Letter pubs.acs.org/NanoLett
Bending Two-Dimensional Materials To Control Charge Localization and Fermi-Level Shift Liping Yu,* Adrienn Ruzsinszky, and John P. Perdew Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, United States S Supporting Information *
ABSTRACT: High-performance electronics requires the fine control of semiconductor conductivity. In atomically thin two-dimensional (2D) materials, traditional doping technique for controlling carrier concentration and carrier type may cause crystal damage and significant mobility reduction. Contact engineering for tuning carrier injection and extraction and carrier type may suffer from strong Fermi-level pinning. Here, using first-principles calculations, we predict that mechanical bending, as a unique attribute of thin 2D materials, can be used to control conductivity and Fermi-level shift. We find that bending can control the charge localization of top valence bands in both MoS2 and phosphorene nanoribbons. The donor-like in-gap edge-states of armchair MoS2 ribbon and their associated Fermi-level pinning can be removed by bending. A bending-controllable new in-gap state and accompanying direct−indirect gap transition are predicted in armchair phosphorene nanoribbon. We demonstrate that such emergent bending effects are realizable. The bending stiffness as well as the effective thickness of 2D materials are also derived from first principles. Our results are of fundamental and technological relevance and open new routes for designing functional 2D materials for applications in which flexuosity is essential. KEYWORDS: Flexible electronics, two-dimensional materials, first-principles calculations, bending stiffness, strain engineering, Fermi-energy pinning tomically thin and highly flexible layered 2D materials offer a unique and compelling capability for a wide variety of applications,1−6 particularly in flexible nanoelectronics.7−9 To realize such applications, it is critical to understand the interplay between flexural bending and the electronic properties of 2D materials. For a 2D material consisting of two or more atomic layers (e.g., phosphorene and MoS2), it is known that bending induces compressive and tensile strains in its inner and outer layers, respectively. However, it is largely unknown how flexural bending transforms to local in-plane strains (small or large, slightly nonuniform or highly nonuniform) at the atomic scale within each constituent layer, and how such local strains couple with other physical properties (e.g., flexoelectricity, the coupling between polarization and strain gradients10,11). The coexistence of compressive and tensile strains is expected to have different physical effects than those of uniform uniaxial or biaxial strains, which are either compressive or tensile in all constituent layers. The latter have been studied extensively in the past,12−16 whereas the former have been largely unexplored at least from first-principles, especially in the 2D materials beyond graphene. In this work, we use Kohn−Sham density functional orbitals and orbital energies to study the bending effects on the structural and electronic properties, and their consequences for the conductivity behavior. We consider monolayer MoS2 in the 1H structure, phosphorene, and graphene, representing 2D materials of three-, two-, and one-atomic layers thin,
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© 2016 American Chemical Society
respectively. They have the same hexagonal in-plane structure (Figure 1a−c) and can be cut into nanoribbons with either armchair or zigzag edges on both sides (Supplementary Figure S1). Our study is based on first-principles density functional methods with the generalized gradient approximation17 (justified because the nanoribbons we study have only covalent and not van der Waal bonds.) The bending is applied to the nanoribbon width direction. The ribbon axial direction (parallel to the edges, with atomistic periodicity) is free of bending, making the band structure one-dimensional. The overall bending deformation is quantified by an effective bending curvature (κ) or curvature radius (R = 1/κ), which is defined by the width of the ribbon before bending and the distance between its two edges after bending (Figure 1d). This bending scheme is similar to the setup for a two-point bending stiffness measurement. Deleterious Edge-States and Fermi-Level Pinning. Consistent with previous studies,18,19 we find that the flat undoped armchair 1H-MoS2 nanoribbon (denoted as AMoS2R) is a direct-gap semiconductor (Figure 2a). The band gap is formed between two 2-fold-degenerate edge-state bands, which are well separated from the nonedge-state bands. The occupied edge-state bands can donate electrons and hence are Received: December 29, 2015 Revised: March 2, 2016 Published: March 3, 2016 2444
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bending. Figure 2a−c (also Supplementary Figure S2) shows that the donor-like edge states decrease almost linearly in energy as bending curvature κ increases. At a critical bending curvature, κc = 0.072 Å−1 (Rc = 13.9 Å), the donor-like edge states shift down to the non-edge valence bands (Supplementary Note 1). Hence the Fermi-level pinning due to this donor-like in-gap state is removed when κ > κc. For acceptorlike edge-states, their band energy also decreases with increasing κ, but they still remain as in-gap states. This means that bending can tune the Fermi-energy pinning level in n-type A-MoS2R, although it cannot be removed. A similar bending-induced Fermi-level shift is also observed in metallic zigzag MoS2R nanoribbons (Supplementary Figure S3) and zigzag phosphorene nanoribbons (Supplementary Figure S5). Bending Effects on Charge Localization and Conductivity. The majority carriers of a p-type semiconductor are the holes produced mainly from the top of the valence bands, when electrons are thermally excited into localized acceptor states. For A-MoS2R, Figure 2a shows that the partial charge density at the top of the valence bands is delocalized over the whole ribbon width before bending. As κ increases, these topvalence charges shrink to the ribbon middle region and get depleted near the ribbon edge region. In other words, externally applied bending localizes the hole-carriers in the ribbon center region. As a result, the conductivity of p-type A-MoS2R from one edge to the other will be turned off or significantly reduced as bending curvature increases. A similar bending effect on charge localization and conductivity is also found in p-type armchair phosphorene nanoribbons (denoted as A-PR). Figure 3a (also Supplemen-
Figure 1. Crystal structure and bending scheme. (a) 1H-MoS2. (b) Phosphorene. (c) Graphene. The armchair and zigzag directions are denoted as X and Y respectively. (d) Schematic bending diagram, where d0 and d are the ribbon width before and after bending, and κ and R are the effective bending curvature and curvature radius, respectively.
