Stacking Fault Enriching the Electronic and ... - ACS Publications

Letter pubs.acs.org/NanoLett

Stacking Fault Enriching the Electronic and Transport Properties of Few-Layer Phosphorenes and Black Phosphorus Shuangying Lei,†,‡ Han Wang,‡ Lan Huang,† Yi-Yang Sun,*,‡ and Shengbai Zhang*,‡ †

Key Laboratory of Microelectromechanical Systems of the Ministry of Education, Southeast University, Nanjing 210096, China Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, United States

‡

S Supporting Information *

ABSTRACT: Interface engineering is critical for enriching the electronic and transport properties of two-dimensional materials. Here, we identify a new stacking, named Aδ, in few-layer phosphorenes (FLPs) and black phosphorus (BP) based on first-principles calculation. With its low formation energy, the Aδ stacking could exist in FLPs and BP as a stacking fault. The presence of the Aδ stacking fault induces a direct to indirect transition of the band gap in FLPs. It also affects the carrier mobilities by significantly increasing the carrier effective masses. More importantly, the Aδ stacking enables the fabrication of a whole spectrum of lateral junctions with all the type-I, II, and III alignments simply through the manipulation of the van der Waals stacking without resorting to any chemical modification. This is achieved by the widely tunable electron affinity and ionization potential of FLPs and BP with the Aδ stacking. KEYWORDS: Black phosphorus, phosphorene, stacking fault, transport, lateral junction ∼105 have been reported.9,11,20 The carrier mobility measured in experiment shows clear thickness-dependence ranging from ∼10 cm2/V·s for FLPs to ∼1000 cm2/V·s for thicker layers of above 10 nm.9,11 Theoretically, much higher mobility (up to 105 cm2/V·s) has been predicted for monolayer phosphorene.21 With improved defect control and interface engineering, the mobility is expected to be further improved. Given that the issue with their stability in air7 can be adequately addressed,22 the FLPs and BP could be potential candidates for nextgeneration nanoelectronic applications. The electronic structures of FLPs have been the subject of a number of theoretical studies.10,19,21,23−30 Most of the previous studies have focused on the stable stacking sequence as derived from the bulk BP, called AB stacking. Recent experimental and theoretical works have shown that metastable structures of 2D materials exhibit interesting properties. For example, TMDs could exhibit the 2H-to-1T transition, which is accompanied by semiconductor-to-metal transition.31 Monolayer phosphorene could exhibit a change in stacking that dramatically changes the optical absorption onset from infrared to blue region.32 In this paper, we identify a new stacking (named Aδ here) between monolayer phosphorenes, which is found to be the only metastable stacking besides the stable AB stacking. Other stackings, such as AA and AC, are found to be unstable and will spontaneously transform into either AB or Aδ stacking if symmetry constraint is lifted. The new Aδ stacking is found to

I

n the past decade, two-dimensional (2D) materials have attracted significant interest from researchers worldwide. Such materials exhibit new physical phenomena that are not observed in their bulk counterparts, such as massless Dirac Fermion in graphene.1,2 Meanwhile, intensive research has been devoted to exploring real-world applications of the 2D materials. An important area of research is to fabricate fieldeffect transistors (FETs) for replacing silicon in microelectronic (or nanoelectronic) applications.3 Two types of 2D materials have been extensively studied for such applications, namely graphene3 and monolayer transition metal dichalcogenides (TMDs),4−6 both of which have their pros and cons. While the high carrier mobility of graphene is well suited for making FETs, the on−off ratio is too low due to the semimetallicity of graphene.7 In contrast, the on−off ratio of FETs made of TMDs is sufficiently high.7 But the carrier mobility of TMDs is relatively low,7 partly due to the localized d-electrons of the transition metal elements.8 Alternative 2D materials are still constantly sought for addressing these issues. Monolayer and few-layer phosphorenes (FLPs), which are fabricated from the layered material, black phosphorus (BP), have recently emerged as a candidate material for making FETs,9−15 as well as other electronic devices such as photodetectors16 and even terahertz detectors.17 Monolayer phosphorene has a direct optical bandgap of 1.45 eV at room temperature10 and an electric band gap of about 2.05 eV.18 FLPs exhibit a drastic reduction of band gap as a function of thickness, while preserving the direct gap.19 This feature is different from TMDs, which undergo a direct-to-indirect transition.4 Phosphorene FETs with high on−off ratios up to © XXXX American Chemical Society

