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Quantum wells formed in transition-metal dichalcogenide nanosheet-superlattices: stability and electronic structures from first principles† Xiangying Su,ab Ruizhi Zhang,*a Chongfeng Guo,*a Meng Guoc and Zhaoyu Rena The possibility of forming quantum wells (QWs) in transition-metal dichalcogenide nanosheet assembled superlattices (SLs) was investigated by using the first principles calculations. The interfacial binding energies and electronic structures of MoS2/MX2 (MX2 = MoSe2, WS2, and WSe2) SLs were calculated. The interfacial binding energies show that all the SLs are stable, and the most stable atomic configuration is that where M atoms are located right above S atoms. By calculating the band offsets in the SLs, it was found that a QW with a depth of 0.17 eV can be formed in the MoS2 layer in MoS2/WSe2 SLs. The calculated band structure shows that this SL has an indirect band gap due to the tensile strained state of the MoS2 layer. The charge
Received 12th October 2013, Accepted 12th November 2013
transfer between the two layers is very small, which is in favor of the QWs’ formation. In particular, the
DOI: 10.1039/c3cp54080d
different strain dependencies of the two materials’ band gaps. These findings will guide the choice of
depth of the QW in the SLs can be adjusted by strain engineering, which can be attributed to the future nanosheet assembled SLs to work on and suggest a new route to facilitate the design of QW
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based optoelectronic devices.
Introduction Quantum wells (QWs) are widely used in optoelectronic devices, such as GaN based light emitting diodes and semiconductor lasers.1 In QWs, the ‘quantum confinement’ occurs and is responsible for the optoelectronic performance enhancement. To form QWs, usually superlattices (SLs) or heterostuctures, consisting of two (or more) materials with different band gaps, e.g. GaN/InGaN, should be fabricated. The fabrication needs nanoscale precision techniques, such as pulsed laser deposition (PLD), molecular beam epitaxy (MBE) or metal–organic chemical vapor deposition (MOCVD). The recent progress in 2-dimensional nanosheets offers an opportunity to fabricate SLs by using a low cost technique. a
Institute of Photonics & Photon-Technology and Department of Physics, National Key Laboratory of Photoelectric Technology and Functional Materials (Culture Base) in Shaanxi Province, National Photoelectric Technology and Functional Materials & Application of Science and Technology International Cooperation Base, Northwest University, Xi’an, 710069, China. E-mail: [email protected], [email protected]; Fax: +86-29-8830-2661; Tel: +86-29-8830-2661 b School of Physics and Engineering, Henan University of Science and Technology, Luoyang, 471023, China c National Supercomputer Center in Jinan, Shandong Computer Science Center, Jinan 250101, P. R. China † Electronic supplementary information (ESI) available: The partial density of states and the layer thickness dependence of MoS2/WSe2 superlattices, local density of states of strained MoS2/WSe2 and MoS2/WS2 superlattices. See DOI: 10.1039/c3cp54080d
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If two (or more) kinds of nanosheets can be alternately stacked layer-by-layer, a nanosheet SL can be obtained, as shown in Fig. 1(a). Recently, a self-assembled superlattice was successfully fabricated from two types of oxide nanosheets, and the ferromagnetism properties were investigated.2 This inspired us to find more nanosheet materials to build SLs and to evaluate the possibility of forming QWs. The most widely studied nanosheet has been graphene since its isolation in 2004.3–5 However, graphene is not suitable for building quantum wells due to its gapless nature. Alternatively, the MoS2 nanosheet with a certain band gap is a good choice. It can be easily obtained by chemical or mechanical exfoliation due to the weak van der Waals forces between the layers.6–8 MoS2 nanosheets have many advantages over graphene. On one hand, they all have band gaps, which makes them promising semiconductor materials for a range of applications, e.g. transistors.9 The band gap can also be tuned over a wide range by either the intrinsic mismatch strain10,11 or external electrical field.12,13 On the other hand, in contrast to sp2-bonded graphene, MoS2 with strongly-correlated d electrons could show distinctively new physics. For example, MoS2 was found to undergo a transition from indirect to direct band gap behavior as layering is reduced to a monolayer.14,15 Giant enhancement (104) in photoluminescence quantum yield emerges because of the direct band gap of MoS2 monolayers.16 All these prove that MoS2 nanosheets have great potential to act as optoelectronic materials. Furthermore, the ample compositional variation of transition-metal dichalcogenides
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Fig. 1 (a) Illustration of the nanosheet-superlattice structure. The cyan and rose colored layers represent MX2 and MoS2 nanosheets, respectively. (b) Side and (c) top views of the MoS2/MX2 superlattice. The big and small spheres denote the metal and dichalcogenide atoms, respectively. By periodic translational symmetry, the position of the grayer atoms can be anywhere within a red dashed irreducible parallelogram.
