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Tunable band gaps in graphene/GaN van der Waals heterostructures
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 295304 (http://iopscience.iop.org/0953-8984/26/29/295304) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 295304 (7pp)
doi:10.1088/0953-8984/26/29/295304
Tunable band gaps in graphene/GaN van der Waals heterostructures Le Huang1, Qu Yue2, Jun Kang1, Yan Li1 and Jingbo Li1 1
State Key Laboratory for Superlattice and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, People’s Republic of China 2 School of Science, National University of Defense Technology, Changsha 410073, People’s Republic of China E-mail: [email protected] and [email protected] Received 21 March 2014, revised 4 May 2014 Accepted for publication 23 May 2014 Published 30 June 2014 Abstract
Van der Waals (vdW) heterostructures consisting of graphene and other two-dimensional materials provide good opportunities for achieving desired electronic and optoelectronic properties. Here, we focus on vdW heterostructures composed of graphene and gallium nitride (GaN). Using density functional theory, we perform a systematic study on the structural and electronic properties of heterostructures consisting of graphene and GaN. Small band gaps are opened up at or near the Γ point of the Brillouin zone for all of the heterostructures. We also investigate the effect of the stacking sequence and electric fields on their electronic properties. Our results show that the tunability of the band gap is sensitive to the stacking sequence in bilayer-graphene-based heterostructures. In particular, in the case of graphene/graphene/GaN, a band gap of up to 334 meV is obtained under a perpendicular electric field. The band gap of bilayer graphene between GaN sheets (GaN/graphene/graphene/GaN) shows similar tunability, and increases to 217 meV with the perpendicular electric field reaching 0.8 V Å − 1. Keywords: vdW heterostructures, density functional theory, tunable band gap, electric field, electronic properties (Some figures may appear in colour only in the online journal)
1. Introduction
various approaches have been proposed. For instance, a tunable band gap can be created by patterning two-dimensional layered materials into a nanometer-wide nanoribbon [6–10] or by using uniaxial strain [11, 12]. Interaction with a substrate can also open a band gap in graphene [13, 14]. In the graphene/BN bilayer, graphene and single-layer BN bind together via van der Waals (vdW) interaction and a band gap of 30 meV is created [15]. A band gap of 53 meV was generated if graphene was deposited on h-BN [16]. Ruge Quhe and colleagues reported a tunable band gap of graphene when it was properly sandwiched between two hexagonal boron nitride layers. The gap can be up to 0.16 eV without an electric field and 0.34 eV in the presence of a strong electric field [17]. Moreover, much attention has been paid to graphene bilayers, especially AB-stacked BLG (bilayer graphene) [18–21]. It is reported that a tunable band gap with the maximum of a few hundred meV [8, 22, 23] can be created in a graphene bilayer structure by applying a
Graphene, a two-dimensional allotrope of carbon, has been widely explored due to its unique electronic and transport properties [1–5]. Extensive computations and accurate experimental measurements on the band structure of graphene have shown that two bands with linear dispersion appear around the Fermi level. Carriers in graphene, therefore, behave like massless Dirac particles and exhibit very high mobility. A pristine graphene monolayer is a semiconductor without an energy gap, and its electronic valence and conduction bands cross at Dirac points in the reciprocal space. The absence of a band gap in graphene, which results from the equivalence of its two carbon sublattices, impedes its applications in electronic devices such as field effect transistors. It is possible to open a band gap if the equivalence of its two carbon sublattices is removed. Much effort has been devoted to opening a band gap in graphene, and 0953-8984/14/295304+7$33.00
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J. Phys.: Condens. Matter 26 (2014) 295304
perpendicular electric field, which has been shown both experimentally and theoretically [19, 24, 25]. Generally, the graphene layer and other layered materials bind together via weak vdW interaction. So it is difficult to open a wide band gap in graphene-based vdW heterostructures. Nevertheless, by applying an external electric field, one can enhance the Coulomb interaction between two layers and create a tunable wide band gap. Very recently, graphene-based vdW heterostructures have been a subject of extensive investigation due to their novel properties and potential applications [26–29]. In the configurations consisting of graphene and other two-dimensional sheets, results show that their lattice mismatch is between 1.9% and 8% [17, 26, 27, 30]. However, It is unrealistic to put two different layered materials in a supercell during theoretical computation if their lattice mismatch is too large. In this work, the calculated lattice constant of free-standing graphene is 2.464 Å(agraphene = 2.464 Å), which agrees well with the experimental value [31]. The optimized lattice constant of a GaN monolayer is 3.197 Å(aGaN = 3.197 Å), which is consistent with the reported value of 3.185 Å [32, 33, 49]. To minimize the lattice mismatch between the stacking sheets, we used a ( 3 × 3 )R30° cell as the unit cell of graphene, whose lattice constant is 4.267 Å [34]. The ( 3 × 3 ) R30° graphene can be generated on an SiC substrate by epitaxial growth [35, 36]. For the all the heterostructures considered in this work, we employed supercells consisting of 3 × 3 unit cells of graphene and 4 × 4 unit cells of GaN. Thus 3 3 agraphene = 12.801 Å and 4aGaN = 12.788 Å, which leads to a small lattice mismatch of 0.10% in the supercell. In our computation, the lattice of graphene are set to match that of GaN. The supercells are then fully relaxed for the atomic geometry. It has been reported that the GaN thin films on graphene layers show excellent optical and electric characteristics, and may have good applications in optoelectronic and electronic devices [37–40]. However, few theoretical works have been carried out to study the properties of vdW heterostructures consisting of GaN and graphene. Therefore, it is crucial to predict and understand their properties. The objective of this work is to exploit the possibility of tuning band gaps in graphene/GaN vdW heterostructures [41]. Using density functional theory (DFT), we show that band gaps are created in all the vdW heterostructures—although graphene/GaN has a small band gap and is not a suitable candidate for graphene/ GaN device use. The band gaps of graphene/graphene/GaN and GaN/graphene/graphene/GaN structures with specific stacking order of the two graphene layers show good tunability in the presence of a perpendicular electric field. It is also found that the tunable band gap is sensitive to the stacking order of the two graphene layers.
The generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof (PBE) functional [46] is adopted for electron exchange and correlation. A pairwise force field in the DFT-D2 method of Grimme [47] is used to described the vdW interlayer interactions. A vacuum larger than 10 Å is used to eliminate the interaction between layers in neighboring supercells. We have addressed the effect of the k-mesh density on the band gap, as shown in figure 4(a). Taking AB-stacked graphene/graphene/GaN as an example, our results show that there exists a band gap in all the cases of k-mesh density considered. The band gap remains unchanged if the k-mesh density is more than 4 × 4 × 1. In this work, the Brillouin zone is sampled with a (5 × 5 × 1) Monkhorst–Pack grid [48] and 60 k-points along the high symmetry line for band structure computations. Because band gap opening occurs at or near high symmetry points of the Brillouin zone (the Γ point in the case discussed in our paper), the k-mesh density of the relaxation and the self-consistent computation exhibits little influence on the band gap. The cutoff energy for the plane-wave basis set is set to 400 eV. All the structures are fully relaxed with a force tolerance of 0.02 eV Å − 1. To investigate the effect of the external electric field on the properties of the heterostructures, an electric field with strength ranging from 0 to 1 V Å − 1 is applied along the direction perpendicular to the graphene plane. The 2s and 2p electrons of nitrogen and carbon, and 3d, 4s and 4p electrons of Ga are considered as valence electrons in the electronic structure calculations. 3. Results and discussion 3.1. Structural properties
Top views of the crystal structures of the GaN monolayer, graphene monolayer and graphene/GaN are shown in figure 1. A unit cell of ( 3 × 3 )R30° graphene is indicated by a red rhombus in panel (a). The optimized lattice constants of pristine graphene and hexagonal GaN are 2.467 Å and 3.197 Å, respectively. There is no obvious buckling of the graphene or GaN single layer after full relaxation. Figure 2 presents the crystal structures of all the configurations considered in this paper. The bilayer-graphene-based vdW heterostructure is characterized by the stacking sequence of the two graphene layers. For example, AB-stacked G/G/GaN denotes the graphene/graphene/GaN for which the stacking order of the two graphene layers is AB. Table 1 presents the stacking order, interlayer spacing and binding energy for the considered structures. The interlayer spacing between the graphene and the GaN layer, d1, is about 3.28 Å in all the heterostructures considered, and it is almost independent of the stacking order and the number of layers. In G/G/GaN and GaN/G/G/GaN systems, AA-stacked configurations show a larger interlayer spacing between two graphene layers, d2 (about 3.45 Å), as compared with that (about 3.20 Å) of the AB-stacked configurations. In order to better understand the strength of the vdW interaction between layers, we compute the interlayer binding energy of the structures discussed above. The binding energy
2. Methods and computational details Calculations are performed using the projector augmented plane-wave (PAW) method [42, 43] within the framework of DFT in the ‘Vienna Ab Initio Simulation Package’ [44, 45]. 2
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J. Phys.: Condens. Matter 26 (2014) 295304
(a)
(b)
(c)
a 3a Figure 1. The crystal structures of (a) single-layer graphene, (b) GaN monolayer and (c) graphene Ga/N of top views. The balls shown in brown, silver and green represent carbon, nitrogen and gallium atoms respectively.
Figure 2. The crystal structures of (a) BLG, (b) bilayer GaN, (c) graphene/GaN, (d) G/GaN/G, (e) AA-stacked G/G/GaN, (f) AB-stacked G/G/GaN, (g) AA-stacked GaN/G/G/GaN and (h) AB-stacked GaN/G/G/GaN. Here AA and AB represent the stacking orders of two graphene layers. The graphene layer is denoted by G. Table 1. Structure properties of vdW heterostructures at the vdW DFT level of theory. The band gap without an electric field and the
maximum band gap under an external electric field are also given; the corresponding strength of external electric field for generating the maximum band gap is written in brackets at the side in units of V Å − 1.
