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Thermal transport and thermoelectric properties of beta-graphyne nanostructures
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Nanotechnology Nanotechnology 25 (2014) 245401 (10pp)
doi:10.1088/0957-4484/25/24/245401
Thermal transport and thermoelectric properties of beta-graphyne nanostructures Tao Ouyang1,3 and Ming Hu1,2 1
Institute of Mineral Engineering, Division of Materials Science and Engineering, Faculty of Georesources and Materials Engineering, Rheinisch-Westfaelische Technische Hochschule (RWTH Aachen University), 52064 Aachen, Germany 2 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, 52062 Aachen, Germany E-mail: [email protected] Received 13 March 2014, revised 9 April 2014 Accepted for publication 14 April 2014 Published 23 May 2014 Abstract
Graphyne, an allotrope of graphene, is currently a hot topic in the carbon-based nanomaterials research community. Taking beta-graphyne as an example, we performed a comprehensive study of thermal transport and related thermoelectric properties by means of nonequilibrium Green’s function (NEGF). Our simulation demonstrated that thermal conductance of beta-graphyne is only approximately 26% of that of the graphene counterpart and also shows evident anisotropy. Meanwhile, thermal conductance of armchair beta-graphyne nanoribbons (A-BGYNRs) presents abnormal stepwise width dependence. As for the thermoelectric property, we found that zigzag beta-graphyne nanoribbons (Z-BGYNRs) possess superior thermoelectric performance with figure of merit value achieving 0.5 at room temperature, as compared with graphene nanoribbons (∼0.05). Aiming at obtaining a better thermoelectric coefficient, we also investigated ZBGYNRs with geometric modulations. The results show that the thermoelectric performance can be enhanced dramatically (figure of merit exceeding 1.5 at room temperature), and such enhancement strongly depends on the width of the nanoribbons and location and quantity of geometric modulation. Our findings shed light on transport properties of beta-graphyne as high efficiency thermoelectrics. We anticipate that our simulation results could offer useful guidance for the design and fabrication of future thermoelectric devices. Keywords: graphyne, thermal transport, thermoelectrics, nonequilibrium Green’s function, energy conversion (Some figures may appear in colour only in the online journal) 1. Introduction
conversion efficiency of thermoelectric materials [2], ZT = S2σ/k, where S is the Seebeck coefficient, σ the electronic conductance, and k (k = ke + kp) the thermal conductance composed of the contributions from electrons (ke) and phonons (kp), respectively. However, it is very challenging to obtain high thermoelectric performance in a conventional bulk system, due to the strong coupling effect between the electronic and phononic transport in ZT [3]. Low-dimensional materials as proposed by Hicks and Dresselhaus [4], on the other hand, provided us a promising route to decouple the relationship between electronic and phononic conductance and thus led to the significant increase in ZT. Therefore, a tremendous amount of research in the past decade [5–8] has
Recently, recycling waste heat from thermal engines has aroused widespread interest for both theoretical and technological research, owing to the gradual exhaust of fossil fuels [1]. To achieve this goal, one of the most efficient approaches involves utilizing thermoelectric effect, which can convert a temperature gradient to electricity directly. The dimensionless figure of merit (ZT) is defined to measure the energy 3
On leave from Hunan Key Laboratory for Micro-Nano Energy Materials and Device and Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China.
0957-4484/14/245401+10$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
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the thermoelectric performance of zigzag β-graphyne nanoribbons. In section 5, we present a summary and conclusion.
focused on fabricating and optimizing nanomaterials for high thermoelectric performance. Graphene and its derivatives, as representatives of lowdimensional carbon materials with hexagonal lattice structure, possess a wide range of super physical properties and have attracted exceptional attention recently. Although the electronic properties of graphene and its related nanostructures are fascinating (including record high value of carrier mobility and giant Seebeck coefficient) [9–11], they are not efficient thermoelectric materials (ZT ∼ 0.05) [12], primarily due to their extremely high thermal conductance [13–15]. In order to overcome this drawback, significant measures have been taken to decrease the thermal conductance and thus enhance the thermoelectric performance [16–24], e.g., creating scattering by edge roughness [16], isotope engineering [17, 18], defect engineering [19–21], and introducing electronic resonance in periodically patterned nanoholes and superlattice structures [12, 22–24]. However, little previous work can improve the figure of merit of graphene and its derivatives to exceed 1.0 at room temperature (for very narrow graphene ribbons with complex structure, the ZT can achieve 2.0 at room temperature) [18]. Graphyne, a new type of planar carbon allotrope containing both sp and sp2 hybridized states, is a new topic in the present carbon nanomaterials research communities [25–27]. Due to the presence of acetylenic fragments, graphyne holds four typical geometrical structures, namely, alpha(α)-graphyne, beta(β)-graphyne, gamma(γ)-graphyne, and (6,6,12)graphyne [26]. These novel carbon allotropes not only share unique mechanical and chemical properties with graphene [28–31], but also exhibit a rich variety of amazing electronic and thermal transport performance [32–38], which qualify them as powerful competitors of graphene. Malko et al [39] predicted that Dirac cones and their associated electronic transport properties exist in α-and β-graphyne as well. In addition, (6,6,12)-graphyne is found to have two self-doped nonequivalent distorted Dirac cones, indicating that it naturally contains conducting charge carriers and could conduct electrons in a preferred direction. The intrinsic carrier mobility of the γ-graphyne and (6,6,12)-graphyne can reach as high as 105 cm2 Vm−1 at room temperature, which is comparable to that of graphene [40, 41]. More interestingly, the recent studies showed that the phonon-contributed thermal conductance of graphyne is fundamentally lower than that of graphene, benefiting from the insertion of acetylenic linkages [37, 38]. These works suggest that graphyne is very promising for high thermoelectric performance [42, 43]. In this paper, taking β-graphyne as an example, we performed a comprehensive study of thermal transport and related thermoelectric properties of β-graphyne by using nonequilibrium Green’s function (NEGF) method. In section 2, we briefly describe our model structure and NEGF formulas for both electron and ballistic phonon transport. In section 3, we report thermal transport properties of β-graphyne nanoribbons. The corresponding thermoelectric properties of β-graphyne nanoribbons are shown in section 4. In section 5, we discuss the effect of geometric modulation on
2. Model structure and simulation methodology By cutting infinite β-graphyne sheet along different directions, one can obtain two typical β-graphyne nanoribbons with different edges: zigzag β-graphyne nanoribbons (ZBGYNRs) (figure 1(a)) and armchair β-graphyne nanoribbons (A-BGYNRs) (figures 1(b) and (c)). Unlike the cases of ZBGYNRs, the A-BGYNRs could possess two types of edge terminal named as A-BGYNR-I and A-BGYNR-II, which are depicted in figures 1(b) and (c). The width of A-BGYNRs (ZBGYNRs) is denoted by NA (NZ), which is determined by the number of sp2 state dimer lines along the non-periodic direction. It should be mentioned that the configuration of ABGYNR-I will be identical to that of A-BGYNR-II when NA is an even number. To calculate electronic transport properties of these nanostructures, the Hamiltonian is described by an atomistic π-orbital tight-binding (TB) model with nearest-neighbor coupling, which is defined as H=
∑ εici+ci + ∑ ti,jci+cj , i
(1)
i, j
where εi is the site energy at the ith atom and ti, j is nearestneighbor hopping energy. The corresponding parameters are taken from [44] and [45]. Based on this Hamiltonian, the retarded Green’s function is expressed as [46] G r ( E ) = ⎡⎣ ( E + i0+) I − Hc −
∑L − ∑R ⎤⎦ r
r
−1
,
(2)
where E is the electron energy, I is the identity matrix, and Hc is the Hamiltonian matrix of the central region. The r ∑β = V Cβgβr V βC ( β = L, R, corresponding to the left and right electronic leads, respectively) denotes the self-energy of T
( )
electronic lead β, where V βC = V Cβ
is the coupling matrix
of the lead β to the center region and gβr is the lead surface Green’s function calculated via an iterative procedure. Once the retarded Green’s function Gr ( E ) is obtained, we can calculate electronic transmission coefficient Te [E ] [46] Te [E ] = Tr { G r ( E ) ΓLG a ( E ) ΓR},
(
r
a
(3)
)
where Γβ = i ∑β − ∑β = −2 Im V Cβgβr V βC is the coupling function of the β lead. Using the electronic transmission coefficient Te [E ], the electronic conductance σ, the Seebeck coefficient S, and the electron-contributed thermal conductance ke can be calculated based on the Onsager’s relations and the Landauer’s theory of quantum transport [20] σ ( u) = e 2 L 0 ( u , T ) , S ( u) = 2
1 L1 ( u , T ) , eT L 0( u , T )
(4a) (4b)
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Figure 1. Schematic of infinite β-graphyne sheet (gray meshes): (a) zigzag-beta(β)-graphyne (Z-BGYNR), (b) armchair-beta(β)-graphyne (A-
BGYNR) with edge terminal of type I (A-BGYNR-I), and (c) A-BGYNR with edge terminal of type II (A-BGYNR-II). The yellow circles are carbon atoms. The widths of the Z-BGYNR and A-BGYNR are denoted by NZ and NA, respectively.
