Enhanced thermoelectric properties of armchair graphene nanoribbons with defects and magnetic field

W. Zhao, Z. X. Guo, J. X. Cao, and J. W. Ding'

Citation: AIP Advances 1, 042135 (2011); doi: 10.1063/1.3660787 View online: http://dx.doi.org/10.1063Z1.3660787 View Table of Contents: http://aip.scitation.org/toc/adv/1M Published by the American Institute of Physics

Enhanced thermoelectric properties of armchair graphene nanoribbons with defects and magnetic field

W. Zhao,12 Z. X. Guo,12 J. X. Cao,12 and J. W. Ding12 3

1 Department of Physics & Institute for Nanophysics and Rare-earth Luminescence, Xiangtan University, Xiangtan 411105, People's Republic of China

2Key Laboratory of Low Dimensional Materials & Application Technology of Ministry of Education, Xiangtan University, Xiangtan 411105, People's Republic of China

(Received 10 July 2011; accepted 19 October 2011; published online 2 November 2011)

We have investigated the thermoelectric properties of armchair graphene nanoribbons with defects and magnetic field by using non-equilibrium Green's function method. For perfect armchair graphene nanoribbons, it is shown that with its width increasing, the maximum of the figure of merit ZT is monotonously decreased while the phononic thermal conductance increases linearly. In the presence of defects, the phononic thermal conductance decreases monotonously with the defect number increasing. Interestingly, the maximum of ZT values is proportional to the defect number in longitudinal direction, but inversely proportional to that in transversal direction. In the presence of magnetic field, very remarkable enhancement of ZT value is further obtained at the bottom of conduction band. Copyright 2011 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License. [doi: 10.1063/1.3660787]

Since the graphene was successfully fabricated in experiments, the transport properties of graphene-based nanodevices have attracted great interests for electronic applications owing to their unique band structure.1-3 As the quasi-one-dimensional (Q1D) graphene nanostructures, the electronic and thermal properties of graphene nanoribbons (GNRs) have been extensively studied experimentally and theoretically.4-9 Recently, the thermoelectric properties of GNRs became a new focus due to their potential applications in the thermoelectrics.10 The performance of a thermoelectric device is characterized by the figure of merit ZT=GeS2T/K, where Ge is the electronic conductance, S is the Seebeck coefficient, T is the temperature, and k is the total thermal conductance including both the electronic contribution ke and phononic contribution kp. However, due to the high thermal conductivity, perfect GNRs usually have a very small ZT value.5 For actual applications, some efforts have been done to enhance the ZT value through artificial manipulation such as introducing disorder11,12 and nanojuctions.13 However, since the electronic conductance can also be destroyed by these treatments, very little progress was obtained. Recently, the experimental14,15 and theoretical16 studies of silicon nanowires (NWs) indicate that the ZT value can be largely increased because the thermal conductivity can be sufficiently decreased while the electronic conductivity keeps nearly unchanged.17 Compared with silicon NWs, the GNRs have more perfect Q1D structures. Its electronic conductance is expected to be not seriously changed by defects. This means that the introduction of defects may be a promising way to enhance the thermoelectric efficiency of GNRs. In the presence of magnetic field, also, the electronic conductivity and Seebeck coefficient can be largely enhanced but the thermal conductivity is nearly keeping unchanged.18-20 Therefore, it is also expected that the ZT value can be enhanced by external magnetic field.

In this paper, within the tight-binding approximation, we have investigated theoretically the thermoelectric properties of armchair GNRs (AGNRs) with defects (vacancies) by using non-equilibrium Green's function (NEGF) method. With width increasing, the maximum of ZT values of

aE-mail: jwding@xtu.edu.cn

2158-3226/2011/1 (4)/042135/6 1, 042135-1 © Author(s) 2011 r M ■

FIG. 1. Schematic diagram of a defected 27-AGNR atomistic structure of the width ^=3nm and length L=10nm, for which the defect numbers are of n=3 and m=8.

perfect AGNRs decreases monotonously, while the phononic thermal conductance increases linearly. The thermal conductivity of graphene is estimated to be 3200W/mK, which is consistent with the recent experimental results.7 For defected AGNRs, the phononic thermal conductance decreases monotonously with the defect numbers n and m in both transversal and longitudinal direction. Interestingly, the maximum of ZT values is proportional to the defect number in longitudinal direction, but inversely proportional to that in transversal direction. In the presence of magnetic field, on the other hand, large enhancement of ZT value is obtained, which is more sensitive to magnetic field at the bottom of energy band than at the top. Especially, with the number of defects increasing, the ZT value manipulated by magnetic field can be more susceptible. Thus, the results provide an effective way for enhancing the thermoelectric properties of GNRs materials.

