Jie Suna,
Na Lin*a,
Zhenyu Li*b,
Hao Renc,
Cheng Tanga and
Xian Zhao*a
aState Key Laboratory of Crystal Materials, Shandong University, 250100 Jinan, Shandong, PR China. E-mail: linnakth@gmail.com; zhaoxian@sdu.edu.cn
bHefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei 230026, China. E-mail: zyli@ustc.edu.cn
cCenter for Bioengineering and Biotechnology, China University of Petroleum (East China), 266580 Qingdao, PR China
First published on 15th December 2015
To understand the effect of grain boundaries (GBs) on the electronic transport properties of graphene, we have performed first principles studies on the electronic structure and transport properties of graphene with four single GBs and two crossed GBs. Calculations indicate that the zero band gap nature of graphene is not destroyed by introducing GBs. The localized states introduced by GBs have a positive contribution to the transport ability in the vicinity of the Fermi level. The transport properties across and along the GBs show an obvious discrepancy. Compared with the pristine graphene, an at least 50% current suppression is found in the transport across the GBs, while less current degradation is present along the GBs. The transport along other directions shows that the transmission ability can be efficiently enhanced by avoiding the transport direction across the GBs. Moreover, the transport behavior of the crossed GB is akin to that of the transport across a single GB.
Early theoretical studies on electronic structure of graphene have suggested that the van Hove singularity states close to the Fermi energy exist in certain ordered GBs.14,15 A recent experiment well verified these peculiar states by using scanning tunneling microscopy (STM) and confirmed that the conductance around certain ordered GBs could be significantly enhanced by these states.16 For the transport investigations, there is still a debate whether the GB substantially degrade the electronic transport properties of graphene. Yu et al. found the inter-grain (across the GB) resistance largely increased compared with intra-grain (along the grain) using four-probe measurement.13 Nevertheless, Huang et al. declared that GBs were not strongly resistive by electrical measurement.17 Instead, Clark et al. measured electronic transport near specific GBs and found the grain boundary conductance was less than one-third of the bulk conductance.18 The discrepancy in these findings is mainly caused by a lack of knowledge of the precise morphology of GBs for the measured samples. In spite of the fact that precise atomic configuration at the local region of a GB can be revealed by aberration corrected high-resolution transmission electron microscope (AC-HRTEM),19 the electronic transport measurement corresponding to a definite structure of GBs is still lacking.
Despite the limitation in experiment, theoretical investigations have gained some insight into graphene with GBs. Yazyev and Louie found that the transport behaviors of GBs, either high transparency or perfect reflection of charge carriers, depended on the translation vector of the two adjacent grains based on the momentum conservation.20 The rationality of this statement was also verified by first principles studies.21 Vancsó et al. showed the misorientation angle of the two adjacent graphene grains and the atomic structures of the GBs were two main factors governing the transport.22 In addition, Zhang et al. predicted that the transmission was not severely degraded by ordered GB compared with that of perfect graphene and they attributed this degradation to the adsorbates.23 These previous theoretical works mainly concentrated on the transport direction across a single ordered GB but little involved with the transport direction along or with other angles relative to GB. The transport through the graphene with GBs should be direction-dependent due to the structure anisotropy of the system.24 Furthermore, false-colour dark-field TEM (DF-TEM) image of polycrystalline graphene clearly displayed crossed GBs in samples.11 However, there is still no study on the electronic structure and transport properties of graphene with crossed GBs. A recent STM study also presented two identical GBs joined with a 120° turn.25 It is interesting the states existed in two GBs vanished at the 120° junction while its mechanism is still unknown.
Based on the unexplored issues mentioned above, in present study we carry out first principles calculations on the electronic structure and transport properties of graphene with GBs, in which four types of single GBs and two types of crossed GBs are taken into account. We consider the transport through the single GB with the directions including across and along the GB as well as other angles relative to the GB. We hope our calculations, to some extent, contribute to the design of practical electronic devices.
I = G0∫T(E, V)[fL(E) − fR(E)]dE |
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Fig. 1 Atomic structure of graphene with GBs. The rectangular supercells used in this work depicted in red line. |
(2,0)|(2,0) | (2,1)|(2,1) | (3,1)|(3,1) | (5,0)|(3,3) | (2,0)|(2,0)D | (2,1)|(2,1)D | (3,1)|(3,1)D | ||
---|---|---|---|---|---|---|---|---|
T1(Å) | 4.9619 | 6.5939 | 8.9776 | 12.5622 | 39.3300 | 26.1572 | 35.9401 | |
T2(Å) | 31.4839 | 29.3414 | 31.1452 | 32.1596 | 21.0459 | 30.9972 | 124.6430 | |
Eform (eV nm−1) | This work | 4.91 | 3.23 | 2.74 | 4.56 | 5.38 | 3.22 | 2.70 |
Previous studies | 3.3,36 3.38 (ref. 14) | 2.84 (ref. 14) | 5.03 (ref. 29) | |||||
3.44,29 4.0 (ref. 32) | 2.91 (ref. 29) |
In the case of (2,0)|(2,0) GB, it shows an almost flat band along the Γ–Y (L–X) line in the vicinity of Fermi level. The flat-band character is similar to that of the zigzag-edged graphene nanoribbons.39 In addition, the extended van Hove regions in the band structure of (2,0)|(2,0) GB are located at about ±0.1 eV near the Fermi level. This phenomenon is confirmed by a recent experiment, which has observed the van Hove singularity states were close to the Fermi energy in certain ordered GBs by using scanning tunnelling microscopy.16
These local band states also exist in (2,1)|(2,1) and (3,1)|(3,1) GB corresponding to the region within 0.5 eV below and above the Dirac point.14 Similar to (2,0)|(2,0) GB, for the more complex structure of (5,0)|(3,3) GB, there is also a flat band near the Fermi level.