unit cells of the thermal lead in all of our calculations. As for the systems with structural defects, we have also added buffer layers (about three unit cells) between the central region and thermal leads to avoid the interference originated from the defect on the thermal leads. The phonon–phonon and electron–phonon interaction are neglected in our calculation. The phonon contributed thermal conductance is given by [49]
2
ke ( u) =
L1( u , T ) 1 , L 2 ( u, T ) − L 0( u , T ) T
(4c)
where Ln ( u, T ) =
2 h
+∞
∫ ( E − u) +∞
n
⎡ ∂f ( E , u , T ) ⎤ ⎥ Te [E ] dE . (5) ⎢− e ⎥⎦ ⎢⎣ ∂E
kp (T ) =
is Lorenz integral with fe ( E , u , T ) is the Fermi–Dirac distribution function at the chemical potential u and temperature T. For thermal transport, phonon transmission coefficient Te [ ω] of β-graphyne nanostructures can be calculated in harmonic approximation, which is similar to the way the electrons are calculated [47–49]. One only needs to substitute E by ω2 and HC by KC in the equation (2), and then to calculate the self-energies and retarded Green’s function accordingly. Here ω is the frequency of phonons, and KC is the mass-weighted force constant matrix of the central region. The second-generation reactive empirical bond order potential [50], which was proven to give excellent description of carbon–carbon bonding interactions [37, 51], is adopted to optimize the geometric structure and to obtain the force constants of β-graphyne nanostructures. Since this potential holds long-range character, it leads to unphysical results if the length of the central region is too small (i.e., there exists interaction between the left and right thermal leads). To avoid this, we have tested the convergence of the thermal conductance as a function of the central region’s size. The results show that when the length of the central region exceeds five unit cells of the thermal lead, the thermal conductance of betagraphyne nanoribbons does not change with further increase in size. Therefore, the length of the central region is set as five
ℏ 2π
∫
0
∞
T [ ω] ω
∂fp ( ω) ∂T
dω ,
(6)
where fp ( ω) is the Bose–Einstein distribution function of phonons. 3. Thermal transport properties of β-graphyne nanoribbons In figure 2 we depict the phonon part of thermal conductance scaled by cross-sectional area (kp/A) as a function of width (W) for A-BGYNRs and Z-BGYNRs at room temperature (300 K). Herein the cross-sectional area A is defined as A = Wh, where h = 0.34 nm is chosen as the layer separation of bulk β-graphyne. One can find from figure 2 that the scaled thermal conductance of A-BGYNR-I and A-BGYNR-II shares a similar phenomenon, although their edge terminals are quite different. The thermal conductance of A-BGYNRs first decreases rapidly with width increasing for narrow ribbons (W < 3 nm) and then levels off for wider nanoribbons (W ⩾ 3 nm). More interestingly, the scaled thermal conductance of A-BGYNR exhibits an oscillatory behavior as the width increases. The oscillation mainly originates from the stepwise width dependence of the thermal conductance, as shown in the inset of figure 2. In contrast, for Z-BGYNRs, the 3
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DOS/PDOS (arbitrary units)
A-BGYNR-I A-BGYNR-II Z-BGYNR
κP (nW/K)
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PDOS-Center PDOS-Edge Overall DOS
(a)
PDOS-Center PDOS-Edge Overall DOS
(b)
PDOS-Center PDOS-Edge-I PDOS-Edge-II Overall DOS
(c)
500
1000
Frequency
Figure 2. Phonon part of thermal conductance scaled by cross-
1500 -1 (cm )
2000
Figure 3. Phonon projected density of states (PDOS) and overall
sectional area (kp/A) as a function of nanoribbon width (W) for ABGYNRs and Z-BGYNRs at room temperature. The inset shows unscaled thermal conductance as a function of width for A-BGYNRs at room temperature.
density of states (DOS) vs phonon frequency for (a) A-BGYNR-I with NA = 5, (b) A-BGYNR-II with NA = 5, and (c) A-BGYNR with NZ = 6.
scaled thermal conductance increases gradually with the width, implying that the zigzag edge makes a weak contribution to the thermal transport. When the width is about 10 nm, the scaled thermal conductance of A-BGYNR and ZBGYNR is approximately 1.14 and 0.94 nW (Knm−2), respectively. The scaled thermal conductance of BGYNRs is found to be only 26% of that for graphene nanoribbons [GNR, about 3.6–4.2 nW (Knm−2)] [51]. The large reduction is mainly attributed to the inefficient thermal transport in the acetylenic fragments. Similar phenomena have also been observed in previous studies using molecular dynamic simulation [38]. Meanwhile, the different values of the scaled thermal conductance of BGYNRs for different edges also indicate the anisotropic thermal transport property in β-graphyne, which is consistent with previous studies for GNRs and γ-graphyne nanoribbons [37, 51]. In order to elucidate the unique stepwise width dependence of thermal conductance of A-BGYNRs, the phonon density of states (DOS) and projected density of states (PDOS) versus the phonon frequency are plotted in figure 3. One can find that the phonon DOS of A-BGYNR-I is mainly contributed by the atoms in the central region. That is to say, owing to the edge effect, the atoms in the edge region could not propagate the thermal transport effectively, which results in weak contribution to the overall phonon DOS. For ABGYNR-II, shown in figure 3(b), however, the phonon PDOS from the edge region is more or less similar to that from the central region, especially for frequencies smaller than 750 cm−1, indicating that the edge has a weak influence on the phonon transport of A-BGYNRs with type-II edge terminal. This different edge effect can be seen more clearly when comparing two cases in figure 3(c). Obviously, the phonon PDOS of edge region with type-II terminal is larger than that with type-I edge terminal. The different edge effect can be
simply understood in terms of the special geometric structure of A-BGYNRs (figure 1). Figure 1(c) shows that passivated carbon atoms embed in the edge transport channel of ABGYNR-I, which will damage the continuity of the thermal transport, while the passivated carbon atoms of A-BGYNR-II remain away from the edge channel, thus protecting the thermal transport path from interruption. Therefore, the thermal conductance of A-BGYNR-I is similar to that of ABGYNR-II although it possesses more edge channels and thus leads to the edge terminal independence of thermal conductance. It should be noted that the difference in the scaled thermal conductance between A-BGYNR-I and ABGYNR-II shown in figure 2 is a result of the differing widths. For the stepwise width dependence of thermal conductance in A-BGYNRs, it can be explained as follows: when the width of A-BGYNR (NA) increases from an odd number to even number, the increased width will bring in more phonon modes. However, the transmission of these new phonon modes is restricted by the width of the narrowest part of the nanoribbons and thus they cannot contribute to the thermal transport effectively. As a result, the thermal conductance of A-BGYNRs exhibits interesting stepwise width dependence. In figure 4 the phonon spectra and phonon transmission of A-BGYNR-I (NA = 5, W = 1.98 nm) and Z-BGYNR (NZ = 4, W = 2.07 nm) are depicted to explain the anisotropic thermal transport property of BGYNRs. Only the low frequency region is provided because the phonons in this region dominate the thermal transport at room temperature. By comparing figures 4(a) and (b), it can be found that in general the phonon branches in A-BGYNR-I possess larger slope (therefore larger phonon group velocity) than those in ZBGYNR, where there are many flat branches corresponding to very low phonon group velocity. In other words, the 4
T Ouyang and M Hu
1.2 A-BGYNRs Type I Type II 300k 500k 700k
Max ZT
1.0
0.8
Thermal power (pW/K2)
Nanotechnology 25 (2014) 245401 3.0 2.5
A-BGYNR-I A-BGYNR-II
2.0 1.5 1.0 0.5 0.0 0
2
µ (ev)
4
6
0.6
0.4
(a) 0.2 Z-BGYNRs
Figure 4. Phonon spectra for (a) A-BGYNR-I with NA = 5, (b) ZBGYNR with NZ = 4. (c) Corresponding phonon transmission coefficient Tp [ ω].
300k 500k 700k
Max ZT
1.6
phonon spectrum in A-BGYNR-I is more dispersive than that in Z-BGYNR. This discrepancy mainly originates from the unique geometric structure and large edge effect in ZBGYNR, which generates more localized phonon states than A-BGYNR. Owing to these dispersive phonon spectra, there are more phonon branches at a given frequency for ABGYNR-I. According to Landauer's theory of quantum transport, the phonon transmission coefficient is essentially equal to the number of phonon branches at a given frequency in the ballistic transport region [49]. Therefore, as shown in figure 4(c), A-BGYNR-I has a larger phonon transmission than Z-BGYNR, which leads to anisotropy in the thermal transport property of β-graphyne.