In order to enhance the ZT value of AGNRs, we introduce vacancies as shown in Fig. 1. The system is composed of a central junction of width W and length L and two semi-infinite ideal leads of the same width. The central junction is formed by removing hexagonal carbon rings from perfect AGNRs, in which the numbers of hexagons along the transversal and longitudinal directions are denoted by index n and m. Following a common convention, we refer to the AGNR with N dimer lines in width as N-AGNR.21

For electronic transport, by using an atomistic pz orbital tight binding basis, the device can be described by the Hamiltonian22

H = c+c — t J] el 6'¡c¡ cj (1)

U) H, j)

where ci+ and ci are the creation and annihilation operators on site i and summation in the second term is performed over all available nearest-neighbors (NN). The NN hopping integral t is given by 2.7eV,4 and the on-site energy e equals to 0. In the presence of a perpendicular magnetic field B, the hopping integral requires the Peierls phase factor, 0ij=2n<jj/<o, in which <ij = f/ A ■ dl is the line integral of the vector potential A from site i to neighbor site j, where the vector potential A= (-By, 0, 0) and <p0=h/e is the flux quantum, with the Plank constant h. For incident electronic energy E, the electronic transmission per spin though junction region is calculated as Te (E) = Tr [TLGrTRG"], where the linewidth function TLr(E) is defined as rLr (E) — i [XL r — r], and the retarded (advanced) Green's function is given by Gr'a (E) = [(E ± in) I — HC — XrLa — £Ra], in which the retarded (advanced) self energy due to the coupling to all leads can be obtained numerically.23 Due to the low scattering rates and large mean free paths of electrons in GNRs on room temperature, which is induced by suppression for the backscattering of electrons and phonons with large wave vector, the electron-phonon interaction and electron-electron interaction are neglected.24

10 20 N

FIG. 2. (Color online) (a) Phononic thermal conductance and maxima of ZT values as a function of the width N of perfect AGNR. (b) ZT values of perfect 27-AGNR as a function of the chemical potential /.

Define an intermediate function In(ß,T) as25'26

In(ß, T) = (2/h) I (E - ß)n[-dfFD(E, ß, T)/dE]Te(E)dE

where the factor 2 counts the spin degeneracy, fFD(E,ß,T ) is the Fermi-Dirac distribution function and ß is the chemical potential. The electronic conductance, thermopower, and the electronic contribution to the thermal conductance can be conveniently derived from

Ge(ß, T) = e2Io(ß, T)

S(ß, T) = Ii(ß, T)/[qTIo(ß, T)]

Ke(ß, T ) = (1/T )[I2(ß, T ) - Ii2(ß, T )/Io(ß, T )]

where q is the electric charge of carriers, which is positive for holes and negative for electrons.

The Green's function and transmission can be calculated similarly for phonon transport.27 One only needs to change E to m2M, H to D, and compute the self-energies accordingly, where m is the phonon frequency, M is the mass of carbon atom, H is the Hamiltonian and D is the dynamic matrix. The dynamic matrix only includes carbon atoms since the hydrogen atoms at the edges are not important in forming vibration modes.28 The dynamic matrix is constructed by using a spring mass model29 and 4NN interaction is considered. Phonon-phonon scattering and a harmonic lattice vibration are ignored. The phononic contribution to the thermal conductance is given by

Kp(T) = (1/2n) I hœ[dfBE(œ, T)/dT]Tp(œ)drn

where Tp(o>) is the phononic transmission and fBE(m,T ) is the Bose-Einstein distribution function. The temperature T is fixed to 300K.

In Fig. 2, we first calculate the thermoelectric properties of the perfect AGNRs. As shown in Fig. 2(a), the phononic thermal conductance increases linearly with the width N increasing, which results from the increase in number of effective phononic channels. Taking the phononic mean free path and thickness of graphene as 0.8^m, 0.34nm,32 the thermal conductivity of graphene is

-■-n=1 -•—n=2 n=3 -▼— n=4

FIG. 3. (Color online) (a) Phononic thermal conductance and (b) maxima of ZT values of 27-AGNRs with defect numbers m in longitudinal direction and n in transversal direction.

estimated to be 3200 W/mK by the relation between conductance and conductivity. The result is in agreement with the experimental values in the range of 3000-5000 W/mK of graphene, depending on the specific sizes varied from 1 to 5 /j,m.9 Also, the maximum of ZT values decreases monotonously with N increasing, which indicates that the thinner AGNRs have the higher thermoelectric efficiency. Here we can estimate the ZT value of graphene in the order of 10-3, which is too small to actual application. To illustrate the tuning effect by gating or doping, we calculate the ZT values as a function of the chemical potential \x in Fig. 2(b). It is shown that the maximum of ZT values occurs at the bottom or top of energy band, which is due to the higher Seebeck coefficient and less electronic thermal conductance. This is different from the previous work11 in which the maximum of ZT values appears only at the bottom of bands. These results indicate that the higher ZT value can be obtained by controlling the electronic behavior in the edges of energy bands.

