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Enhanced thermoelectric performance of defected silicene nanoribbons
Solid State Communications 227 (2016) 1–8
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Enhanced thermoelectric performance of defected silicene nanoribbons W. Zhao a, Z.X. Guo b, Y. Zhang b, J.W. Ding b,n, X.J. Zheng a,n a b
School of Mechanical Engineering, Xiangtan University, Xiangtan 411105, China Department of Physics and Institute for Nanophysics and Rare-earth Luminescence, Xiangtan University, Xiangtan 411105, China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 November 2015 Accepted 16 November 2015 Communicated by: Prof. A.H. MacDonald
Based on non-equilibrium Green's function method, we investigate the thermoelectric performance for both zigzag (ZSiNRs) and armchair (ASiNRs) silicene nanoribbons with central or edge defects. For perfect silicene nanoribbons (SiNRs), it is shown that with its width increasing, the maximum of ZT values (ZTM) decreases monotonously while the phononic thermal conductance increases linearly. For various types of edges and defects, with increasing defect numbers in longitudinal direction, ZTM increases monotonously while the phononic thermal conductance decreases. Comparing with ZSiNRs, defected ASiNRs possess higher thermoelectric performance due to higher Seebeck coefficient and lower thermal conductance. In particular, about 2.5 times enhancement to ZT values is obtained in ASiNRs with edge defects. Our theoretical simulations indicate that by controlling the type and number of defects, ZT values of SiNRs could be enhanced greatly which suggests their very appealing thermoelectric applications. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Thermoelectric materials which can convert dissipated heat into electric energy at nanoscale have attracted increasing attention from both theoretical and experimental research due to their potential technological applications [1–3]. The performance of thermoelectric materials is measured by the figure of merit ZT¼S2GT/(κe þ κp), where S, G, T, κe and κp represent the Seebeck coefficient, electronic conductance, absolute temperature, electronic and phononic thermal conductance, respectively. Explicitly, an optimal thermoelectric material should possess high Seebeck coefficient, high electronic conductance, and low thermal conductance. Both experimental and theoretical research have indicated that nanoscale materials could exhibit much higher ZT values compared with those of bulk materials, which lead to a very important prospect of thermoelectric applications. Experimentally, in particular, it was found that the thermal conductivity of Si nanowires (SiNWs) can be 100 times smaller than that of bulk silicon, which suggests the possibility of using silicon-based nanostructures as efficient thermoelectric materials [4,5]. Theoretically, improved thermoelectric performance of SiNWs was predicated [6–9], as a result of reduced thermal conductivity caused by phonon surface n
Corresponding authors. E-mail addresses: [email protected] (J.W. Ding), [email protected] (X.J. Zheng). http://dx.doi.org/10.1016/j.ssc.2015.11.012 0038-1098/& 2015 Elsevier Ltd. All rights reserved.
scattering and enhanced power factor due to quantum confinement effect. Subsequently, other silicon-based nanostructures like nanotubes [10] and nanomembranes [11,12] have been suggested, and their interesting thermoelectric properties have also been reported. Very recently, new two-dimensional materials like graphene and silicene, especially one-dimensional graphene (GNRs) and silicene (SiNRs) nanoribbons, have been attracting a great interest due to their unique properties [13–16]. Though the thermopower of pristine graphene is not very high, it can be considerably enhanced in GNRs, especially in nanostructures consisting of nanoribbons of various types. Indeed, in a properly designed nanoribbon with alternating zigzag and armchair sections, thermoelectric figure of merit exceeding unity at room temperature has been found [17]. The efficiency can be also enhanced by randomly distributed hydrogen vacancies in almost completely hydrogenated GNRs [18]. Furthermore, structural defects, especially in the form of antidots, also appear a promising way to enhance thermoelectric efficiency [19–21]. Meanwhile, giant spin related thermoelectric phenomena have been predicted for ferromagnetic ZGNRs with antidotes [22]. It is generally accepted that decreasing the characteristic size of nanostructures and introducing defects are two effective ways to further improve their ZT values. So higher thermoelectric performance can be expected for SiNRs with defects, which is due to lower thermal conductance than GNRs. In this paper, we investigate the thermoelectric performance for both ZSiNRs and ASiNRs
2
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
with central or edge defects by using non-equilibrium Green's function method. For perfect silicene nanoribbons (SiNRs), it is shown that with its width increasing, the maximum of ZT values (ZTM) decreases monotonously while the phononic thermal conductance increases linearly. For various types of edges and defects, with increasing defect numbers in longitudinal direction, ZTM increases monotonously while the phononic thermal conductance decreases. Comparing with ZSiNRs, defected ASiNRs possess higher thermoelectric performance due to higher Seebeck coefficient and lower thermal conductance. In particular, about 2.5 times enhancement to ZT values is obtained in ASiNRs with edge defects. Our theoretical simulations indicate that by controlling the type and number of defects, ZT values of SiNRs could be enhanced greatly which suggests their very appealing thermoelectric applications.
2. Model and method In order to enhance the ZT values of SiNRs, we introduce central and edge cavities in ZSiNRs and ASiNRs respectively, as shown in Fig. 1. Following a common convention, we refer to the SiNRs with N dimer lines in width as N-SiNRs. The system is composed of a central junction of length L and width W and two semi-infinite ideal leads of the same width. The central junction is formed by removing hexagonal carbon rings from perfect SiNRs, in which the numbers of hexagons along the transversal and longitudinal directions are denoted by index n and m.
