Lattice Strain Effect on the Band Offset in Single-Layer MoS2

Feb 16, 2017 - address the band offset in single-layer MoS2 modulated by the uniaxial tensile strain based on atomic-bond-relaxation consideration...
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Lattice Strain Effect on the Band Offset in Single-Layer MoS2: An Atomic-Bond-Relaxation Approach Yipeng Zhao, Zhe Zhang, and Gang Ouyang* Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China ABSTRACT: Two-dimensional molybdenum disulfide (MoS2) attracts a great deal of interest owing to its potential application in the next generation of electronic devices in recent years. However, the physical mechanism on the strain engineering for the band offset in single-layer MoS2 from the atomistic origin is still a challenge. Herein, we propose an analytical model to address the band offset in single-layer MoS2 modulated by the uniaxial tensile strain based on atomic-bond-relaxation consideration. It was found that the bandgap of single-layer MoS2 shows an approximately linearly red shift with a rate of ∼53.4 meV/% strain under uniaxial tensile strain. The underlying mechanism can be attributed to the variation of crystal potential induced by the changes of bond identities such as bond length, strength, and angle. The results were validated by comparing them with the available evidence, suggesting that the proposed model can be an effective method to clarify the modulation mechanism of relevant electronic properties in two-dimensional semiconductor nanostructures. single-layer MoS2 at ∼10% tensile or ∼15% compressive biaxial strain has been found within the framework of first-principles density functional calculations.13,16 Kang et al.17 revealed that the bandgap energy saturates to 0.45 eV when the strain along the x-direction becomes larger than 0.2. López-Suárez et al.19 found a transition from semiconductor to metal at 0.13 compression strain for monolayer MoS2 and 0.10 for the bulk. Very recently, Nayeri et al.20 presented a atomistic tight binding approach on the optical properties of the monolayer MoS2 and found that the uniaxial tensile strain can lead to a red shift of peaks in the spectrum of dielectric function. Ouyang et al.21 demonstrated that the interface electronic coupling enhanced by in-plane strain will affect the electronic properties in IV TMD within the framework of deformation potential theory. Furthermore, mechanical stress experiments show that the single-layer MoS2 possesses very high strength and can sustain isotropic strain up to 11%,22−24 allowing exceptional control of material properties by strain engineering. Physically, as the dimension of a specimen reduces to a few nanometers, the high coordination imperfection of edge atoms as well as the quantum potential depression would play a significant effect on their performances, which can make some related properties in single-layer MoS2 differ from their corresponding bulk. Although a number of achievements on the modulation of electronic properties in single-layer MoS2 have been obtained, the relevant physical mechanism on the strain engineering for the band offset from the atomistic origin is still a challenge. The theoretical relationship for the change of

1. INTRODUCTION Two-dimensional (2D) materials, such as transition metal dichalcogenides (TMDs), attract much attention owing to their promising potential for applications in transistors, photodetectors, and optoelectronic nanodevices.1,2 MoS2, the most studied ultrathin 2D TMD nanosystem, shows a transition from an indirect gap with a bandgap of ∼1.29 eV to a direct gap semiconductor with a bandgap of ∼1.8 eV when the thickness changes from bulk to monolayer.3 Single-layer MoS2, a semiconducting TMD with the absence of inversion symmetry, offers excellent opportunities for band structure modulation via external stimuli,4,5 allowing one to modulate the electronic properties in a desired behavior at a highly controllable level. In general, the modulation of physical properties in electronic materials can be approached by the method of “strain engineering”. Currently, a series of experimental6−12 and theoretical13−19 attempts demonstrated that the optical and electronic properties of single-layer MoS2 can be achieved by applied strain. For example, the bandgap of monolayer MoS2 will be reduced under the condition of uniaxial strain, and the bandgap shows a strain-induced red shift of ∼50 meV/% strain.6−9 Recently, Lloyd et al.10 observed the effect of strain on the bandgap of 99 meV/% strain in single-layer MoS2 under the approach of biaxial strain. He et al.11 reported that the monolayer MoS2 shows a linear red shift under strain with a rate of 56 and 55 meV/% strain by using microphotoluminescence spectroscopy measurements and first-principles calculations, respectively. Besides, strain engineering on the bandgap of MoS2 nanosheets shows that the out-of-plane uniaxial tensile strain gives rise to a blue shift whereas the inplane uniaxial tensile strain results in a red shift.12 Moreover, in theoretical a crossover from semiconducting to metallic of © XXXX American Chemical Society

Received: December 16, 2016 Revised: February 16, 2017 Published: February 16, 2017 A

