J Mater Sci (2014) 49:6762–6771 DOI 10.1007/s10853-014-8370-5
Theoretical study on strain-induced variations in electronic properties of monolayer MoS2 Liang Dong • Raju R. Namburu • Terrance P. O’Regan Madan Dubey • Avinash M. Dongare
•
Received: 9 February 2014 / Accepted: 28 May 2014 / Published online: 20 June 2014 Ó Springer Science+Business Media New York 2014
Abstract Ultrathin MoS2 sheets and nanostructures are promising materials for electronic and optoelectronic devices as well as chemical catalysts. To expand their potential in applications, a fundamental understanding is needed of the electronic structure and carrier mobility as a function of strain. In this paper, the effect of strain on electronic properties of monolayer MoS2 is investigated using ab initio simulations based on density functional theory. Our calculations are performed in both infinitely large two-dimensional (2D) sheets and one-dimensional (1D) nanoribbons which are theoretically cut from the sheets with semiconducting ½1100 (armchair) edges. The 2D crystal is studied under biaxial strain, uniaxial strain, and uniaxial stress conditions, while the 1D nanoribbon is studied under a uniaxial stress condition. Our results suggest that the electronic bandgap of the 2D sheet experiences a direct-indirect transition under both tensile and compressive strains. Its bandgap energy (Eg) decreases under tensile strain/stress conditions, while for an in-plane compression, Eg is initially raised by a small amount and then decreased as the strain varies from 0 to -6 %. On the other hand, Eg at the semiconducting edges of monolayer L. Dong A. M. Dongare (&) Department of Materials Science and Engineering and Institute of Materials Science, University of Connecticut, Storrs, CT 06269, USA e-mail:
[email protected] R. R. Namburu Computational and Information Sciences Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, USA T. P. O’Regan M. Dubey Sensors and Electron Devices Directorate, U.S. Army Research Laboratory, Adelphi, MD 20783, USA
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MoS2 nanoribbons is relatively invariant under uniaxial stretches or compressions. The effective masses of electrons at the conduction band minimum (CBM) and holes at the valence band maximum (VBM) are generally decreased as the in-plane extensions or compressions become stronger, but abrupt changes occur when CBM or VBM shifts between different k-points in the first Brillouin zone.
Introduction Since the discovery of graphene [1], there has been rapid growth of research on ultrathin two-dimensional (2D) materials [2]. 2D sheets and nanoflakes have been successfully isolated from crystals with layered structures such as graphene, BN, MoS2, and V2O5 [2, 3]. Charge carriers of these 2D materials are strongly confined within the atomic layers when their thicknesses are reduced to nm-scale or below. As a result of such a quantum confinement effect, 2D materials demonstrate novel mechanical, electronic, transport, and chemical properties that are substantially different from their respective bulk format [3]. Among the aforementioned 2D materials, monolayer MoS2 is a direct bandgap semiconductor that is of particular interest for electronic, optoelectronic, and catalytic applications. The finite intrinsic bandgap energy (Eg = 1.9 eV) under ambient conditions [4] makes it suitable to replace graphene, which is naturally a semimetal [1], in transparent and flexible electronic devices. For example, field effect transistors (FETs) fabricated with a monolayer MoS2 nanosheet as the conductive channel have already been demonstrated to have a high carrier mobility [200 cm2/(V s)] and a large room-temperature current on/ off ratio (108) [5]. Monolayer MoS2 also holds a great potential for light emission diodes (LEDs), solar cells, and