(TMDs) makes them a large family of materials. Actually, complementing graphene, two-dimensional TMDs have attracted growing attention and emerged as promising materials with potential applications in nanoelectronics and nanophotonics.16–19 More importantly, TMDs have similar in-plane lattice constants and different band gaps,14,20 so we have many choices when seeking for suitable building blocks to form nanosheet assembled SLs. Although the MoS2 nanosheets based composite or heterostrctures were theoretically or experimentally investigated,21–23 the formations of nanosheet SLs have been rarely reported so far. In the present work, MoS2, MoSe2, WS2 and WSe2 nanosheets are chosen as building blocks. As the quantum confinement effect in MoS2 has been proven to be effective in enhancing the optoelectric properties,15,16 we would like to form quantum wells in MoS2 layers and use three other materials as potential barriers. By using the first principles calculations, we will find out whether this is achievable by investigating the stablity of the SLs, the possibility of forming QWs, and the influence of strain on these SLs. Our theoretical results will guide the choice of future nanosheet SLs to work on and suggest a new route to facilitate the design of QW based optoelectronic devices.
Calculation details and models All geometry optimizations and electronic properties calculations were performed by using the density functional theory (DFT) as implemented in the Vienna ab-initio simulation package (VASP).24,25 Projector-augmented-wave (PAW) potentials26 were used to account for electron–ion interactions, while the electron exchange–correlation interactions were treated by using local density approximation (LDA).27 The structural relaxations were performed by using a conjugate gradient algorithm with a force tolerance of 0.01 eV Å1. Electronic minimization was performed with a tolerance of 106 eV and electronic convergence was accelerated with a Gaussian smearing of the Fermi surface by 0.05 eV. The kinetic energy cutoff was set at 400 eV for all the calculations. A k-point Monkhorst–Pack sampling of 11 11 5 was used for the relaxation calculations, and a dense 27 27 7 mesh was used for generating accurate charge densities and density of states (DOS).