For electronic transport, by using an atomistic pz orbital basis, the silicene system can be described by the second-nearestneighbor tight binding model [23] H ¼ lEz
X † X † iλSO X † ciα μi ciα t ciα cjα þ pffiffiffi ciα νij σ zαβ cjβ 3 3 iα i;j α i;j αβ h i hh ii
ð1Þ
where ciα þ and ciα are creation and annihilation operators with spin polarization α on site i and the combinations i; j and i; j run overall the nearest and next nearest neighbo rhopping sites respectively. In Hamiltonian, the first term represents the staggered sublattice potential, where l¼0.23 Å and μi ¼ 71 for the A (B)site. It generates a staggered sublattice potential p 2lEz between silicon atoms at A and B sites. The second term is the usual nearest-neighbor hopping term with the transfer energy t¼1.6 eV. The third term represents the effective SO coupling with λSO ¼ 3.9 meV, where σ ¼(σx,σy,σz) is the Pauli matrix of spin, with νij ¼ þ 1 if the next nearest neighboring hopping is anticlockwise and νij ¼ 1 if it is clockwise with respect to the positive z axis. For incident electronic energy E, the electronic transmission per spin though junction region is calculated as T e ðEÞ ¼ Tr Γ L Gr Γ R Ga , where the line width function ΓL,R(E) is defined as Γ L;R ðEÞ ¼ i½Σ rL;R Σ aL;R , andthe retarded (advanced) Green's func r;a r;a tion is given by Gr;a ðEÞ ¼ E 7 iη I H C Σ L Σ R , in which the retarded (advanced) self energy due to the coupling to all leads can be obtained numerically [24]. Define an intermediate function Ln(μ,T) as [25,26] Z FD ð2Þ Ln ðμ; TÞ ¼ ð2=hÞ dET e ðEÞðE μÞn ½ ∂f ðE; μ; TÞ=∂E
Fig. 1. (Color online) Schematic diagrams of defected SiNR atomistic structures with the length L ¼15 nm and width W ¼3 nm, for which the defect numbers are of m¼ 4 and n¼ 3. Figure (a–d) corresponding to 10-ZSiNR with central and edge defects and 15-ASiNR with central and edge defects, respectively.
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
where the factor 2 counts the spin degeneracy,f FD(E,μ,T) is the Fermi–Dirac distribution function and μ is the chemical potential. The electronic conductance, thermopower, and the electronic contribution to the thermal conductance can be conveniently derived from Ge ðμ; TÞ ¼ e2 L0 ðμ; TÞ
ð3Þ
Sðμ; TÞ ¼ L1 ðμ; TÞ=½qTL0 ðμ; TÞ
ð4Þ
κ e ðμ; TÞ ¼ ð1=TÞ½L2 ðμ; TÞ L21 ðμ; TÞ=L0 ðμ; TÞ
ð5Þ
where q is the electric charge of carriers, which is positive for holes and negative for electrons. The Green's function and transmission can be calculated similarly for phonon transport [27]. One only needs to change E to ω2M, H to D, and compute the self-energies accordingly, where ω is the phonon frequency, M is the mass of carbon atom, H is the Hamiltonian and D is the dynamic matrix. The dynamic matrix is constructed by using a spring mass model and 4NN interaction is considered [28,29]. The phononic contribution to the thermal
conductance is given by Z h i κ p ðTÞ ¼ ð1=2π Þ dωT p ðωÞℏω ∂f BE ðω; TÞ=∂T
3
ð6Þ
whereTp(ω) is the phononic transmission and f BE(ω,T) is the Bose– Einstein distribution function. According to the experimental [30,31] and theoretical [32,33] results on Si/Ge superlattices, anharmonic effects are not important for temperatures below 500 K. So we do not include phonon–phonon scattering in calculations. The temperature T is fixed to 300 K.
3. Results and discussions We first calculate the thermoelectric properties of perfect ZSiNRs and ASiNRs, as shown in Fig. 2. The phononic thermal conductance of both ZSiNRs and ASiNRs increases linearly with increasing the width N in Fig. 2(a), which is due to the increase in number of effective phononic channels. Comparing with the thermal conductivity of graphene [34], this result implies that
Fig. 2. (Color online) Phononic thermal conductance (κp) as a function of (a) width N and (b) temperature T. Maximum of ZT values (ZTM) as a function of (c) width N and (d) chemical potential μ.
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silicene devices should have higher thermoelectric performance. In Fig. 2(b), the phononic thermal conductance of SiNRs raises rapidly with the temperature enhancing, which suggests that devices have higher ZT values at lower temperature [35]. As illustrated in Fig. 2(c), the maximum of ZT values decreases monotonously with width N increasing, which clearly indicates that the thinner SiNRs have the higher thermoelectric efficiency. Here we can estimate the ZT value of silicene in the order of 10 2, which is too small to actual application. To illustrate the tuning effect by gating or doping, we calculate the ZT values of 10-ZSiNRs and 15ASiNRs as a function of the chemical potential μ in Fig. 2(d). It is shown that the maximum of ZT values occurs in the vicinity of either the top of valence band or the bottom of conduction band, which results from the high Seebeck coefficient and low electronic thermal conductance. These results indicate that the higher ZT value can be obtained by controlling the electronic behavior in the edges of energy bands. In order to explore the effect of defects, as shown in Fig. 3, we further calculate the phononic thermal conductance (κp) and maximum of ZT values (ZTM) of 10-ZSiNRs with central and edge
defects, as the defect numbers m in longitudinal direction and n in transversal direction increasing. The width of 10-ZSiNRs is corresponding to 3 nm. For a given n, it is shown from Fig. 3(a–b) that the phononic thermal conductance decreases exponentially with the defect number m increasing, converging to a limit value of the ZSiNR superlattice with periodically arranged defects. These limit values decrease with n increasing. The lowest phononic thermal conductance 0.15 nW/K is obtained at n ¼ 3 in ZSiNR with edge defects, much less than 1.15 nW/K in the case of no defects. It is due to the fact that the high frequency phononic modes are destroyed by the defects, leading to a large reduction of phononic thermal conductance at room temperature. For ZT characteristic of the ZSiNRs with central and edge defects, the results are illustrated in Fig. 3(c–d). It is observed that for a given n, the maximum of ZT increases with m increasing, converging to a limit value of the 10-ZSiNR superlattice with periodically arranged defects. Especially, the highest ZT limit value 0.86 can be obtained at n ¼3 in 10-ZSiNR with edge defects, 245% higher than 0.35 in the case of no defects. This means that the thermoelectric properties of SiNRs can be modulated remarkably
Fig. 3. (Color online) Phononic thermal conductance as a function of defect numbers m and n for 10-ZSiNRs with (a) central and (b) edge defects. ZTM as a function of defect numbers m and n for 10-ZSiNRs with (c) central and (d) edge defects.