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photo detectors, Et Cetera, due to its direct band structure [6]. Electrically excited luminescence at around 1.9 eV and photo-induced electrical current have been realized using FETs based on monolayer MoS2 [7, 8]. The edges of monolayer MoS2 nanoflakes are chemically active sites that have been considered as a high-efficiency and low-cost catalyst for the hydrogen evolution reaction (which generates hydrogen during water splitting) [9] and the hydrodesulfurization process (which extracts sulfur from fossil fuels) [10]. The promising applications of 2D materials have triggered research that aims at understanding as well as modifying the physical and chemical properties under diverse conditions and requirements. To achieve this goal, a convenient way is through inducing internal strain (e.g., lattice mismatch strain) and external stress (e.g., hydrostatic pressure) that may tailor the crystal structure and electronic structure of materials [11, 12]. For example, strain engineering has been applied to graphene through mechanical deformation of flexible substrates [13], patterning of the substrate morphology [14], utilizing piezoelectronic actuators as a substrate [15], and thermal cycling of graphene on substrates with mismatched thermal expansion coefficients [16]. Given the structural similarity between graphene and monolayer MoS2, these substrate treatments have led to the investigations of strain effect in monolayer MoS2 as well. For instance, photoluminescence (PL) and absorption spectroscopy studies suggest a significant red shift at tensile uniaxial strains up to 2 % [17, 18]. Furthermore, the PL intensity decreases significantly at a uniaxial strain of 1 %, indicating a crossover from a direct bandgap electronic structure to an indirect one [18]. These experiments evidently show that strain engineering is an effective tool to tune Eg of monolayer MoS2, but they are confined to relatively small uniaxial strains (B2 %). Earlier measurements of the mechanical properties of single-layer MoS2 suggest that much larger tensile strains up to 11 % can still be maintained in this material without fracture [19]. Therefore, in order to fully explore the potential of monolayer MoS2, it is necessary to investigate the effect of various types of strains/stresses in wider variation ranges on the electronic and optoelectronic properties. Parallel to experiments, the effect of strain in monolayer MoS2 has been addressed by first-principles simulations based on density functional theory (DFT) [20–25]. These studies agree with the aforementioned experiments that Eg of monolayer MoS2 decreases with an increasing uniaxial tensile strain [20, 21]. It has also been found that the effect of a biaxial tensile strain is even stronger than that of a uniaxial one, and a semiconductor-to-metal transition is induced when the strain is around 10 % [20, 22]. These studies have been extended to a wide compressive strain region, however, only for biaxial strains [22, 23]. In this
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Fig. 1 (Color online) a Top view and b side view of the crystal structure of monolayer MoS2
paper, the effect of strain on monolayer MoS2 sheets is studied using DFT computations in both tensile and compressive strain domains under three types of strain and stress conditions: a biaxial strain, a uniaxial strain, and a uniaxial stress. Furthermore, we notice that previous studies are limited to infinitely large 2D sheets, and there is not much discussion about the effect of strain in MoS2 nanostructures, such as one-dimensional (1D) nanoribbons. Such a topic is also addressed in this study by applying a uniaxial strain to a nanoribbon with semiconducting ½1100 (armchair) edges. The results show that Eg and carrier effective masses of the infinitely large 2D sheets of monolayer MoS2 can be effectively tuned with the application of strains or stresses. In the 1D nanoribbon, Eg of the central part displays a similar variation trend as that of an infinitely large monolayer under a uniaxial strain. Eg of the edges of the nanoribbon, however, is substantially smaller than that of the 2D crystal and is relatively insensitive to uniaxial stretches or compressions. This study demonstrates the possibility of strain engineering in devices based on monolayer MoS2 constructs to acquire desired electronic, optoelectronic, and electrical transport properties.
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yielding two stoichiometric ½1100(armchair) edges (Fig. 2). In such a 1D structure, only uniaxial strain/stress along the y direction can be induced in a way that we consider a fourth MBC: (4)
Uniaxial stress condition with ex = 0, ey = 0, rx = 0, and ry = 0.