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Table 1 The calculated lattice constants and band gaps of bulk materials and the SLs. Experimental data and previous theoretical values are also shown as reference
LDA results a (Å) MoS2 WS2 MoSe2 WSe2 MoS2/WS2 SL MoS2/MoSe2 SL MoS2/WSe2 SL a
3.135 3.130 3.262 3.254 3.132 3.196 3.194
c (Å) 12.052 12.092 12.690 12.816 12.039 12.605 12.474
Experimental data Eg (eV) a (Å) 0.825 1.022 0.798 0.990 0.765 0.747 0.680
3.160 3.153 3.299 3.286
Eg (eV)
c (Å) a
12.294 1.290a 12.323a 12.938a 1.100a 12.976a 1.200a
Experimental lattice constants and Eg data of bulk MX2.28–32 Eg data of reference.20
b
0.875b 0.984b 0.851b 0.940b
Calculated
The calculated lattice constants and energy band gaps of the bulk materials are listed in Table 1. The experimental and theoretical values reported in the previous literature20,28–32 are also shown as references. The agreement is very good and shows the reliability of our calculation method. The optimized values of lattice constants are all smaller than corresponding experimental data within 2% due to the shortcoming of semilocal functionals in DFT.27 All these materials have similar in-plane lattice constants as can be seen from Table 1. As the interaction between the layers is the weak van der Waals force, we also introduce the vdWs interaction by adding a semiempirical dispersion potential (through a pairwise force field following Grimme’s DFT-D2 method33) as implemented in VASP. The comparison between the LDA and vdWs results is shown in Table S1, Fig. S1 and S2 (ESI†). It can be seen that LDA well reproduces all the results from vdWs, including the lattice parameters, band gaps, the positions of band edges and the depth of the QW. This also agrees with a previous work where LDA and vdWs gave the same calculation results.22 It should also be noted that although the band gaps were underestimated in our calculations due to the well known shortcoming of the semi-local functional, and the GW approach gives more accurate band gap value by considering the many body effect, it has been reported that the semilocal function and GW give the same trends and similar values when calculating the relative positions of the band edges in TMDs.34
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We have three types of SLs, i.e. MoS2/MoSe2, MoS2/WS2, and MoS2/WSe2. The structures of the SLs are shown in Fig. 1(b). Hereafter these SLs consisting of one monolayer of MoS2 and one monolayer of another TMD nanosheet will be investigated unless otherwise stated. Then the most energy favorable configuration of these SLs, i.e. the relative position of the two layers, should be determined. This was done by using the following methods: fix the MoS2 layer and move the MX2 layer both in the in-plane and out-of-plane direcitons, and calculate the total energy at each step. The results are shown in Fig. 2. It should be noted that due to the periodic conditions, in the in-plane direction only the irreducible region35 is considered, as indicated by red lines in Fig. 1(c). Fig. 2(a) shows the energy landscape in the irreducible region of the MoS2/WSe2 SL. There are 3 special points, as indicated by A, B and C, which are the configurations of X atoms located right above Mo atoms, M atoms located right above Mo atoms, and M atoms located right above S atoms, respectively. As shown in Fig. 2(a), A has the highest total energy and is the most unstable, while C is the most stable configuration. For the other two SLs, the C configuration is also the most stable one. Therefore in the following calculations, only C configuration is considered. The total energy of MoS2/ WSe2 as functions of the a and c lattice constants is shown in Fig. 2(b). Then the optimum values can be obtained as listed in Table 1. Those for the other two SLs are calculated by using the same method and are also listed in Table 1.
Results and discussion 1. Stability of the superlattices The stability of the SLs relates to three types of energy: the strain energy, the adhesive energy and the binding energy. The last one is the sum of the former two energies, and determines the energy cost when forming the nanosheet SLs. The strain energy Estrain and adhesive energy Ead can be calculated by Estrain = (EMX2 + EMoS2) (E 0 MX2 + E 0 MoS2) Ead = ESL (EMX2 + EMoS2) where ESL, EMX2 and EMoS2 are energies of the SL, MoS2 and MX2 nanosheets with the same in-plane lattice constant as the SLs, respectively; E 0 MoS2 and E 0 MX2 are the energies of the freestanding MoS2 and MX2 nanosheets, respectively. The calculated energies are all listed in Table 2. From Table 2, it can be seen that all the strain energies are positive, but the values are all very small. This indicates that all the nanosheets are in the strained state in the SLs, but the lattice mismatch between different nanosheets is small. The adhesive energies are all negative, indicating that the nanosheets tend to assemble to form SLs. Although the strain energies are positvie, they can be easily compensated for by the negative adhesive energies,19 therefore the binding energies are all negative. This means that the nanosheet SLs are all stable.
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Fig. 2 (a) The total energy landscape of the MoS2/WSe2 SL within the irreducible region shown in Fig. 1(c). (b) The total energy as a functions of a and c lattice constants of the MoS2/WSe2 SL.