System Graphene/GaN G/GaN/G G/G/GaN G/G/GaN GaN/G/G/GaN GaN/G/G/GaN
Stacking order
Interlayer spacing d1 d2 (Å)
Binding energy/supercell (eV)
Band gap without electric field (meV)
AA AB AA AB
3.28 3.27 3.28 3.46 3.29 3.20 3.28 3.44 3.28 3.20
− 3.11 − 6.43 − 6.22 − 6.86 − 9.67 − 10.31
3.5 8.5 15 47 7.9 43
is defined as the difference between the total energy of an assembled system and that of the corresponding individual total energies of the two isolated parts: Eb = Esupercell − Epart 1 − Epart 2. Here Esupercell, Epart 1 and Epart 2 are the total energies per unit cell of the supercell and of the two isolated parts of the supercell respectively. We have performed a series of computations on the effect of the separation distance on the binding energy. Also, we take AB-stacked graphene/graphene/
Maximum band gap under electric field (meV (V Å − 1)) 3.9(1.0) 334(1.0) 125(0.8) 217(0.7)
GaN as an example. Figure 4(b) presents the energy versus separation distance curves. d1 denotes the interlayer spacing between the middle graphene layer and the GaN layer; d2 represents the interlayer spacing between the two graphene layers. The two curves show minima at 3.29 Å and 3.20 Å respectively, which means that AB-stacked graphene/graphene/GaN will relax to the most stable configuration with the lowest total energy when d1 and d2 are equal to 3.29 Å 3
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J. Phys.: Condens. Matter 26 (2014) 295304
c1
b1
b2 K
(b)
(a)
c2
M
(c)
(d)
(e)
Figure 3. The regular hexagons in black and red represent the Brillouin zones of (3 × 3)R30° graphene and (1 × 1) graphene. b1, b2 and c1, c2 are their reciprocal lattice vectors. Figure 4. All the results are obtained from AB-stacked graphene/ graphene/GaN structure. (a) shows the relation between k-mesh and band gap without the presence of electric field. The relations between binding energy and interlayer spacing d1 and d2 are given in (b). (c) and (d) give the band structure with energy ranging from –1 to 1 eV around the Fermi energy. (e) presents a complete band structure at the presence of electric field with its strength being 0.6 V/Å
and 3.20 Å respectively. These results are in good accordance with our results in this paper (table 1). All the binding energies given in table 1 in our paper are equilibrium binding energies. From table 1, it is clear that the binding energy of AB-stacked configurations is numerically larger than that of AA-stacked configurations in G/G/GaN and GaN/G/G/GaN systems. This indicates a stronger interaction between graphene layers with AB stacking than those with AA stacking. Therefore we can conclude that AB-stacked G/G/GaN and GaN/G/G/GaN are more stable than the corresponding AA-stacked structures.
The presence of a band gap implies lifting of the degeneracy of the bands at the Dirac points. The AB-stacked G/G/GaN configuration shows a smaller interlayer spacing between two graphene layers and a wider band gap as compared with those of AA-stacked G/G/GaN, because of the spontaneous symmetry breaking in AB-stacked BLG [21, 51]. Figures 4(c) and (d) give the band structure of AB-stacked graphene/graphene/GaN with energy ranging from − 1 to 1 eV around the Fermi energy. They show clearly the appearance of the band gap. The G/GaN/G system shows the narrowest band gap of 9 meV at the Γ point of the BZ because of the negligible interaction between the two graphene layers. The band structures of AA-stacked GaN/G/G/GaN and AB-stacked GaN/G/G/GaN show similar differences as compared with those of G/G/GaN systems; these are presented in figure 5(e) and (f). Band gaps of 8 meV and 43 meV appear in the band structures of AA-stacked and AB-stacked GaN/G/G/GaN obtained by vdW PBE computation. The band structure of graphene-based vdW heterostructures is sensitive to the lattice symmetry. The chirality of the π and π* states is doubly degenerate in AB-stacked bilayer graphene. The interlayer interaction and an external potential induced by the GaN layer can cause nonequivalence of the two carbon sublattices. The symmetry with respect to the c-axis can also be broken by the alternate stacking sequence of the two graphene layers. As a consequence, band gap opening should occur at the Dirac point of the Brillouin zone in the band structure. A wider band gap can be expected in AB-stacked structures than in AA-stacked structures because of the stronger interaction between the two graphene layers. There is some additional symmetry broken in AB-stacked structures. What is more, the coupling between π and π* states of carbon atoms can be enhanced by an external e lectric field.
3.2. Electronic properties and the effect of the stacking order
Figure 3 gives the first Brillouin zones of ( 3 × 3 )R30° graphene and (1 × 1) graphene. Both of them are hexagonal. The length of the reciprocal lattice vectors of the latter is 3 times larger than that of the reciprocal lattice vectors the former, and there is a relative rotation of 30° between the two Brillouin zones. Due to the increase of the real space unit cell from (1 × 1) to ( 3 × 3 )R30°, there will be a consequent folding of bands. In figure 3, the bands at the K point for the large Brillouin zone will be folded such that they will appear at the Gamma point for the small Brillouin zone. Thus Dirac points will appear at the Gamma point in the Brillouin zone of ( 3 × 3 )R30° graphene. So we can conclude safely that a linear dispersion relation appears at the Γ point in the band structure of ( 3 × 3 )R30° graphene. It has been proved that ( n 3 × n 3 )R30° and (3n × 3n) constructions, with n a positive integer, will also share this property [50]. The electronic band structure along the high symmetry points in k-space of graphene/GaN is plotted in figure 5(a). The bands dominated by graphene and GaN are plotted using red and blue circles respectively. It is clear that the states of the graphene/ GaN structure near the Fermi level are dominated by the bands associated with the carbon atoms. A very small band gap of about 3.5 meV appears at the Γ point. Linear dispersion characteristic of graphene is retained in the bands near the Fermi level. Figure 5 also shows the band structures of G/GaN/G, AA-stacked G/G/GaN and AB-stacked G/G/GaN. Band gaps of 15 meV and 47 meV appear near the Fermi level in AA-stacked and AB-stacked G/G/GaN systems respectively. 4
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Figure 5. The projected band structures of (a) graphene/GaN, (b) AB-stacked G/GaN/G, (c) AA-stacked G/G/GaN, (d) AB-stacked G/G/GaN, (e) AA-stacked GaN/G/G/GaN and (f) AB-stacked GaN/G/G/GaN. The bands dominated by the graphene layer and the GaN layer are plotted with red circles and blue circles respectively. The Fermi level is set at zero.