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
by increasing m. The 10-ZSiNRs with central and edge defects show similar highest ZT limit values. Interestingly, for n ¼2, the 10ZSiNRs with central defects has highest thermoelectric performance, while the case with edge defects has lowest performance. To understand the difference in ZT behavior, for 10-ZSiNRs with central and edge defects at n ¼2, we calculate electronic conductance, Seebeck coefficient, electronic thermal conductance and ZT respectively as a function of the chemical potential μ, as shown in Fig. 4(a–d). 10-ZSiNRs with central defects show more metallicity, which displays narrower band gap and higher electronic conductance. Within the rigid-band picture, μ 40 corresponds to the n-type doping while μ o0 corresponds to the p-type doping. For the case with central defects, the ZTM locates p-type region, but for the case with edge defects, the ZTM locates n-type region. The former is more close to the band edge and possess higher Seebeck coefficient, which induces higher ZTM. It is well known that the thermal properties are very sensitive to defects [36–38],
5
but the electronic properties not. So we expect to enhance the ZT value of SiNRs by introducing some defects periodically, which could decrease the phononic thermal conductance greatly, but not change the electronic conductance severely. Meanwhile, as shown in Fig. 5, we calculate the phononic thermal conductance (κp) and maximum of ZT values (ZTM) of 15ASiNRs with central and edge defects, as the defect numbers m in longitudinal direction and n in transversal direction increasing. In Fig. 5(a–b), similar to ZSiNRs, it is shown that for a given n, the phononic thermal conductance decreases exponentially with the defect number m increasing, converging to a limit value of the ASiNR superlattice with periodically arranged defects. These limit values decrease with n increasing. The lowest phononic thermal conductance 0.16 nW/K is obtained at n ¼3 in ASiNR with edge defects, much less than 0.81 nW/K in the case of no defects. Fig. 5(c–d) characterizes the ZT features of ASiNRs with central and edge defects. It is shown that for a given n, ZTM increases with
Fig. 4. (Color online) (a) Electronic conductance, (b) Electronic thermal conductance, (c) Seebeck coefficient and (d) ZT values as a function of chemical potential μ for 10-ZSiNRs with central and edge defects, for which defect numbers are of m¼ 8 and n¼3.
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m increasing, converging to a limit value of the 15-ASiNR superlattice with periodically arranged defects. These limit values increase with n increasing. Especially, the highest ZT limit value 1.19 can be obtained at n ¼3 in 15-ASiNR with edge defects, 248% higher than 0.48 in the case of no defects. This means that the thermoelectric properties of SiNRs can be modulated remarkably by increasing m and n simultaneously. The 15-ASiNRs with edge defects show higher ZT limit values, in contrast to the case with central defects. To understand the difference in ZT behavior for 15-ASiNRs with central and edge defects at n ¼3, we calculate electronic conductance, electronic thermal conductance, Seebeck coefficient and ZT values as a function of the chemical potential μ in Fig. 6(a–d), respectively. 15-ASiNRs with edge defects show more insulativity, which displays lower electronic conductance and wider band gap. So the high ZTM for the case with edge defects should origin from high Seebeck coefficient and low electronic thermal conductance.
4. Conclusions In summary, by using NEGF method, we have investigated numerically the the thermoelectric performance for both ZSiNRs and ASiNRs with central or edge defects within tight-binding approximation. For perfect SiNRs, it is shown that with its width increasing, ZTM decreases monotonously while the phononic thermal conductance increases linearly. For various types of edges and defects, with increasing defect numbers in longitudinal direction, ZTM increases monotonously while the phononic thermal conductance decreases. Comparing with ZSiNRs, defected ASiNRs possess higher thermoelectric performance due to higher Seebeck coefficient and lower thermal conductance. In particular, about 2.5 times enhancement to ZT values is obtained in ASiNRs with edge defects. Our theoretical simulations indicate that by controlling the type and number of defects, ZT values of SiNRs could be enhanced greatly which suggests their very appealing thermoelectric applications.
Fig. 5. (Color online) Phononic thermal conductance as a function of defect numbers m and n for 15-ASiNRs with (a) central and (b) edge defects. ZTM as a function of defect numbers m and n for 15-ASiNRs with (c) central and (d) edge defects.
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
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Fig. 6. (Color online) (a) Electronic conductance, (b) Electronic thermal conductance, (c) Seebeck coefficient and (d) ZT values as a function of chemical potential μ for 15-ASiNRs with central and edge defects, for which defect numbers are of m¼ 8 and n ¼3.