1.2
0.8
(b) 0.4
0
2
4
W (nm)
6
8
10
Figure 5. Peak value of ZT as a function of nanoribbon width for (a) A-BGYNRs and (b) Z-BGYNRs at different temperatures. The inset shows thermal power for A-BGYNR-I and A-BGYNR-II with width NA = 13 (corresponding width is 5.34 nm) at room temperature.
4. Thermoelectric properties of β-graphyne nanoribbons
BGYNRs-II have an oscillatory behavior around a constant value, due to the oscillatory width dependence of the scaled phonon-contributed thermal conductance shown in figure 2. By comparing these two cases, the thermoelectric performance of A-BGYNRs-I is found to be better than that of ABGYNRs with type-II edge terminal. This is mainly attributed to the better electronic property (thermal power S2σ) of ABGYNR-I, as illustrated in the inset of figure 5(a). Figure 5(b) shows the peak ZT value for Z-BGYNRs. In contrast to ABGYNRs, the ZT value of Z-BGYNRs with wide width does not show an obvious oscillatory behavior. The stable ZT value at room temperature is about 0.48 and 0.27 for Z-BGYNR and A-BGYNR, respectively. This value is substantially higher than that observed in graphene nanoribbons (about 0.05 at room temperature) [12], indicating that β-graphyne is a potential candidate for good thermoelectric materials. It is worth pointing out that, considering the high Debye temperature of the carbon-based materials [52], the thermoelectric properties of BGYNRs predicted in this paper are valid without further quantum corrections.
After calculating the thermal conductance (kp and ke), the electronic conductance, and the Seebeck coefficient, we can obtain the ZT of β-graphyne nanoribbons. In figure 5, we present the peak value of ZT for A-BGYNRs and ZBGYNRs as a function of width. There are some common features when comparing the results between A-BGYNRs and Z-BGYNRs. First of all, one can find that the ZT values for narrow BGYNRs (W < 3 nm) decrease rapidly as width increases, which is consistent with previous findings for graphene nanoribbons [12]. For wider nanoribbons (W ⩾ 3 nm), the ZT values level off and do not change significantly with further increases in width. Secondly, it can be seen that the ZT values of both A-BGYNRs and ZBGYNRs increase with an increase in temperature. For example, the maximum ZT value of A-BGYNR-I with width of 4.5 nm increases from 0.35 to 0.67 as the temperature increases from 300 K to 700 K. We also compare the ZT results between A-BGYNRs-I and A-BGYNRs-II in figure 5(a). Both A-BGYNRs-I and A5
Nanotechnology 25 (2014) 245401
(a)
T Ouyang and M Hu
number of unit cells of Z-BGYNR as shown in figure 6(a). From figure 6(b) one can clearly see that the thermoelectric properties are enhanced by the edge modulation. For example, the ZT value at room temperature increases from 0.51 with perfectly smooth edge to ∼0.67 with edge modulation. With temperature increasing, the ZT value gradually increases and exhibits an evident oscillatory behavior as the length of edge modulation increases. When the similar edge modulations are applied to narrower nanoribbons (NZ = 4, W = 2.07 nm), the thermoelectric performance of Z-BGYNRs is enhanced as well. The ZT value can be even elevated to 1.0 at room temperature as shown in figure 6(c). The similar oscillatory behavior of ZT has also been observed in the narrow ZBGYNRs with edge modulation, especially for high temperatures, e.g., the ZT value can vary from 1.4 to 1.9 at 700 K. By comparing the two cases of different nanoribbon widths in figures 6(b) and (c), we also noticed that the thermoelectric performance is enhanced more notably for Z-BGYNRs with width NZ = 4. This is because the edge modulation can affect both the electronic and thermal (mainly phononic) transport properties more effectively for narrower nanoribbons. According to the results presented in figure 6, it can be expected that thermoelectric property of Z-BGYNRs can be further improved by increasing the quantity of edge modulation. In figure 7, the electronic transmission, Seebeck coefficient, phonon-contributed thermal conductance, and thermal power are depicted to elucidate the enhancement and oscillatory behavior of thermoelectric coefficient of Z-BGYNRs with edge modulation. We see that, once the edge modulation is introduced into the system, the electronic transmission spectrum of Z-BGYNRs decreases dramatically and the quantization platforms are destroyed simultaneously (figure 7(a)). In the meantime, band gaps emerge in the energy range from −0.5–−1.2 eV to 0.5–1.2 eV. With the length of edge modulation increasing, more intense transmission peaks and pits occur in the transmission spectrum. Such suppression of electronic transmission arises from the region of the edge modulation acting as a scattering center, and the mode mismatch between this area and the electronic leads. Meanwhile, the transmission peaks originate from the electronic resonance generated in the center of Z-BGYNR with edge modulation, in which the region of edge modulation acts as potential barrier and the center region acts as potential well. According to the equation (4b), the Seebeck coefficient is related to the derivative of the electronic transmission. Generally speaking, a larger logarithmic derivative of the electronic transmission stems from the contributions of numerical fluctuations and leads to higher Seebeck coefficient. Just because of the sharp peaks and large band gaps in the electronic transmission, the Z-BGYNRs with edge modulation exhibit higher Seebeck coefficient than the unmodulated Z-BGYNRs [figure 7(b)]. Moreover, the Seebeck coefficient is insensitive to the length of edge modulation. For phonons, due to the destroyed edges, the phonon boundary scattering is more severe when phonons transmit through the system. Therefore, the phononic thermal conductance of ZBGYNR with edge modulation is dramatically reduced as
LEM
1.1 300K 500K 700K
1.0
Nz=6
(b)
Nz=4
(c)
Max ZT
0.9 0.8 0.7 0.6 0.5
300K 500K 700K
1.8
Max ZT
1.6 1.4 1.2 1.0 0.8 0.6
0
2
4
6
8
10
LEM (unit cell) Figure 6. (a) Schematic of Z-BGYNR with edge modulation. The
length of edge modulation (LEM) is calculated by the number of unit cell of Z-BGYNR, which is denoted by the dashed box. (b), (c) peak value of ZT as a function of length of edge modulation for ZBGYNRs with different width: (b) NZ = 6 and (c) NZ = 4. The horizontal dash-dotted lines represent ZT values of corresponding nanoribbons at room temperature with perfectly smooth edges, i.e. without edge modulation.
5. Improving thermoelectric properties of βgraphyne nanoribbons by nanostructuring With the aim of achieving a large ZT value, we now nanoengineer the structure (mainly geometric modulation) and investigate the effect on the thermoelectric properties of BGYNRs. Only Z-BGYNRs are considered in this section due to their better thermoelectric performance as compared to A-BGYNRs. First, we study the edge modulation of the nanoribbon. In figure 6 the peak ZT value for Z-BGYNRs with different lengths of edge modulation is depicted. The length of edge modulation (LEM) is characterized by the 6
Nanotechnology 25 (2014) 245401
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Figure 7. Comparison of transport coefficients of Z-BGYNR with and without different edge modulation: (a) electronic transmission, (b) Seebeck coefficient at 700 K, (c) thermal conductance, and (d) thermal power at 700 K.
thermoelectric coefficient of Z-BGYNRs with geometric modulation. We now consider the influence of number of edge modulation (NEM) on the thermoelectric performance of ZBGYNRs. Based on the findings in figure 6, the system with the shortest length of edge modulation (LEM = 1) generally gives the best thermoelectric performance. Then we take LEM = 1 as the repeating unit in this NEM study. The structure is shown in figure 8(a). Figure 8(b) shows the peak ZT value achieved for the Z-BGYNRs with different widths (NZ = 4 and 6) as a function of number of edge modulation. It is clearly seen that the ZT value increases dramatically as the number of edge modulation goes up to 4. After that, the ZT value gradually approaches a stable value for more edge modulations (NEM > 4). The reason for this behavior can be explained as follows. When the number of edge modulation is sufficiently large, the entire system can be approximately regarded as a superlattice structure composed of a thin ZBGYNR slice with perfect edge and an edge-modulated Z-
compared to the unmodulated Z-BGYNR, as shown in figure 7(c). It is worth pointing out that the geometric modulation could also yield phonon resonance in this system. However, the phononic resonance plays little contribution to the thermal transport, which leads to the insensitivity of the thermal conductance to the length of edge modulation. Figure 7(d) shows the thermal power of Z-BGYNR as a function of chemical potential. One can find that the thermal power of Z-BGYNRs with LEM = 4 is lower than that of the nanoribbons with LEM = 6, although both of them share a similar value of Seebeck coefficient. This discrepancy mainly stems from the different electronic conductance due to the resonant behavior. Based on the results presented in figure 7, we conclude that (1) the electronic resonance in the central region induces the oscillatory behavior of ZT in the ZBGYNRs with different length of edge modulation, (2) the combined effect of the enhanced Seebeck coefficient and the largely reduced thermal conductance finally boosts the 7
Nanotechnology 25 (2014) 245401 1
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2
3
NEM
(a)
(a)
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2.4
(b)
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(b) Nz=4 Nz=6
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Nz=4 Nz=6 300K 500K 700K
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Figure 8. (a) Schematic of Z-BGYNR with multi edge modulations.