DOI: 10.1021/acs.jpcc.6b12679 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C bond identities, including bond length, bond angle, and bond energy, in single-layer MoS2 under uniaxial stress and the related band offset is still lacking. Moreover, the deformation potential of band offset in single-layer MoS2 is scattered in the range of 40−65 meV/% strain. Therefore, in this contribution we put forward an analytical model to address the straindependent band offset of single-layer MoS2 in terms of the functional dependence on the bonding identities based on atomic-bond-relaxation (ABR) consideration.25−28

2. THEORETICAL METHODOLOGIES In general, with the size down to nanoscale the coordination deficiency and atomic bond contraction at the surface can significantly influence the related properties and generate completely new effects such as solubility or luminescence, resulting in some distinctive features from their corresponding bulk.29 In the light of ABR method, the reduced coordination numbers (CNs) of atoms at edges will shrink spontaneously to a new self-equilibrium state and the bond strength will be stronger than that of the interior bulk, leading to the generation of lattice strain.30,31 Li et al.32 indicated that the Mo−S bond lengths of 8-Z MoS2 nanoribbons are 2.41, 2.42, and 2.45 Å in the inner sites and 2.37 and 2.39 Å in the Mo-terminated and Sterminated edges after full relaxation, verifying that the remaining bonds among the undercoordinated atoms will shrink spontaneously. Generally, the lattice strain in a selfequilibrium state can be expressed by ε0 = h*/h0 − 1, where h* and h0 denote the average bond length and that of bulk, respectively. Single-layer MoS2 consists of a hexagonal plane of Mo atoms sandwiched between two planes of S atoms and coordinated through ionic−covalent interactions, as shown in Figure 1. Evidently, the Mo−S bond 2 ⎛ h1 ⎞ a c 2 + ( 2 ) , and the bond angle lengthh = ⎜ h ⎟ = 3 ⎝ 2⎠ ⎛ cos θ1 ⎞ cos ψ1 a2 c2 ⎜ ⎟ = 1 − = 1 − 2 , where h1 and h2 2 and cos ψ cos θ 2 h h 2 2 ⎝ 2⎠ denote the bond length along x- and y-direction, a and c are the S−S distance, and θi and ψi (i = 1,2) are the in-plane and outof-plane bond angles, respectively. When the uniaxial tensile strain is approached, lattice parameters along the stretching direction will be elongated, and the vertical directions will shrink and the shrinkage depends on the Poisson’s ratio. As expected, not only the MoS bond lengths, but also the S MoS bond angles vary with tensile strain. Naturally, the variations of bond length and bond angle under tensile strain would induce the enhancement of deformation energy. For the case of single-layer MoS2, the interaction potential between Mo and S can be expressed as the summation of the bond-stretching energy Ubond, the bond angle variation energy Uangle and the Coulomb electrostatic energy Uc33,34

Figure 1. Schematic showing the (a) top and (b) side view of singlelayerMoS2. The yellow and blue balls represent S and Mo atoms, respectively. θ is the SMoS angle vertical to the plane, while ψ is the in-plane SMoS angle. (c,d) The schematic of a unit cell under uniaxial tensile strain along x-direction and y-direction, respectively. The lattice parameters are listed as a = 3.160 Å an c = 3.160 Å for S S distance in-plane and out-of-plane.41

( )

(

U=

MoS bond induced by uniaxial tensile strain can be obtained as ΔE = −ΔU/n, where n is the number of bonds in a unit cell. Theoretically, the average single bond energy determines the crystal potential of the system and is primarily used to evaluate the binding strength of a material. The cohesive energy is defined as the energy required breaking the atoms into isolated atomic species. Also, the cohesive energy of an isolated atom can be approached as the sum of its bond energy over all the nearest neighbors, that is, EC = zEb, where EC and Eb are cohesive energy and single bond energy, and z is the CNs. Subsequently, the cohesive energy of a unit cell under uniaxial tensile strain can be obtained as

)

∑ Ubond + ∑ Uangle + ∑ Uc

EC(ε) = Nz1(E1 + ΔE)

(2)

where N is the number of atoms, and z1 = 4 is the CNs of single-layer MoS2. Remarkably, the bond order loss of an atom in the surface shell causes the remaining bonds of the undercoordinated atom to contract spontaneously, leading the intraatomic potential well depression from Eb to E1 = c1−mEb, where c1 = 2/(1 + exp((12 − z1)/8z1)) is the bond contraction coefficient, and the index m is the bond nature factor.25 Furthermore, the bandgap energy (Eg) between the valence band and the conduction band is from the crystal potential over the entire solid, and the width bandgap is proportional to the first Fourier coefficient of the crystal potential.35,36 Moreover, the bandgap energy is proportional to the single bond energy Eg ∝ ⟨E0⟩ = EC/Nz. Consequently, combining with the relationships shown above, the strain-dependent bandgap energy of single-layer MoS2 under uniaxial tensile strain is

