Effect of topological line defects on electron-derived thermal transport in zigzag graphene nanoribbons

Accepted Manuscript Effect of topological line defects on electron-derived thermal transport in zigzag graphene nanoribbons Zhong-Xiang Xie, Yong Zhang, Li-Fu Zhang, Dian-Yuan Fan PII:

S0008-6223(16)31039-9

DOI:

10.1016/j.carbon.2016.11.065

Reference:

CARBON 11504

To appear in:

Carbon

Received Date: 3 September 2016 Revised Date:

9 November 2016

Accepted Date: 23 November 2016

Please cite this article as: Z.-X. Xie, Y. Zhang, L.-F. Zhang, D.-Y. Fan, Effect of topological line defects on electron-derived thermal transport in zigzag graphene nanoribbons, Carbon (2016), doi: 10.1016/ j.carbon.2016.11.065. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Near the intrinsic Fermi level, the normalized thermal conductance for both 4-TLD

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and 8-TLD is the same as that pristine ZGNRs regardless of the introduction of 4-TLD and 8-TLD. However, both 558-TLD and 5757-TLD can enhance the electron-derived thermal conductance due to the additional contribution of the defective state. When the Fermi level is far above the top of the first subband, the normalized thermal conductance for these four TLDs is much lower than the

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corresponding value for pristine ZGNRs, since the high-energy electronic structure

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for the TLDs is less dispersive compared to pristine ZGNRs.

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Effect of topological line defects on electron-derived thermal transport in zigzag graphene nanoribbons Zhong-Xiang Xie a, b∗, Yong Zhang b, Li-Fu Zhang a∗, Dian-Yuan Fan a SZU-NUS Collaborative Innovation Center for Optoelectronic Science Technology,

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a

Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University,

Department of Mathematics and Physics, Hunan Institute of Technology, Hengyang

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b

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Shenzhen 518060, China

421002,China

Abstract

We investigate electron-derived thermal transport in zigzag graphene nanoribbons

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(ZGNRs) with topological line defects (TLDs) including tetragon-TLD (4-TLD), octagon-TLD (8-TLD), double pentagons and octagon-TLD (558-TLD), as well as double pentagons and double heptagons-TLD (5757-TLD). Results show that these

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TLDs can efficiently modify the band structure and electron-derived thermal

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conductance. Near the intrinsic Fermi level, the normalized thermal conductance for both 4-TLD and 8-TLD (both 558-TLD and 5757-TLD) is the same as (higher than) that of pristine ZGNRs. When the Fermi level is far above the top of the first subband, the normalized thermal conductance for four TLDs is much lower than the pristine value. We also show that the TLDs’ position dependence of the normalized thermal conductance exhibits similar trends for different types of TLDs, but this trend strongly ∗ Corresponding author. Tel: +86 0755-26010620. E-mail address: [email protected] (Z.-X. Xie), [email protected] (L.-F. Zhang)

ACCEPTED MANUSCRIPT depends on the Fermi level. A brief analysis of these results is given. 1.

Introduction Graphene, a single atomic layer of graphite, has drawn much interest due to its

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outstanding structural, mechanical, electronic, as well as thermal properties [1-8]. What is more interesting is that graphene can be tailored into graphene nanoribbons (GNRs) with well-controlled size and edge shape by the mechanical cutting. It has

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been demonstrated that the band structures of GNRs can be modulated by the edge

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shape and ribbon width [9]. However, several defects are inevitably introduced into graphene during the growth process or during the irradiation with electrons or ions. Much effort has been devoted to exploring effects related to the defects such as Stone-Wales defects, vacancies, substitutional doping, and so on, in graphene-based

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structures [10-14]. It is well known that these defects will result in the localized states at particular energies, and thus modify the electronic, thermal, and thermoelectric properties of graphene and GNRs [13].

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Recently, a class of peculiar topological line defects (TLDs) embedded in

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graphene were experimentally and theoretically reported [15-19]. Lahiri et al. firstly synthesized a TLD with repeating octagons and double-pentagons (558-TLD) created by alternating divacancies and Stone-Thrower-Wales defect [15]. Following this work, Huang et al. observed a different type of the TLD containing a pair of pentagon-heptagon rings by stitching different grains [16]. These pentagons, heptagons, and octagons of carbon within the TLDs are theoretically predicted to result from the reconstruction of periodic divacancies [17]. Very recently, a new type

ACCEPTED MANUSCRIPT of TLD with tetragonal carbon rings (4-TLD) as the initial geometry to stabilize other types of TLDs, was theoretically predicted [18] and experimentally confirmed later [19]. It is proposed that TLDs may present a new opportunity to tune the band

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structures and the transport properties due to the introduction of non-hexagonal carbon rings. Experimentally, it is shown that the 558-TLD embedded in graphene exhibits the metallic characteristics [15]. Using a tight-binding model, Song et al.