(c)
PDOS-Center PDOS-Edge Overall DOS
DOS/PDOS
(b) Peak value of ZT as a function of number of edge modulation (NEM) for Z-BGYNRs with width NZ = 6 (hollow symbols) and NZ = 4 (solid symbols).
BGYNR slice. The electrons and phonons transmitting through this system only encounter scattering at the interface between the electronic/thermal leads and the superlattice structure, which leads to the gradually steady states and stable ZT values. As temperature increases, the enhancement of thermoelectric performance is more evident. For example, when the number of edge modulation is 5, the ZT value increases from 0.77 to 1.14 for Z-BGYNR with NZ = 6 as the temperature increases from 300 K to 700 K, while it increases from 1.21 to 2.14 for the nanoribbons with NZ = 4. That is to say, narrower nanoribbons could achieve more remarkable improvement in thermoelectric coefficient than the sample with wider width. This arises from the more notable edge effect on the transport property of narrower nanoribbons. We also investigated the effect of central geometric modulation on the thermoelectric property of Z-BGYNRs. The geometric structure we considered is shown in figure 9(a). From the figure 9(b) one can see that for ZBGYNR with NZ = 4 and 6, the ZT value increases gradually with the number of central modulation increasing (NCM < 4), while it saturates as the NCM is greater than 5. By comparing the two types of geometric modulation, i.e., edge and central modulation, it can be found that the central modulation can improve the thermoelectric performance of Z-BGYNR more significantly than edge modulation. For the case of ZBGYNR with width NZ = 4, the ZT value can be enhanced to about 1.21 by edge modulation, while it can be boosted to 1.75 for central modulation even at room temperature. This improvement can be observed more evidently at high temperatures. To understand the better thermoelectric coefficient
0
500
1000
Frequency
1500 (cm-1)
2000
Figure 9. (a) Schematic of Z-BGYNR with central region
modulation. (b) Peak value of ZT as a function of number of central modulation (NCM) for Z-BGYNRs with width NZ = 6 (hollow symbols) and NZ = 4 (solid symbols). (c) Phonon projected density of states (PDOS) and overall density of states (DOS) vs. frequency for Z-BGYNR with width NZ = 4.
enhancement in Z-BGYNRs with central modulation, in figure 9(c) we depict the phonon DOS and PDOS for the case of Z-BGYNRs with width NZ = 4. One can clearly see that the PDOS of the central region is substantially larger than that of the edge region due to the stronger localized phonons in this area, indicating that phonons in the central region of nanoribbons dominate the thermal transport of the modulated ZBGYNRs. Therefore, when central geometric modulation is introduced to Z-BGYNRs, the primary phonon transport channels are destroyed. In contrast, these channels can be retained perfectly in the case of edge modulation. Consequently, a lower thermal conductance exists in Z-BGYNR with central modulation and thus leads to the higher value of ZT than that of Z-BGYNR with edge modulation. This result could provide a new route to enhance thermoelectric performance of β-graphyne by engineering the structures. Before closing, it is worth pointing out that, as discussed above, the improvement of thermoelectric performance of 8
Nanotechnology 25 (2014) 245401
T Ouyang and M Hu
and provide useful guidance for synthesizing nanostructured thermoelectric devices in experiments.
beta-graphyne nanostructures originates from the largely enhanced Seebeck coefficient and drastically reduced thermal conductance that compensate for the depression in electronic transmission. Therefore, one can expect that this enhancement behavior could also exist in the beta-graphyne nanostructures with other types of vacancy. However, the present work focuses more on the influences of location and size of vacancy on the thermoelectric efficiency of beta-graphyne nanostructures. The results can be generalized for extended systems, i.e. the results do not depend on the specific type of the vacancy considered.
Acknowledgments Simulations were performed with computing resources granted by the Jülich Aachen Research Alliance–High Performance Computing (JARA–HPC) from RWTH Aachen University under Project No. jara0073. This work is supported by the National Natural Science Foundation of China (Grant No. 11304262), the Hunan Provincial Natural Science Foundation of China (No. 14JJ3074), and the Scientific Research Fund of Hunan Provincial Education Department (No. 13C927).
6. Concluding remarks In summary, using nonequilibrium Green’s function method, we systematically studied thermal transport and thermoelectric properties of β-graphyne nanoribbons and related nanostructures. Our simulation revealed that thermal conductance of β-graphyne nanoribbons is only 26% of that of their graphene counterparts and shows evident directional dependence, where armchair-edged nanoribbons hold better thermal conductance than those with a zigzag edge. The anisotropic thermal transport property of β-graphyne originates from the more dispersive low-frequency phonon spectrum present in armchair β-graphyne nanoribbons. Moreover, the thermal conductance of A-BGYNRs presents abnormal stepwise width dependence. By analyzing the phonon-projected density of states, we attribute this unexpected thermal transport phenomenon to the weaker influence induced by edges with type-II terminal than those with type-I terminal. For thermoelectric property, the simulation results show that, as compared to the graphene counterparts (ZT 0.05), β-graphyne possesses superior thermoelectric performance with figure of merit ZT achieving 0.5 at room temperature. For the cases of Z-BGYNRs, we also investigate the effect of edge and central geometric modulations on the thermoelectric coefficient. It is found that, when edge modulation is introduced, the thermoelectric performance of Z-BGYNRs is significantly enhanced, and such enhancement can be further improved by increasing the number of edge modulation. The enhancement stems from the largely enhanced Seebeck coefficient and drastically reduced thermal conductance that compensate the depression in electronic transmission. Meanwhile, change of edge modulation length in Z-BGYNRs results in oscillatory behavior of ZT value, which originates from the induced electronic resonance in the structure. Furthermore, when central region modulation is applied to Z-BGYNRs, the enhancement of thermoelectric performance becomes more distinct than edge modulation (ZT exceeding 1.5 even at room temperature), especially for narrow ribbons at high temperatures. The superior enhancement by geometric modulation is explained in terms of the dominant contribution of the central region to the overall thermal transport of Z-BGYNRs. The findings presented in this paper qualify β-graphyne as a promising candidate for high performance thermoelectric applications