3.3. The effect of a perpendicular electric field
It has been demonstrated that a large band gap of up to 0.25 eV can be opened in BLG upon the application of a perpendicular electric field. We wonder whether similar behavior also occurs in the vdW heterostructures consisting of graphene and GaN. Herein we focus on the tunability of the band gaps in graphene/GaN vdW heterostructures, via the external perpendicular electric field. The evolution curves of the band gaps with a perpendicular electric field are shown in figure 6. It is clear from this figure that the band gaps of all the AB-stacked BLG-based heterostructures are modulated by the external field. The band gaps of trilayers (AB-stacked G/GaN/G and G/G/GaN) and quadrilayers (AB-stacked GaN/G/G/GaN) show increasing trends with increasing strength of the external electric field. A band gap of up to 334 meV can be generated under the positive bias voltage of 1.0 eV Å1 in the case of the AB/GaN configuration, which is much larger than that of the graphene bilayer on the SiO2 substrate induced by a dual-gated electric field [20]. Figure 6 gives an insight into the influence of the external electric field on the band gaps of heterostructures consisting of graphene and GaN. In the case of the graphene/GaN structure, a band gap of 3.5 meV is opened at the Fermi level. This band gap remains almost unchanged with increasing external electric field. It reaches its maximum value of 3.9 meV at 1.0 V Å − 1, whereas in the AB-stacked G/GaN/G configuration,
(a)
(b)
(c)
(d)
Figure 6. The changes in band gaps for (a) graphene/GaN, (b) ABstacked G/GaN/G, (c) AB-stacked G/G/GaN and (d) AB-stacked GaN/G/G/GaN due to the external electric field. 5
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J. Phys.: Condens. Matter 26 (2014) 295304
(a)
(b)
(c)
Figure 7. Band structure in the vicinity of the K point as a function of the external perpendicular electric field for (a) AB-stacked G/G/GaN under a positive bias voltage, (b) AB-stacked G/G/GaN under a reverse-biased electric field, (c) AB-stacked GaN/G/G/GaN.
point for each configuration and increases with the strength of the electric field.
the band gap is about 9 meV without an external electric field and it increases rapidly to 39 meV under a perpendicular electric field of 0.6 V Å − 1; then it remains unchanged with increasing electric field. In this work, a positive bias voltage is applied from top to bottom in the configurations of figure 2, and a negative bias voltage is applied reversely. The changes in the band gaps of AB-stacked G/G/GaN and GaN/G/G/GaN due to the externally applied perpendicular electric field are also depicted in figures 6(c) and (d). Band gaps of 47 meV and 43 meV are opened at the Γ point of the Brillouin zone in the band structure of AB-stacked G/G/GaN and GaN/G/G/ GaN respectively at zero electric field, and they increase linearly with the strength of the electric field. In the case of a positive bias voltage, the band gap of AB-stacked G/G/GaN reaches 334 meV at 1.0 V Å − 1, and in case of a negative bias voltage, the gap shows a maximum of 125 meV at 0.8 V Å − 1; while the band gap of AB-stacked GaN/G/G/GaN structure attains its maximum of 217 meV as the external electric field increases to 0.7 V Å − 1. As the external electric field increases to above 0.7 V Å − 1 (or 0.8 V Å − 1 reverse bias), the band gap of AB-stacked GaN/G/G/GaN (or AB-stacked G/G/GaN) decreases. It can be seen that AB-stacked G/G/GaN shows a wider band gap opening and a larger rate of increase than AB-stacked GaN/G/G/GaN in the presence of a positive bias voltage. This happens because there is some extra symmetry broken in AB-stacked G/G/GaN because of its lack of mirror symmetry. Panel (c) of figure 6 reveals different rates of increase for the cases of positive bias voltage and negative bias voltage for AB-stacked G/G/GaN structure. This difference should be attributed to the different interactions of the dipole moment caused by the external electric field. There is almost no difference in the changes of band gaps in the presence of positively biased and negatively biased electric fields for other heterostructures. Band structures in the vicinity of the Γ point for AB-stacked G/G/GaN and GaN/G/G/GaN systems under 0.0 V Å − 1, 0.2 V Å − 1 and 0.6 V Å − 1 perpendicular electric fields are shown in figure 7. It is clear that the valence bands are affected more strongly than the conduction bands by the external electric field for all the structures considered. Parabolic dispersive relations appear near the Fermi level in the band structures. A band gap opening happened at the Γ
4. Conclusion In summary, we have investigated the geometric and electronic structure of vdW heterostructures consisting of graphene and GaN using vdW DFT calculations. The interlayer spacing of the two graphene layers in the AB-stacked systems is smaller and the binding energy is numerically larger than those in AA-stacked systems. AB-stacked G/G/GaN and GaN/G/G/GaN show wider band gaps than the corresponding AA-stacked systems. A band gap is opened near the Fermi level for all vdW heterostructures considered. Then we studied the effect of the electric field on the electronic structure of graphene/GaN vdW heterostructures. The band gaps of AB-stacked G/G/GaN and GaN/G/G/GaN show a linear increase with the strength of the external electric field. In particular, a maximum band gap of 334 meV is generated in AB-stacked G/G/GaN systems in the presence of a 1.0 V Å − 1 external electric field. It is expected that our calculated results may provide a fundamental basis for applications of BLG in optoelectronic devices and field effect transistors. Acknowledgments This work was supported by the National Natural Science Foundation of China under grant no. 91233120 and the National Basic Research Program of China (2011CB921901). References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666 [2] Novoselov K, Geim A K, Morozov S, Jiang D, Grigorieva M K I, Dubonos S and Firsov A 2005 Nature 438 197 [3] Neto A C, Guinea F, Peres N, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [4] Jiang Z, Zhang Y, Stormer H and Kim P 2007 Phys. Rev. Lett. 99 106802 [5] Peres N M R 2009 J. Phys.:Condens. Matter 21 323201 [6] Han M Y, Özyilmaz B, Zhang Y and Kim P 2007 Phys. Rev. Lett. 98 206805 6
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Tunable band gaps in graphene/GaN van der Waals heterostructures
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 295304 (http://iopscience.iop.org/0953-8984/26/29/295304) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 131.215.225.9 This content was downloaded on 23/08/2014 at 22:28
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 295304 (7pp)
doi:10.1088/0953-8984/26/29/295304