Received: November 18, 2015 Revised: January 21, 2016

A

DOI: 10.1021/acs.nanolett.5b04719 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters give rise to a whole zoo of enriched electronic and transport properties for FLPs and BP, including a direct-to-indirect transition in band gap, oppositely behaved effective masses as the layer thickness increases, and widely tunable band gap, electron affinity (EA), and ionization potential (IP). Important applications in nanoelectronics are expected by employing such enhanced properties. In particular, lateral junctions with type-I, II, and III alignments could all be realized by a single material simply by manipulation of van der Waals (vdW) stacking without any chemical modification. Our first-principles calculations were based on density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP).33 Projector-augmented wave method34 was used to describe the interaction between ion cores and valence electrons. Atomic structures were optimized using the exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE)35 combined with the DFT-D3 method of Grimme36 for describing the vdW interaction. Plane-waves with a cutoff energy of 544 eV for the kinetic energy were used as the basis set. Monkhorst−Pack 10 × 14 × 1 k-point grid was used to sample the Brillouin zone (BZ). The total energy convergence criterion was set to 10−7 eV. The criterion for structure optimization, including lattice constants and ionic positions, was set to be 0.005 eV/Å. A vacuum space of at least 10 Å was used to separate adjacent slabs. All properties related to electronic structure, including band gap, EA, IP and effective mass, were obtained using the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06) with a mixing parameter of 0.25 and a screening parameter of 0.2 Å−1 for the Hartree− Fock exchange. A cutoff energy of 272 eV for the plane-wave basis set and a 5 × 7 × 1 k-grid were used in the HSE06 calculations. We performed an exhaustive search of the possible stacking of bilayer phosphorene. We started from the AA stacking as shown in Figure 1a, where the two monolayers have the same coordinates along lattice vectors a and b and the relative shifts, δa and δb, between the two monolayers are defined to be zero. Our search in the translational degrees of freedom was performed by varying δa and δb in the range of [−a/2, a/2] and [−b/2, b/2], respectively, with a 28 × 20 grid. At each grid point, all atomic coordinates were allowed to relax except that the relative shifts δa and δb were kept fixed. The lattice constants a and b were fixed at the values optimized using monolayer. The potential energy surface (PES) as a function of δa and δb is shown in Figure 1e. Five extrema exist on the PES. Two of them are maxima and the other three are minima. The maxima at (0, 0) and (0.5, 0.5) correspond to the AA and AC stackings shown in Figure 1a,c, respectively, which are unstable. If the structural optimization were done with the symmetry constrains lifted, AA and AC will relax to one of the minima. The minimum at (0, 0.5) corresponds to the AB stacking, shown in Figure 1b. The other two degenerate minima, called Aδ stacking here, are not intuitive and to our best knowledge have not been reported before. The Aδ stacking is obtained by shifting the second layer from AA stacking along the a direction by about 0.281 a. The symmetry is reduced from the Pbcm (or D11 2h) space group of the AB stacking to the P2/m (or C12h) space group. Taking the structures as found above, we reoptimized the structures of AB and Aδ by allowing the lattice constants to change. The optimized structural parameters are listed in Supporting Information (SI) Table S1. It is found that a decreases from

Figure 1. Atomic structures of phosphorene bilayers with (a) AA, (b) AB, (c) AC, and (d) Aδ stacking, where the upper and lower panels show the top and side views, respectively. The dashed lines mark the unit cells. (e) Contour plot of potential energy surface of phosphorene bilayer as a function of the relative shifts, δa and δb, along the lattice vectors a and b.