(1) 2

Here Ubond = D × [1 − e−α(hij−h) ]2, 1 1 Uangle = 2 kθ(hΔθ )2 + 2 k ψ (hΔψ )2 , Uc = C·qiqj/hij, where D, α, kθ, and kψ are the bond potential parameters and C is the Coulomb electrostatic potential parameter, hij is the distance between atoms i and j, and qi and qj are the partial electrostatic charges for atoms i and j. Thus, the average energy gain ΔE of B

DOI: 10.1021/acs.jpcc.6b12679 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Eg (ε) = Egb(1 + Δε)

uniaxial tensile strain ε is defined as ε = (a′ − a)/a and ε = (b′ − b)/b for strain applied along x- and y-directions, where a and b are the equilibrium in-plane lattice parameters shown in Figure 1a,b, and a′ and b′ are the corresponding strained inplane lattice parameters. Furthermore, Figure 1c,d shows the schematic of a unit cell under uniaxial tensile strain along x- and y-directions, respectively, thus the relationship among bond identities (bond length and bond angles) and ε satisfies

(3)

where Δε = (c1−m) + ΔE/Eb, the previous term and latter term in the formula denote the perturbation induced by size and uniaxial tensile strain, respectively. Ebg is the bandgap of the bulk counterpart. In order to elucidate the strain effect on the band offset in single-layer MoS2, we first explore the geometry relationship between the bond lengths, bond angles, as well as the uniaxial tensile strain applied along x- and y-directions, respectively. The

⎧ ⎞2 ⎛ c(1 + ε ·ν⊥) ⎞2 ⎛a ⎞2 ⎛ a ⎪ ⎜ ⎟ ⎟ ⎪ h1′ = ⎝ 2 (1 + ε)⎠ + ⎜⎝ 2 3 (1 + ε ·ν//)⎟⎠ + ⎜⎝ ⎠ 2 ⎪ ⎪ ⎛ a ⎞2 ⎛ c(1 + ε ·ν⊥) ⎞2 ⎪ h ′ = (1 + ε · ν ) x ‐direction ⎜ ⎟ // ⎟ + ⎜ ⎪ 2 ⎝ 3 ⎠ ⎝ ⎠ 2 ⎨ ⎪ ⎛ 2⎤ ⎞ ⎡ 3 a(1 + ε·ν//) 2 a(1 + ε) ⎪ ⎜ h1′2 + h2′2 − ⎢ + ⎥⎟ ⎪ 2 2 ⎦⎟ ⎣ ⎜ ⎪ θ1′ = arccos⎜ ⎟ 2·h1′·h2′ ⎪ ⎟⎟ ⎜ ⎜ ⎪ ⎠ ⎝ ⎩

(4)

⎧ ⎞2 ⎛ c(1 + ε ·ν⊥) ⎞2 ⎛a ⎞2 ⎛ a ⎪ ⎜ ⎟ ⎟ ⎪ h1′ = ⎝ 2 (1 + ε ·ν//)⎠ + ⎜⎝ 2 3 (1 + ε)⎟⎠ + ⎜⎝ ⎠ 2 ⎪ ⎪ ⎛ a ⎞2 ⎛ c(1 + ε ·ν⊥) ⎞2 ⎪ h ′ = (1 + ε ) y‐direction ⎜ ⎟ +⎜ ⎟ ⎪ 2 ⎝ 3 ⎠ ⎝ ⎠ 2 ⎨ ⎪ 2⎤ ⎞ ⎛ 2 ⎡ 3 a(1 + ε) 2 a(1 + ε·ν//) ⎪ + ⎜ h1′ + h2′2 − ⎢ ⎥⎟ ⎪ 2 2 ⎦⎟ ⎣ ⎜ ⎪ θ1′ = arccos ⎜ ⎟ 2·h1′·h2′ ⎪ ⎜ ⎟ ⎪ ⎝ ⎠ ⎩

(5)

(

) (

) (

(

)

