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Thermoelectric Transport in Graphyne Nanotubes Xiao-Ming Wang, and Shu-shen Lu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp406536e • Publication Date (Web): 29 Aug 2013 Downloaded from http://pubs.acs.org on September 2, 2013

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The Journal of Physical Chemistry

Thermoelectric Transport in Graphyne Nanotubes Xiao-Ming Wang, Shu-Shen Lu*

School of Chemistry and Chemical Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong, China

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Abstract: The thermoelectric transport properties of graphyne nanotubes (GNTs) are investigated by using the non-equilibrium Green’s function (NEGF) method, as implemented in the density functional based tight-binding (DFTB) framework. We find that both the band gap and thermoelectric figure of merit ZT show a damped oscillation with the tube diameter increased. Sign changes can be observed for the Seebeck coefficient, as the absolute value of the chemical potential increased, which is attributed to the minibands formation in the GNTs. In addition, the thermoelectric performance is reduced due to hydrogenation. Unexpectedly, the phonon thermal conductance of one of the hydrogenated GNTs is maintained, which is due to the counteraction of the surface roughness and the strain relaxation. Our investigations and findings are very important and of broad interest for expanding the graphyne community.

Keywords: thermoelectric, graphyne nanotube, tight-binding, Green’s function

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I.

INTRODUCTION Carbon nanotubes (CNTs)1 and graphene2 have attracted enormous attentions of the

science community over the past two decades due to their excellent electrical, mechanical, and thermal properties3-6 and tremendous findings of new physics and technical advances have been achieved.7-10 Both of the materials are constituted by sp2 hybridized carbon atoms, which form the fundamental hexagons. However, other forms of carbon have never been snubbed, for example, graphynes, constituted by sp and sp2 carbons and proposed by Baughman et al11 in 1987, have recently been of increasing interest.12-13 Compared with graphene, graphynes have many superior properties such as higher carrier mobility,14 larger thermoelectric figure of merit ZT,15 and preferred chemical properties.16 As CNTs can be considered as formed by rolling up graphene sheets to cylinders, one can obtain graphyne based nanotubes (GNTs)17 in the same way. Diversity in the graphynes results in various types of GNTs.17 Thus, a variety of properties and corresponding applications should be predicted accordingly. However, there are very limited investigations of GNTs in literature. Theoretical investigations18-19 show that there are rich variations in the electronic properties of GNTs. In addition, experimental success has been achieved in the synthesis of graphdiyne nanotube (GDNT) arrays through an anodic aluminum oxide template,20 which shows the GDNT arrays with high-performance field emission properties. Though the wall thickness of the GDNT obtained is 15 nm, far from single-walled, it is believed that the synthesis of single-walled GNTs may be soon achieved. 3



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In this paper, a non-equilibrium Green’s function (NEGF) method, as implemented in the density functional based tight-binding (DFTB) framework, or called DFTB+NEGF, is utilized to investigate the thermoelectric transport properties of one kind of GNTs, γ-GNT, which is predicted to be more stable. For convenience, hereafter, GNT is used instead. The band gap is found to show a damped oscillation with the tube diameter increased, contrary to the diameter independence characteristic predicted in the literature.17 An interesting finding is that sign changes can be observed for the Seebeck coefficient (S) with the absolute value of the chemical potential (µ) increased. By carefully analyzing the band structure and electron transmittance, we illustrate the origin of this phenomenon. Similarly to that of the band gap, the dependence of the ZT on the tube diameter is also damped oscillatory. In addition, we have tried to improve the thermoelectric performance of GNTs by hydrogenations, which is believed to be very effective for CNTs.21 However, the ZT of hydrogenated GNTs is found to be reduced instead of to be raised unexpectedly. The reason for this abnormal behavior is carefully addressed.

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Figure 1. Schematic of (a) zigzag and (b) armchair GNTs. Both of the GNTs are periodic along the z direction. The red dashed rectangles denote the unit cells. (c) The universal LCR configuration described in the NEGF method. The L and R leads are semi-infinitely extended.