Here, ey is given by Eq. (2), while ex is determined by ex ¼ ð~ a a0 Þ=a0 ;
Fig. 2 (Color online) Top view of the relaxed 1D nanoribbon structure of monolayer MoS2 with armchair edges along the y direction. The width of this ribbon is around 2.5 nm, corresponding to 8 periods of unit cells along the x direction
Crystal structure and methodology The structure of a monolayer MoS2 crystal is shown in Fig. 1. This structure is characterized by the in-plane lattice pffiffiffi parameters a0 and b0 (b0 ¼ 3a0 ) and the Mo-S bond length d0 along z direction (Fig. 1). In the presence of strain/stress conditions, this crystal structure is subject to new in-plane lattice parameters a and b and a corresponding out-of-plane lattice parameter d. The strains along the x and y directions (ex and ey) are given by ex ¼ ða a0 Þ=a0 ;
ð1Þ
and ey ¼ ðb b0 Þ=b0 :
ð2Þ
In our study, the effect of strain/stress in an infinitely large sheet of monolayer MoS2 is investigated under three in-plane mechanical boundary conditions (MBCs): (1)
(2) (3)
Biaxial strain condition with ex = ey = 0 and rx = ry = 0, where rx and ry are the stresses along the x and y directions in Fig. 1, respectively; Uniaxial strain condition with ex = 0, ey = 0, rx = 0, and ry = 0; Uniaxial stress condition with ex = 0, ey = 0, rx = 0, and ry = 0 (ey is the commensurate stressfree strain for a given magnitude of ex, and our results show that ey/ex & - 0.22).
Under these three MBCs, ex is taken as the independent parameter of the calculations and varied from -6 % (compressive) to 6 % (tensile). The nanoribbon of monolayer MoS2 is constructed by cutting the structure of a 2D sheet along the y direction,
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ð3Þ
where a~ is the Mo–Mo distance along the x direction at the center of the nanoribbon (see Fig. 2). First-principles DFT calculations are carried out using projector augmented wave pseudo-potentials [26] and plane-wave expansions with a cutoff energy of 500 eV as implemented in the VASP code [27]. The electronic configurations of Mo and S are 4p65s14d5 and 3s23p4, respectively. To prevent the unphysical interactions between periodic images of the ultrathin 2D sheet, a vac˚ thick is used along the z direction in uum space of 20 A Fig. 1. A Monkhorst-Pack k-point mesh of 15 9 15 9 1 in the first Brillouin zone is found to yield well-converged results. For the 1D nanoribbon structure (Fig. 2), an additional vacuum space of the same size is placed along the x direction, and the k-point mesh is reduced to 1 9 9 9 1. The atomic positions are relaxed until all components of ˚. the forces on each atom are reduced below 0.01 eV/A The exchange–correlation functional is treated within the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximations (GGA) [28]. We note that GGA generally underestimates the absolute bandgap energy of semiconductors [29], and modern time-intensive hybrid functionals (e.g., HSE06 [30]) and the quasiparticle GW approximation [31] usually lead to better agreements with the experimental results [32]. For an MoS2 monolayer crystal, however, PBE yields Eg of the relaxed structure (1.76 eV) that is close to the experimental value (1.9 eV) [4], whereas previous calculations using HSE06 or GW overestimate Eg by 0.4–0.85 eV [25, 33, 34]. Despite the deviations in the absolute bandgap energy, relative variations of the electronic properties using different levels of theory are actually in mutual agreement with each other. Due to this reason, the relatively simpler PBE is employed to determine the electronic structure of 2D sheets and 1D nanoribbons of monolayer MoS2.