Table 2
The adhesive, strain and binding energies of SLs (eV per unit-cell)
Strain energy Adhesive energy Binding energy
MoS2/MoSe2
MoS2/WS2
MoS2/WSe2
0.068 0.241 0.173
0.001 0.218 0.217
0.025 0.240 0.215
2. Formation of the quantum wells The electronic structures of the SLs are calculated to find out whether QWs can be formed. Band offset should exsit in order to form QWs. The valence band offset (VBO) or the conduction band offset (CBO) is defined by the difference between the positions of the valence band maximum (VBM) or the conduction band minimum (CBM) of the two materials, respectively. There are two methods to calculate the band offset: planaraveraged electrostatic potential and the local density of states (LDOS). These two methods give nearly the same results as reported in the literature.36 Here we use the LDOS method, and the results of the three SLs are shown in Fig. 3(a). It is clear that band offsets at the CBM in MoS2/WSe2 and MoS2/MoSe2 SLs are 0.17 eV and 0.02 eV, respectively, and in both cases the QWs are formed in MoS2 layers. Therefore electrons will be confined in MoS2 layers. Compared to MoS2 bulk material, stronger optoelectric effects will emerge in these SLs due to the quantum confinement effect in the QWs.16 However, in the MoS2/WS2 SL there is no band offset, i.e. no QW can be formed. This indicates that a large lattice mismatch and different transition metals are both needed to form a deep QW, which will be further discussed below. As the band offset of the MoS2/WSe2 SL is the largest, in the following calculations we only focus on the electronic properties of this SL. The formation of QWs in the conduction band is illustrated in Fig. 3(b). For the MoS2/WSe2 SL, the QWs are formed in the MoS2 layers, and the WSe2 layers act as potential barriers. So by alternately stacking MoS2 and WSe2 nanosheets, as shown in Fig. 3(c), QWs can be formed due to the CBO of the two materials, or equally, due to the CBM difference of the two layers. The CBO is mainly contributed by the Mo 4d orbitals, as shown in Fig. S3 in the ESI.†
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Fig. 3 (a) Local density of states of MoS2/MX2 SLs. (b) Illustration of the multiple quantum wells formation at the conduction band bottom. (c) The optimal stacking pattern of MoS2 and WSe2. MoS2 and WSe2 are potential well and barrier materials, respectively.
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To verify this, the influence of the barrier thickness on the type of band gap was examined, and the results are shown in Fig. S4 in the ESI.† It can be seen that the valence band maximum at the K point does approach the Fermi level in MoS2/2WSe2, compared with that in MoS2/WSe2. This supports our scenario that increasing the barrier thickness will lead to a direct band gap in the SL. However, as the barrier thickness further increases, the SL still shows an indirect band gap. This is because the band structures are calculated at the equilibrium lattice constant of the SLs, in which MoS2 is in tensile strain due to the larger in-plane lattice constant of WSe2, and the strain can turn MoS2 monolayers into an indirect band gap seimconductor.11,20 Even by fixing the in-plane lattice parameter of the SL equal to that of the free standing MoS2 (i.e. no strain for the MoS2 layer but a compressive strain for the SL), we still cannot obtain the direct band gap. This is because the depth of the QW is highly sensitive to the strain of the SL, as shown in Fig. 6(a) and discussed in detail below. When the SL is under compressive strain, the QW vanishes and the conduction band, which makes the MoS2 layer bulk like: the CBM leaves the K point, as shown in Fig. S5 in the ESI.† Based on the analysis above, we ´ structure can be formed, such as that in propose that if a Morie MoS2/MoTe2,22 and when the barrier is thick enough, the direct band gap can be obtained in the SLs. However, this cannot be done in our calculation because a large supercell is needed, and the Brillouin zone folding in such a large supercell will make it difficult to find whether the material is direct band gap or not.37 So this still needs an experimental verification. When two nanosheets are assembled to form a SL, it is important to know whether there is charge transfer between the two constituents. It can be calculated as Dr = rSL rMoS2 rWse2
Fig. 4 Band structures of (a) the MoS2 monolayer, (b) the MoS2/WSe2 SL, and (c) the WSe2 monolayer. The Fermi energy is set to 0 eV.