that of monolayer, while b increases. However, the change in a is much more significant than b because phosphorene is much stiffer along b (or zigzag) than a (or armchair) direction, as shown in SI Figure S1. The optimized a of bilayer with the Aδ stacking is 4.537 Å, which is larger than that with the AB stacking (4.504 Å). For comparison, the optimized a of monolayer is 4.595 Å. The difference in a is mainly a result of the change in bond angles, while the difference in bond lengths is less than 0.004 Å between AB and Aδ stackings. The interlayer distance for the Aδ stacking is 5.408 Å, larger than that of the AB stacking (5.344 Å), which is consistent with the metastable nature of the Aδ stacking. As found from the PES, the new Aδ stacking is higher than the AB stacking by about 31 meV per unit cell (or about 4 meV per atom). The energy barrier between the two minima can be estimated from the PES to be 40 and 9 meV per unit cell from the AB and Aδ sides, respectively. We checked the energy difference between the two minima using the local density approximation (LDA) to the exchange-correlation functional, which yields a slightly larger energy difference of 40 meV per unit cell. The comparison with LDA also suggests that the metastability is not changed with different functionals used in the calculation. A similar PES for the bulk BP was also obtained by using a supercell containing eight monolayers with one δ layer sandwiched between two A layers, which will be further discussed later. The resulting stacking fault in bulk BP has a formation energy of 68 meV per unit cell, which is about two times the value for bilayer due to the presence of two interfaces. It is slightly higher because we did not allow the lattice constants to relax in bulk calculations. Given the small formation energy, the Aδ stacking fault is expected to exist in FLPs and BP or can be obtained in experiment by delicate B

DOI: 10.1021/acs.nanolett.5b04719 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters sample transfer techniques37 or controlled growth38 if deposition method can be developed for making FLPs. Once the stacking fault is formed, the energy barrier could hinder the transition back to the AB stacking as long as the sample is reasonably large. We also searched in the rotational degree of freedom for stacking by constructing bilayers with mutually twisted layers from the stable AB stacking. We considered twist angles of 15°, 30°, 45°, 60°, 75° and 90°. The structures and supercells used in these calculations are shown in SI Figure S2. In general, we found that the twisting energy per atom are larger than the case of translational shifts. Within all the rotational angles studied, the bilayer with a twist angle of 75° is the most stable but is still higher than the Aδ stacking by about 5 meV per atom (see SI Figure S2). In this work, we will only study the effect of the Aδ stacking fault. Figure 2 shows a comparison of the band structures of monolayer (1L), bilayer (2L), and trilayer (3L) phosphorenes

consistently show the direct-to-indirect transition (see SI Figure S3). Considering time-reversal symmetry, there are two degenerate peaks of the top valence band in the whole BZ. A threedimensional plot of the top valence band for the case of 2L-Aδ is shown in SI Figure S4, where the topography near the Γ point can be better seen. A zoom-in view (SI Figure S4) for the 3L cases shows more complicated landscape near the Γ point, where there are three peaks with two located at about ±0.07 kx and the third one right at Γ point. In the case of 4L with all δ stacking, there are four peaks located at about ±0.08 kx and ±0.03 kx, respectively (SI Figure S4). These small bumps, as also seen in Figure 2c,e,f, effectively flatten the landscape of top valence band near Γ point and make the effective mass larger as discussed below. Figure 3a shows the effective masses calculated for electrons and holes along the Γ-X direction. Note that we will focus our

Figure 3. (a) Effective masses of holes (solid circles) and electrons (open circles) of FLPs with AB (brown) and Aδ (blue) stackings calculated as a function of thickness. The unit m0 stands for electron rest mass. (b) VBM (solid circles) and CBM (open circles) of FLPs with AB (brown) and Aδ (blue) stackings as a function of thickness. The vacuum level is used in (b) as the reference potential meaning that the CBM and VBM correspond to EA and IP, respectively. All results were obtained using the HSE06 functional.

Figure 2. Band structures calculated using HSE06 functional for phosphorenes of monolayer (1L) (a), bilayer (2L) with AB (b) and Aδ (c) stackings, and trilayer (3L) with ABA (d), AδA (e), and Aδδ (f) stackings. X and Y have coordinates π/a and π/b corresponding to the armchair and zigzag directions, respectively. The reference potential is taken to be the vacuum level. Dashed circles in (b) and (c) highlight the opposite trends in the splitting of VBM and CBM levels at Γ point (see text).