Strain engineering in 2D sheets In our study, the calculated lattice parameters a0 and d0 (Fig. 1) in the strain-free crystal of monolayer MoS2 are ˚ and 1.566 A ˚ , respectively. These results are con3.158 A sistent with the experimental values for both monolayer
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Fig. 4 (Color online) Variations in lattice parameter d of monolayer MoS2 crystal under the three mechanical boundary conditions
Fig. 3 (Color online) a Band structure and b density of states of the strain-free monolayer MoS2 crystal. In both figures, the Fermi energy is set to be 0 eV
˚ ) [35] and bulk (a0 = 3.160 A ˚ and (a0 = 3.2 A ˚ c0 = 1.586 A) [36] crystals. The electronic band structure of this material features a direct bandgap at the K point in the reciprocal lattice (Fig. 3a). Analysis of density of states (DOS) (Fig. 3b) shows that the valence band (VB) and the conduction band (CB) are primarily occupied by hybridized orbitals of Mo - 4d and S - 3p electrons. In particular, valence band maximum (VBM) has contributions from Mo-4dx2 y2 , Mo - 4dxy, S - 3px, and S - 3py orbitals, while conduction band minimum (CBM) has contributions from Mo-4dz2 , S - 3px, and S - 3py orbitals. Magnitudes of Eg calculated from the band diagram (1.76 eV) and DOS (1.79 eV) are consistent with each other and close to the experimental value (1.90 eV) [4]. Under the three in-plane strain/stress conditions (MBCs 1, 2, and 3), the out-of-plane lattice parameter (d) decreases almost linearly as ex increases from -6 to 6 % (Fig. 4) to minimize the change in the overall Mo-S bonding length from the equilibrium value. The slope of Dd/De under ˚ / %) is nearly twice as large as MBC 1 (-1.08 9 10-2 A ˚ / % and that under MBC 2 and MBC 3 (-5.4 9 10-3 A ˚ / %, respectively). Such changes of inter-4.2 9 10-3 A atomic distances have a strong bearing on the electronic band structure in semiconductor materials. In Fig. 5, we
demonstrate the variations in band structure of monolayer MoS2 crystal in the presence of four strain conditions, namely ex = 1, 5, -1, and -5 % under MBC 1. Similar variations in the band diagram occur under the uniaxial strain and stress conditions (MBCs 2 and 3) as well (not shown). Comparing to the equilibrium condition (Fig. 3a), the profiles of the VB and CB in Fig. 5 are greatly changed. For the VB, the local energy maximum at the C point is raised with respect to the VBM at the K point by a tensile biaxial strain (Fig. 5a). For ex = 1 %, the VB energy at the C point almost equals that at the K point; while for ex = 5 % (Fig. 5b), the former is much higher than the latter, indicating a transition of the VBM from the K point to the C point. For a compressive strain (Fig. 5c, d), however, the VBM remains at the K point because of the relative lowering of the VB energy at the C point. On the other hand, the CBM remains at the K point for a tensile biaxial strain (Fig 5a ,b) but experiences a transition at around ex = - 1 % from the K point to the Rmin point, which is nearly halfway between the K and C points (Fig. 5c). We note that a recent DFT study predicts a direct K–K bandgap structure even for a strong biaxial compression at ex = - 8 % [37]. Their band structure, however, seems problematic because the CB profile for strainfree monolayer MoS2 crystal does not contain a local energy minimum at the Rmin point. On the contrary, our study agrees with all other DFT studies using different levels of theory (GGA, HSE06, and GW) [20, 22, 23, 25, 33, 34, 38] that this energy minimum does exist in monolayer MoS2 (Fig. 3a). From Fig. 5, it is clear that there are two local energy maximums on the VB (K and C) and two local energy minimums on the CB (K and Rmin). Therefore, we calculate the energy differences between the four points (i.e., C–K, C–Rmin, K–Rmin, and K–K) to obtain a comprehensive
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Fig. 5 (Color online) Band diagram of monolayer MoS2 crystal for biaxial strain at a ex = 1 %, b ex = 5 %, c ex = - 1 %, and d ex = - 5 %. The arrow in each diagram indicates the direction of the electron transition from valence band maximum to conduction band minimum