The band structure of the MoS2/WSe2 SL is shown in Fig. 4. The band structures of the MoS2 and WSe2 monolayer nanosheets are also shown as references. It can be seen that MoS2 and WSe2 monolayers both have a direct band gap with the VBM and CBM located at the same high symmetry point K, consistent with previously reported results.15 While MoS2/WSe2 SL has an indirect band gap, with the CBM locating at K and the VBM locating at G. This is because the thickness of potential barriers, i.e. one layer of WSe2, is not enough to block off the interaction between the neighbouring MoS2 layers. The direct band gap of MoS2 monolayers originates from the quantized out-of-plane electronic momenta k.16 In a SL when QW is formed, the energy levels are quantized. But if the interaction between two quantum wells is strong enough, the electronic monenta cannot be quantized and the direct band gap cannot be achieved.16
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where rSL, rMoS2 and rWse2 are the charge densities of the MoS2/ WSe2 SL, the free standing MoS2 and WSe2 nanosheets with the same in-plane lattice constant as the SL, respectively. The results are shown in Fig. 5(a) and (b). The interaction between the two layers is rather weak, as evidenced by the small amount of electron transfer. From a Bader population analysis, we find that the S atom gains 0.008 e, while the Se atom losses 0.018 e. These results indicate that the interlayer bonding of MoS2 and WSe2 should be rather weak, which comprises long-range van der Waals forces. 3. Adjustable quantum well depth through strain engineering Strain engineering is a powerful approach to tune the electronic properties of monolayer or bilayer nanosheets.10,11 For example,
Fig. 5 Charge transfer of the MoS2/WSe2 SL, where the isosurface value is 5 104 e Å3. (a) positive charge and (b) negative charge.
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QW depth in the SLs can be tuned by strain, which is a universial behaviour of the nanosheets and SLs. To illustrate this, Fig. S7 in the ESI† shows that a CBO of 0.16 eV can be formed in the MoS2/WS2 SL under a 2% tensile strain, while this SL shows no band offset at its equilibrium lattice constant.
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Conclusions
Fig. 6 (a) Band offset of the MoS2/WSe2 SL as a function of the strain e, which varies from 4% to 5%. The negative and positive values of e represent compressive and tensile strains, respectively. (b) The band gap of strained MoS2 and WSe2 layers in the SL as a function of strain e.
the strain can induce a semiconductor-to-metal transition,10,11 and can induce a direct-to-indirect band gap transition in MoS2 monolayers.11,20 Therefore in these SLs, the electronic properties can also be tuned by mechanical strain. Strain is introduced by adjusting the in-plane lattice constants of the SLs, and the out-of-plane lattice constant and atomic positions are fully relaxed by using the same criteria as mentioned above. After the geomety optimization, the electronic structures of the strained SLs are calculated. Fig. 6(a) shows the band offset of the MoS2/WSe2 SL as a function of the mechanical strain e, which varies from 4% to 5%. The negative and positive values of e correspond to the compressive and tensile strain, respectively. It can be seen at the equilibrium state (e = 0), there is a CBO of 0.17 eV. The value of CBO increases with the increasing tensile strain, and a linear relationship is clear. Therefore the depth of the QWs also increases with the increasing tensile strain. This means that the confinement of electrons in the QWs becomes stronger, which is in favor of the quantum confinement effect. For compressive strain, the band offset vanishes at e = 1%. However, as the compressive strain further increases, a VBO forms. The value of the VBO increases with the increasing compressive strain. The right and left insets in Fig. 6(a) illustrate the formation of CBO and VBO, respectively. If the strain exceeds a certain value, the SL becomes metallic. This is similar to the findings in monolayer and bilayer nanosheets.11 The LDOS values of the MoS2/ WSe2 SL at e = 1%, 1%, 3% are shown in Fig. S6 in the ESI.† From Fig. S6 (ESI†), it can be seen that the QWs locate in MoS2 layers when a CBO is formed, while the QWs locate in WSe2 layers when a VBO is formed. To facilitate a closer inspection of such a behavior of band offsets as a function of strain, the band gaps of the two layers are calcuated by using the LDOS, and the results are shown in Fig. 6(b). It can be seen that the band gap dependence on the strain of MoS2 and WSe2 layers is different, similar to the trends in the corresponding freestanding nanosheets.11 Therefore the band gap difference varies as a function of the biaxial strain. In other words, the change of the band offset and the QW depth can be attributed to the different behaviours of the MoS2 and WSe2 layers under strain. That’s why the band offset and the