Tunable band gaps in graphene/GaN van der Waals heterostructures Le Huang1, Qu Yue2, Jun Kang1, Yan Li1 and Jingbo Li1 1
State Key Laboratory for Superlattice and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, People’s Republic of China 2 School of Science, National University of Defense Technology, Changsha 410073, People’s Republic of China E-mail: [email protected] and [email protected] Received 21 March 2014, revised 4 May 2014 Accepted for publication 23 May 2014 Published 30 June 2014 Abstract
Van der Waals (vdW) heterostructures consisting of graphene and other two-dimensional materials provide good opportunities for achieving desired electronic and optoelectronic properties. Here, we focus on vdW heterostructures composed of graphene and gallium nitride (GaN). Using density functional theory, we perform a systematic study on the structural and electronic properties of heterostructures consisting of graphene and GaN. Small band gaps are opened up at or near the Γ point of the Brillouin zone for all of the heterostructures. We also investigate the effect of the stacking sequence and electric fields on their electronic properties. Our results show that the tunability of the band gap is sensitive to the stacking sequence in bilayer-graphene-based heterostructures. In particular, in the case of graphene/graphene/GaN, a band gap of up to 334 meV is obtained under a perpendicular electric field. The band gap of bilayer graphene between GaN sheets (GaN/graphene/graphene/GaN) shows similar tunability, and increases to 217 meV with the perpendicular electric field reaching 0.8 V Å − 1. Keywords: vdW heterostructures, density functional theory, tunable band gap, electric field, electronic properties (Some figures may appear in colour only in the online journal)
1. Introduction
various approaches have been proposed. For instance, a tunable band gap can be created by patterning two-dimensional layered materials into a nanometer-wide nanoribbon [6–10] or by using uniaxial strain [11, 12]. Interaction with a substrate can also open a band gap in graphene [13, 14]. In the graphene/BN bilayer, graphene and single-layer BN bind together via van der Waals (vdW) interaction and a band gap of 30 meV is created [15]. A band gap of 53 meV was generated if graphene was deposited on h-BN [16]. Ruge Quhe and colleagues reported a tunable band gap of graphene when it was properly sandwiched between two hexagonal boron nitride layers. The gap can be up to 0.16 eV without an electric field and 0.34 eV in the presence of a strong electric field [17]. Moreover, much attention has been paid to graphene bilayers, especially AB-stacked BLG (bilayer graphene) [18–21]. It is reported that a tunable band gap with the maximum of a few hundred meV [8, 22, 23] can be created in a graphene bilayer structure by applying a
Graphene, a two-dimensional allotrope of carbon, has been widely explored due to its unique electronic and transport properties [1–5]. Extensive computations and accurate experimental measurements on the band structure of graphene have shown that two bands with linear dispersion appear around the Fermi level. Carriers in graphene, therefore, behave like massless Dirac particles and exhibit very high mobility. A pristine graphene monolayer is a semiconductor without an energy gap, and its electronic valence and conduction bands cross at Dirac points in the reciprocal space. The absence of a band gap in graphene, which results from the equivalence of its two carbon sublattices, impedes its applications in electronic devices such as field effect transistors. It is possible to open a band gap if the equivalence of its two carbon sublattices is removed. Much effort has been devoted to opening a band gap in graphene, and 0953-8984/14/295304+7$33.00
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perpendicular electric field, which has been shown both experimentally and theoretically [19, 24, 25]. Generally, the graphene layer and other layered materials bind together via weak vdW interaction. So it is difficult to open a wide band gap in graphene-based vdW heterostructures. Nevertheless, by applying an external electric field, one can enhance the Coulomb interaction between two layers and create a tunable wide band gap. Very recently, graphene-based vdW heterostructures have been a subject of extensive investigation due to their novel properties and potential applications [26–29]. In the configurations consisting of graphene and other two-dimensional sheets, results show that their lattice mismatch is between 1.9% and 8% [17, 26, 27, 30]. However, It is unrealistic to put two different layered materials in a supercell during theoretical computation if their lattice mismatch is too large. In this work, the calculated lattice constant of free-standing graphene is 2.464 Å(agraphene = 2.464 Å), which agrees well with the experimental value [31]. The optimized lattice constant of a GaN monolayer is 3.197 Å(aGaN = 3.197 Å), which is consistent with the reported value of 3.185 Å [32, 33, 49]. To minimize the lattice mismatch between the stacking sheets, we used a ( 3 × 3 )R30° cell as the unit cell of graphene, whose lattice constant is 4.267 Å [34]. The ( 3 × 3 ) R30° graphene can be generated on an SiC substrate by epitaxial growth [35, 36]. For the all the heterostructures considered in this work, we employed supercells consisting of 3 × 3 unit cells of graphene and 4 × 4 unit cells of GaN. Thus 3 3 agraphene = 12.801 Å and 4aGaN = 12.788 Å, which leads to a small lattice mismatch of 0.10% in the supercell. In our computation, the lattice of graphene are set to match that of GaN. The supercells are then fully relaxed for the atomic geometry. It has been reported that the GaN thin films on graphene layers show excellent optical and electric characteristics, and may have good applications in optoelectronic and electronic devices [37–40]. However, few theoretical works have been carried out to study the properties of vdW heterostructures consisting of GaN and graphene. Therefore, it is crucial to predict and understand their properties. The objective of this work is to exploit the possibility of tuning band gaps in graphene/GaN vdW heterostructures [41]. Using density functional theory (DFT), we show that band gaps are created in all the vdW heterostructures—although graphene/GaN has a small band gap and is not a suitable candidate for graphene/ GaN device use. The band gaps of graphene/graphene/GaN and GaN/graphene/graphene/GaN structures with specific stacking order of the two graphene layers show good tunability in the presence of a perpendicular electric field. It is also found that the tunable band gap is sensitive to the stacking order of the two graphene layers.
The generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof (PBE) functional [46] is adopted for electron exchange and correlation. A pairwise force field in the DFT-D2 method of Grimme [47] is used to described the vdW interlayer interactions. A vacuum larger than 10 Å is used to eliminate the interaction between layers in neighboring supercells. We have addressed the effect of the k-mesh density on the band gap, as shown in figure 4(a). Taking AB-stacked graphene/graphene/GaN as an example, our results show that there exists a band gap in all the cases of k-mesh density considered. The band gap remains unchanged if the k-mesh density is more than 4 × 4 × 1. In this work, the Brillouin zone is sampled with a (5 × 5 × 1) Monkhorst–Pack grid [48] and 60 k-points along the high symmetry line for band structure computations. Because band gap opening occurs at or near high symmetry points of the Brillouin zone (the Γ point in the case discussed in our paper), the k-mesh density of the relaxation and the self-consistent computation exhibits little influence on the band gap. The cutoff energy for the plane-wave basis set is set to 400 eV. All the structures are fully relaxed with a force tolerance of 0.02 eV Å − 1. To investigate the effect of the external electric field on the properties of the heterostructures, an electric field with strength ranging from 0 to 1 V Å − 1 is applied along the direction perpendicular to the graphene plane. The 2s and 2p electrons of nitrogen and carbon, and 3d, 4s and 4p electrons of Ga are considered as valence electrons in the electronic structure calculations. 3. Results and discussion 3.1. Structural properties
Top views of the crystal structures of the GaN monolayer, graphene monolayer and graphene/GaN are shown in figure 1. A unit cell of ( 3 × 3 )R30° graphene is indicated by a red rhombus in panel (a). The optimized lattice constants of pristine graphene and hexagonal GaN are 2.467 Å and 3.197 Å, respectively. There is no obvious buckling of the graphene or GaN single layer after full relaxation. Figure 2 presents the crystal structures of all the configurations considered in this paper. The bilayer-graphene-based vdW heterostructure is characterized by the stacking sequence of the two graphene layers. For example, AB-stacked G/G/GaN denotes the graphene/graphene/GaN for which the stacking order of the two graphene layers is AB. Table 1 presents the stacking order, interlayer spacing and binding energy for the considered structures. The interlayer spacing between the graphene and the GaN layer, d1, is about 3.28 Å in all the heterostructures considered, and it is almost independent of the stacking order and the number of layers. In G/G/GaN and GaN/G/G/GaN systems, AA-stacked configurations show a larger interlayer spacing between two graphene layers, d2 (about 3.45 Å), as compared with that (about 3.20 Å) of the AB-stacked configurations. In order to better understand the strength of the vdW interaction between layers, we compute the interlayer binding energy of the structures discussed above. The binding energy
2. Methods and computational details Calculations are performed using the projector augmented plane-wave (PAW) method [42, 43] within the framework of DFT in the ‘Vienna Ab Initio Simulation Package’ [44, 45]. 2
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(a)
(b)
(c)
a 3a Figure 1. The crystal structures of (a) single-layer graphene, (b) GaN monolayer and (c) graphene Ga/N of top views. The balls shown in brown, silver and green represent carbon, nitrogen and gallium atoms respectively.
Figure 2. The crystal structures of (a) BLG, (b) bilayer GaN, (c) graphene/GaN, (d) G/GaN/G, (e) AA-stacked G/G/GaN, (f) AB-stacked G/G/GaN, (g) AA-stacked GaN/G/G/GaN and (h) AB-stacked GaN/G/G/GaN. Here AA and AB represent the stacking orders of two graphene layers. The graphene layer is denoted by G. Table 1. Structure properties of vdW heterostructures at the vdW DFT level of theory. The band gap without an electric field and the
maximum band gap under an external electric field are also given; the corresponding strength of external electric field for generating the maximum band gap is written in brackets at the side in units of V Å − 1.