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show that a quantum channel can be constructed by using a 558-TLD, and such

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channel can be controlled by tuning the gate voltage [20]. Based on the first-principles calculations, Guan et al. present a study of the electronic structure in ZGNRs with the 558-TLD, where the transformation among the spin-polarized metal, metal with Dirac point, and half-metal can be observed by varying the position of 558-TLD [21]. The

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electron transport through GNRs with various TLDs has been intensively investigated [22-25]. Some intriguing electron-transport properties such as the controllable 100% valley polarization [22], the Fano effect [25], and so on, are revealed. On the other

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hand, some researchers have predicted the influence of TLDs on the phonon-derived

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thermal transport in graphene-based nanostructures [26,27]. In their studies, it is reported that the thermal conductance contributed by phonons can be tuned by over wide ranges by controlling the direction and bond configuration of TLDs [26,27]. More recently, molecular-dynamics simulations show that the phonon thermal conductivity at room temperature displays the similar behaviors for 5757-TLD and 558-TLD, and in both systems the thermal conductance in the direction of the TLDs is better than that perpendicular to the TLDs [28].

ACCEPTED MANUSCRIPT In spite of these above progresses in exploring TLDs-induced effects in graphene-based nanostructures, a scientific study of the electron-derived thermal transport in GNRs with TLDs, along with the effect of TLDs (such as the type and the

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position of TLDs) on the thermal conductance of electrons, is still absent. Previous studies have demonstrated that the electronic contribution to thermal conductance becomes comparable to the phonon contribution in gated graphene at low

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temperatures when the Fermi level is large enough [29]. In heavy doped graphene, the

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electronic thermal conductance is even higher than the phonon thermal conductance. More recently, it has been also predicted that in stanene, as a counterpart of graphene, the electronic thermal conductance is as important as the phonon thermal conductance, and even it becomes dominant at room temperature [30]. Thus, the electron-derived

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thermal transport in GNRs with the TLDs is worthy of particular attention. In a recent study, the thermoelectric properties in zigzag GNRs (ZGNRs) with the TLD have been reported by Karamitaheri et al [12]. In their study, however, only the 558-TLD is

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considered. In the present work, we systematically investigate the effects of various

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types of TLDs on the electron-derived thermal transport in ZGNRs. Here, several representative TLDs including 4-TLD, 8-TLD, 558-TLD, as well as 5757-TLD, are considered. It has been theoretically predicted that these TLDs can be formed by the reconstruction at the interface of two adjacent zigzag edges of GNRs [17-19], and some of them have been observed experimentally [15,16]. Our results show that these TLDs play an important role in influencing the electron-derived thermal transport in ZGNRs. Near the intrinsic Fermi level, the normalized thermal conductance for both

ACCEPTED MANUSCRIPT 4-TLD and 8-TLD is the same as that for pristine ZGNRs. However, the normalized thermal conductance for both 558-TLD and 5757-TLD is much higher than that for pristine ZGNRs due to the additional contribution of the defective state. When the

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Fermi level is far above the top of the first subband, the normalized thermal conductance for four TLDs is lower than the pristine value, since the high-energy band structure for the TLDs is less dispersive compared to pristine ZGNRs. It is also

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show that the TLDs’ position dependence of the normalized thermal conductance

the Fermi level. 2.

Method and model

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exhibits similar trends for different types of TLDs, but this trend strongly depends on

We consider four types of TLDs, as shown in Fig. 1. Figs. 1(a)-(d) correspond to

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4-TLD (tetragonal carbon rings), 8-TLD (octagonal carbon rings), 558-TLD (octagonal and double-pentagonal carbon rings), and 5757-TLD (double-pentagonal and double-heptagonal carbon rings), respectively. In each figure, the position of the

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TLDs embedded in ZGNRs is denoted by a pair of integers (m, n), where m and n are

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the number of the zigzag carbon chair above and below the TLD, respectively. In the following numerical calculations, the electron-derived thermal transport can be described by the tight-binding model with the nearest neighbor approximation. Within this model, the hopping parameter t is set to -2.7 eV and the on-site parameter is taken as 0 [12]. This model has been recently employed to describe the electron transmission in graphene with TLDs, and the results are consistent with first-principle calculations [12,20,23]. It has been also proved that although the rearrangement of