3. RESULTS AND DISCUSSION In the light of relationships mentioned above, we calculate the strain-dependent MoS bond lengths, as shown in Figure 2a. Clearly, for the strain applied along x-direction the MoS distance elongates along the strain direction and shrinks at the vertical direction. The MoS bond length along x-direction h1′ increases up to 2.5 Å at the strain of 10%, and the bond length along y-direction h′2 decreases down to 2.37 Å. While for strain applied along y-direction, the MoS distance h′2 increases with strain and reaches up to 2.55 Å at 10% strain, but h1′ remains almost unchanged at the vertical direction. Similarly, Miao et al.18 demonstrated that the average bond length along xdirection is increased up to 2.5835 Å, while along y-direction the bond length is decreased down to 2.3965 Å at 22% strain. Under tensile strain, the original crystal symmetry will be broken, and the deformation involves not only the in-plane lattice but also the out-of-plane SS distance due to contraction. As expected, not only the MoS bond lengths, but also the SMoS bond angles vary with strain. In terms of eqs 4−6, the strain-dependent SMoS bond angle can be

and ⎧ 2 2⎞ ⎛ ⎪ θ2′ = arccos⎜1 − a (1 + ε) ⎟ 2 ⎪ 2·h1′ ⎠ ⎝ ⎪ ⎛ ⎪ c 2(1 + ε ·ν⊥)2 ⎞ ⎨ ψ1′ = arccos⎜1 − ⎟ 2·h1′2 ⎠ ⎝ ⎪ ⎪ ⎛ ⎪ c 2(1 + ε ·ν⊥)2 ⎞ ⎟ ⎪ ψ2′ = arccos⎜1 − 2·h2′2 ⎠ ⎝ ⎩

)

(6)

where h′1 and h′2 denote the strained bond length along x and y directions, θi and ψi (i = 1, 2) are the strained in-plane and outof-plane SMoS bond angles, respectively, as also shown in Figure 1c,d. Note that v∥ = 0.21 and v⊥ = 0.27 are the in-plane and out-of-plane Poisson’s ratio of single-layer MoS2.23,24 C

DOI: 10.1021/acs.jpcc.6b12679 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 3. Strain-dependent cohesive energy of a unit cell of singlelayer MoS2 under uniaxial tensile strain, and the inset shows straindependent energy changes in bond stretching, bond-angle variation, and electrostatic interaction.

nonlinearly with uniaxial tensile strain for strain applied x- or ydirection, and the variation tendency becomes steeper with increasing strain. Particularly, the inset in Figure 3 shows the strain-dependent energy changes in bond stretching, bondangle variation, and electrostatic interaction. It is obvious that the bond-stretching energy is much larger than the bond-angle variation energy and electrostatic energy, and the variations of bond-angle variation energy and electrostatic energy are almost the same for strain applied along x- or y-direction, while the bond-stretching energy is direction dependence. Interestingly, the change of energy remains almost the same for strain applied along x- and y-directions under the condition of small strain, indicating that single-layer MoS2 is virtually isotropic at small strain. Figure 4a depicts the bandgap energy of single-layer MoS2 with respect to 0−10% uniaxial tensile strain. It is evident that

Figure 2. Strain-dependent MoS bond length (a), SMoS bond angles of θ (b) and ψ (c).

calculated, as shown in Figure 2b,c. It is evident that the bond angle of SMoS varies almost linearly with strain. For the case of x-direction, θ2′ and ψ2′ increase with increasing strain, while θ1′ and ψ1′ decrease with increasing strain. In addition, the bond angles decrease with uniaxial tensile strain except θ′1 for the y-direction. Figure 3 shows the strain-dependent cohesive energy in single-layer MoS2 under uniaxial tensile strain. Note that the necessary parameters are D = 19.945, α = 0.858, kθ = 0.9387, kψ = 0.8631, and the partial electrostatic charges of 0.76e and −0.38e for Mo and S atoms, respectively.34 The atomic cohesive energy of unstrained single-layer MoS2 is 1.35 eV.37 In the case of single-layer MoS2, the adjacent Mo and S atoms bond together through ionic−covalent interactions. Thus, it is necessary to consider the variation of both Coulomb electrostatic potential and elastic energy under uniaxial tensile strain. As shown in Figure 3, the cohesive energy decreases

Figure 4. Strain-dependent bandgap of single-layer MoS2 under 0− 10% (a) and 0−2% (b) uniaxial tensile strain.

the bandgap energy decreases with tensile strain and the variations are very similar for strain applied along x- or ydirection. Notably, the bandgap reduced from 1.83 to 0.77 or 0.84 eV at 10% uniaxial tensile strain. Johariet al.14 indicated that the bandgap of single-layer MoS2 approximately reduces by half at 10% uniaxial tensile strain for both x- and y-directions based on the density functional theory. Moreover, Lu et al.15 D

DOI: 10.1021/acs.jpcc.6b12679 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

well in agreement with the experimental observations and calculations, demonstrating that the single-layer MoS2 is quite sensitive to strain, which has great potential for engineering tunable electronic devices through strain engineering.