References [1] Bell L 2008 Science 321 1457 [2] Nolas G S, Sharp J and Goldsmid J 2001 Thermoelectrics: Basic Principles and New Materials Developments (New York: Springer) [3] Venkatasubramanian R, Siivola E, Colpitts T and O’Quinn B 2001 Nature 413 597 [4] Hicks L D and Dresselhaus M S 1993 Phys. Rev. B 47 12727 [5] Majumdar A 2004 Science 303 777 [6] Hochbaum A, Chen R, Delgado R, Liang W J, Garnett E C, Najarian M, Majumdar A and Yang P D 2008 Nature 451 163 [7] Harman T C, Taylor P J, Walsh M P and LaForge B E 2002 Science 297 2229 [8] Balandin A and Wang K L 1198 J. Appl. Phys. 84 6149 [9] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [10] Morozov S V, Novoselov K S, Katsnelson M I, Schedin F, Elias D C, Jaszczak J A and Geim A K 2008 Phys. Rev. Lett. 100 016602 [11] Dragoman D and Dragoman M 2007 Appl. Phys. Lett. 91 203116 [12] Chen Y, Jayasekera T, Calzolari A, Kim K W and Buongiorno Nardelli M 2010 J. Phys.: Condens. Matter 22 372202 [13] Balandin A A, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F and Lau C N 2008 Nano Lett. 8 902 [14] Balandin A A 2011 Nat. Mater. 10 569 [15] Wang Z, Xie R, Bui C T, Liu D, Ni X, Li B and Thong J T 2011 Nano Lett. 11 113 [16] Sevinçli H and Cuniberti G 2010 Phys. Rev. B 81 113401 [17] Ouyang T, Chen Y P, Yang K K and Zhong J X 2010 Europhys. Lett. 91 46006 [18] Sevinçli H, Sevik C, Çaǧın T and Cuniberti G 2013 Sci Rep. 3 1228 [19] Ni X, Liang G, Wang J S and Li B 2009 Appl. Phys. Lett. 95 192114 [20] Ouyang Y and Guo J 2009 Appl. Phys. Lett. 94 263107 [21] Karamitaheri H, Neophytou N, Pourfath M, Faez R and Kosina H 2012 J. Appl. Phys. 111 054501 [22] Mazzamuto F, Nguyen V H, Apertet Y, Caer C, Chassat C, Saint-Martin J and Dollfus P 2011 Phys. Rev. B 83 235426 [23] Liang L B, Cruz-Silva E, Costa Girao E and Meunier V 2012 Phys. Rev. B 86 115438 [24] Karamitaheri H, Pourfath M, Faez R and Kosina H 2011 J. Appl. Phys. 110 054506 [25] Hirsch A 2010 Nat. Mater. 9 868 9
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[39] Malko D, Neiss C, Vines F and Gorling A 2012 Phys. Rev. Lett. 108 086804 [40] Long M Q, Tang L, Wang D, Li Y L and Shuai Z G 2011 ACS Nano 5 2593 [41] Chen J M, Xi J Y, Wang D and Shuai Z G 2013 J. Phys. Chem. Lett. 4 1443 [42] Ouyang T, Xiao H P, Xie Y E, Wei X L, Chen Y P and Zhong J X 2013 J. Appl. Phys. 114 073710 [43] Wang X M, Mo D C and Lu S S 2013 J. Chem. Phys. 138 204704 [44] Liu Z, Yu G D, Yao H B, Liu L, Jiang L W and Zheng Y S 2012 New J. Phys. 14 113007 [45] Yu G D, Liu Z, Gao W Z and Zheng Y S 2013 J. Phys.: Condens. Matter 25 285502 [46] Datta S 1995 Electronic Transport in Mesoscopic Systems (Cambridge: Cambridge University Press) [47] Mingo N 2006 Phys. Rev. B 74 125402 [48] Yamamoto T and Watanabe K 2006 Phys. Rev. Lett. 96 255503 [49] Wang J S, Wang J and Lu J T 2008 Eur. Phys. J. B 62 381 [50] Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni B and Sinnott S B 2002 J. Phys.: Condens. Matter 14 783 [51] Xu Y, Chen X B, Gu B L and Duan W H 2009 Appl. Phys. Lett. 95 233116 [52] Pop E, Varshney V and Roy A K 2012 MRS Bull. 37 1273
[26] Baughman R H, Eckhardt H and Kertesz M 1987 J. Chem. Phys. 87 6687 [27] Ivanovskii A L 2013 Prog. Solid State Chem. 41 1 [28] Cranford S W and Buehler M J 2011 Carbon 49 4111 [29] Zhang Y Y, Pei Q X and Wang C M 2012 Appl. Phys. Lett. 101 081909 [30] Zhang H, Zhao M, He X, Wang Z, Zhang X and Liu X 2011 J. Phys. Chem. C 115 8845 [31] Narita N, Nagai S, Suzuki S and Nakao K 1998 Phys. Rev. B 58 11009 [32] Ni Y, Yao K L, Fu H H, Gao G Y, Zhu S C, Luo B, Wang S L and Li R X 2013 Nanoscale 5 4468 [33] Kang J, Li J B, Wu F M, Li S S and Xia J B 2011 J. Phys. Chem. C 115 20466 [34] Zhou J, Lv K, Wang Q, Chen X S, Sun Q and Jena P 2011 J. Chem. Phys. 134 174701 [35] Pan L D, Zhang L Z, Song B Q, Du S X and Gao H J 2011 Appl. Phys. Lett. 98 173102 [36] Luo G, Qian X, Liu H, Qin R, Zhou J, Li L, Gao Z, Wang E, Mei W N, Lu J, Li Y and Nagase S 2011 Phys. Rev. B 84 075439 [37] Ouyang T, Chen Y P, Liu L M, Xie Y E, Wei X L and Zhong J X 2012 Phys. Rev. B 85 235436 [38] Zhang Y Y, Pei Q X and Wang C M 2012 Comp. Mat. Sci. 65 406
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Thermal transport and thermoelectric properties of beta-graphyne nanostructures
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Nanotechnology Nanotechnology 25 (2014) 245401 (10pp)
doi:10.1088/0957-4484/25/24/245401
Thermal transport and thermoelectric properties of beta-graphyne nanostructures Tao Ouyang1,3 and Ming Hu1,2 1
Institute of Mineral Engineering, Division of Materials Science and Engineering, Faculty of Georesources and Materials Engineering, Rheinisch-Westfaelische Technische Hochschule (RWTH Aachen University), 52064 Aachen, Germany 2 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, 52062 Aachen, Germany E-mail: [email protected] Received 13 March 2014, revised 9 April 2014 Accepted for publication 14 April 2014 Published 23 May 2014 Abstract
Graphyne, an allotrope of graphene, is currently a hot topic in the carbon-based nanomaterials research community. Taking beta-graphyne as an example, we performed a comprehensive study of thermal transport and related thermoelectric properties by means of nonequilibrium Green’s function (NEGF). Our simulation demonstrated that thermal conductance of beta-graphyne is only approximately 26% of that of the graphene counterpart and also shows evident anisotropy. Meanwhile, thermal conductance of armchair beta-graphyne nanoribbons (A-BGYNRs) presents abnormal stepwise width dependence. As for the thermoelectric property, we found that zigzag beta-graphyne nanoribbons (Z-BGYNRs) possess superior thermoelectric performance with figure of merit value achieving 0.5 at room temperature, as compared with graphene nanoribbons (∼0.05). Aiming at obtaining a better thermoelectric coefficient, we also investigated ZBGYNRs with geometric modulations. The results show that the thermoelectric performance can be enhanced dramatically (figure of merit exceeding 1.5 at room temperature), and such enhancement strongly depends on the width of the nanoribbons and location and quantity of geometric modulation. Our findings shed light on transport properties of beta-graphyne as high efficiency thermoelectrics. We anticipate that our simulation results could offer useful guidance for the design and fabrication of future thermoelectric devices. Keywords: graphyne, thermal transport, thermoelectrics, nonequilibrium Green’s function, energy conversion (Some figures may appear in colour only in the online journal) 1. Introduction
conversion efficiency of thermoelectric materials [2], ZT = S2σ/k, where S is the Seebeck coefficient, σ the electronic conductance, and k (k = ke + kp) the thermal conductance composed of the contributions from electrons (ke) and phonons (kp), respectively. However, it is very challenging to obtain high thermoelectric performance in a conventional bulk system, due to the strong coupling effect between the electronic and phononic transport in ZT [3]. Low-dimensional materials as proposed by Hicks and Dresselhaus [4], on the other hand, provided us a promising route to decouple the relationship between electronic and phononic conductance and thus led to the significant increase in ZT. Therefore, a tremendous amount of research in the past decade [5–8] has
Recently, recycling waste heat from thermal engines has aroused widespread interest for both theoretical and technological research, owing to the gradual exhaust of fossil fuels [1]. To achieve this goal, one of the most efficient approaches involves utilizing thermoelectric effect, which can convert a temperature gradient to electricity directly. The dimensionless figure of merit (ZT) is defined to measure the energy 3
On leave from Hunan Key Laboratory for Micro-Nano Energy Materials and Device and Department of Physics, Xiangtan University, Xiangtan 411105, Hunan, China.
0957-4484/14/245401+10$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
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T Ouyang and M Hu
the thermoelectric performance of zigzag β-graphyne nanoribbons. In section 5, we present a summary and conclusion.