understanding of the variation in bandgap energy of a monolayer MoS2 crystal (Fig. 6). Three general trends can be observed from Fig. 6 under the three mechanical conditions considered for the 2D crystal. First, the K–Rmin energy gap increases, while the other three energy gaps decrease from ex = - 6 to ex = 6 %. In the tensile strain domain (ex C 0), the K–K and C–K energy gaps, which correspond to the A and I peaks in PL and absorption spectroscopies, decrease linearly as a function of ex. The slopes of A and I energy gaps with respect to ex in Fig. 5 are listed in Table 1 and are found to agree well with the available experimental and theoretical results [17, 18, 20, 22–25]. Second, comparing the slopes of A and I energy gaps among the three MBCs as listed in Table 1, it is obvious that the effect of biaxial strain is much stronger than that of uniaxial strain/stress. This is correlated to the fact that the Mo-S bonding distance changes much more under the biaxial strain condition than the uniaxial strain/ stress conditions (Fig. 4). Third, MoS2 is a direct bandgap semiconductor only within a narrow domain of strain. The intrinsic (optical) bandgap of monolayer MoS2 is transferred from the direct type (K–K) in the relaxed structure to an indirect one (C–K or K–Rmin) in the presence of tensile or compressive strain/stress conditions. The transitions occur at ex & ± 1, ± 1.5, and ± 2 % under MBCs 1, 2, and 3, respectively. The direct–indirect crossover for a tensile uniaxial strain at ex & 1.5 % (Fig. 6b) is close to the experimental result which shows a sudden decrease in
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PL intensity as the uniaxial strain goes beyond 1 % [18]. Due to such transitions, Eg of monolayer MoS2 crystal changes at segmented slopes in the domain of strain/stress considered in our study. Taking the biaxial strain condition as an example (Fig. 6a), Eg increases from 1.53 eV to 1.86 eV for ex from -6 to -1 % (69.2 meV/ %) and then decreases from 1.86 to 1.59 eV for ex from -1 to 1 % (-116.3 meV/ %) and from 1.59 to 0.58 eV for ex from 1 to 6 % (-93.2 meV/ %). The effect of strain/stress on the band structure of monolayer MoS2 crystal is not limited to the bandgap energy. It also modifies the curvature of the profiles of the CB and VB, which ultimately determine the effective mass of electrons (me) and holes (mh), respectively. The carrier effective masses are defined as 1 me ¼ h2 oEC2 =ok2
ð4Þ
and 1 mh ¼ h2 oEV2 =o2 k ;
ð5Þ
where h = 1.05 9 10-34 kg m2/s, EC (EV) is the conduction (valence) band energy, and k is the wavevector in the reciprocal lattice. It is well known that electrons and holes accumulate primarily around the CBM and VBM, respectively. Therefore, the effective masses at the CBM and VBM are more important than those at other points of the band diagram. Since there are two candidate locations for
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Fig. 6 (Color online) Variations of C–K, C–Rmin, K–Rmin, and K–K bandgaps with a a biaxial strain, b a uniaxial strain, and c a uniaxial stress. The arrows in each figure indicate the approximate positions where the intrinsic bandgap crosses over between the indirect and direct types
Table 1 Slopes of the C–K(I) and K–K (A) energy gaps with respect to ex in the domain of tensile strain/stress conditions (Units: meV/ %) MBC
This work
Experiments
Other DFT works
1 I
209.3
195a, 203b, 174c, 183d, 232e
A
93.2
86b, 94.5e
I
115.4
2 A
57.0
105a f
64 , 45
g
59
g
3
a
I
89.8
A
48.1
compressive biaxial strains. From ex = - 1 % (Fig. 5c) to ex = - 5 % (Fig. 5d), the rising of CB energy at the K point gradually diminishes the parabolic curvature surrounding this point, resulting in a significant increase in me (K) according to Eq. (4). From ex = - 5 to ex = - 6 %, this trend is reversed so that me (K) is reduced. From Fig. 7, one can conclude that a tensile strain/stress tends to reduce me at the CBM (the K point) but introduces an abrupt increase in mh when the VBM is transferred from the K point to the C point. On the other hand, a compressive strain/stress tends to reduce mh at the VBM (the K point), but introduces an abrupt increase in me when the CBM is transferred from the K point to the Rmin point.