II. MODEL AND METHOD Like CNTs, the nomenclature (n,m) can also be applied for GNTs.19 The tube diameter of a GNT is given by D = a n 2 + m 2 + nm / π , where a is the lattice constant of the graphyne sheet.

With this notation, (n,n) and (n,0) represent zigzag and

armchair GNTs, respectively, as shown in Figures 1(a) and (b). In the present work, we only investigate these two kinds of GNTs.

A. The DFTB framework DFTB22 is based on a second-order expansion of the Kohn-Sham total energy in Density-Functional Theory (DFT) with respect to charge density fluctuations. However, it is highly computationally efficient, hence, suited to large systems and has been implemented in DFTB+.23 In this scheme a non-orthogonal tight-binding like representation of the single-particle wavefunction ψi(r) is constructed by a linear combination of the confined Slater-type atomic orbitals φµ(r), which are determined by solving a modified Schrödinger equation for a free atom within SCF-LDA calculations:22, 24-25

ψ i ( r ) = ∑ cµi ϕ µ ( r )

(1)

µ

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In the atomic orbital basis, the Hamiltonian and overlap matrix elements are expressed as:

0 H µν

ε µneutral freea tom  =  ϕ µA Tˆ + V ( n0A + n0B ) ϕνB  0

if µ =ν if A ≠ B

, S µν = ϕ µ ϕν

(2)

otherwise

where µ and ν are atomic orbital indexes. A and B denote the atoms on which the atomic orbitals φµ and φν are centered. Considering the self-consistent charge (SCC) contribution,22 the Kohn-Shame equation results in:

H µν cν = ε ∑ S µν cν ∑ ν ν i

i

(3)

i

where H is the SCC-DFTB Hamiltonian with the self-consistent correction H1: 0 1 H µν = H µν + H µν ,

1 H µν =

1 S µν ∑ C ( γ AC + γ BC )∆qC 2

∀ µ∈A, ν∈B

(4)

∆q is the Mulliken charge and γ is appropriate functional taking into account the Hartree and exchange-correlation interactions. Finally, the expression for the total DFTB energy reads:22

i E tot = ∑ cµi cνi H µν + i µν

1 ∑ ∆qA ∆qBγ AB + Erep 2 AB

(5)

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where the short-range repulsive pair potential Erep can be obtained by taking the difference of the SCF-LDA cohesive energy and the corresponding SCC-DFTB electronic energy for a suitable reference structure.22

The interatomic force constants (IFC) matrix can be derived by taking the second derivative of the total energy with respect to the nuclear coordinates:

K AB ( R ) =

∂E tot ∂R A ∂R B

(6)

Phonon frequencies ω(q) are solutions of the secular equation:

det

1

MAMB

K% AB ( q ) − ω 2 ( q ) = 0

(7)

where K% AB ( q ) = ∑ e− iq⋅R K AB ( R ) is the Fourier transformation of the IFC. In the R

present work, we use a 1×1×4 supercell to construct the IFC of GNTs and 50 q points along the G-X line of the first Brillouin zone to calculate the corresponding phonon dispersion relations.

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Figure 2. (a) (b) Band structures and (c) (d) phonon dispersion relations of the (2,2) GNT obtained by DFT and DFTB, respectively. The red dashed line denotes the Fermi energy.

We compare the band structures and phonon dispersion relations of the (2,2) GNT obtained from DFTB with those from first-principles calculations (see the Supporting Information for details), as shown in Figure 2. The band gap is further opened to 1.38 eV within DFTB compared with that of 0.59 eV calculated from DFT. It is believed that DFT usually underestimates the band gap of semiconductors. The band gap of the graphyne sheet calculated by DFTB is 1.25 eV, which is also larger than the value of 0.46 eV within DFT.15 More accurate evaluation of the band gap needs GW calculations.26 The quasiparticle band gap of graphdiyne, whose electronic 8