System Graphene/GaN G/GaN/G G/G/GaN G/G/GaN GaN/G/G/GaN GaN/G/G/GaN
Stacking order
Interlayer spacing d1 d2 (Å)
Binding energy/supercell (eV)
Band gap without electric field (meV)
AA AB AA AB
3.28 3.27 3.28 3.46 3.29 3.20 3.28 3.44 3.28 3.20
− 3.11 − 6.43 − 6.22 − 6.86 − 9.67 − 10.31
3.5 8.5 15 47 7.9 43
is defined as the difference between the total energy of an assembled system and that of the corresponding individual total energies of the two isolated parts: Eb = Esupercell − Epart 1 − Epart 2. Here Esupercell, Epart 1 and Epart 2 are the total energies per unit cell of the supercell and of the two isolated parts of the supercell respectively. We have performed a series of computations on the effect of the separation distance on the binding energy. Also, we take AB-stacked graphene/graphene/
Maximum band gap under electric field (meV (V Å − 1)) 3.9(1.0) 334(1.0) 125(0.8) 217(0.7)
GaN as an example. Figure 4(b) presents the energy versus separation distance curves. d1 denotes the interlayer spacing between the middle graphene layer and the GaN layer; d2 represents the interlayer spacing between the two graphene layers. The two curves show minima at 3.29 Å and 3.20 Å respectively, which means that AB-stacked graphene/graphene/GaN will relax to the most stable configuration with the lowest total energy when d1 and d2 are equal to 3.29 Å 3
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c1
b1
b2 K
(b)
(a)
c2
M
(c)
(d)
(e)
Figure 3. The regular hexagons in black and red represent the Brillouin zones of (3 × 3)R30° graphene and (1 × 1) graphene. b1, b2 and c1, c2 are their reciprocal lattice vectors. Figure 4. All the results are obtained from AB-stacked graphene/ graphene/GaN structure. (a) shows the relation between k-mesh and band gap without the presence of electric field. The relations between binding energy and interlayer spacing d1 and d2 are given in (b). (c) and (d) give the band structure with energy ranging from –1 to 1 eV around the Fermi energy. (e) presents a complete band structure at the presence of electric field with its strength being 0.6 V/Å
and 3.20 Å respectively. These results are in good accordance with our results in this paper (table 1). All the binding energies given in table 1 in our paper are equilibrium binding energies. From table 1, it is clear that the binding energy of AB-stacked configurations is numerically larger than that of AA-stacked configurations in G/G/GaN and GaN/G/G/GaN systems. This indicates a stronger interaction between graphene layers with AB stacking than those with AA stacking. Therefore we can conclude that AB-stacked G/G/GaN and GaN/G/G/GaN are more stable than the corresponding AA-stacked structures.
The presence of a band gap implies lifting of the degeneracy of the bands at the Dirac points. The AB-stacked G/G/GaN configuration shows a smaller interlayer spacing between two graphene layers and a wider band gap as compared with those of AA-stacked G/G/GaN, because of the spontaneous symmetry breaking in AB-stacked BLG [21, 51]. Figures 4(c) and (d) give the band structure of AB-stacked graphene/graphene/GaN with energy ranging from − 1 to 1 eV around the Fermi energy. They show clearly the appearance of the band gap. The G/GaN/G system shows the narrowest band gap of 9 meV at the Γ point of the BZ because of the negligible interaction between the two graphene layers. The band structures of AA-stacked GaN/G/G/GaN and AB-stacked GaN/G/G/GaN show similar differences as compared with those of G/G/GaN systems; these are presented in figure 5(e) and (f). Band gaps of 8 meV and 43 meV appear in the band structures of AA-stacked and AB-stacked GaN/G/G/GaN obtained by vdW PBE computation. The band structure of graphene-based vdW heterostructures is sensitive to the lattice symmetry. The chirality of the π and π* states is doubly degenerate in AB-stacked bilayer graphene. The interlayer interaction and an external potential induced by the GaN layer can cause nonequivalence of the two carbon sublattices. The symmetry with respect to the c-axis can also be broken by the alternate stacking sequence of the two graphene layers. As a consequence, band gap opening should occur at the Dirac point of the Brillouin zone in the band structure. A wider band gap can be expected in AB-stacked structures than in AA-stacked structures because of the stronger interaction between the two graphene layers. There is some additional symmetry broken in AB-stacked structures. What is more, the coupling between π and π* states of carbon atoms can be enhanced by an external e lectric field.
3.2. Electronic properties and the effect of the stacking order
Figure 3 gives the first Brillouin zones of ( 3 × 3 )R30° graphene and (1 × 1) graphene. Both of them are hexagonal. The length of the reciprocal lattice vectors of the latter is 3 times larger than that of the reciprocal lattice vectors the former, and there is a relative rotation of 30° between the two Brillouin zones. Due to the increase of the real space unit cell from (1 × 1) to ( 3 × 3 )R30°, there will be a consequent folding of bands. In figure 3, the bands at the K point for the large Brillouin zone will be folded such that they will appear at the Gamma point for the small Brillouin zone. Thus Dirac points will appear at the Gamma point in the Brillouin zone of ( 3 × 3 )R30° graphene. So we can conclude safely that a linear dispersion relation appears at the Γ point in the band structure of ( 3 × 3 )R30° graphene. It has been proved that ( n 3 × n 3 )R30° and (3n × 3n) constructions, with n a positive integer, will also share this property [50]. The electronic band structure along the high symmetry points in k-space of graphene/GaN is plotted in figure 5(a). The bands dominated by graphene and GaN are plotted using red and blue circles respectively. It is clear that the states of the graphene/ GaN structure near the Fermi level are dominated by the bands associated with the carbon atoms. A very small band gap of about 3.5 meV appears at the Γ point. Linear dispersion characteristic of graphene is retained in the bands near the Fermi level. Figure 5 also shows the band structures of G/GaN/G, AA-stacked G/G/GaN and AB-stacked G/G/GaN. Band gaps of 15 meV and 47 meV appear near the Fermi level in AA-stacked and AB-stacked G/G/GaN systems respectively. 4
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Figure 5. The projected band structures of (a) graphene/GaN, (b) AB-stacked G/GaN/G, (c) AA-stacked G/G/GaN, (d) AB-stacked G/G/GaN, (e) AA-stacked GaN/G/G/GaN and (f) AB-stacked GaN/G/G/GaN. The bands dominated by the graphene layer and the GaN layer are plotted with red circles and blue circles respectively. The Fermi level is set at zero.