reported that the bandgap of single-layer MoS2 decreases from 1.68 to 0.85 eV or to 0.83 eV under 8% uniaxial tensile strain in the x- or y-direction. In fact, when the tensile strain is applied to the single-layer MoS2, the orbital overlap between the Mo and S atoms will decrease due to increasing distance between Mo and S atoms. In addition, Figure 4b describes the strain dependence of bandgap in the range of 0−2% strain. The bandgap energy reveals an approximately linear red shift under uniaxial tensile strain with a rate of ∼53.4 meV/% strain, and the variations for strain applied along x- or y-directions are almost the same. Conley et al.6 and Zhu et al.7 measured a decrease in the bandgap of single-layer MoS2 that is approximately linear with strain, that is, ∼45 and ∼48 meV/ % strain by means of Raman spectroscopy. Meanwhile, a directto-indirect transition will occur at ∼1−2% strain. Also, firstprinciple calculations show the bandgap decreases as the tensile strain increases, accompanying a shift of valence band maximum from K to Γ point and resulting in a direct-toindirect bandgap transition.14,38,39 Essentially, the valence band maximum (VBM) at K point is dominated by dxy + dx2 + y2 (inplane) states, while the local maxima at Γ is dominated by dz2 (out-of-plane) states. The overlap of dz2 and chalcogen p states is reduced while the coupling of dxy + dx2+y2 and chalcogen p states is strengthened when the tensile strain is applied.38 The increased overlap of in-plane orbitals with tension can explain the decrease in energy of the VBM around K with respect to those around Γ. As the conduction band minimum around the K and Γ points does not experience a similar shift and on the contrary remains totally unaffected by the tensile strain, a direct-to-indirect bandgap transformation occurs, which can be reflected on the change of atomic interaction. According to the band theory,25,30,36 the single body Hamiltonian can be with expressed as Ĥ = Ĥ 0 + Ĥ ′, 2 2 ℏ ∇ Ĥ 0 = − + Va(r ) + Vc(r + R c) and Ĥ ′ = Vc(r + R c)Δ,

4. CONCLUSION In summary, we perform a theoretical analysis on the band offset modulated by the uniaxial tensile strain in the single-layer MoS2 based on ABR mechanism. It has been found that the bandgap of single-layer MoS2 shows an approximately linearly red shift with a rate of ∼53.4 meV/% strain under uniaxial tensile strain. The physical mechanism can be attributed to the variation of crystal potential induced by the changes of bond identities such as bond length, strength, and angle. The theoretical predictions are well consistent with the available evidence, suggesting that the proposed model can be an effective method to clarify the modulation of electronic properties in single-layer MoS2 from the perspective of atomistic origin. Importantly, our approach provides information on the measurable quantities and the atomic bond identities on the 2D nanomaterials, which infers that the developed method is helpful for strain design on tunable electronic properties of 2D semiconductor nanostructures.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Gang Ouyang: 0000-0003-2589-9641 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grants 91233203 and 11574080).

2μ

where μ is the effective mass of an electron, Va(r) is the intra-atomic trapping potential of an isolated atom, Vc(r) is the periodic potential of the crystal, and Rc is the lattice constant. Note that the interatomic potential Vc(r) forms the key to the band structures. In the case of monolayer MoS2 under the condition of uniaxial tensile strain, the perturbation to the Vc(r) will modify the entire band structure. Also, the bandgap energy depends on the crystal field or a sum of the binding energies over the entire solid. The binding energy, or interatomic potential, depends on the atomic distance and charge quantity of the atoms. Thus, the relaxation of bond length and bond angle induced by tensile strain play a significant effect on bandgap between the valence band and the conduction band. In our case, we focus on the effect of strain on the bandgap shift in monolayer MoS2 from the atomistic origin in terms of ABR method and move forward to a theoretical relationship among bond identities and applied strain. We find that the underlying mechanism regarding strain-dependent bandgap shift can be attributed to the variation of crystal potential induced by the changes of bond parameters. In addition, it is feasible to analyze the strain effects from the response of Raman spectra. Theoretically, Yang et al.40 reported that an incorporation of the bond order−length−strength correlation into the Raman frequencies has led to the formulation of the Raman shifts depending functionally only on the oriented strain in terms of the response of the length and energy of the bond, being independent of the process of phonon or electron scattering. Importantly, our predictions are

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REFERENCES

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DOI: 10.1021/acs.jpcc.6b12679 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.jpcc.6b12679 J. Phys. Chem. C XXXX, XXX, XXX−XXX