Ref. [20]
b
Ref. [22]
c
Ref. [23]
d
Ref. [24]
e
Ref. [25]
f
Ref. [17]
g
Ref. [18]
both the CBM (K and Rmin) and VBM (C and K) under different strain/stress levels (Fig. 6), me at the K and Rmin points [me (K) and me (Rmin)] as well as mh at the C and K points [mh (C) and mh (K)] are calculated along the K–C direction. As listed in Table II, the obtained effective masses of the strain-free monolayer MoS2 agree well with the previous DFT simulations using PBE, local density approximations (LDA), GW, and HSE06 [22, 23, 34, 38]. The strain dependences of me (K), me (Rmin), mh (C), and mh (K) are shown in Fig. 7 for MBCs 1, 2, and 3. As ex varies from -6 to 6 %, me (Rmin) and mh (K) generally increase, whereas me (K) and mh (C) decrease. An exception here is me (K) peaks at ex = - 5 % under MBC 1 (upper panel of Fig. 7a). This peak is associated with the relative variations in CB profile at the K point under
Strain engineering in 1D nanoribbons The effect of strains on the electronic properties of 1D nanoribbons of monolayer MoS2 is described here. The nanoribbon is theoretically cut from an infinitely large 2D sheet along the armchair directions. As seen in Fig. 2, in a strain-free ribbon, the Mo atoms at the two edges move substantially toward the inner side. Such a structural relaxation agrees well with previous DFT simulations [39, 40]. The unpaired dangling bonds of the edge atoms introduce large amounts of surface states that are located within the bandgap energy region but do not penetrate the Fermi level (Fig. 8a); therefore, the entire ribbon is still semiconducting. As the width of this 1D construct increases along the x direction in Fig. 2, its Eg is initially oscillating but finally converges to 0.56 eV after the width goes ˚ (corresponding to 8 periods of units along the beyond 25 A x direction) [39]. This behavior is confirmed by the calculations here that suggest a bandgap of 0.57 eV for the edge atoms (Fig. 8a). At this width, however, the surface states still appear within the local bandgap of central
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Fig. 7 (Color online) Variations of electron effective mass (me, upper panel) and hole effective mass (mh, lower panel) with a a biaxial strain, b a uniaxial strain, and c a uniaxial stress. The vertical dashed line marks the transition of CBM from the Rmin point to the K point in upper panel of a, b, and c, while in lower panel, c it marks the transition of VBM from the K point to the C point
Fig. 8 (Color online) Partial density of states of Mo and S atoms at a the edge and b the center of the semiconducting 1D monolayer MoS2 nanoribbon with armchair edges. The Fermi level of this construct is set to be 0 eV
atoms, meaning that this region is affected by the surface reconstruction as well. To recover Eg of a 2D monolayer MoS2 crystal at the center of this 1D ribbon, we repeat the computations with an increasing nanoribbon width along the x direction. The surface states are observed to be completely eliminated from the bandgap of the central ˚ (Fig. 8b, corresponding to 19 atoms at a width of 60 A periods of units along the x direction). Eg at the center (1.79 eV) is the same as the value that is obtained from DOS calculation of the 2D crystal (Fig. 3b). Orbital˚ wide nanodecomposed DOS is performed on the 60 A ribbon to further distinguish between the local band
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structure at the center and that at the two edges. It is found that in the central region, the VBM has contributions from Mo-4dx2 y2 , Mo - 4dxy, S - 3px, and S - 3py orbitals, and the CBM has contributions from Mo-4dz2 , S - 3px, and S - 3py orbitals, similar to the 2D monolayer MoS2 sheet. For comparison, at the edges, the dangling bonds induced VBM has contributions from Mo-4dx2 y2 , Mo - 4dxy, Mo-4dz2 , and S - 3pz orbitals, and the dangling bonds induced CBM has contributions from Mo-4dx2 y2 , Mo - 4dxy, Mo-4dz2 , S - 3px, and S - 3py orbitals. Upon the application of a uniaxial strain ey along the y direction in Fig. 2, the width of the 1D nanoribbon changes commensurately to relax the stress in the x direction, resulting in a uniaxial stress condition (MBC 4). Under