discussion below on the transport properties along the Γ-X direction (or armchair direction) because FLPs and BP exhibit strong anisotropy in transport with significantly larger effective masses, hence lower mobility, along the Γ-Y direction (or zigzag direction)11,21 with an exception of monolayer, which could have large hole mobility along Γ-Y due to a small deformation potential.21 All effective masses were obtained by fitting the band around the VBM or conduction band minimum (CBM) within the range of ±0.025 kx using 11 data points. HSE06 functional was used in these calculations, which is known to yield more accurate effective masses than the PBE functional.42 As can be seen from Figure 3a, the hole effective masses with AB and Aδ stackings show opposite trends as a function of layer thickness. The AB stacking gives rise to decreasing hole effective mass as layer thickness increases, while Aδ stacking gives rise to increasing hole effective mass. The electron effective masses with the Aδ stacking are also significantly larger than those in the AB stacking. But different from the case of holes, the electron effective mass exhibits a large increase from 1L to 2L, then starts to decrease for thicker

with AB and Aδ stackings. A main difference observed for the cases with Aδ stacking is that the band gaps become indirect, which has not been reported for any phosphorenes before. In the case of 2L-Aδ shown in Figure 2c, the valence band maximum (VBM) occurs at about 0.05 kx away from the Γ point, where kx = π/a. For 3L, we considered three different stacking sequences: ABA, AδA, and Aδδ, where AδA means that one δ layer is sandwiched between two A layers and Aδδ means that both the second and third layers have a δ stacking with respect to the lower layer. Both AδA and Aδδ stackings give rise to indirect band gap with the VBM occurring at about 0.07 kx away from the Γ point. We have performed calculations on the band structure of 2L-Aδ using the quasi-particle G 0W 0 method39 and the nonlocal vdW-DF functional,40,41 which C

DOI: 10.1021/acs.nanolett.5b04719 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters layers. The large differences in effective masses yields appreciable difference in carrier mobilities in AB- and Aδstacked FLPs as shown in SI Table S2, which were estimated by also considering the effect of deformation potential.21,43 For 3L-AδA, the hole effective mass, 0.159 m0 (m0 stands for electron rest mass), is smaller than that of the 3L-Aδδ stacking (0.170 m0), but still larger than that of the 3L-ABA stacking (0.154 m0). The bottom of conduction band of 3L-AδA is rather flat, as can be seen in Figure 2e, suggesting a large effective mass and a shift of the CBM from Γ point to an off-Γ position. Effective mass cannot be obtained reliably from a quadratic fitting for this case. Another striking difference between the band structures of AB and Aδ stackings in Figure 2 is that the level splitting at bottom of conduction bands and top of valence bands shows opposite trends for the two stackings. With AB stacking, the VBM splitting is larger than the CBM splitting, while with Aδ stacking the CBM splitting is larger than the VBM splitting, as can been seen in the circled regions in Figure 2b,c for bilayers. For thicker layers, similar trends are observed. A direct consequence of the opposite trends in level splitting is that both VBM and CBM of Aδ-stacked FLPs are lower than those of the AB-stacked FLPs. Figure 3b shows the calculated band edge positions of the FLPs, where we used the vacuum level as an absolute reference so that the CBM and VBM correspond to the EA and IP, respectively. It can be seen that for AB stacking, as the layer thickness increases the band gap reduction is mainly a result of the increase of the VBM. In contrast, for the Aδ stacking, the band gap reduction is mainly a result of the decrease of the CBM. This result suggests that FLPs with the same thickness and different stackings could form lateral junctions of type-II alignment, while FLPs with the same stacking but different thickness could form type-I alignment. As the thickness increases, the CBM from Aδ stacking will eventually become lower than the VBM from the AB stacking so that type-III alignment can be formed as will be discussed below for bulk BPs. We constructed a tight-binding (TB) model with a minimum set of hopping parameters to understand the observed difference in the electronic structures of FLPs with AB and Aδ stackings. We used two intramonolayer hopping parameters t1 and t2, as illustrated in Figure 4a,b, which were solved for using the band gaps of monolayer at Γ and X points from HSE06 calculations. For intermonolayer hopping, we first considered the nearest neighbors only, which were represented by parameters T1 (see Figure 4). However, for both AB and Aδ stackings, T1 only gives rise to symmetric splitting of the VBM and CBM levels, as shown in the left panels of Figure 4c,d. Only after considering the second nearest neighbor hopping T2, the asymmetry of the splitting can be reproduced, as can be seen from the right panels of Figure 4c,d. Key features from the DFT calculations, such as the indirect gap in Aδ stacking, can also be reproduced from this minimum TB model. The significant stacking-dependent level splitting seen in both the DFT and TB results, which affects the EA and IP, can be understood based on an orbital interaction picture. We found that for an eigenstate φ (e.g., the VBM or CBM) of monolayer, when stacking into bilayer, the splitting energy Δ corresponding to this state is approximately proportional to η = ⟨φlower|φupper⟩, which is the overlap (or coupling) of the state φ in the lower (φlower) and upper layer (φupper). Table 1 shows the calculated η for the VBM and CBM states with AB and Aδ stackings. It can be seen that the magnitudes of η follow the