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properties are believed to be similar to graphyne, is increased to 1.1 eV within GW approximation from 0.44 eV predicted by DFT.27 Thus, the band gap of GNTs obtained by DFTB should be superior to that from DFT. In addition, the phonon modes calculated from DFTB are in excellent agreement with the results from DFT. Therefore, the electronic and thermal properties can be described very well within the DFTB framework. B. The NEGF method The NEGF method is widely used to evaluate the quantum transport properties of electrons28-29 and phonons30-31 for nanoscale systems. Here, we only focus on the ballistic transport, while the inelastic effects such as electron-phonon32 and phonon-phonon33 interactions can also be included in this method, which are out of the scope of the present work. In this method, the whole system is divided into three parts as a LCR configuration shown in Fig. 1(c). In homogeneous systems like GNTs, one only needs to know the intralayer and interlayer interactions, with sufficient unit cells in one principal layer. With a non-orthogonal basis Hamiltonian as in the DFTB framework, the retarded Green’s function reads: G r = ( ε SC − H C − Σ rL − Σ rR )

−1

(8)

where ε is the electron energy, H and S are the Hamiltonian and overlap matrix of the central region, respectively. Σ r denotes the self-energies for the leads: † † Σ rL = H LC g Lr H LC , Σ rR = H CR g Rr H CR

(9)

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where g Lr and g Rr are the retarded surface Green’s functions of the semi-infinite left and right leads, which can be calculated via an iterative procedure.34 Once the retarded Green’s function is obtained, we can calculate the electron transmittance through: T (ε ) = Tr(G r Γ LG a Γ R )

(10)

where G a = ( G r )† is the advanced Green’s function and Γ β = i(Σrβ − Σaβ ) describes the interaction between the leads and the central reigon. By using the electron transmittance, one can obtain the linear coherent transport of different physical quantities under linear response approximation: G ( µ , T ) = e 2 L0 ( µ , T )

S ( µ,T ) =

(11)

1 L1 ( µ , T ) × eT L0 ( µ , T )

(12)

2 L1 ( µ , T )  1   κ e ( µ , T ) = ×  L2 ( µ , T ) − T  L0 ( µ , T ) 

Ln ( µ , T ) =

(13)

 ∂f ( ε , µ , T )  2 n dε T (ε ) × (ε − µ ) × −  ∫ h ∂ε  

(14)

where G, S, κe denote the electron conductance, Seebeck coefficient, and electron thermal conductance, respectively. Ln ( µ , T ) is Lorenz function, e is the charge of electron, h is the Planck constant. µ is the chemical potential, which can be varied by doping or gating. T is the absolute temperature, and f (ε, µ, T) is Fermi-Dirac distribution function. For phonon transport, one only needs to substitute εS by 10


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(ω+iη)2I and HC by KC in Eq. (8). Once the phonon transmittance is obtained, the phonon thermal conductance (κph) can be calculated:

κ ph (T ) =

h 2π

∫

∞

0

T (ω)ω

∂f (ω, T ) dω ∂T

(15)

where ω is the phonon frequency, f (ω, T) is the Bose-Einstein distribution function. Finally we can obtain the thermoelectric figure of merit:

ZT =

GS 2T κ e + κ ph

(16)

III. RESULTS AND DISCUSSIONS

Figure 3. Band gaps of GNTs vs tube diameter. The red squares and blue circles denote armchair and zigzag GNTs, respectively. The solid dashed line denotes the graphyne sheet.

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Figure 3 displays the band gap variations of GNTs as a function of the tube diameter. The band gaps of both of the armchair and zigzag GNTs show a damped oscillation with the tube diameter increased, and are gradually converged to that of the graphyne sheet. In addition, the gap positions are located at the Gamma point and the edge of the Brillouin zone for the armchair and zigzag GNTs when n is odd and even, respectively. It is to be noted, that our result is different from that of ref. 19, which claimed that the band gap, with a constant value that of the graphyne sheet, is diameter independent. Though DFT usually underestimates the band gap as mentioned in the previous section, it can give the result qualitatively or predict the trend of the band gap variation. Thus, we refer to the DFT calculations. The band gaps of the (2,2) and (2,0) GNTs calculated from DFT are 0.59 eV and 0.93 eV, respectively, which indicates the band gap dependence on the tube diameter. Therefore, the band gap variation is reasonable.

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Figure 4. (a) (b) Electron and (c) (d) phonon transmittance of GNTs.