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Fig. 1 Schematic diagram of four types of TLDs embedded in ZGNRs. (a)-(d) denote 4-TLD

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(tetragonal carbon rings), 8-TLD (octagonal carbon rings), 558-TLD (double-pentagonal and

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octagonal carbon rings), and 5757-TLD (double-pentagonal and double-heptagonal carbon rings), and, respectively. In each figure, the position of TLDs is characterized by a pair of integers (m, n), where m and n denote the number of carbon chairs above and below the TLD, respectively. The red box stands for the unit cell.

carbon atoms near ELDs will lead to the change of bond distance, the hopping energy t is nearly unaffected with respect to defect-free GNRs [31]. According to the Landauer formula [29], the ballistic thermal current by electrons along the x direction

ACCEPTED MANUSCRIPT can be described as follows: 2 ∞ ( E - m)[ f ( E, µ , Thot ) - f ( E, µ , Tcold )]τ ( E)dE , h ∫0

J=

(1)

where τ ( E) denotes the transmission coefficient corresponding to the number of

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available electron-states, f ( E, µ , Thot ( cold ) ) is the Fermi distribution function with the Fermi level µ , and h is the Planck constant. In the linear response regime, the

2k T h

∫

∞

0

E−µ ) kBT E−µ 2 τ ( E)dE . ( ) E−µ kBT 2 [exp( ) + 1] kBT exp(

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κ=

2 B

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thermal conductance κ is given by κ = J / ∆T ( ∆T = Thot − Tcold → 0) , and as below

(2)

Introducing the scaled energy variable x = ( E − µ ) / kBT gives the below expression

κ=

2 kB2T ∞ 2 exp( x) x τ ( xkBT + µ )dx . h ∫0 [exp( x) + 1]2

(3)

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Note that the above equation shows that kB2T / h plays the role of the quantum unit of the electron-derived thermal conductance, similar to the phonon case [10]. Numerical results and discussion

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3.

To elucidate the effects of the TLDs on the electron-derived thermal transport,

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we firstly compare the band structure between ZGNRs with and without TLDs, as shown in Fig. 2. Due to the electron-hole symmetry of the Hamiltonian, pristine ZGNRs exhibit the symmetric band structure with respect to the charge neutrality point (CNP) with E=0, in agreement with the previous result [12]. The symmetric band structure can be also observed in ZGNRs with 4-TLD or 8-TLD, while it cannot be observed in ZGNRs with 558-TLD or 5757-TLD. Intriguingly in Figs. 2(b)-(e), it is found that a special band (red curve) around the CNP can be observed in ZGNRs

ACCEPTED MANUSCRIPT with TLDs. This special band is different from other bands in ZGNRs, and is different for different types of TLDs. For example, the special band for both 4-TLD and 8-TLD is absolutely flat, while it for both 558-TLD and 5757-TLD is upward-sloping. These

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behaviors can be understood by the following two aspects. One aspect is that the

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TLDs embedded in ZGNRs can induce the special band, corresponding to the

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Fig. 2 The electron-energy disperse relations for ZGNRs (a) without the TLD, (b) with 4-TLD, (c) with 8-TLD, (d) with 558-TLD, and (e) with 5757-TLD. Here, we take m=6 and n=6.

defective state. The other aspect, as mentioned in Ref. 12 is that the profile of the special band is determined by the carbon-atom arrangement in ZGNRs with TLDs. In general, the carbon-atom arrangement in graphene-based structures can be split into sublattices "A" and "B". For ZGNRs with 4-TLD and 8-TLD, sublattices "A" and "B" are coupled with each other, which results in the special band being absolutely flat.

ACCEPTED MANUSCRIPT However, for ZGNRs with 558-TLD and 5757-TLD, "A" is connected to "A" or "B" is connected to "B" in the defective region, and so the special band is upward-sloping. Note that the defective state (namely the special band) for 558-TLD and 5757-TLD

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can make the additional contribution to the thermal conductance of electrons, while it for 4-TLD and 8-TLD has no contribution. This will lead to the fact that near the intrinsic Fermi level ( µ =0 eV), the thermal conductance for 558-TLD and 5757-TLD

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is higher than that for 4-TLD, 8-TLD, and even pristine ZGNRs, as discussed below.

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It is also shown that when the TLDs are introduced, the band structure becomes less

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dispersive than that for pristine ZGNRs, and it for 5757-TLD is least dispersive. It is

Fig. 3 The transmission coefficient of electrons versus the electron-energy. The top-right inset shows the detailed behavior of the transmission coefficient. The parameters employed here are the same as those in Fig. 2.