Figure 4. Atomic structure of bilayer phosphorenes with AB (a) and Aδ (b) stackings showing the tight-binding hopping parameters. The intramonolayer parameters t1 and t2, which were set to −1.16 and 3.09 eV, respectively, are the same for both stackings and only shown in (a). (c,d) Band structures of the bilayers obtained by appending t1 and t2 with T1 only (left panels) for intermonolayer hopping and with both T1 and T2 (right panels). T1 and T2 were respectively set to −0.31 and 0.10 eV for AB stacking and −1.07 and −0.26 eV for Aδ stacking.

Table 1. Correlation of the Splitting Energy Δ of the VBM and CBM States in Bilayer with the Calculated η, Which Is the Overlap of the VBM or CBM State of Monolayer in the Geometry of Bilayer with Corresponding Stackinga AB stacking Δ (eV) η Δ/η a

Aδ stacking

VBM

CBM

VBM

CBM

0.820 0.069 11.9

0.426 0.037 11.5

0.034 0.007 4.9

1.008 0.096 10.5

The ratio Δ/η is shown in the last row.

same order as the splitting energy Δ. The ratios of Δ/η are similar except for the VBM state of Aδ stacking possibly because both Δ and η for this state are significantly smaller than the other states. This result clearly demonstrates that even though the stacking is through the weak vdW interaction, the overlap of wave functions from two neighboring layers could still significantly modify the electronic structures of each other. More discussion and calculation details about the orbital interaction analysis are given in the SI.) As discussed above, the Aδ stacking with the low formation energy could exist in bulk BP as a stacking fault. We studied its effect on the electronic structure of bulk BP. We started from a perfect supercell containing eight monolayers with the standard AB stacking. One of the B layers was shifted to become a δlayer as shown in Figure 5a. The band structure of this stacking fault-containing supercell is shown in Figure 5b. Different from the cases in FLPs discussed above, the band gap remains to be direct gap. The effect of the stacking fault on effective masses, however, remains. Along the Γ-X direction, the electron and D

DOI: 10.1021/acs.nanolett.5b04719 Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

■

Letter

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b04719. Optimized structural parameters for FLPs and BP, total energy of monolayer phosphorene as a function of strain along the armchair and zigzag directions, results for twisted 2L phosphorenes, results from nonlocal vdW-DF functional and G0W0 calculations, topography of top valence bands for 2L, 3L, and 4L phosphorenes, calculated carrier mobilities, and discussion on the orbital interaction analysis. (PDF)

■

Figure 5. (a) Atomic structure (a) and band structure (b) of ABstacked BP with a δ-layer stacking fault. (c,d) The band structure and atomic structure of a polymorph of BP, called δ-BP here, where all monolayers have an Aδ stacking relative to each other. The dashed lines in the atomic structures mark the supercells. The green frames mark individual monolayer unit cells to guide the eyes.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Y.-Y.S.). *E-mail: [email protected] (S.Z.). Author Contributions