focused on fabricating and optimizing nanomaterials for high thermoelectric performance. Graphene and its derivatives, as representatives of lowdimensional carbon materials with hexagonal lattice structure, possess a wide range of super physical properties and have attracted exceptional attention recently. Although the electronic properties of graphene and its related nanostructures are fascinating (including record high value of carrier mobility and giant Seebeck coefficient) [9–11], they are not efficient thermoelectric materials (ZT ∼ 0.05) [12], primarily due to their extremely high thermal conductance [13–15]. In order to overcome this drawback, significant measures have been taken to decrease the thermal conductance and thus enhance the thermoelectric performance [16–24], e.g., creating scattering by edge roughness [16], isotope engineering [17, 18], defect engineering [19–21], and introducing electronic resonance in periodically patterned nanoholes and superlattice structures [12, 22–24]. However, little previous work can improve the figure of merit of graphene and its derivatives to exceed 1.0 at room temperature (for very narrow graphene ribbons with complex structure, the ZT can achieve 2.0 at room temperature) [18]. Graphyne, a new type of planar carbon allotrope containing both sp and sp2 hybridized states, is a new topic in the present carbon nanomaterials research communities [25–27]. Due to the presence of acetylenic fragments, graphyne holds four typical geometrical structures, namely, alpha(α)-graphyne, beta(β)-graphyne, gamma(γ)-graphyne, and (6,6,12)graphyne [26]. These novel carbon allotropes not only share unique mechanical and chemical properties with graphene [28–31], but also exhibit a rich variety of amazing electronic and thermal transport performance [32–38], which qualify them as powerful competitors of graphene. Malko et al [39] predicted that Dirac cones and their associated electronic transport properties exist in α-and β-graphyne as well. In addition, (6,6,12)-graphyne is found to have two self-doped nonequivalent distorted Dirac cones, indicating that it naturally contains conducting charge carriers and could conduct electrons in a preferred direction. The intrinsic carrier mobility of the γ-graphyne and (6,6,12)-graphyne can reach as high as 105 cm2 Vm−1 at room temperature, which is comparable to that of graphene [40, 41]. More interestingly, the recent studies showed that the phonon-contributed thermal conductance of graphyne is fundamentally lower than that of graphene, benefiting from the insertion of acetylenic linkages [37, 38]. These works suggest that graphyne is very promising for high thermoelectric performance [42, 43]. In this paper, taking β-graphyne as an example, we performed a comprehensive study of thermal transport and related thermoelectric properties of β-graphyne by using nonequilibrium Green’s function (NEGF) method. In section 2, we briefly describe our model structure and NEGF formulas for both electron and ballistic phonon transport. In section 3, we report thermal transport properties of β-graphyne nanoribbons. The corresponding thermoelectric properties of β-graphyne nanoribbons are shown in section 4. In section 5, we discuss the effect of geometric modulation on
2. Model structure and simulation methodology By cutting infinite β-graphyne sheet along different directions, one can obtain two typical β-graphyne nanoribbons with different edges: zigzag β-graphyne nanoribbons (ZBGYNRs) (figure 1(a)) and armchair β-graphyne nanoribbons (A-BGYNRs) (figures 1(b) and (c)). Unlike the cases of ZBGYNRs, the A-BGYNRs could possess two types of edge terminal named as A-BGYNR-I and A-BGYNR-II, which are depicted in figures 1(b) and (c). The width of A-BGYNRs (ZBGYNRs) is denoted by NA (NZ), which is determined by the number of sp2 state dimer lines along the non-periodic direction. It should be mentioned that the configuration of ABGYNR-I will be identical to that of A-BGYNR-II when NA is an even number. To calculate electronic transport properties of these nanostructures, the Hamiltonian is described by an atomistic π-orbital tight-binding (TB) model with nearest-neighbor coupling, which is defined as H=
∑ εici+ci + ∑ ti,jci+cj , i
(1)
i, j
where εi is the site energy at the ith atom and ti, j is nearestneighbor hopping energy. The corresponding parameters are taken from [44] and [45]. Based on this Hamiltonian, the retarded Green’s function is expressed as [46] G r ( E ) = ⎡⎣ ( E + i0+) I − Hc −
∑L − ∑R ⎤⎦ r
r
−1
,
(2)
where E is the electron energy, I is the identity matrix, and Hc is the Hamiltonian matrix of the central region. The r ∑β = V Cβgβr V βC ( β = L, R, corresponding to the left and right electronic leads, respectively) denotes the self-energy of T
( )
electronic lead β, where V βC = V Cβ
is the coupling matrix
of the lead β to the center region and gβr is the lead surface Green’s function calculated via an iterative procedure. Once the retarded Green’s function Gr ( E ) is obtained, we can calculate electronic transmission coefficient Te [E ] [46] Te [E ] = Tr { G r ( E ) ΓLG a ( E ) ΓR},
(
r
a
(3)
)
where Γβ = i ∑β − ∑β = −2 Im V Cβgβr V βC is the coupling function of the β lead. Using the electronic transmission coefficient Te [E ], the electronic conductance σ, the Seebeck coefficient S, and the electron-contributed thermal conductance ke can be calculated based on the Onsager’s relations and the Landauer’s theory of quantum transport [20] σ ( u) = e 2 L 0 ( u , T ) , S ( u) = 2
1 L1 ( u , T ) , eT L 0( u , T )
(4a) (4b)
Nanotechnology 25 (2014) 245401
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Figure 1. Schematic of infinite β-graphyne sheet (gray meshes): (a) zigzag-beta(β)-graphyne (Z-BGYNR), (b) armchair-beta(β)-graphyne (A-
BGYNR) with edge terminal of type I (A-BGYNR-I), and (c) A-BGYNR with edge terminal of type II (A-BGYNR-II). The yellow circles are carbon atoms. The widths of the Z-BGYNR and A-BGYNR are denoted by NZ and NA, respectively.
unit cells of the thermal lead in all of our calculations. As for the systems with structural defects, we have also added buffer layers (about three unit cells) between the central region and thermal leads to avoid the interference originated from the defect on the thermal leads. The phonon–phonon and electron–phonon interaction are neglected in our calculation. The phonon contributed thermal conductance is given by [49]
2
ke ( u) =
L1( u , T ) 1 , L 2 ( u, T ) − L 0( u , T ) T
(4c)
where Ln ( u, T ) =
2 h
+∞
∫ ( E − u) +∞
n
⎡ ∂f ( E , u , T ) ⎤ ⎥ Te [E ] dE . (5) ⎢− e ⎥⎦ ⎢⎣ ∂E
kp (T ) =
is Lorenz integral with fe ( E , u , T ) is the Fermi–Dirac distribution function at the chemical potential u and temperature T. For thermal transport, phonon transmission coefficient Te [ ω] of β-graphyne nanostructures can be calculated in harmonic approximation, which is similar to the way the electrons are calculated [47–49]. One only needs to substitute E by ω2 and HC by KC in the equation (2), and then to calculate the self-energies and retarded Green’s function accordingly. Here ω is the frequency of phonons, and KC is the mass-weighted force constant matrix of the central region. The second-generation reactive empirical bond order potential [50], which was proven to give excellent description of carbon–carbon bonding interactions [37, 51], is adopted to optimize the geometric structure and to obtain the force constants of β-graphyne nanostructures. Since this potential holds long-range character, it leads to unphysical results if the length of the central region is too small (i.e., there exists interaction between the left and right thermal leads). To avoid this, we have tested the convergence of the thermal conductance as a function of the central region’s size. The results show that when the length of the central region exceeds five unit cells of the thermal lead, the thermal conductance of betagraphyne nanoribbons does not change with further increase in size. Therefore, the length of the central region is set as five
ℏ 2π
∫
0
∞
T [ ω] ω
∂fp ( ω) ∂T
dω ,
(6)
where fp ( ω) is the Bose–Einstein distribution function of phonons. 3. Thermal transport properties of β-graphyne nanoribbons In figure 2 we depict the phonon part of thermal conductance scaled by cross-sectional area (kp/A) as a function of width (W) for A-BGYNRs and Z-BGYNRs at room temperature (300 K). Herein the cross-sectional area A is defined as A = Wh, where h = 0.34 nm is chosen as the layer separation of bulk β-graphyne. One can find from figure 2 that the scaled thermal conductance of A-BGYNR-I and A-BGYNR-II shares a similar phenomenon, although their edge terminals are quite different. The thermal conductance of A-BGYNRs first decreases rapidly with width increasing for narrow ribbons (W < 3 nm) and then levels off for wider nanoribbons (W ⩾ 3 nm). More interestingly, the scaled thermal conductance of A-BGYNR exhibits an oscillatory behavior as the width increases. The oscillation mainly originates from the stepwise width dependence of the thermal conductance, as shown in the inset of figure 2. In contrast, for Z-BGYNRs, the 3
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1.6
3
DOS/PDOS (arbitrary units)
A-BGYNR-I A-BGYNR-II Z-BGYNR
κP (nW/K)
κP/A (nW/Knm2)
2.0
2 1 0
2
4
6
8
10
W (nw)
1.2
0.8
0.4
2
4
6
8
10
0
W (nm)
PDOS-Center PDOS-Edge Overall DOS
(a)
PDOS-Center PDOS-Edge Overall DOS
(b)
PDOS-Center PDOS-Edge-I PDOS-Edge-II Overall DOS
(c)
500
1000
Frequency
Figure 2. Phonon part of thermal conductance scaled by cross-
1500 -1 (cm )
2000
Figure 3. Phonon projected density of states (PDOS) and overall
sectional area (kp/A) as a function of nanoribbon width (W) for ABGYNRs and Z-BGYNRs at room temperature. The inset shows unscaled thermal conductance as a function of width for A-BGYNRs at room temperature.
density of states (DOS) vs phonon frequency for (a) A-BGYNR-I with NA = 5, (b) A-BGYNR-II with NA = 5, and (c) A-BGYNR with NZ = 6.