We shift our focus now to the crossed GBs. The zero band gap nature is unexpected in the intersection of two GBs. The (2,1)|(2,1)D GB keeps the Dirac point on the Γ–Y line, nevertheless, the (3,1)(3,1)D GB shifts to the Y point. A striking band-split along Γ–Y line is shown in (3,1)|(3,1)D GB. In addition, there is still a van Hove region around the L point in the vicinity of the Fermi level in the crossed GB (3,1)|(3,1)D.
In order to further investigate the properties of the band states lying close to the Fermi level, we represent isosurface contour plots of wave functions of the highest valence band (HVB) and lowest conductance band (LCB) at Γ point for (2,0)|(2,0), (3,1)|(3,1), (5,0)|(3,3) GBs and Y point for (2,1)|(2,1), (2,1)|(2,1)D GBs (see Fig. 3). We can see that the states close to Fermi level are mainly localized along the GB region. For (2,0)|(2,0) GB, it shows the typical zigzag edge states found in zigzag nanoribbons. However, the GB states hybridized with the bulk states are found in (2,1)|(2,1) and (3,1)|(3,1) GB. These hybridized local states take place within a region of ∼5 Å from the geometric center of the GB.15,16 In the case of (5,0)|(3,3) GB, the asymmetric wave functions of both HVB and LCB show that some states in the armchair region are present in addition to the states in the GB region. As for the crossed GB (2,1)|(2,1)D, the localized states distribute along the GBs with the exception of the crossed region. Both the HVB and LCB states in the crossed region have a great difference with the states in other GB regions. The difference is probably due to the interaction of two GBs. Moreover, a recent experiment shows the states existed in two GBs vanished at the 120° junction in (3,1)|(3,1)D GB.25
To clarify the discrepancy of the electronic states in the crossed region and other GB regions, we project the total density of states (DOS) of (3,1)|(3,1)D GB onto the carbon atoms, which belong to two heptagons, one in the crossed region and another in other GB regions (Fig. 4(a)). The partial DOS (PDOS) of the atoms and the heptagon core (sum over the PDOS of each atom) are shown in Fig. 4(b) and (c), respectively. We can see that the heptagon in crossed region has a positive (negative) contribution to the total DOS below (above) the Fermi level compared with that in other GB regions. However, the PDOS of each atom shows more complex contribution in two different regions. For example, the atoms in position 1 and 5 in two regions have similar contributions to the total DOS, while large changes in other positions. In the case of the atom in position 6, we notice that a prominent peak in other GB region is located above the Fermi level but not in the crossed region. This phenomenon agrees well with the STM experimental observations that the states existed in two GBs vanished at the 120° junction.25
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Fig. 4 (a) Two heptagons in crossed region and other GB regions. The partial DOS, projected onto each atom (b) and heptagon core atoms (c), in two different regions. |
In addition to the electronic structures, electronic transport across and along the single GBs are simulated by first principles quantum transport calculations based on density functional theory and the non-equilibrium Green's function formalism (see ESI† section 4 for more details). The two-probe system has been adopted and a comparable size of the scattering region of each GB has been chosen in our calculations. The calculated zero-bias transmission spectra are collected in Fig. 5. One can see that the transmission probabilities mainly decrease as the electrons across the GBs in the range of |E| > 0.5 eV. This decreased transmission probabilities have also been found in other theoretical works.20,23 However, an increased transmission intensity appears in the range of |E| < 0.2 eV of (2,0)|(2,0) GB and (5,0)|(3,3) GB and |E| < 0.5 eV of (2,1)|(2,1) GB and (3,1)|(3,1) GB. This is probably due to the existence of extended van Hove band states in this energy region, as shown in our calculated band structures, which has been verified to enhance the conductance around ordered grain boundaries.16
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Fig. 5 The transmission spectra at zero-bias. (a) Transport across the GBs (b) transport along the GBs. |
By comparing the transmission spectra of these GBs, it is worth to stress that (2,1)|(2,1) GB has less effect to impede the electronic transport compared with other GBs. On the other hand, a distinct transport gap located at about −1.0 eV is induced by (3,1)|(3,1) and (5,0)|(3,3) GBs. Therefore, the electronic transport properties depend on the specific GB structures. However, the GB structure is nubilous as the electronic transport properties measured in experiment.13,17 We consider this is the origin of the debate about whether the GB largely degrades the electronic transport.