We further calculate the phononic thermal conductance and maximum of ZT values with the defect numbers m in longitudinal direction and n in transversal direction in Figs. 3(a) and 3(b), respectively. The width of AGNR is fixed to 3nm, corresponding to an index of N=27. For a given n, it is seen from Fig. 3(a) that the phononic thermal conductance decreases exponentially with the defect number m increasing, converging to a limit value of the AGNR superlattice with periodically arranged defects. The limit value is decreased with n increasing. The thermal conductivity of 0.6 nW/K is obtained at n=4, much less than 3.6 nW/K in the case of no defects. It is due to the fact that the high frequency phononic modes are destroyed by the defects, leading to a large reduction of phononic thermal conductance at room temperature.

For ZT characteristics of the defected AGNR, the results are shown in Fig. 3(b). It is seen that for a given m, the maximum of ZT increases with n decreasing. Interestingly, it is observed that for a given n, the maximum of ZT increases with m increasing, converging to a limit value of the AGNR superlattice with periodically arranged defects. Especially, a large enhancement in ZT limit value can be obtained at n=1, 300% higher than that in the case of no defects. This means that the thermoelectric properties of GNR can be modulated by both increasing m and decreasing n. The result is very different from the results in Ref. 11, in which the ZT value decreases greatly with defects increasing. Interestingly, the maximum of ZT values increases with m increasing in longitudinal direction, while it decreases with n increasing in transversal direction. This phenomenon can be understood from the following consideration. The number of electronic modes is increased

q> (h/e)

20 30 <P (h/e)

FIG. 4. (Color online) Magnetic flux dependence of the maxima of ZT values (a) at various chemical potential j of 1.0, 4.0 and 7.0eV, located at the bottom, middle and top of energy band, and (b) for different defect numbers n in the transversal direction. The length of AGNRs is fixed to 20nm, corresponding to an index of m=16.

by decreasing n, which results in an increase of the electronic conductance. On the other hand, the phononic thermal conductance is decreased by increasing m, leading to an increase of ZT value. It is well known that the thermal properties of GNRs are very sensitive to defects,30-32 but the electronic properties not. So we expect to enhance the ZT value of GNRs by introducing some defects periodically, which could decrease the phononic thermal conductance greatly, but not change the electronic conductance severely.

The electronic conductance and Seebeck coefficient and thus ZT value can be greatly modulated by the external magnetic field. For defected AGNRs, in Fig. 4(a) we consider the magnetic field dependence of ZT value at chemical potential 1.0, 4.0 and 7.0eV, located at the bottom, middle and top of energy band, respectively. The length of AGNRs is fixed to 20nm, corresponding to an index of m=16. It is found that the ZT value oscillates with the magnetic flux, and reach the maximum ZT=0.27 at <p=30^o. It is worth notice that the peak width of ZT value extends about 5^0, which shows that the state formed by magnetic field is very stable. In addition, the enhancement of ZT value is most obvious at the bottom of band, which only needs small gate voltage. For a given chemical potential of 1.0eV corresponding to the bottom of band, also, we calculate the ZT value for different defect number n. It is shown that with the defect number n increasing in the transversal direction, the tunable effect of magnetic field is strengthened largely. It is due to the Landau levels induced by magnetic field, leading an enhanced electronic conductance.

In summary, by using NEGF method, we have investigated numerically the thermoelectric properties of AGNRs with periodic defects and magnetic field within the tight-binding approximation. For perfect AGNRs, with the width increasing, the phononic thermal conductance increases linearly and the maximum of ZT values decreases monotonously. For defected AGNRs, with the number of defects increasing, the phononic thermal conductance decreases monotonously. Especially, the maximum of ZT is proportional to the defect number in longitudinal direction, but inversely proportional to that in transversal direction. In the presence of magnetic field, a remarkable increase of ZT value is also obtained, which is more sensitive to the magnetic field at the bottom of energy bands than at the band top. Furthermore, with the number of defects increasing, the tunable effect of magnetic field becomes more important. Our results provide an effective mean for enhancing the thermoelectric and thermospintronic properties of GNR materials.33

This work is supported by National Natural Science Foundation of China (No. 10674113), and the Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 200726), partially by Start-up funds (No. 10QDZ11) and Scientific Research Fund (10XZX05) of Xiangtan University.

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