3.3. The effect of a perpendicular electric field
It has been demonstrated that a large band gap of up to 0.25 eV can be opened in BLG upon the application of a perpendicular electric field. We wonder whether similar behavior also occurs in the vdW heterostructures consisting of graphene and GaN. Herein we focus on the tunability of the band gaps in graphene/GaN vdW heterostructures, via the external perpendicular electric field. The evolution curves of the band gaps with a perpendicular electric field are shown in figure 6. It is clear from this figure that the band gaps of all the AB-stacked BLG-based heterostructures are modulated by the external field. The band gaps of trilayers (AB-stacked G/GaN/G and G/G/GaN) and quadrilayers (AB-stacked GaN/G/G/GaN) show increasing trends with increasing strength of the external electric field. A band gap of up to 334 meV can be generated under the positive bias voltage of 1.0 eV Å1 in the case of the AB/GaN configuration, which is much larger than that of the graphene bilayer on the SiO2 substrate induced by a dual-gated electric field [20]. Figure 6 gives an insight into the influence of the external electric field on the band gaps of heterostructures consisting of graphene and GaN. In the case of the graphene/GaN structure, a band gap of 3.5 meV is opened at the Fermi level. This band gap remains almost unchanged with increasing external electric field. It reaches its maximum value of 3.9 meV at 1.0 V Å − 1, whereas in the AB-stacked G/GaN/G configuration,
(a)
(b)
(c)
(d)
Figure 6. The changes in band gaps for (a) graphene/GaN, (b) ABstacked G/GaN/G, (c) AB-stacked G/G/GaN and (d) AB-stacked GaN/G/G/GaN due to the external electric field. 5
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(a)
(b)
(c)
Figure 7. Band structure in the vicinity of the K point as a function of the external perpendicular electric field for (a) AB-stacked G/G/GaN under a positive bias voltage, (b) AB-stacked G/G/GaN under a reverse-biased electric field, (c) AB-stacked GaN/G/G/GaN.
point for each configuration and increases with the strength of the electric field.
the band gap is about 9 meV without an external electric field and it increases rapidly to 39 meV under a perpendicular electric field of 0.6 V Å − 1; then it remains unchanged with increasing electric field. In this work, a positive bias voltage is applied from top to bottom in the configurations of figure 2, and a negative bias voltage is applied reversely. The changes in the band gaps of AB-stacked G/G/GaN and GaN/G/G/GaN due to the externally applied perpendicular electric field are also depicted in figures 6(c) and (d). Band gaps of 47 meV and 43 meV are opened at the Γ point of the Brillouin zone in the band structure of AB-stacked G/G/GaN and GaN/G/G/ GaN respectively at zero electric field, and they increase linearly with the strength of the electric field. In the case of a positive bias voltage, the band gap of AB-stacked G/G/GaN reaches 334 meV at 1.0 V Å − 1, and in case of a negative bias voltage, the gap shows a maximum of 125 meV at 0.8 V Å − 1; while the band gap of AB-stacked GaN/G/G/GaN structure attains its maximum of 217 meV as the external electric field increases to 0.7 V Å − 1. As the external electric field increases to above 0.7 V Å − 1 (or 0.8 V Å − 1 reverse bias), the band gap of AB-stacked GaN/G/G/GaN (or AB-stacked G/G/GaN) decreases. It can be seen that AB-stacked G/G/GaN shows a wider band gap opening and a larger rate of increase than AB-stacked GaN/G/G/GaN in the presence of a positive bias voltage. This happens because there is some extra symmetry broken in AB-stacked G/G/GaN because of its lack of mirror symmetry. Panel (c) of figure 6 reveals different rates of increase for the cases of positive bias voltage and negative bias voltage for AB-stacked G/G/GaN structure. This difference should be attributed to the different interactions of the dipole moment caused by the external electric field. There is almost no difference in the changes of band gaps in the presence of positively biased and negatively biased electric fields for other heterostructures. Band structures in the vicinity of the Γ point for AB-stacked G/G/GaN and GaN/G/G/GaN systems under 0.0 V Å − 1, 0.2 V Å − 1 and 0.6 V Å − 1 perpendicular electric fields are shown in figure 7. It is clear that the valence bands are affected more strongly than the conduction bands by the external electric field for all the structures considered. Parabolic dispersive relations appear near the Fermi level in the band structures. A band gap opening happened at the Γ
4. Conclusion In summary, we have investigated the geometric and electronic structure of vdW heterostructures consisting of graphene and GaN using vdW DFT calculations. The interlayer spacing of the two graphene layers in the AB-stacked systems is smaller and the binding energy is numerically larger than those in AA-stacked systems. AB-stacked G/G/GaN and GaN/G/G/GaN show wider band gaps than the corresponding AA-stacked systems. A band gap is opened near the Fermi level for all vdW heterostructures considered. Then we studied the effect of the electric field on the electronic structure of graphene/GaN vdW heterostructures. The band gaps of AB-stacked G/G/GaN and GaN/G/G/GaN show a linear increase with the strength of the external electric field. In particular, a maximum band gap of 334 meV is generated in AB-stacked G/G/GaN systems in the presence of a 1.0 V Å − 1 external electric field. It is expected that our calculated results may provide a fundamental basis for applications of BLG in optoelectronic devices and field effect transistors. Acknowledgments This work was supported by the National Natural Science Foundation of China under grant no. 91233120 and the National Basic Research Program of China (2011CB921901). References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666 [2] Novoselov K, Geim A K, Morozov S, Jiang D, Grigorieva M K I, Dubonos S and Firsov A 2005 Nature 438 197 [3] Neto A C, Guinea F, Peres N, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [4] Jiang Z, Zhang Y, Stormer H and Kim P 2007 Phys. Rev. Lett. 99 106802 [5] Peres N M R 2009 J. Phys.:Condens. Matter 21 323201 [6] Han M Y, Özyilmaz B, Zhang Y and Kim P 2007 Phys. Rev. Lett. 98 206805 6
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