Figure 2. Bending effects in armchair 1H-MoS2 nanoribbons. (a) Band structures of A-MoS2R at three different bending curvature radii. The bands highlighted in red are 2-fold degenerate edge-state bands (Supplementary Figure S2). The lower and upper horizontal dashed lines mark the valence band maximum and conduction band minimum, respectively. (b) Isosurfaces of partial charge densities of states I, II, and III, as marked in panel a.
donor-like. The unoccupied edge state bands can accept electrons and hence are acceptor-like. These highly dense ingap edge-states can absorb a large quantity of charge injected from contact metals, pinning the Fermi-level to a location around the edge states. The strong Fermi-level pinning effect, like that in many commercially important semiconductors (e.g., Si, Ge, GaAs),20 can be very frustrating for the design of semiconductor devices (e.g., solar cells21). For example, it may largely disenable contact engineering22,23 for controlling carrier injection and extraction and carrier-types. (Contract engineering seeks to control the Fermi-level in the semiconductor by contact with a metal of a given work function.) Bending Effects on Fermi-Level Pinning. We find that the in-gap edge-state induced Fermi-level pinning in A-MoS2R can be removed or adjusted by externally applied mechanical
Figure 3. Bending effects in armchair phosphorene nanoribbons. (a) PBE-calculated band structures. (b) Isosurfaces of partial charge densities of states I, II, III, and IV as marked in panel a.
tary Figure S4) shows that flat and undoped A-PR is a direct gap semiconductor with no in-gap edge-states. The partial charge density of the top two valence bands is also delocalized over the full ribbon width. As bending curvature κ reaches a critical value, κc = ∼0.073 Å−1, these charges get depleted in the ribbon middle region and the corresponding holes thus get localized at edge regions (Supplementary Note 3). Accordingly, the conductivity of p-type A-PR can thus be turned on/off or changed significantly near κc by bending. The conductivity of n-type A-PR can also be tuned by bending. Figure 3a indicates that bending induces empty in-gap 2445
DOI: 10.1021/acs.nanolett.5b05303 Nano Lett. 2016, 16, 2444−2449
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Nano Letters states in A-PR when κ > ∼0.098 Å−1 (or R < ∼10.2 Å). The term “in-gap” is defined here with respect to the band gap of the flat ribbon. These in-gap states are unoccupied and localized only within the top region of A-PR’s outer layer, differing from those originating from the defects or the edges with dangling bonds.24,25 They behave like deep acceptor defect states and are immobile along the ribbon width direction. Consequently, once these in-gap states appear, the conductivity from one edge to the other is also turned off or significantly reduced. Along the ribbon axial direction (zigzag) of n-type APR (Supplementary Figure S4), the in-gap states have heavier electron effective mass than higher-energy conduction bands. Therefore, the electron mobility is also expected to have a sudden decrease along the ribbon axial direction when these ingap states appear. Features of Emerging New In-Gap States. The appearance of such in-gap states is also indicative of a directindirect gap transition in A-PR. In fact, such a direct−indirect gap transition appears before the in-gap states completely separate out from other bands. This direct−indirect transition thus differs from that triggered by uniform uniaxial or biaxial strains.14 We further find that the direct optical transitions from top valence bands to these in-gap state bands are forbidden. The calculated dipolar matrix elements for the transition between them are essentially zero at any k-points along the Γ− X line. So these in-gap states do not contribute to optical absorption across the band gap. As can be seen from Supplementary Figure S4i, the optical absorption thus takes off at a threshold energy determined by the energy gap between the valence band maximum and the conduction band minimum above the in-gap states, which is insensitive to bending. Bending Effects on Electronic Structure of Graphene Nanoribbons. Different from what we find in MoS2 and phosphorene nanoribbons, we find no sizable bending effects in the electronic properties of graphene (Supplementary Note 4). Without bending, both armchair and zigzag graphene nanoribbons are direct-gap semiconductors.26 As the bending curvature increases, their band gap changes very little ( ∼0.11 Å−1 (Supplementary Figure S6).