The electron and phonon transmittance calculated according to eq. (10) are shown in Figure 4. The transmittance curves display stepwise behavior which implies that the electrons and phonons of the leads can pass through the central region without any scattering. The gaps in the electron transmittance correspond to the band gaps in the band structures. The phonon transmittance at zero frequency is 4, which is physical and indicates the well localized matrix elements of the IFC in DFTB. In addition, the transmittance of both electron and phonon increases with the tube diameter increased, which is due to the introduction of more transport channels.

Figure 5. Thermoelectric factors of armchair GNTs: (a) G (b) S (c) κe (d) κph. The first three factors are obtained at room temperature. G0 is the conductance quantum

G0=2e2/h. A is the cross area, A=πdδ for GNTs and A=aδ for graphyne and graphene, 13



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where d is the tube diameter, a is the lattice constant, and δ=0.335 nm. Inset of (d) shows the corresponding logarithmic plot.

Figure 6. Thermoelectric factors of zigzag GNTs. Similar notations as Figure 5. The inset of (b) shows the enlargement of the low chemical potential part.

Figures 5 and 6 show the thermoelectric factors, namely, G, S, κe, and κph , of armchair and zigzag GNTs, respectively. If µ is in the middle of the band gap, nearly no electrons can be excited into the conduction channels, thus a zero value of G can be observed. When µ moves into the valence and conduction subbands, the electrons are excited into the conduction channels, which are corresponding to the electron transmittance. Thus, G and κe increase correspondingly.

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Figure 7. The band structure and transmittance of the (2,0) GNT.

Compared with that of zigzag GNTs, the S of armchair GNTs shows more obvious variation due to larger curvature. It is three to five times larger the S of GNTs than that of graphyne.15 In addition, an interesting finding is that one or more sign changes of the S can be observed with the absolute value of µ increased. The phenomenon is more obvious for armchair GNTs, especially the (2,0) GNT, which is also the smallest in diameter. It is well known that N-type materials have negative S, while P-type materials have positive S. A sign change of S can be observed in the thermoelectric transport of superlattices due to minibands formation.35-36 To clarify why the sign of the S changes in GNTs, we refer to the band structures and corresponding transmittance, as shown in Figure 7. There are two obvious minibands formation nearly 1 eV away from the Fermi energy, where sign changes of S can be observed. Minibands can form in multi-barrier systems. Carefully analyzing the optimized structure of the (2,0) GNT, we find that the acetylenic linkages along the 15



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circumferencial direction deviate from the tube wall to form barriers due to the large curvature, as shown in Figure 8. With the tube diameter increased, the deviation is reduced and the S becomes monotonic.

Figure 8. Barriers formation in the (2,0) GNT. The red solid line is a sketch of the barriers along the tube length.

Figures 5(d) and 6(d) show the κph of armchair and zigzag GNTs, respectively, as a function of temperature. The κph of graphene and graphyne are also plotted for comparison. The κph of GNTs is reduced to approximately 40% that of graphene compared with that of graphyne, which is 46%15 that of graphene, at room temperature. Above about 100 K, the κph slightly increases with the tube diameter increased, while the diameter dependence is reversed at low temperatures, as shown in the inset, which is attributed to that the phonon transport at low temperatures is dominated by the flexural phonon modes15. In addition, all the GNTs have the linear T dependence of κph, which is similar to that of CNTs.37

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Figure 9. The room temperature ZT of (a) armchair and (b) zigzag GNTs as a function of chemical potential.

Figure 10. The diameter dependence for the optimized ZT of GNTs. α and β denote two hydrogenated GNTs.

By using the thermoelectric factors, we get the ZT of the GNTs, as shown in Figure 9. The maximized ZT value of the armchair GNTs, 0.83 for the (3,0) tube which is 17



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slightly smaller than that of the commercial thermoelectrics, is larger than that of the zigzag ones. The ZT peaks for both the electrons and holes conductions are located near the corresponding conduction and valence subbands, respectively. A damped oscillation of the optimized ZT on the tube diameter, similar to that of the band gap, can be observed, as shown in Figure 10, which reflects the weak dependence of the

κph on diameter. It is to be expected that when the tube diameter grows larger, the ZT of the GNTs will be converged to that of the graphyne sheet, as shown by the black dashed line.