such a condition, the stress-free strain ex can be estimated by Eq. (3). For very small uniaxial stretching or extension (-1 B ey B 1 %), ex changes little from 0 %, whereas beyond the two thresholds at ey = ± 1 %, ex decreases rapidly as ey increases from -6 to -2 % and from 2 to 6 % (Fig. 9a). This behavior is significantly different from the uniaxial stress condition for 2D monolayer MoS2 sheet (MBC 3), wherein ex and ey are linearly related. Under MBC 4, the bandgaps of the edges and the center of the nanoribbon are both varied as a response to the changes in the interatomic distances (Fig. 9b). In the central region, Eg shows segmented slopes versus the strain with two turning points at ey = - 2 and ey = 1 % (Fig. 9b). For -6 \ ey \ - 2 %, -2 \ ey \ 1 %, and 1 \ ey \ 6 %, Eg of the central atoms varies at a slope of 30, -49, and -84 meV/ %, respectively. This trend is similar to the case
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6769 Table 2 Electron effective mass (me) at the K and Rmin points and hole effective mass (mh) at the C and K points along the K–C direction of the first Brillouin zone of strain-free monolayer MoS2 [Units: free electron mass (m0)] This Work
PBEa
LDAb
GWc
HSE06d
0.35
0.37
me K
0.458
0.483
0.48
Rmin
0.605
0.569
0.58
0.57
mh
a
Fig. 9 (Color online) Variations in a ex and b Eg (center and edges) of the semiconducting 1D monolayer MoS2 nanoribbon with armchair edges as a function of ey. The arrows in b indicate the positions where the slope of Eg at the central region changes
of a 2D monolayer MoS2 crystal under MBC 3, implying that atoms at the center of the 1D nanoribbon behave as in an infinitely large 2D sheet. Therefore, it can also be inferred that the two turning points at ey = - 2 and ey = 1 % in Eg of the central atoms are related to direct–indirect bandgap crossovers. At the edges, Eg is relatively insensitive to the inplane mechanical deformations regardless of the degree of strain (Fig. 9b). It is raised from 0.57 eV in a strain-free nanoribbon to 0.59 eV for a relatively moderate tensile strain at ey = 3 %. For stronger elongations along the y direction, Eg of the edge atoms starts to decrease slowly and reaches 0.5 eV for ey = 6 %. On the other hand, for ey from 0 to -6 %, Eg of the edge atoms is monotonically reduced with respect to the strain-free value at a small slope of approximately 18 meV/ %. In general, strain effect in the nanoribbon is more obvious in the central part than at the edges. This is understandable because the atoms at the edges maintain the strain-free interatomic distances though atomic relaxations more easily than those at the center.
Discussion Bandgap energy and carrier effective masses are important parameters that determine the performance of electronic and optoelectronic devices based on semiconductor
C
3.774
3.524
3.55
3.108
2.80
K
0.547
0.637
0.62
0.428
0.44
Ref. [23]
b
Ref. [22]
c
Ref. [34]
d
Ref. [38]
materials. Therefore, our results regarding strain effect on Eg, me, and mh in monolayer MoS2 sheets and nanoribbons have several implications to the development of devices based on this material. For infinitely large 2D sheets of monolayer MoS2, the direct K–K bandgap structure can only be reserved within a narrow strain/stress domain (-2 % \ ex \ 2 %) around the strain-free region. Out of this region, Eg becomes indirect (C–K for tensile strain/ stress and K–Rmin for compressive strain/stress). In the tensile strain domain, our simulations are consistent with recent PL and absorption measurements [17, 18], which show that Eg (the direct K–K or I peak) of this material is significantly decreased by a uniaxial in-plane extension. Furthermore, the biaxial strain, uniaxial strain, and uniaxial stress reduce the indirect C–K bandgap of monolayer MoS2 crystal at the rate of 209.3, 115.4, and 89.8 meV/ %, respectively (Table 2). Provided that the strain-free C–K bandgap value is 1.832 eV, a semiconductor–metal transition is expected at ex & 8.8, 15.9, and 20.4 % under the three MBCs, respectively. While the latter two values (15.9 and 20.4 %) may break a monolayer MoS2 sheet, a tensile strain of 8.8 % is still sustainable for this material [19]. On the other