in the local potential profile of a nanoribbon. The different charge-localization regions in bent A-MoS2R (near the ribbon center) and bent A-PR (near the ribbon edge) here may be attributed to the different nonuniform local potentials and different top-valence band characters in these two materials. In graphene nanoribbons, the bending-induced local in-plane strains are also found to be nonuniform, but less than 0.3% (Figure 4a). Such in-plane strains are too small to hybridize σ and π orbitals and to induce sizable change in the electronic properties of graphene. Realizability of Bending. Bending curvature in a singlelayer 2D material can be easily induced during material growth, manifesting as the formation of nanotubes, scrolls, ripples, and bubbles.29−31 It can also be intentionally introduced by many methods such as using lattice-mismatched substrate, defects and doping, and mechanical bending via atomic force microscope and electron beam.32 Indeed, subnanometer-diameter singlewalled carbon nanotubes33 and MoS2 nanotubes34 have already been synthesized. Their corresponding curvatures are both much larger than those shown in Figures 2 and 3. Figure 4b shows our first-principles calculated areal bending energy density, i.e., the total energy difference between the bent and flat ribbons divided by the area of the flat ribbon. We find that the bending energies of graphene nanoribbon are smaller than weak van der Waals interlayer binding energy of bulk graphite (18.32 meV Å−2).35 For 1H-MoS2 ribbons, the bending energies are also smaller than the van der Waals binding energy of bulk MoS2 (20.53 meV/Å−2)36 when R > ∼13 Å. Such a small bending energy explains why bending of 2D materials can be easily induced during growth or applied after growth. Nonlinear Elastic Behavior and Bending Stiffness. From classical linear elastic theory, the bending energy of a pure bent ribbon is quadratic in the bending curvature.37 Here we find that our calculated areal bending energy densities shown in Figure 4b fit well to a third-order polynomial function of curvature, indicating nonlinear elastic behavior. The bending 2446
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Concluding Remarks. From first-principles calculations, we find that mechanical bending can change the Fermi-energy level and charge localization in A-MoS2R and A-PR. In ZMoS2R and Z-PR, we find a similar bending effect on the Fermi-energy level shift (Supplementary Figures S3 and S5), but no bending-induced charge localization. The latter is not surprising since zigzag and armchair ribbons of phosphorene and MoS2 have different electronic structures.13,45 In addition to Fermi-energy level shift and charge localization, mechanical bending also introduces sizable effects on the band gap, band edge, and effective mass in both ziagzag and armchair nanoribbons of MoS2 (Supplementary Figures S2 and S3) and phosphorene (Supplementary Figures S4 and S5). Notably, we find that small bending (κ < 0.091 Å−1) can enhance the photocatalytic capability of MoS2 for water splitting. The large bending (κ > 0.091 Å−1) in MoS2 nanoribbons can disenable water reduction (H+/H2) but enhance water oxidation (H2O/ O2) (see Supplementary Note 2). The results open a door to take advantage of the unique flexural bending attribute of 2D materials to control the physical properties of interest and to design new functional 2D materials by bending. Mechanical bending offers a new tool to control conductivity and Fermilevel shift, complementary to the conventional doping method and contact engineering technique. Finally, note that all our calculations are based on DFTGGA, which underestimates the band gap.46 However, we expect that bending effects on the charge-localization of top valence bands in both phosphorene and MoS2 will be welldescribed by DFT-GGA, which yields accurate ground-state electron densities in solids. The more accurate GW study of the quasiparticle band gap of phosphorene47 shows that manyelectron self-energy mainly enlarges the band gap without varying the energy spacing between conduction bands significantly. Hence, our GGA-predicted unoccupied in-gap states in bent A-PR should also be valid, though their energy levels with respect to the valence bands are shifted. Methods. All calculations were performed using density functional theory (DFT) and the plane-wave projector augmented-wave (PAW)48 method as implemented in the VASP code.49 An energy cutoff of 400 eV was used for all three 2D materials. All considered nanoribbons were passivated with hydrogen atoms on both edge sides (Supplementary Figure S1). In the calculations, periodic boundary conditions were used in all directions and a 15 Å vacuum space was inserted in both the ribbon-width direction and the out-of-plane direction. Along the ribbon axial direction, the smallest bulk unit period was used. An 8 × 1 × 1 or 1 × 8 × 1 k-point grid was used to sample the Brillouin zone during structure relaxation. For the flat nanoribbon, we relaxed all atomic coordinates. A bent nanoribbon structure was obtained by fixing the distance between the two edges of a ribbon (Figure 1) during structure relaxation. Specifically, this was achieved by fixing the outermost edge atoms (i.e., carbon in graphene, phosphorus in phosphorene, and Mo atom in MoS2) in the ribbon-width direction and the out-of-plane direction during structure relaxation. Along the ribbon edge (or axial) direction, these edge atoms were relaxed. All other atoms were fully relaxed in all directions until their atomic forces were less than 0.01 eV/Å. The lattice parameter along the ribbon axial direction was also fully relaxed.