Figure 11. Top view of the two kinds of hydrogenated (3,0) GNTs. The big black and small red circles denote carbon and hydrogen atoms, respectively.

Hydrogenation is believed to be able to improve the thermoelectric performance of CNTs significantly.21 As graphyne is more chemical reactive than graphene,16 it is natural to modify the surface of GNTs by hydrogenation. Though different hydrogen coverage ratios can result in different transport properties, for simplicity, only two kinds of hydrogenated GNTs are considered, as shown in Figure 11. One is the (3,0) GNT with two hydrogen atoms added to one of the acetylenic bonds along the 18


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circumferencial direction, denoted by α, which are more favored according to our calculations. The other is the (3,0) GNT with all the acetylenic bonds along the circumferencial direction hydrogenated, as denoted by β. The main structure of α keeps tubular with slight deviation around the hydrogen atoms. However, the tube wall of β is reconstructed due to great hydrogenation. Figure 12 shows the power factor S2G and κph of α and β compared with the pristine (3,0) GNT. The S2G of hydrogenated GNTs are significantly reduced, especially for β, which simply implies that the electronic transport is greatly affected by hydrogenation. As for α, the κph is reduced due to the surface roughness. After hydrogenation, the C-H bond is perpendicular to the tube wall, which introduces the surface roughness compared with the pristine GNT. The surface roughness can affect the phonon transport effectively. However, the κph is maintained for β, unexpectedly, which we attribute to the counteraction of the surface roughness and the strain relaxation due to hydrogenation. The reconstruction of the tube wall in β relaxes the strains due to the curvature, which can enhance the thermal conductance.38 Finally, the two effects counteract each other, and the κph is maintained. Thus, the ZT of the (3,0) GNT is reduced due to hydrogenation, as shown in Figure 10. Therefore, hydrogenation cannot improve the thermoelectric performance of GNTs, effectively. Other band engineering and structure design scheme should be searched to improve the thermoelectric performance of GNTs.

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Figure 12. The (a) power factor and (b) κph of the two hydrogenated (3,0) GNTs.

IV. CONCLUSION In conclusion, the thermoelectric transport properties of graphyne nanotubes (GNTs) are systematically investigated. A non-equilibrium Green’s function (NEGF) method as implemented in the density functional based tight-binding (DFTB) framework is utilized. The DFTB+NEGF approach can generate accurate results compared with the first-principles calculations. The results show that the band gap of the GNTs shows a damped oscillation with the tube diameter increased. The same behavior can also be observed for the diameter dependence of the ZT. An interesting finding is that sign changes can be observed for the Seebeck coefficient of GNTs with the absolute value of the chemical potentials increased, especially for the smallest (2,0) GNT, which is attributed to the minibands formation by the deviation of the acetylenic linkage from the tube wall. The phonon thermal conductance increases slightly with the tube diameter increased. The maximum value of the ZT is 0.83 for the (3,0) GNT at room temperature, which is slightly smaller than that of the commercial thermoelectics. In addition, the hydrogenated GNTs are also investigated. 20


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The thermoelectric power factors are significantly reduced due do hydrogenation. Unexpectedly, the phonon thermal conductance of one of the hydrogenated GNTs is maintained, which is attributed to the counteraction of the surface roughness and the strain relaxation. Thus, the ZT of GNTs by hydrogenation is reduced. Our investigations and findings are very important and of broad interest for expanding the graphyne community.

AUTHOR INFORMATION Corresponding authors: *

E-mail address: [email protected]

ACKNOWLEGEMENTS Financial supports from the Key Program of the National Natural Science Foundation of China-Guangdong Joint Fund (Grant No. U1034004) and the National Natural Science Foundation of China (Grant Nos. 51276202 and J1103305) are gratefully acknowledged.

Supporting Information Available: The first-principles calculations are performed on a plane wave basis method in the framework of density functional theory within the local density approximation (LDA). This material is available free of charge via the Internet at http://pubs.acs.org.

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