Acknowledgments This work is supported by National Natural Science Foundation of China (Grant no. 11204259 and No. 11374252), and partially by SPCSIRT (IRT1080).
References [1] M.S. Dresselhaus, G. Chen, M.Y. Tang, R.G. Yang, H. Lee, D.Z. Wang, Z.F. Ren, J. P. Fleurial, P. Gogna, Adv. Mater. 19 (2007) 1043. [2] G.J. Snyder, E.S. Toberer, Nat. Mater. 7 (2008) 105. [3] J.F. Li, W.S. Liu, L.D. Zhao, M. Zhou, NPG Asia Mater. 2 (2010) 152. [4] A.I. Hochbaum, R. Chen, R.D. Delgado, W. Liang, E.C. Garnett, M. Najarian, A. Majumdar, P. Yang, Nature 451 (2008) 163. [5] A.I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W.A. Goddard III, J.R. Heath, Nature 451 (2008) 168. [6] T.T.M. Vo, A.J. Williamson, V. Lordi, G. Galli, Nano Lett. 8 (2008) 1111. [7] T. Markussen, A.-P. Jauho, M. Brandbyge, Phys. Rev. B 79 (2009) 035415. [8] T. Markussen, A.-P. Jauho, M. Brandbyge, Phys. Rev. Lett. 103 (2009) 055502. [9] F. Sansoz, Nano Lett. 11 (2011) 5378. [10] J. Chen, G. Zhang, B.W. Li, Nano Lett. 10 (2010) 3978.
[11] J. Tang, H.-T. Wang, D.H. Lee, M. Fardy, Z. Huo, T.P. Russell, P. Yang, Nano Lett. 10 (2010) 4279. [12] A. Mittal, S. Mazumder, J. Heat. Transf. 132 (2010) 052402. [13] H.B. Heersche, P. Jarillo-Herrero, J.B. Oestinga, L.M. Vandersypen, A. F. Morpurgo, Nature 446 (2007) 56. [14] B. Aufray, A. Kara, S. Vizzini, H. Oughaddou, C. Leandri, B. Ealet, G. Le Lay, Appl. Phys. Lett. 96 (2010) 183102. [15] P.D. Padova, C. Quaresima, B. Olivieri, P. Perfetti, G. Le. Lay, Appl. Phys. Lett. 98 (2011) 081909. [16] P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M.C. Asensio, A. resta, B. Ealet, G. Le Lay, Phys. Rev. Lett. 108 (2012) 155501. [17] F. Mazzamuto, V.H. Nguyen, Y. Apertet, C. Caer, C. Chassat, J. Saint-Martin, P. Dollfus, Phys. Rev. B 83 (2011) 235426. [18] Y. Ding, J. Ni, Appl. Phys. Lett. 95 (2009) 083115. [19] T.G. Pedersen, C. Flindt, J. Pedersen, N.A. Mortensen, A.P. Jauho, K. Pedersen, Phys. Rev. Lett. 100 (2008) 136804. [20] W. Zhao, Z.X. Guo, J.X. Cao, J.W. Ding, AIP Adv. 1 (2011) 042135. [21] Y. Yan, Q.F. Liang, H. Zhao, C.Q. Wu, B. Li, Phys. Lett. A 376 (2012) 2425. [22] M. Wierzbicki, R. Swirkowicz, J. Barnas, Phys. Rev. B 88 (2013) 235434. [23] M. Ezawa, Phys. Rev. Lett. 109 (2012) 055502. [24] D.H. Lee, J.D. Joannopoulos, Phys. Rev. B 23 (1980) 4997. [25] U. Sivan, Y. Imry, Phys. Rev. B 33 (1986) 551. [26] K. Esfarjani, M. Zebarjadi, Y. Kawazoe, Phys. Rev. B 73 (2006) 085406. [27] T. Yamamoto, K. Watanabe, Phys. Rev. Lett. 96 (2006) 255503. [28] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, UK, 2004.
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[29] K. Zberecki, M. Wierzbicki, J. Barnas, R. Swirkowicz, Phys. Rev. B 88 (2013) 115404. [30] S.M. Lee, D.G. Cahill, R. Venkatasubramanian, Appl. Phys. Lett. 70 (1997) 2957. [31] T. Borca-Tasciuc, W.L. Liu, J.L. Liu, T.F. Zeng, D.W. Song, C.D. Moore, G. Chen, K. L. Wang, M.S. Goorsky, T. Radetic, R. Gronsky, T. Koga, M.S. Dresselhaus, Superlattices Microstruct. 28 (2000) 199. [32] J. Garg, N. Bonini, N. Marzari, Nano Lett. 11 (2011) 5135. [33] E.S. Landry, A.J.H. McGaughey, Phys. Rev. B 79 (2009) 075316.