scaled thermal conductance increases gradually with the width, implying that the zigzag edge makes a weak contribution to the thermal transport. When the width is about 10 nm, the scaled thermal conductance of A-BGYNR and ZBGYNR is approximately 1.14 and 0.94 nW (Knm−2), respectively. The scaled thermal conductance of BGYNRs is found to be only 26% of that for graphene nanoribbons [GNR, about 3.6–4.2 nW (Knm−2)] [51]. The large reduction is mainly attributed to the inefficient thermal transport in the acetylenic fragments. Similar phenomena have also been observed in previous studies using molecular dynamic simulation [38]. Meanwhile, the different values of the scaled thermal conductance of BGYNRs for different edges also indicate the anisotropic thermal transport property in β-graphyne, which is consistent with previous studies for GNRs and γ-graphyne nanoribbons [37, 51]. In order to elucidate the unique stepwise width dependence of thermal conductance of A-BGYNRs, the phonon density of states (DOS) and projected density of states (PDOS) versus the phonon frequency are plotted in figure 3. One can find that the phonon DOS of A-BGYNR-I is mainly contributed by the atoms in the central region. That is to say, owing to the edge effect, the atoms in the edge region could not propagate the thermal transport effectively, which results in weak contribution to the overall phonon DOS. For ABGYNR-II, shown in figure 3(b), however, the phonon PDOS from the edge region is more or less similar to that from the central region, especially for frequencies smaller than 750 cm−1, indicating that the edge has a weak influence on the phonon transport of A-BGYNRs with type-II edge terminal. This different edge effect can be seen more clearly when comparing two cases in figure 3(c). Obviously, the phonon PDOS of edge region with type-II terminal is larger than that with type-I edge terminal. The different edge effect can be
simply understood in terms of the special geometric structure of A-BGYNRs (figure 1). Figure 1(c) shows that passivated carbon atoms embed in the edge transport channel of ABGYNR-I, which will damage the continuity of the thermal transport, while the passivated carbon atoms of A-BGYNR-II remain away from the edge channel, thus protecting the thermal transport path from interruption. Therefore, the thermal conductance of A-BGYNR-I is similar to that of ABGYNR-II although it possesses more edge channels and thus leads to the edge terminal independence of thermal conductance. It should be noted that the difference in the scaled thermal conductance between A-BGYNR-I and ABGYNR-II shown in figure 2 is a result of the differing widths. For the stepwise width dependence of thermal conductance in A-BGYNRs, it can be explained as follows: when the width of A-BGYNR (NA) increases from an odd number to even number, the increased width will bring in more phonon modes. However, the transmission of these new phonon modes is restricted by the width of the narrowest part of the nanoribbons and thus they cannot contribute to the thermal transport effectively. As a result, the thermal conductance of A-BGYNRs exhibits interesting stepwise width dependence. In figure 4 the phonon spectra and phonon transmission of A-BGYNR-I (NA = 5, W = 1.98 nm) and Z-BGYNR (NZ = 4, W = 2.07 nm) are depicted to explain the anisotropic thermal transport property of BGYNRs. Only the low frequency region is provided because the phonons in this region dominate the thermal transport at room temperature. By comparing figures 4(a) and (b), it can be found that in general the phonon branches in A-BGYNR-I possess larger slope (therefore larger phonon group velocity) than those in ZBGYNR, where there are many flat branches corresponding to very low phonon group velocity. In other words, the 4
T Ouyang and M Hu
1.2 A-BGYNRs Type I Type II 300k 500k 700k
Max ZT
1.0
0.8
Thermal power (pW/K2)
Nanotechnology 25 (2014) 245401 3.0 2.5
A-BGYNR-I A-BGYNR-II
2.0 1.5 1.0 0.5 0.0 0
2
µ (ev)
4
6
0.6
0.4
(a) 0.2 Z-BGYNRs
Figure 4. Phonon spectra for (a) A-BGYNR-I with NA = 5, (b) ZBGYNR with NZ = 4. (c) Corresponding phonon transmission coefficient Tp [ ω].
300k 500k 700k
Max ZT
1.6
phonon spectrum in A-BGYNR-I is more dispersive than that in Z-BGYNR. This discrepancy mainly originates from the unique geometric structure and large edge effect in ZBGYNR, which generates more localized phonon states than A-BGYNR. Owing to these dispersive phonon spectra, there are more phonon branches at a given frequency for ABGYNR-I. According to Landauer's theory of quantum transport, the phonon transmission coefficient is essentially equal to the number of phonon branches at a given frequency in the ballistic transport region [49]. Therefore, as shown in figure 4(c), A-BGYNR-I has a larger phonon transmission than Z-BGYNR, which leads to anisotropy in the thermal transport property of β-graphyne.
1.2
0.8
(b) 0.4
0
2
4
W (nm)
6
8
10
Figure 5. Peak value of ZT as a function of nanoribbon width for (a) A-BGYNRs and (b) Z-BGYNRs at different temperatures. The inset shows thermal power for A-BGYNR-I and A-BGYNR-II with width NA = 13 (corresponding width is 5.34 nm) at room temperature.
4. Thermoelectric properties of β-graphyne nanoribbons
BGYNRs-II have an oscillatory behavior around a constant value, due to the oscillatory width dependence of the scaled phonon-contributed thermal conductance shown in figure 2. By comparing these two cases, the thermoelectric performance of A-BGYNRs-I is found to be better than that of ABGYNRs with type-II edge terminal. This is mainly attributed to the better electronic property (thermal power S2σ) of ABGYNR-I, as illustrated in the inset of figure 5(a). Figure 5(b) shows the peak ZT value for Z-BGYNRs. In contrast to ABGYNRs, the ZT value of Z-BGYNRs with wide width does not show an obvious oscillatory behavior. The stable ZT value at room temperature is about 0.48 and 0.27 for Z-BGYNR and A-BGYNR, respectively. This value is substantially higher than that observed in graphene nanoribbons (about 0.05 at room temperature) [12], indicating that β-graphyne is a potential candidate for good thermoelectric materials. It is worth pointing out that, considering the high Debye temperature of the carbon-based materials [52], the thermoelectric properties of BGYNRs predicted in this paper are valid without further quantum corrections.
After calculating the thermal conductance (kp and ke), the electronic conductance, and the Seebeck coefficient, we can obtain the ZT of β-graphyne nanoribbons. In figure 5, we present the peak value of ZT for A-BGYNRs and ZBGYNRs as a function of width. There are some common features when comparing the results between A-BGYNRs and Z-BGYNRs. First of all, one can find that the ZT values for narrow BGYNRs (W < 3 nm) decrease rapidly as width increases, which is consistent with previous findings for graphene nanoribbons [12]. For wider nanoribbons (W ⩾ 3 nm), the ZT values level off and do not change significantly with further increases in width. Secondly, it can be seen that the ZT values of both A-BGYNRs and ZBGYNRs increase with an increase in temperature. For example, the maximum ZT value of A-BGYNR-I with width of 4.5 nm increases from 0.35 to 0.67 as the temperature increases from 300 K to 700 K. We also compare the ZT results between A-BGYNRs-I and A-BGYNRs-II in figure 5(a). Both A-BGYNRs-I and A5
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number of unit cells of Z-BGYNR as shown in figure 6(a). From figure 6(b) one can clearly see that the thermoelectric properties are enhanced by the edge modulation. For example, the ZT value at room temperature increases from 0.51 with perfectly smooth edge to ∼0.67 with edge modulation. With temperature increasing, the ZT value gradually increases and exhibits an evident oscillatory behavior as the length of edge modulation increases. When the similar edge modulations are applied to narrower nanoribbons (NZ = 4, W = 2.07 nm), the thermoelectric performance of Z-BGYNRs is enhanced as well. The ZT value can be even elevated to 1.0 at room temperature as shown in figure 6(c). The similar oscillatory behavior of ZT has also been observed in the narrow ZBGYNRs with edge modulation, especially for high temperatures, e.g., the ZT value can vary from 1.4 to 1.9 at 700 K. By comparing the two cases of different nanoribbon widths in figures 6(b) and (c), we also noticed that the thermoelectric performance is enhanced more notably for Z-BGYNRs with width NZ = 4. This is because the edge modulation can affect both the electronic and thermal (mainly phononic) transport properties more effectively for narrower nanoribbons. According to the results presented in figure 6, it can be expected that thermoelectric property of Z-BGYNRs can be further improved by increasing the quantity of edge modulation. In figure 7, the electronic transmission, Seebeck coefficient, phonon-contributed thermal conductance, and thermal power are depicted to elucidate the enhancement and oscillatory behavior of thermoelectric coefficient of Z-BGYNRs with edge modulation. We see that, once the edge modulation is introduced into the system, the electronic transmission spectrum of Z-BGYNRs decreases dramatically and the quantization platforms are destroyed simultaneously (figure 7(a)). In the meantime, band gaps emerge in the energy range from −0.5–−1.2 eV to 0.5–1.2 eV. With the length of edge modulation increasing, more intense transmission peaks and pits occur in the transmission spectrum. Such suppression of electronic transmission arises from the region of the edge modulation acting as a scattering center, and the mode mismatch between this area and the electronic leads. Meanwhile, the transmission peaks originate from the electronic resonance generated in the center of Z-BGYNR with edge modulation, in which the region of edge modulation acts as potential barrier and the center region acts as potential well. According to the equation (4b), the Seebeck coefficient is related to the derivative of the electronic transmission. Generally speaking, a larger logarithmic derivative of the electronic transmission stems from the contributions of numerical fluctuations and leads to higher Seebeck coefficient. Just because of the sharp peaks and large band gaps in the electronic transmission, the Z-BGYNRs with edge modulation exhibit higher Seebeck coefficient than the unmodulated Z-BGYNRs [figure 7(b)]. Moreover, the Seebeck coefficient is insensitive to the length of edge modulation. For phonons, due to the destroyed edges, the phonon boundary scattering is more severe when phonons transmit through the system. Therefore, the phononic thermal conductance of ZBGYNR with edge modulation is dramatically reduced as
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length of edge modulation (LEM) is calculated by the number of unit cell of Z-BGYNR, which is denoted by the dashed box. (b), (c) peak value of ZT as a function of length of edge modulation for ZBGYNRs with different width: (b) NZ = 6 and (c) NZ = 4. The horizontal dash-dotted lines represent ZT values of corresponding nanoribbons at room temperature with perfectly smooth edges, i.e. without edge modulation.