ACCEPTED MANUSCRIPT worthy pointing out that 4-TLD makes the energy scope wider compared to pristine ZGNRs, indicating the transmission coefficient for 4-TLD is larger than that for the pristine case at very high energy, as shown in Fig. 3. From Fig. 3, it is also shown that

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the transmission coefficients exhibit a series of stair-stepping platforms corresponding to the available channels of electrons. For all the cases, the maximum of the transmission coefficient at E ≈ 3 eV can be observed. When the electron-energy is

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below 1eV, the transmission coefficient for 558-TLD and 5757-TLD is higher than the

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corresponding value for pristine ZGNRs, while it for 4-ELD and 8-ELD is the same as the pristine value. Note that for ZGNRs with 5757-TLD, zero transmission coefficient occurs at E ≈ ±1.5 eV, indicating two gaps will be opened by 5757-TLD, as shown in the inset of Fig. 3.

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Now, we turn to elucidate the effects of 4-TLD, 8-TLD, 558-TLD, and 5757-TLD on the thermal conductance of electrons, respectively. Fig. 4 plots the normalized thermal conductance divided by the quantum value κ 0 (κ 0 = π 2 kB2T / 3h) ,

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as a function of the Fermi level at different temperatures. When the Fermi level is

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increased, the normalized thermal conductance for ZGNRs with 4-TLD and 8-TLD is increased steplikely due to the increasing available electron-channels. On the other hand, the normalized thermal conductance for ZGNRs with 558-TLD and 5757-TLD shows an increasing trend with the Fermi level, but its Fermi level dependence exhibits the nonmonotonic behavior with some valleys at the Fermi level ranging from 1.2 eV to 1.8 eV. As mentioned above, this nonmonotonic behavior is due to the transmission dips associated with Van Hove Singularities. Near the intrinsic Fermi

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Fig. 4 The normalized thermal conductance κ / κ 0 of electrons as a function of the Fermi level

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µ at (a) T=1 K, (b) T=50 K, (c) T=300 K, and (d) T=500 K. The solid, dashed, dotted, and dot-dashed curves correspond to 4-TLD, 8-TLD, 558-TLD, and 5757-TLD, respectively. For the

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sake of comparison, the case without the TLD is also calculated. Here, m=12 and n=12.

level, the step-height of the normalized thermal conductance for 4-TLD and 8-TLD is

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2, while it for 558-TLD and 5757-TLD is 4. These indicate that when the Fermi level is very small, both 588-TLD and 5757-TLD can enhance the thermal conductance of electrons compared to pristine GNRs, while both 4-TLD and 8-TLD cannot. When the Fermi level is far above the top of the first subband, the normalized thermal conductance for four TLDs is smaller than that of the pristine case, since the band structure for the TLDs is less dispersive than that for pristine ZGNRs, as shown in Fig. 2. Similar behaviors can be also observed at higher temperatures. Note that at higher

ACCEPTED MANUSCRIPT temperatures, the staircases of the normalized thermal conductance become indistinct,

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and even absent, due to the wider energy integration in Eq. (3).

Fig. 5 The normalized thermal conductance κ / κ 0 of electrons as a function of the temperature T

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at (a) µ = 0 eV, (b) µ = 0.5 eV, (c) µ = 1 eV, and (d) µ = 3 eV. Other parameters employed here are the same as those in Fig. 4.

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Fig. 5 plots the normalized thermal conductance as a function of the temperature

T at different Fermi level. In the zero temperature limit, the electron-derived thermal conductance becomes an integral multiple of the quantized thermal conductance κ 0 . This is consistent with the phonon case, where the phonon-derived thermal conductance present three multiple of the quantized thermal conductance due to the contribution of three branches of acoustic phonon modes [10,32,33]. But at low temperature, the electronic thermal conductance is much higher than the phonon

ACCEPTED MANUSCRIPT thermal conductance when the Fermi level takes a larger value. Near the intrinsic Fermi level, the normalized thermal conductance for 4-TLD and 8-TLD is independent of the variation of the temperature because the contribution to the

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electron-derived thermal conductance only results from the first subband with the large bandwidth. When the Fermi level lies near the edge of the first subband, the normalized thermal conductance for four TLDs monotonically increases with

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temperature, since more bands above and below the Fermi level contribute to the

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thermal conductance with increasing the temperature. Interestingly when µ =3eV, the normalized thermal conductance for 4-TLD and 8-TLD (558-TLD and 5757-TLD) firstly increases (decreases), then tends to be a constant value. Moreover, the constant value for 4-TLD and 8-TLD is more than 558-TLD and 5757-TLD. In the temperature

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dependence of the thermal conductance instead of the normalized thermal conductance divided by κ 0 , the thermal conductance κ

for all the cases

monotonically increases with increasing the temperature.