[34] Z. Guo, D. Zhang, X.-G. Gong, Appl. Phys. Lett. 95 (2009) 163103. [35] L. Pan, H.J. Liu, X.J. Tan, H.Y. Lv, J. Shi, X.F. Tang, G. Zheng, Phys. Chem. Chem. Phys. 14 (2012) 13588. [36] M. Morooka, T. Yamamoto, K. Watanabe, Phys. Rev. B 77 (2008) 033412. [37] J.-W. Jiang, J.-S. Wang, B. Li, Phys. Rev. B 79 (2009) 205418. [38] J. Lan, J.-S. Wang, C.K. Gan, S.K. Chin, Phys. Rev. B 79 (2009) 115401.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Enhanced thermoelectric performance of defected silicene nanoribbons W. Zhao a, Z.X. Guo b, Y. Zhang b, J.W. Ding b,n, X.J. Zheng a,n a b
School of Mechanical Engineering, Xiangtan University, Xiangtan 411105, China Department of Physics and Institute for Nanophysics and Rare-earth Luminescence, Xiangtan University, Xiangtan 411105, China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 November 2015 Accepted 16 November 2015 Communicated by: Prof. A.H. MacDonald
Based on non-equilibrium Green's function method, we investigate the thermoelectric performance for both zigzag (ZSiNRs) and armchair (ASiNRs) silicene nanoribbons with central or edge defects. For perfect silicene nanoribbons (SiNRs), it is shown that with its width increasing, the maximum of ZT values (ZTM) decreases monotonously while the phononic thermal conductance increases linearly. For various types of edges and defects, with increasing defect numbers in longitudinal direction, ZTM increases monotonously while the phononic thermal conductance decreases. Comparing with ZSiNRs, defected ASiNRs possess higher thermoelectric performance due to higher Seebeck coefficient and lower thermal conductance. In particular, about 2.5 times enhancement to ZT values is obtained in ASiNRs with edge defects. Our theoretical simulations indicate that by controlling the type and number of defects, ZT values of SiNRs could be enhanced greatly which suggests their very appealing thermoelectric applications. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Thermoelectric materials which can convert dissipated heat into electric energy at nanoscale have attracted increasing attention from both theoretical and experimental research due to their potential technological applications [1–3]. The performance of thermoelectric materials is measured by the figure of merit ZT¼S2GT/(κe þ κp), where S, G, T, κe and κp represent the Seebeck coefficient, electronic conductance, absolute temperature, electronic and phononic thermal conductance, respectively. Explicitly, an optimal thermoelectric material should possess high Seebeck coefficient, high electronic conductance, and low thermal conductance. Both experimental and theoretical research have indicated that nanoscale materials could exhibit much higher ZT values compared with those of bulk materials, which lead to a very important prospect of thermoelectric applications. Experimentally, in particular, it was found that the thermal conductivity of Si nanowires (SiNWs) can be 100 times smaller than that of bulk silicon, which suggests the possibility of using silicon-based nanostructures as efficient thermoelectric materials [4,5]. Theoretically, improved thermoelectric performance of SiNWs was predicated [6–9], as a result of reduced thermal conductivity caused by phonon surface n
Corresponding authors. E-mail addresses: [email protected] (J.W. Ding), [email protected] (X.J. Zheng). http://dx.doi.org/10.1016/j.ssc.2015.11.012 0038-1098/& 2015 Elsevier Ltd. All rights reserved.
scattering and enhanced power factor due to quantum confinement effect. Subsequently, other silicon-based nanostructures like nanotubes [10] and nanomembranes [11,12] have been suggested, and their interesting thermoelectric properties have also been reported. Very recently, new two-dimensional materials like graphene and silicene, especially one-dimensional graphene (GNRs) and silicene (SiNRs) nanoribbons, have been attracting a great interest due to their unique properties [13–16]. Though the thermopower of pristine graphene is not very high, it can be considerably enhanced in GNRs, especially in nanostructures consisting of nanoribbons of various types. Indeed, in a properly designed nanoribbon with alternating zigzag and armchair sections, thermoelectric figure of merit exceeding unity at room temperature has been found [17]. The efficiency can be also enhanced by randomly distributed hydrogen vacancies in almost completely hydrogenated GNRs [18]. Furthermore, structural defects, especially in the form of antidots, also appear a promising way to enhance thermoelectric efficiency [19–21]. Meanwhile, giant spin related thermoelectric phenomena have been predicted for ferromagnetic ZGNRs with antidotes [22]. It is generally accepted that decreasing the characteristic size of nanostructures and introducing defects are two effective ways to further improve their ZT values. So higher thermoelectric performance can be expected for SiNRs with defects, which is due to lower thermal conductance than GNRs. In this paper, we investigate the thermoelectric performance for both ZSiNRs and ASiNRs
2
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
with central or edge defects by using non-equilibrium Green's function method. For perfect silicene nanoribbons (SiNRs), it is shown that with its width increasing, the maximum of ZT values (ZTM) decreases monotonously while the phononic thermal conductance increases linearly. For various types of edges and defects, with increasing defect numbers in longitudinal direction, ZTM increases monotonously while the phononic thermal conductance decreases. Comparing with ZSiNRs, defected ASiNRs possess higher thermoelectric performance due to higher Seebeck coefficient and lower thermal conductance. In particular, about 2.5 times enhancement to ZT values is obtained in ASiNRs with edge defects. Our theoretical simulations indicate that by controlling the type and number of defects, ZT values of SiNRs could be enhanced greatly which suggests their very appealing thermoelectric applications.
2. Model and method In order to enhance the ZT values of SiNRs, we introduce central and edge cavities in ZSiNRs and ASiNRs respectively, as shown in Fig. 1. Following a common convention, we refer to the SiNRs with N dimer lines in width as N-SiNRs. The system is composed of a central junction of length L and width W and two semi-infinite ideal leads of the same width. The central junction is formed by removing hexagonal carbon rings from perfect SiNRs, in which the numbers of hexagons along the transversal and longitudinal directions are denoted by index n and m.