It can be argued that our calculated transmission spectra are not consistent with those reported in other literatures. Take (5,0)|(3,3) GB as an example, a transport gap was found near the Fermi level by Yazyev and Louie,20 while not present in our result. We have repeated their result and attributed this discrepancy to the different electrode (we use the graphene with GB and they used the graphene without GB) applied in the calculations.
In sharp contrast to the transport across the GBs, the transmission probabilities along the GBs are not degraded. Especially for (2,0)|(2,0) GB, which has been regarded as a good type of embedded metallic wire,30 presents strong transmission intense in the electrons region. Moreover, we still could clearly see the contribution of the extended van Hove band states to the transmission along the GB.
In addition to the zero-bias transmission spectra, we have also calculated the current–voltage (I–V) characteristics of the transport across and along the GBs, the calculated results are summarized in Fig. 6(a) and (b), respectively. As for the transport across the GBs, the declining currents induced by GBs correspond well with the zero-bias transmission analysis. We have noticed that the currents are suppressed at least by 50% compared with the pristine graphene, even up to 80% for (3,1)|(3,1) GB and (5,0)|(3,3) GB at about E > 0.5 eV. Our results are much lower compared with other theoretical predicts which found the conductance of a GB would be nearly 70–80% of the bulk graphene conductance.20,23 In the case of the transport along the GBs, there is no large current decline compared with the transport across the GBs. For example, the reduction of current along the GB in asymmetric (5,0)|(3,3) GB, which is the largest current decline in all GBs, is about half of the pristine graphene, while it is still much smaller than that across the GB. We also notice that the current along the GB is larger than that across the GB. This property is also revealed by previous experimental results,13 which displayed the resistance of the inter-grain (across the GB) was larger than that of the intra-grain (along the grain) by using four-probe measurement. The anisotropic transport properties of graphene with GBs originate from its anisotropic electronic band structures40 (see Fig. 2). For all the structures we study here, both the top of the valence bands and the bottom of the conduction bands have much more significant dispersions along the Γ–Y direction, which is along GB in the real space, however, these bands are more flat along the Γ–X direction, which is across GB direction in the real space. Therefore, the corresponding effective mass of electrons and holes is also anisotropic because it is proportional to the inverse of the curvature of the band dispersion. This directly leads to the anisotropic I–V curve and the associated resistance.
To gain further understanding of the current suppression, we have analysed the bias-dependent transmission spectra (Fig. 6(c) and (d)). Here, we only plot the two directions transmission spectra at 1.0 V bias as an example. It is well known that the current is mainly attributed to the transmission coefficient around the Fermi level within the bias window. We have noticed that a large transport gap exists in (3,1)|(3,1) and (5,0)|(3,3) GB, respectively, which has no contribution to the current across the GBs. However, such a transport gap is not found in the transport along the GBs.
From the above discussions, we can at least conclude that the transport properties of graphene with GBs not only depend on the specific atomic GB structure, but also on the transport direction. Besides the transport across and along the GB, the case of the transport direction with a certain angle relative to (2,0)|(2,0) GB has also been investigated. Due to the large calculation burden, a reduced supercell unit is employed to construct (2,0)|(2,0)D and (2,0)|(2,0)S (the letter S stands for a smaller (2,0)|(2,0) GB supercell is employed to construct (2,0)|(2,0)S) system (Fig. 7(a)). (2,0)|(2,0)D GB can be obtained by making a certain angle rotation from (2,0)|(2,0)S. The reduced supercell can not avoid period GB image interaction, which introduces the states in Fermi level, but this interaction will not affect the main conclusion for the comparison of transport properties. The zero-bias transmission spectra of these two systems are shown in Fig. 7(b). One can see that the electron transmission probabilities have been enlarged by avoiding transport across the GB. Although the full transport-direction relationship is not given, we still think it is a useful method to design more efficient electronic device based on graphene material with GBs by altering transport direction.
In the following part, we pay our attention to the transport of crossed GB (2,1)|(2,1)D, which contains two GB square crossing with each other. The transport direction is along one GB but across another. The zero-bias transmission spectrum is shown in Fig. 7(c). We notice that the transmission probabilities are decreased in the whole region except at |E| < 0.5 eV, which is similar to the transport across a single GB. This indicates the GB vertical to the transport direction plays an important role in electronic transmission reduction. Moreover, the interaction of two GBs may not have great impact on the electronic transport properties compared with the presence of only single GB. Investigation of the crossed GB with other angles is outside the scope of our paper, nevertheless the significance of crossed GBs on the transport properties is demonstrated by comparison with pristine graphene.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra16323d |
This journal is © The Royal Society of Chemistry 2016 |