S.L. and H.W. contributed equally to this work. Notes

hole effective masses in the presence of an Aδ stacking fault are 0.241 and 0.129 m0, respectively, as calculated using the HSE06 functional, while the corresponding values in perfect BP are 0.094 and 0.086 m0, respectively. This result suggests that free carriers close to a stacking fault in BP will experience low mobility due to the increased effective masses. We also studied the case where all monolayers in bulk BP have the metastable Aδ stacking. This case corresponds to a new polymorph of BP, called δ-BP here, as shown in Figure 5d. Because the shift along the a direction is about 0.281 a, we used a supercell containing seven monolayers that approximately recovers the periodicity. The δ-BP polymorph has an indirect gap, as can be seen from the band structure in Figure 5c. Both VBM and CBM of δ-BP are away from the Γ point. The effect of increasing EA and IP for FLPs is preserved for δ-BP. The calculated EA and IP using HSE06 functional are 4.71 and 5.38 eV, respectively. The corresponding values for AB-stacked BP is 4.35 and 4.61 eV, respectively. The alignment with vacuum was obtained using the calculated electrostatic potential at a P atom in bulk and that in the middle layer from 3L-ABA. The error was found to be smaller than 0.02 eV. Thus, the EA of δ-BP is even larger than the IP of AB-stacked BP, meaning that the two polymorphs could form a lateral junction with type-III alignment. The electron and hole effective masses of δ-BP were calculated to be 0.113 and 0.170 m0, respectively, also higher than those of AB-stacked BP. To conclude, using first-principles calculation we identify a metastable stacking, named Aδ, for few-layer phosphorenes and black phosphorus. We show that the Aδ stacking is the only metastable stacking between monolayer phosphorenes besides the stable AB stacking. The Aδ stacking is higher in energy than the AB stacking by about 4 meV per atom and could exist in few-layer phosphorenes and black phosphorus as a stacking fault. The presence of the Aδ stacking fault induces a direct to indirect transition of band gap and can significantly affect the carrier mobilities by increasing the carrier effective masses. The Aδ stacking also offers a means of widely tuning the band edge positions without resorting to any chemical modification and makes it possible to fabricate all-BP-based lateral junctions with all types of band edge alignments.

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE) under Grant No. DE-SC0002623. S.L. was also supported by the National Science Foundation of Jiangsu Province of China (Nos. BK201320668 and BK20151409) and the National Basic Research Program of China (No. 2015CB352100). The supercomputer time was provided by the National Energy Research Scientific Computing Center (NERSC) under DOE Contract No. DE-AC02-05CH11231 and the Center for Computational Innovations (CCI) at RPI.

■

REFERENCES

(1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197−200. (2) Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. Nature 2005, 438, 201−204. (3) Schwierz, F. Nat. Nanotechnol. 2010, 5, 487−496. (4) Wang, Q. H.; Kalantar-Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S. Nat. Nanotechnol. 2012, 7, 699−712. (5) Butler, S. Z.; Hollen, S. M.; Cao, L.; Cui, Y.; Gupta, J. A.; Gutiérrez, H. R.; Heinz, T. F.; Hong, S. S.; Huang, J.; Ismach, A. F.; Johnston-Halperin, E.; Kuno, M.; Plashnitsa, V. V.; Robinson, R. D.; Ruoff, R. S.; Salahuddin, S.; Shan, J.; Shi, L.; Spencer, M. G.; Terrones, M.; Windl, W.; Goldberger, J. E. ACS Nano 2013, 7, 2898−2926. (6) Jariwala, D.; Sangwan, V. K.; Lauhon, L. J.; Marks, T. J.; Hersam, M. C. ACS Nano 2014, 8, 1102−1120. (7) Ling, X.; Wang, H.; Huang, S.; Xia, F.; Dresselhaus, M. S. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 4523−4530. (8) Yoon, Y.; Ganapathi, K.; Salahuddin, S. Nano Lett. 2011, 11, 3768−3773. (9) Li, L.; Yu, Y.; Ye, G. J.; Ge, Q.; Ou, X.; Wu, H.; Feng, D.; Chen, X. H.; Zhang, Y. Nat. Nanotechnol. 2014, 9, 372−377. (10) Liu, H.; Neal, A. T.; Zhu, Z.; Luo, Z.; Xu, X.; Tománek, D.; Ye, P. D. ACS Nano 2014, 8, 4033−4041. (11) Xia, F.; Wang, H.; Jia, Y. Nat. Commun. 2014, 5, 4458. (12) Zhu, W.; Yogeesh, M. N.; Yang, S.; Aldave, S. H.; Kim, J.-S.; Sonde, S.; Tao, L.; Lu, N.; Akinwande, D. Nano Lett. 2015, 15, 1883− 1890. (13) Wang, H.; Wang, X.; Xia, F.; Wang, L.; Jiang, H.; Xia, Q.; Chin, M. L.; Dubey, M.; Han, S. -j. Nano Lett. 2014, 14, 6424−6429. E