hand, in the compressive in-plane strain/stress domain, Eg (K–K) is initially raised by * 0.1 eV from ex = 0 to ex = - 1 % under a biaxial strain condition and to ex = - 2 % under uniaxial strain and stress conditions. Beyond that, Eg is switched to the K–Rmin bandgap energy and decreases slowly for stronger in-plane compressions. A relatively stable domain exists for -3 % \ ex \ 0 under MBC 1 or for -5 % \ ex \ 0 under MBC 2 and MBC 3, wherein increases in Eg are smaller than 5.7 % (Fig. 6). Similar transitions and bandgap variations in the presence of uniaxial stress are observed in the central region of a 7-nm-wide monolayer MoS2 nanoribbon with semiconducting armchair edges, but Eg at the edges of this 1D
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nanoribbon is less affected by mechanical deformations; it is almost unchanged in the tensile stress domain for 0 \ ey \ 3 % and is slowly reduced from the strain-free value for ey \ 0 or ey [ 3 %. Therefore, the armchair edges may be favored for catalytic applications. In addition to changes in bandgap energy, the effective masses of electrons and holes are also tuned by the strain/ stress applied. In the tensile strain/stress domain, me at the K point (CBM) decreases with an increasing strain, but mh increases abruptly at ex = 1 or 2 % as the VBM shifts from the K point to the C point and then decreases for stronger extensions. Conversely, if the strain/stress is compressive, me increases abruptly at ex = - 1 or -2 % as the CBM shifts from the K point to the Rmin point and then decreases for larger compressions, while mh at the K point (VBM) is gradually reduced. In general, changes are moderate (\ 20 %) under compressive strain/stress conditions for -6 % \ ex \ 0 (Fig. 7). Therefore, a substrate with slightly smaller in-plane lattice parameter may be favored to achieve relatively stable transport properties in electronic transistors based on monolayer MoS2.
Conclusions First-principles DFT calculations are carried out to investigate the effect of mechanical strains on the electronic structure of monolayer MoS2 in the formats of 2D sheets and 1D nanoribbons. The results indicate that the bandgap energy and carrier effective masses of monolayer MoS2 can be effectively tuned by in-plane biaxial or uniaxial strain/ stress conditions. For an infinitely large MoS2 sheet, both tensile and compressive strain/stress conditions introduce a transition from a direct K–K band structure to an indirect C–K or K–Rmin band structure. In the tensile strain/stress domain, Eg is monotonically decreased with an increasing value of ex. In the compressive stain/stress domain, Eg is initially increased by 0.1 eV for ex = - 1 % (biaxial strain) or -2 % (uniaxial strain/stress) and is then decreased for stronger compressions. To study the effect of strain in finite sized MoS2 structures, a nanoribbon of monolayer MoS2 is constructed with semiconducting armchair edges. The bandgap in the central region of a 7-nm-wide nanoribbon shows similar variations under uniaxial stress condition as in an infinitely large sheet. As a comparison, Eg at the edges is relatively insensitive to the mechanical deformation. In addition to the changes in bandgap energy, the effective masses of electrons and holes are also modified significantly under the mechanical boundary conditions. These results imply that strain engineering can effectively tune the electronic properties of monolayer MoS2 to satisfy various application requirements for devices based on this material.
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J Mater Sci (2014) 49:6762–6771 Acknowledgements This research was supported in part by an appointment of A. M. Dongare to the Faculty Research Participation Program at the U.S. Army Research Laboratory (USARL) administered by the Oak Ridge Institute for Science and Education through an interagency between the U.S. Department of Energy and ASARL. The authors R. R. Namburu, T. P. O’Regan, and M. Dubey acknowledge the support of the US Army Research Laboratory (ARL) Director’s Strategic Initiative (DSI) program on interfaces in stacked 2D atomic layered materials.
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