For electronic transport, by using an atomistic pz orbital basis, the silicene system can be described by the second-nearestneighbor tight binding model [23] H ¼ lEz
X † X † iλSO X † ciα μi ciα t ciα cjα þ pffiffiffi ciα νij σ zαβ cjβ 3 3 iα i;j α i;j αβ h i hh ii
ð1Þ
where ciα þ and ciα are creation and annihilation operators with spin polarization α on site i and the combinations i; j and i; j run overall the nearest and next nearest neighbo rhopping sites respectively. In Hamiltonian, the first term represents the staggered sublattice potential, where l¼0.23 Å and μi ¼ 71 for the A (B)site. It generates a staggered sublattice potential p 2lEz between silicon atoms at A and B sites. The second term is the usual nearest-neighbor hopping term with the transfer energy t¼1.6 eV. The third term represents the effective SO coupling with λSO ¼ 3.9 meV, where σ ¼(σx,σy,σz) is the Pauli matrix of spin, with νij ¼ þ 1 if the next nearest neighboring hopping is anticlockwise and νij ¼ 1 if it is clockwise with respect to the positive z axis. For incident electronic energy E, the electronic transmission per spin though junction region is calculated as T e ðEÞ ¼ Tr Γ L Gr Γ R Ga , where the line width function ΓL,R(E) is defined as Γ L;R ðEÞ ¼ i½Σ rL;R Σ aL;R , andthe retarded (advanced) Green's func r;a r;a tion is given by Gr;a ðEÞ ¼ E 7 iη I H C Σ L Σ R , in which the retarded (advanced) self energy due to the coupling to all leads can be obtained numerically [24]. Define an intermediate function Ln(μ,T) as [25,26] Z FD ð2Þ Ln ðμ; TÞ ¼ ð2=hÞ dET e ðEÞðE μÞn ½ ∂f ðE; μ; TÞ=∂E
Fig. 1. (Color online) Schematic diagrams of defected SiNR atomistic structures with the length L ¼15 nm and width W ¼3 nm, for which the defect numbers are of m¼ 4 and n¼ 3. Figure (a–d) corresponding to 10-ZSiNR with central and edge defects and 15-ASiNR with central and edge defects, respectively.
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
where the factor 2 counts the spin degeneracy,f FD(E,μ,T) is the Fermi–Dirac distribution function and μ is the chemical potential. The electronic conductance, thermopower, and the electronic contribution to the thermal conductance can be conveniently derived from Ge ðμ; TÞ ¼ e2 L0 ðμ; TÞ
ð3Þ
Sðμ; TÞ ¼ L1 ðμ; TÞ=½qTL0 ðμ; TÞ
ð4Þ
κ e ðμ; TÞ ¼ ð1=TÞ½L2 ðμ; TÞ L21 ðμ; TÞ=L0 ðμ; TÞ
ð5Þ
where q is the electric charge of carriers, which is positive for holes and negative for electrons. The Green's function and transmission can be calculated similarly for phonon transport [27]. One only needs to change E to ω2M, H to D, and compute the self-energies accordingly, where ω is the phonon frequency, M is the mass of carbon atom, H is the Hamiltonian and D is the dynamic matrix. The dynamic matrix is constructed by using a spring mass model and 4NN interaction is considered [28,29]. The phononic contribution to the thermal
conductance is given by Z h i κ p ðTÞ ¼ ð1=2π Þ dωT p ðωÞℏω ∂f BE ðω; TÞ=∂T
3
ð6Þ
whereTp(ω) is the phononic transmission and f BE(ω,T) is the Bose– Einstein distribution function. According to the experimental [30,31] and theoretical [32,33] results on Si/Ge superlattices, anharmonic effects are not important for temperatures below 500 K. So we do not include phonon–phonon scattering in calculations. The temperature T is fixed to 300 K.
3. Results and discussions We first calculate the thermoelectric properties of perfect ZSiNRs and ASiNRs, as shown in Fig. 2. The phononic thermal conductance of both ZSiNRs and ASiNRs increases linearly with increasing the width N in Fig. 2(a), which is due to the increase in number of effective phononic channels. Comparing with the thermal conductivity of graphene [34], this result implies that
Fig. 2. (Color online) Phononic thermal conductance (κp) as a function of (a) width N and (b) temperature T. Maximum of ZT values (ZTM) as a function of (c) width N and (d) chemical potential μ.
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silicene devices should have higher thermoelectric performance. In Fig. 2(b), the phononic thermal conductance of SiNRs raises rapidly with the temperature enhancing, which suggests that devices have higher ZT values at lower temperature [35]. As illustrated in Fig. 2(c), the maximum of ZT values decreases monotonously with width N increasing, which clearly indicates that the thinner SiNRs have the higher thermoelectric efficiency. Here we can estimate the ZT value of silicene in the order of 10 2, which is too small to actual application. To illustrate the tuning effect by gating or doping, we calculate the ZT values of 10-ZSiNRs and 15ASiNRs as a function of the chemical potential μ in Fig. 2(d). It is shown that the maximum of ZT values occurs in the vicinity of either the top of valence band or the bottom of conduction band, which results from the high Seebeck coefficient and low electronic thermal conductance. These results indicate that the higher ZT value can be obtained by controlling the electronic behavior in the edges of energy bands. In order to explore the effect of defects, as shown in Fig. 3, we further calculate the phononic thermal conductance (κp) and maximum of ZT values (ZTM) of 10-ZSiNRs with central and edge
defects, as the defect numbers m in longitudinal direction and n in transversal direction increasing. The width of 10-ZSiNRs is corresponding to 3 nm. For a given n, it is shown from Fig. 3(a–b) that the phononic thermal conductance decreases exponentially with the defect number m increasing, converging to a limit value of the ZSiNR superlattice with periodically arranged defects. These limit values decrease with n increasing. The lowest phononic thermal conductance 0.15 nW/K is obtained at n ¼ 3 in ZSiNR with edge defects, much less than 1.15 nW/K in the case of no defects. It is due to the fact that the high frequency phononic modes are destroyed by the defects, leading to a large reduction of phononic thermal conductance at room temperature. For ZT characteristic of the ZSiNRs with central and edge defects, the results are illustrated in Fig. 3(c–d). It is observed that for a given n, the maximum of ZT increases with m increasing, converging to a limit value of the 10-ZSiNR superlattice with periodically arranged defects. Especially, the highest ZT limit value 0.86 can be obtained at n ¼3 in 10-ZSiNR with edge defects, 245% higher than 0.35 in the case of no defects. This means that the thermoelectric properties of SiNRs can be modulated remarkably
Fig. 3. (Color online) Phononic thermal conductance as a function of defect numbers m and n for 10-ZSiNRs with (a) central and (b) edge defects. ZTM as a function of defect numbers m and n for 10-ZSiNRs with (c) central and (d) edge defects.