DOI: 10.1021/acs.nanolett.5b04719 Nano Lett. XXXX, XXX, XXX−XXX

Letter

Nano Letters (14) Das, S.; Demarteau, M.; Roelofs, A. ACS Nano 2014, 8, 11730− 11738. (15) Xiang, D.; Han, C.; Wu, J.; Zhong, S.; Liu, Y.; Lin, J.; Zhang, X.A.; Hu, W. P.; Ö zyilmaz, B.; Castro Neto, A. H.; Wee, A. T. S.; Chen, W. Nat. Commun. 2015, 6, 6485. (16) Engel, M.; Steiner, M.; Avouris, P. Nano Lett. 2014, 14, 6414− 6417. (17) Viti, L.; Hu, J.; Coquillat, D.; Knap, W.; Tredicucci, A.; Politano, A.; Vitiello, M. S. Adv. Mater. 2015, 27, 5567−5572. (18) Liang, L.; Wang, J.; Lin, W.; Sumpter, B. G.; Meunier, V.; Pan, M. Nano Lett. 2014, 14, 6400−6406. (19) Tran, V.; Soklaski, R.; Liang, Y.; Yang, L. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 235319. (20) Koenig, S. P.; Doganov, R. A.; Schmidt, H.; Castro Neto, A. H.; Ö zyilmaz, B. Appl. Phys. Lett. 2014, 104, 103106. (21) Qiao, J.; Kong, X.; Hu, Z.-X.; Yang, F.; Ji, W. Nat. Commun. 2014, 5, 4475. (22) Kim, J.-S.; Liu, Y.; Zhu, W.; Kim, S.; Wu, D.; Tao, L.; Dodabalapur, A.; Lai, K.; Akinwande, D. Sci. Rep. 2015, 5, 8989. (23) Dolui, K.; Quek, S. Y. Sci. Rep. 2015, 5, 11699. (24) Guo, H.; Lu, N.; Dai, J.; Wu, X.; Zeng, X. C. J. Phys. Chem. C 2014, 118, 14051−14059. (25) Dai, J.; Zeng, X. C. J. Phys. Chem. Lett. 2014, 5, 1289−1293. (26) Xu, Y.; Dai, J.; Zeng, X. C. J. Phys. Chem. Lett. 2015, 6, 1996− 2002. (27) Fei, R.; Yang, L. Nano Lett. 2014, 14, 2884−2889. (28) Rudenko, A. N.; Yuan, S.; Katsnelson, M. I. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 085419. (29) Rudenko, A. N.; Katsnelson, M. I. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 201408. (30) Ç akır, D.; Sevik, C.; Peeters, F. M. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92, 165406. (31) Lin, Y.-C.; Dumcenco, D. O.; Huang, Y.-S.; Suenaga, K. Nat. Nanotechnol. 2014, 9, 391−396. (32) Zhu, Z.; Tománek, D. Phys. Rev. Lett. 2014, 112, 176802. (33) Kresse, G.; Furthmüller, J. Comput. Mater. Sci. 1996, 6, 15−50. (34) Kresse, G.; Joubert, D. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758−1775. (35) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865−3868. (36) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104. (37) Huang, S.; Ling, X.; Liang, L.; Kong, J.; Terrones, H.; Meunier, V.; Dresselhaus, M. S. Nano Lett. 2014, 14, 5500−5508. (38) Brown, L.; Hovden, R.; Huang, P.; Wojcik, M.; Muller, D. A.; Park, J. Nano Lett. 2012, 12, 1609−1615. (39) Hybertsen, M. S.; Louie, S. G. Phys. Rev. B: Condens. Matter Mater. Phys. 1986, 34, 5390−5413. (40) Klimeš, J.; Bowler, D. R.; Michaelides, A. J. Phys.: Condens. Matter 2010, 22, 022201. (41) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. Phys. Rev. Lett. 2004, 92, 246401. (42) Kim, Y.-S.; Marsman, M.; Kresse, G.; Tran, F.; Blaha, P. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 205212. (43) Bruzzone, S.; Fiori, G. Appl. Phys. Lett. 2011, 99, 222108.

F

DOI: 10.1021/acs.nanolett.5b04719 Nano Lett. XXXX, XXX, XXX−XXX