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To further reveal the influence of TLDs on the electron-derived thermal

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conductance in ZGNRs, we also explore the TLDs' position dependence of the normalized thermal conductance at room temperature (T=300 K), as shown in Fig. 6. It is seen that the normalized thermal conductance exhibits similar behaviors for different types of the TLDs. At the intrinsic Fermi level, the normalized thermal conductance for all the TLDs considered here seems to be independent of their position. This is because that the low-energy band structure near the CNP isn't nearly affected by the position of the TLD, similar to the phonon case where defects hardly

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Fig. 6 The normalized thermal conductance κ / κ 0 of electrons as a function of the position m of the ELD at (a) µ = 0 eV and (b) µ = 0.5 eV. Here, we fix m+n=24 and T=300 K.

influence the phonon transmission of ZGNRs in low frequency limit [10]. At the

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Fermi level being 0.5 eV, when the position of the TLDs is shifted from the edge to the middle of ZGNRs, the normalized thermal conductance starts to decrease rapidly,

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then becomes nearly insensitive to the variation of the position of the TLDs. Similar behaviors can be also observed in wider ZGNRs (not shown). Note that in wider

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ZGNRs, the effects of the TLDs are always less complicated due to the decrease of the edge effect. It is envisioned that similar to the phonon case [10], the influence of TLDs on the electron-derived thermal conductance of ZGNRs may be consistent with that of carbon nanotubes when their size becomes large enough. Finally, we discuss the relation between the electronic thermal conductance κ and the electronic conductance G for ZGNRs with TLDs. It is well known that the

ACCEPTED MANUSCRIPT Wiedemann-Franz law with the Lorentz number L0 =

κ GT

=

π 2 kB2 3e 2

, is invalid in 2D

materials that possess massless Dirac fermions [30]. In a recent study, several researchers also observed the breakdown of the Wiedemann-Franz Law in graphene

given by 2e2 h

∫

∞

0

exp( x) [exp( x) + 1]2

τ ( xkBT + µ )dx ,

(4)

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G=

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[34]. For ZGNRs with TLDs, the electronic conductance G in the ballistic region is

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where x and τ are the same as those in Eq. (3). At high temperatures, the behavior of the electronic thermal conductance versus the chemical potential is not parallel to that of the electronic conductance versus the chemical potential, due to the wide energy integration in Eqs. (3) and (4). Thus, it is here envisioned that the ration of the

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electronic thermal conductance and the electric conductance becomes sensitive to the chemical potential, indicating the Wiedemann-Franz law is broke. Note that in the low temperature limit, κ and G are taken as Nκ 0 and NG0 (G0 = e 2 / h) with N

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being the number of available subbands, respectively. This indicates that in the low

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temperature limit, the Wiedemann-Franz law is satisfied for ZGNRs with TLDs, in agreement with that for pristine metallic GNRs [29]. 4.

Conclusion

In conclusion, we have studied ballistic thermal transport of electrons for ZGNRs

with TLDs by using the tight-binding model. Four types of ELDs: 4-TLD, 8-TLD, 558-TLD as well as 5757-TLD, are considered here. It is shown that these TLDs can effectively modify the band structure and electron-derived thermal conductance. Near

ACCEPTED MANUSCRIPT the intrinsic Fermi level, the normalized thermal conductance for both 4-TLD and 8-TLD is the same as that pristine ZGNRs, while it for both 558-TLD and 5757-TLD is higher than the pristine value due to the additional contribution of the defective

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state. These indicate that both 558-TLD and 5757-TLD can enhance the electron-derived thermal conductance at very small Fermi level, while both 4-TLD and 8-TLD cannot. When the Fermi level is far above the top of the first subband, the

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normalized thermal conductance for these four TLDs is lower than the corresponding

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value for pristine ZGNRs, since the high-energy electronic structure for the TLDs is less dispersive compared to pristine ZGNRs. We also show that the normalized thermal conductance exhibits the similar trend with the variation of the TDLs' position for different types of the TLDs, but this trend strongly depends on the Fermi Level.

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Acknowledgements

This work is supported supported by the National Natural Science Foundation of China (Grant Nos. 11404110, 11547205, 11547197), the Natural Science Foundation

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of SZU (Grant No. 000053), the Science and Technology Planning Project of

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Guangdong Province (Grant No. 2016B050501005), and the Outstanding Young Program of Hunan Provincial education department of China (Grant No. 14B046).

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