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by increasing m. The 10-ZSiNRs with central and edge defects show similar highest ZT limit values. Interestingly, for n ¼2, the 10ZSiNRs with central defects has highest thermoelectric performance, while the case with edge defects has lowest performance. To understand the difference in ZT behavior, for 10-ZSiNRs with central and edge defects at n ¼2, we calculate electronic conductance, Seebeck coefficient, electronic thermal conductance and ZT respectively as a function of the chemical potential μ, as shown in Fig. 4(a–d). 10-ZSiNRs with central defects show more metallicity, which displays narrower band gap and higher electronic conductance. Within the rigid-band picture, μ 40 corresponds to the n-type doping while μ o0 corresponds to the p-type doping. For the case with central defects, the ZTM locates p-type region, but for the case with edge defects, the ZTM locates n-type region. The former is more close to the band edge and possess higher Seebeck coefficient, which induces higher ZTM. It is well known that the thermal properties are very sensitive to defects [36–38],
5
but the electronic properties not. So we expect to enhance the ZT value of SiNRs by introducing some defects periodically, which could decrease the phononic thermal conductance greatly, but not change the electronic conductance severely. Meanwhile, as shown in Fig. 5, we calculate the phononic thermal conductance (κp) and maximum of ZT values (ZTM) of 15ASiNRs with central and edge defects, as the defect numbers m in longitudinal direction and n in transversal direction increasing. In Fig. 5(a–b), similar to ZSiNRs, it is shown that for a given n, the phononic thermal conductance decreases exponentially with the defect number m increasing, converging to a limit value of the ASiNR superlattice with periodically arranged defects. These limit values decrease with n increasing. The lowest phononic thermal conductance 0.16 nW/K is obtained at n ¼3 in ASiNR with edge defects, much less than 0.81 nW/K in the case of no defects. Fig. 5(c–d) characterizes the ZT features of ASiNRs with central and edge defects. It is shown that for a given n, ZTM increases with
Fig. 4. (Color online) (a) Electronic conductance, (b) Electronic thermal conductance, (c) Seebeck coefficient and (d) ZT values as a function of chemical potential μ for 10-ZSiNRs with central and edge defects, for which defect numbers are of m¼ 8 and n¼3.
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m increasing, converging to a limit value of the 15-ASiNR superlattice with periodically arranged defects. These limit values increase with n increasing. Especially, the highest ZT limit value 1.19 can be obtained at n ¼3 in 15-ASiNR with edge defects, 248% higher than 0.48 in the case of no defects. This means that the thermoelectric properties of SiNRs can be modulated remarkably by increasing m and n simultaneously. The 15-ASiNRs with edge defects show higher ZT limit values, in contrast to the case with central defects. To understand the difference in ZT behavior for 15-ASiNRs with central and edge defects at n ¼3, we calculate electronic conductance, electronic thermal conductance, Seebeck coefficient and ZT values as a function of the chemical potential μ in Fig. 6(a–d), respectively. 15-ASiNRs with edge defects show more insulativity, which displays lower electronic conductance and wider band gap. So the high ZTM for the case with edge defects should origin from high Seebeck coefficient and low electronic thermal conductance.
4. Conclusions In summary, by using NEGF method, we have investigated numerically the the thermoelectric performance for both ZSiNRs and ASiNRs with central or edge defects within tight-binding approximation. For perfect SiNRs, it is shown that with its width increasing, ZTM decreases monotonously while the phononic thermal conductance increases linearly. For various types of edges and defects, with increasing defect numbers in longitudinal direction, ZTM increases monotonously while the phononic thermal conductance decreases. Comparing with ZSiNRs, defected ASiNRs possess higher thermoelectric performance due to higher Seebeck coefficient and lower thermal conductance. In particular, about 2.5 times enhancement to ZT values is obtained in ASiNRs with edge defects. Our theoretical simulations indicate that by controlling the type and number of defects, ZT values of SiNRs could be enhanced greatly which suggests their very appealing thermoelectric applications.
Fig. 5. (Color online) Phononic thermal conductance as a function of defect numbers m and n for 15-ASiNRs with (a) central and (b) edge defects. ZTM as a function of defect numbers m and n for 15-ASiNRs with (c) central and (d) edge defects.
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Fig. 6. (Color online) (a) Electronic conductance, (b) Electronic thermal conductance, (c) Seebeck coefficient and (d) ZT values as a function of chemical potential μ for 15-ASiNRs with central and edge defects, for which defect numbers are of m¼ 8 and n ¼3.
