the vacancy size N = 2k − 1 (k is a positive integer), it is obvious that there are k channels destroyed for the Z -BNNRs. With respect to the A-BNNRs, because of the effective width (i.e., the lead width subtract the vacancy size) decreasing to NA − (2k − 1), there are also k channels destroyed. Therefore, the thermal conductance shows linear attenuation due to the gradually reducing phonon transport channels. These results imply that the thermal conductance of BNNRs can be significantly modulated by manipulating the geometrical size of triangle vacancy. 4. Conclusions Fig. 5. (Color online) Thermal conductance σ versus triangle vacancy size N for (a) Z -BNNRs and (b) A-BNNRs with various widths at room temperature (300 K), the solid and hollow symbols represent the thermal conductance of BNNRs with boron- and nitrogen-terminated triangle vacancy, respectively. Other parameters are NU = 5 in the Z -BNNRs and NL = 10 in the A-BNNRs.
is analogous to that of Z -BNNR by comparing Figs. 4(a) and 2(a). In the low frequency region, quantum plateaus are found. At high frequency the transmission shows some oscillations. Nevertheless, it is noticed that there is a narrow phonon band-gap located around ω = 400–430 cm−1 in the transmission of A-BNNRs with boronor nitrogen-terminated vacancy because of the armchair edges. In Fig. 4(b) we depict the transmission coefficient for A-BNNR including larger boron-terminated triangle vacancy. One can see that the transmission is sharply suppressed and some dips induced by antiresonance at specific frequencies are shown, such as ω = 443 and 451 cm−1 . Except these dips, it is interesting to find that there is a peak at ω = 396 cm−1 in the transmission of defective A-BNNR, which can also be found in the transmission of A-BNNR with larger nitrogen-terminated triangle vacancy (not shown). This suggests that there exists phononic resonant transport in the structure. Similar to the case of antiresonant dip, this resonant peak also corresponds to a high DOS peak as shown in Fig. 4(c). The inset of Fig. 4(b) depicts the phonon LDOS corresponding to the resonant peak. It can be seen that the state is strongly confined around the vacancy defect. This implies that the resonant peak in the transmission of defective A-BNNR is also originated from a quasibound state. In the inset of Fig. 4(b), the local thermal currents at ω = 396 cm−1 are calculated. It is found that the thermal currents in the center of the nanoribbon are much bigger and form a clear channel along the transport direction, which is distinct from the current distributions of antiresonant zero-transmission dip in the defective Z -BNNR. In this case the thermal currents could flow through the ribbon, and thus the resonant peak is observed. Moreover, our calculations indicate that the number of the resonant peak increases with the triangle vacancy size (not shown). The effect of triangle vacancy on the thermal conductance of BNNRs is discussed in Fig. 5. Fig. 5(a) and (b) respectively show the thermal conductance as a function of triangle vacancy size N for Z -BNNRs and A-BNNRs. The solid and hollow symbols represent the results of BNNRs with boron-terminated and nitrogen-terminated vacancy. It can be seen that the all thermal conductance curves decrease linearly with the increase of N, and the thermal conductance of wide BNNRs is larger than that of narrow nanoribbons for a fixed N. These phenomena can be qualitatively explained by the variation of transport channel. The number of phonon channels in the perfect Z -BNNRs equals the ribbon width NZ . While the number of phonon channels in the perfect A-BNNRs equals (NA − 1)/2 (NA is odd). Thus, the wider the ribbons, the more phonon channels that contribute to the
In conclusion, we have studied the thermal transport properties of BNNRs with a triangle vacancy by using the NEGF approach. It is shown that the effect of triangle vacancy on the phonon transmission of Z -BNNRs is different from that of A-BNNRs. The triangle vacancy induces antiresonant dips in the spectrum of Z BNNRs. Moreover, the boron-terminated triangle vacancy causes antiresonant zero-transmission dip and the number of the zerotransmission dip increases with the geometrical size of the triangle vacancy. As for the A-BNNRs, except some antiresonant dips, an interesting resonant peak is found in the transmission due to the triangle vacancy. The antiresonant and resonant phenomena are explained by analyzing phonon LDOS and local thermal currents. Although the antiresonant dip and the resonant peak are both originated from quasibound states, their distributions of local thermal currents are distinct, which leads to the transport discrepancy. In addition, the thermal conductance of BNNRs decreases linearly with increasing the vacancy size. These results indicate that the thermal transport of BNNRs can be effectively tuned through manipulating the triangle vacancy. Our findings could be useful for building BNNR-based thermoelectric devices. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 51006086, 11074213 and 10774127), the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708068), the Open Fund based on innovation platform of Hunan colleges and universities (No. 09K034), and the Hunan Provincial Innovation Foundation for Postgraduate (No. CX2010B253). References [1] K.S. Novoselov, D. Jiang, F. Schedin, T.J. Booth, V.V. Khotkevich, S.V. Morozov, A.K. Geim, Proc. Natl. Acad. Sci. 102 (2005) 10451. [2] M.Y. Han, B. Özyilmaz, Y. Zhang, P. Kim, Phys. Rev. Lett. 98 (2007) 206805. [3] F. Zheng, G. Zhou, Z. Liu, J. Wu, W. Duan, B.L. Gu, S.B. Zhang, Phys. Rev. B 78 (2008) 205415. [4] Z.G. Chen, J. Zou, G. Liu, F. Li, Y. Wang, L. Wang, X.L. Yuan, T. Sekiguchi, H.M. Cheng, G.Q. Lu, ACS Nano 2 (2008) 2183. [5] K. Watanabe, T. Taniguchi, H. Kanda, Nat. Mater. 3 (2004) 404. [6] T. Ouyang, Y.P. Chen, Y.E. Xie, K.K. Yang, Z.G. Bao, J.X. Zhong, Nanotechnology 21 (2010) 245701. [7] B. Li, L. Wang, G. Casati, Phys. Rev. Lett. 93 (2004) 184301. [8] L. Wang, B. Li, Phys. Rev. Lett. 99 (2007) 177208. [9] J.C. Meyer, A. Chuvilin, G.A. Siller, J. Biskupek, U. Kaiser, Nano Lett. 9 (2009) 2683. [10] C. Jin, F. Lin, K. Suenaga, S. Iijima, Phys. Rev. Lett. 102 (2009) 195505. [11] S. Azevedo, J.R. Kaschny, C.M.C. de Castilho, F. de Brito Mota, Nanotechnology 18 (2007) 495707. [12] S. Okada, Phys. Rev. B 80 (2009) 161404. [13] L.C. Yin, H.M. Cheng, R. Saito, Phys. Rev. B 81 (2010) 153407. [14] M.S. Si, D.S. Xue, Phys. Rev. B 75 (2007) 193409. [15] A. Du, Y. Chen, Z. Zhu, R. Amal, G.Q. Lu, S.C. Smith, J. Am. Chem. Soc. 131 (2009) 17354. [16] S. Tang, Z. Cao, Comput. Mater. Sci. 48 (2010) 648. [17] M. Topsakal, E. Aktürk, S. Ciraci, Phys. Rev. B 79 (2009) 115442.
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