stiffness, which corresponds to the coefficient of the quadratic term,38,39 can thus be directly calculated using data shown in Figure 4b. As expected, our calculated bending stiffness (Table 1) follows the same trend of bending energies: MoS2 > A-PR > graphene > Z-PR at a given bending curvature. Table 1. Mechanical Properties of MoS2, Phosphorene and Graphene; the Numbers in Parentheses Are Experimental Values A/Z-GR bending stiffness (NÅ) 2D in-plane stiffness (Nm−1) 3D in-plane stiffness (TPa) effective thickness (Å) interlayer distance (Å)
A-PR
Z-PR
A/Z-MoS2R
24.81
102.81
24.02
157.59
344.2 ± 0.5 (340)41 3.70
91.6
21.4
0.25
0.06
123.7 ± 0.6 (180 ± 60)42 0.32
0.93
3.67 ± 0.16
3.91 ± 0.5
2.10
3.13
Effective Thickness of 2D Materials. The effective thickness (t) of a rectangular ribbon can be derived to be40 t=
12Sb/Y2D
(1)
where Sb is bending stiffness and Y2D is in-plane stiffness (in units of force/length) that can also be directly derived from first-principles total energy calculations. Table 1 shows that our calculated Y2D values for graphene and MoS2 agree well with experiment.41,42 Our calculated effective thickness of graphene is 0.93 Å, which is larger than that calculated from empirical potential methods. For phosphorene and 1H-MoS2, the effective thicknesses are found to be 3.67 and 3.91 Å, respectively, which as expected are larger than the distance between the upper and bottom atomic layers (2.10 Å in phosphorene and 3.11 Å in MoS2). Using this t, one can then define an intrinsic Young’s modulus of 2D materials via Y3D = Y2D/t (Table 1), which permits us to compare the stiffness against uniaxial stress of various 2D and 3D materials, e.g., graphene versus steel. Distinct Features of Our Bending Scheme. Our study of bending effects is based on a defect-free nanoribbon bent into different curvatures. Our bending scheme is different from that using different sizes of nanotubes43 or nonplanar monolayers with defects.44 The nanoribbons unfolded from different nanotubes have different widths and hence different band gaps due to quantum confinement effect, which is disentangled from the curvature effect in our scheme. The remarkable bending effects predicted from our scheme are mainly on the conductivity behavior in a direction from one edge to the other. Such edges do not exist in nanotubes along the circumferential direction. In addition, the in-plane strains are uniform in nanotubes, while the in-plane strains in our bent ribbons are highly nonuniform. Such highly nonuniform strains may lead to strong flexoelectric effects, which are implicitly included in our density functional calculations and can be extracted explicitly. We also expect that bending effects in a nanoribbon are similar to those in a different-width nanoribbon. Indeed, as shown in Figure 4b, the same trend in mechanical properties is found in A-GR (d0 = 2.83 nm) and Z-GR (d0 = 3.27 nm), and in AMoS2R (d0 = 3.64 nm) and Z-MoS2 (d0 = 3.57 nm), although their widths d0 are different. 2447
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b05303. Crystal structures of nanoribbons; detailed bending effects in A-MoS2R, Z-MoS2R, AP-R, Z-PR, A-GR, and Z-GR; areal bending energy density of A-PR; bending effects on the photocatalytic property of 1H-MoS2; notes on bending effects in graphene nanoribbons and edge− edge interaction (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Author Contributions
L.Y. designed the project, performed the calculations, and wrote the first draft. A.R. and J.P. provided support and discussions and wrote the final draft. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported as part of the Center for the Computational Design of Functional Layered Materials (CCDM), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award #DE-SC0012575. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. This research was also supported in part by the National Science Foundation through major research instrumentation grant number CNS-09-58854. The authors thank Matthaeus Wolak, Laszlo Frazer, Eric Borguet, Michael Zdilla, Goran Karapetrov, Daniel Strongin, Maria Iavarone, and other CCDM members for helpful discussions.
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