Acknowledgments This work is supported by National Natural Science Foundation of China (Grant no. 11204259 and No. 11374252), and partially by SPCSIRT (IRT1080).
References [1] M.S. Dresselhaus, G. Chen, M.Y. Tang, R.G. Yang, H. Lee, D.Z. Wang, Z.F. Ren, J. P. Fleurial, P. Gogna, Adv. Mater. 19 (2007) 1043. [2] G.J. Snyder, E.S. Toberer, Nat. Mater. 7 (2008) 105. [3] J.F. Li, W.S. Liu, L.D. Zhao, M. Zhou, NPG Asia Mater. 2 (2010) 152. [4] A.I. Hochbaum, R. Chen, R.D. Delgado, W. Liang, E.C. Garnett, M. Najarian, A. Majumdar, P. Yang, Nature 451 (2008) 163. [5] A.I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W.A. Goddard III, J.R. Heath, Nature 451 (2008) 168. [6] T.T.M. Vo, A.J. Williamson, V. Lordi, G. Galli, Nano Lett. 8 (2008) 1111. [7] T. Markussen, A.-P. Jauho, M. Brandbyge, Phys. Rev. B 79 (2009) 035415. [8] T. Markussen, A.-P. Jauho, M. Brandbyge, Phys. Rev. Lett. 103 (2009) 055502. [9] F. Sansoz, Nano Lett. 11 (2011) 5378. [10] J. Chen, G. Zhang, B.W. Li, Nano Lett. 10 (2010) 3978.
[11] J. Tang, H.-T. Wang, D.H. Lee, M. Fardy, Z. Huo, T.P. Russell, P. Yang, Nano Lett. 10 (2010) 4279. [12] A. Mittal, S. Mazumder, J. Heat. Transf. 132 (2010) 052402. [13] H.B. Heersche, P. Jarillo-Herrero, J.B. Oestinga, L.M. Vandersypen, A. F. Morpurgo, Nature 446 (2007) 56. [14] B. Aufray, A. Kara, S. Vizzini, H. Oughaddou, C. Leandri, B. Ealet, G. Le Lay, Appl. Phys. Lett. 96 (2010) 183102. [15] P.D. Padova, C. Quaresima, B. Olivieri, P. Perfetti, G. Le. Lay, Appl. Phys. Lett. 98 (2011) 081909. [16] P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M.C. Asensio, A. resta, B. Ealet, G. Le Lay, Phys. Rev. Lett. 108 (2012) 155501. [17] F. Mazzamuto, V.H. Nguyen, Y. Apertet, C. Caer, C. Chassat, J. Saint-Martin, P. Dollfus, Phys. Rev. B 83 (2011) 235426. [18] Y. Ding, J. Ni, Appl. Phys. Lett. 95 (2009) 083115. [19] T.G. Pedersen, C. Flindt, J. Pedersen, N.A. Mortensen, A.P. Jauho, K. Pedersen, Phys. Rev. Lett. 100 (2008) 136804. [20] W. Zhao, Z.X. Guo, J.X. Cao, J.W. Ding, AIP Adv. 1 (2011) 042135. [21] Y. Yan, Q.F. Liang, H. Zhao, C.Q. Wu, B. Li, Phys. Lett. A 376 (2012) 2425. [22] M. Wierzbicki, R. Swirkowicz, J. Barnas, Phys. Rev. B 88 (2013) 235434. [23] M. Ezawa, Phys. Rev. Lett. 109 (2012) 055502. [24] D.H. Lee, J.D. Joannopoulos, Phys. Rev. B 23 (1980) 4997. [25] U. Sivan, Y. Imry, Phys. Rev. B 33 (1986) 551. [26] K. Esfarjani, M. Zebarjadi, Y. Kawazoe, Phys. Rev. B 73 (2006) 085406. [27] T. Yamamoto, K. Watanabe, Phys. Rev. Lett. 96 (2006) 255503. [28] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, UK, 2004.
8
W. Zhao et al. / Solid State Communications 227 (2016) 1–8
[29] K. Zberecki, M. Wierzbicki, J. Barnas, R. Swirkowicz, Phys. Rev. B 88 (2013) 115404. [30] S.M. Lee, D.G. Cahill, R. Venkatasubramanian, Appl. Phys. Lett. 70 (1997) 2957. [31] T. Borca-Tasciuc, W.L. Liu, J.L. Liu, T.F. Zeng, D.W. Song, C.D. Moore, G. Chen, K. L. Wang, M.S. Goorsky, T. Radetic, R. Gronsky, T. Koga, M.S. Dresselhaus, Superlattices Microstruct. 28 (2000) 199. [32] J. Garg, N. Bonini, N. Marzari, Nano Lett. 11 (2011) 5135. [33] E.S. Landry, A.J.H. McGaughey, Phys. Rev. B 79 (2009) 075316.
[34] Z. Guo, D. Zhang, X.-G. Gong, Appl. Phys. Lett. 95 (2009) 163103. [35] L. Pan, H.J. Liu, X.J. Tan, H.Y. Lv, J. Shi, X.F. Tang, G. Zheng, Phys. Chem. Chem. Phys. 14 (2012) 13588. [36] M. Morooka, T. Yamamoto, K. Watanabe, Phys. Rev. B 77 (2008) 033412. [37] J.-W. Jiang, J.-S. Wang, B. Li, Phys. Rev. B 79 (2009) 205418. [38] J. Lan, J.-S. Wang, C.K. Gan, S.K. Chin, Phys. Rev. B 79 (2009) 115401.