5. Improving thermoelectric properties of βgraphyne nanoribbons by nanostructuring With the aim of achieving a large ZT value, we now nanoengineer the structure (mainly geometric modulation) and investigate the effect on the thermoelectric properties of BGYNRs. Only Z-BGYNRs are considered in this section due to their better thermoelectric performance as compared to A-BGYNRs. First, we study the edge modulation of the nanoribbon. In figure 6 the peak ZT value for Z-BGYNRs with different lengths of edge modulation is depicted. The length of edge modulation (LEM) is characterized by the 6
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Figure 7. Comparison of transport coefficients of Z-BGYNR with and without different edge modulation: (a) electronic transmission, (b) Seebeck coefficient at 700 K, (c) thermal conductance, and (d) thermal power at 700 K.
thermoelectric coefficient of Z-BGYNRs with geometric modulation. We now consider the influence of number of edge modulation (NEM) on the thermoelectric performance of ZBGYNRs. Based on the findings in figure 6, the system with the shortest length of edge modulation (LEM = 1) generally gives the best thermoelectric performance. Then we take LEM = 1 as the repeating unit in this NEM study. The structure is shown in figure 8(a). Figure 8(b) shows the peak ZT value achieved for the Z-BGYNRs with different widths (NZ = 4 and 6) as a function of number of edge modulation. It is clearly seen that the ZT value increases dramatically as the number of edge modulation goes up to 4. After that, the ZT value gradually approaches a stable value for more edge modulations (NEM > 4). The reason for this behavior can be explained as follows. When the number of edge modulation is sufficiently large, the entire system can be approximately regarded as a superlattice structure composed of a thin ZBGYNR slice with perfect edge and an edge-modulated Z-
compared to the unmodulated Z-BGYNR, as shown in figure 7(c). It is worth pointing out that the geometric modulation could also yield phonon resonance in this system. However, the phononic resonance plays little contribution to the thermal transport, which leads to the insensitivity of the thermal conductance to the length of edge modulation. Figure 7(d) shows the thermal power of Z-BGYNR as a function of chemical potential. One can find that the thermal power of Z-BGYNRs with LEM = 4 is lower than that of the nanoribbons with LEM = 6, although both of them share a similar value of Seebeck coefficient. This discrepancy mainly stems from the different electronic conductance due to the resonant behavior. Based on the results presented in figure 7, we conclude that (1) the electronic resonance in the central region induces the oscillatory behavior of ZT in the ZBGYNRs with different length of edge modulation, (2) the combined effect of the enhanced Seebeck coefficient and the largely reduced thermal conductance finally boosts the 7
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Figure 8. (a) Schematic of Z-BGYNR with multi edge modulations.
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BGYNR slice. The electrons and phonons transmitting through this system only encounter scattering at the interface between the electronic/thermal leads and the superlattice structure, which leads to the gradually steady states and stable ZT values. As temperature increases, the enhancement of thermoelectric performance is more evident. For example, when the number of edge modulation is 5, the ZT value increases from 0.77 to 1.14 for Z-BGYNR with NZ = 6 as the temperature increases from 300 K to 700 K, while it increases from 1.21 to 2.14 for the nanoribbons with NZ = 4. That is to say, narrower nanoribbons could achieve more remarkable improvement in thermoelectric coefficient than the sample with wider width. This arises from the more notable edge effect on the transport property of narrower nanoribbons. We also investigated the effect of central geometric modulation on the thermoelectric property of Z-BGYNRs. The geometric structure we considered is shown in figure 9(a). From the figure 9(b) one can see that for ZBGYNR with NZ = 4 and 6, the ZT value increases gradually with the number of central modulation increasing (NCM < 4), while it saturates as the NCM is greater than 5. By comparing the two types of geometric modulation, i.e., edge and central modulation, it can be found that the central modulation can improve the thermoelectric performance of Z-BGYNR more significantly than edge modulation. For the case of ZBGYNR with width NZ = 4, the ZT value can be enhanced to about 1.21 by edge modulation, while it can be boosted to 1.75 for central modulation even at room temperature. This improvement can be observed more evidently at high temperatures. To understand the better thermoelectric coefficient
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Figure 9. (a) Schematic of Z-BGYNR with central region
modulation. (b) Peak value of ZT as a function of number of central modulation (NCM) for Z-BGYNRs with width NZ = 6 (hollow symbols) and NZ = 4 (solid symbols). (c) Phonon projected density of states (PDOS) and overall density of states (DOS) vs. frequency for Z-BGYNR with width NZ = 4.
enhancement in Z-BGYNRs with central modulation, in figure 9(c) we depict the phonon DOS and PDOS for the case of Z-BGYNRs with width NZ = 4. One can clearly see that the PDOS of the central region is substantially larger than that of the edge region due to the stronger localized phonons in this area, indicating that phonons in the central region of nanoribbons dominate the thermal transport of the modulated ZBGYNRs. Therefore, when central geometric modulation is introduced to Z-BGYNRs, the primary phonon transport channels are destroyed. In contrast, these channels can be retained perfectly in the case of edge modulation. Consequently, a lower thermal conductance exists in Z-BGYNR with central modulation and thus leads to the higher value of ZT than that of Z-BGYNR with edge modulation. This result could provide a new route to enhance thermoelectric performance of β-graphyne by engineering the structures. Before closing, it is worth pointing out that, as discussed above, the improvement of thermoelectric performance of 8
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and provide useful guidance for synthesizing nanostructured thermoelectric devices in experiments.
beta-graphyne nanostructures originates from the largely enhanced Seebeck coefficient and drastically reduced thermal conductance that compensate for the depression in electronic transmission. Therefore, one can expect that this enhancement behavior could also exist in the beta-graphyne nanostructures with other types of vacancy. However, the present work focuses more on the influences of location and size of vacancy on the thermoelectric efficiency of beta-graphyne nanostructures. The results can be generalized for extended systems, i.e. the results do not depend on the specific type of the vacancy considered.
Acknowledgments Simulations were performed with computing resources granted by the Jülich Aachen Research Alliance–High Performance Computing (JARA–HPC) from RWTH Aachen University under Project No. jara0073. This work is supported by the National Natural Science Foundation of China (Grant No. 11304262), the Hunan Provincial Natural Science Foundation of China (No. 14JJ3074), and the Scientific Research Fund of Hunan Provincial Education Department (No. 13C927).
6. Concluding remarks In summary, using nonequilibrium Green’s function method, we systematically studied thermal transport and thermoelectric properties of β-graphyne nanoribbons and related nanostructures. Our simulation revealed that thermal conductance of β-graphyne nanoribbons is only 26% of that of their graphene counterparts and shows evident directional dependence, where armchair-edged nanoribbons hold better thermal conductance than those with a zigzag edge. The anisotropic thermal transport property of β-graphyne originates from the more dispersive low-frequency phonon spectrum present in armchair β-graphyne nanoribbons. Moreover, the thermal conductance of A-BGYNRs presents abnormal stepwise width dependence. By analyzing the phonon-projected density of states, we attribute this unexpected thermal transport phenomenon to the weaker influence induced by edges with type-II terminal than those with type-I terminal. For thermoelectric property, the simulation results show that, as compared to the graphene counterparts (ZT 0.05), β-graphyne possesses superior thermoelectric performance with figure of merit ZT achieving 0.5 at room temperature. For the cases of Z-BGYNRs, we also investigate the effect of edge and central geometric modulations on the thermoelectric coefficient. It is found that, when edge modulation is introduced, the thermoelectric performance of Z-BGYNRs is significantly enhanced, and such enhancement can be further improved by increasing the number of edge modulation. The enhancement stems from the largely enhanced Seebeck coefficient and drastically reduced thermal conductance that compensate the depression in electronic transmission. Meanwhile, change of edge modulation length in Z-BGYNRs results in oscillatory behavior of ZT value, which originates from the induced electronic resonance in the structure. Furthermore, when central region modulation is applied to Z-BGYNRs, the enhancement of thermoelectric performance becomes more distinct than edge modulation (ZT exceeding 1.5 even at room temperature), especially for narrow ribbons at high temperatures. The superior enhancement by geometric modulation is explained in terms of the dominant contribution of the central region to the overall thermal transport of Z-BGYNRs. The findings presented in this paper qualify β-graphyne as a promising candidate for high performance thermoelectric applications
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