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We investigated the geometric and electronic structures of MoS2/MX2 (MX2 = MoSe2, WS2, and WSe2) nanosheet assembled superlattices (SLs). These SLs can be easily formed due to the same crystal structure and similar in-plane lattice constants of the TMD nanosheets. The calculated binding energies are all negative, suggesting that the SLs are stable. The most stable stacking pattern is the configuration where M atoms are located right above S atoms. MoS2/WSe2 has the largest band offset among these SLs, and a deep QW of 0.17 eV is formed in the MoS2 layer. The depth of the QW can be tuned over a wide range by the mechanical strains, which can be attributed to the different behavior of the different layers under strain. It should be noted that MoS2 nanosheets were investigated in this paper mainly because MoS2 nanosheets have been well studied and the number of the experimental findings increase dramatically, which makes the experimental verification of our theoretical predictions easier. We believe that there should be many other types of nanosheet assembled SLs, which still need a systemic exploration in the further theoretical and experimental work.
Acknowledgements This work was supported by the high-level talent project of Northwest University, National Natural Science Foundation of China (No. 11104220, 11274251), Natural Science Foundation of Shaanxi Province (No. 2011JQ1012), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (excellent), and Foundation of Key Laboratory of Photoelectric Technology in Shaanxi Province (12JS094). We thank the National Super Computing Center in Jinan for making some of the computations possible.
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Quantum wells formed in transition-metal dichalcogenide nanosheet-superlattices: stability and electronic structures from first principles† Xiangying Su,ab Ruizhi Zhang,*a Chongfeng Guo,*a Meng Guoc and Zhaoyu Rena The possibility of forming quantum wells (QWs) in transition-metal dichalcogenide nanosheet assembled superlattices (SLs) was investigated by using the first principles calculations. The interfacial binding energies and electronic structures of MoS2/MX2 (MX2 = MoSe2, WS2, and WSe2) SLs were calculated. The interfacial binding energies show that all the SLs are stable, and the most stable atomic configuration is that where M atoms are located right above S atoms. By calculating the band offsets in the SLs, it was found that a QW with a depth of 0.17 eV can be formed in the MoS2 layer in MoS2/WSe2 SLs. The calculated band structure shows that this SL has an indirect band gap due to the tensile strained state of the MoS2 layer. The charge
Received 12th October 2013, Accepted 12th November 2013
transfer between the two layers is very small, which is in favor of the QWs’ formation. In particular, the
DOI: 10.1039/c3cp54080d
different strain dependencies of the two materials’ band gaps. These findings will guide the choice of
depth of the QW in the SLs can be adjusted by strain engineering, which can be attributed to the future nanosheet assembled SLs to work on and suggest a new route to facilitate the design of QW
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based optoelectronic devices.
Introduction Quantum wells (QWs) are widely used in optoelectronic devices, such as GaN based light emitting diodes and semiconductor lasers.1 In QWs, the ‘quantum confinement’ occurs and is responsible for the optoelectronic performance enhancement. To form QWs, usually superlattices (SLs) or heterostuctures, consisting of two (or more) materials with different band gaps, e.g. GaN/InGaN, should be fabricated. The fabrication needs nanoscale precision techniques, such as pulsed laser deposition (PLD), molecular beam epitaxy (MBE) or metal–organic chemical vapor deposition (MOCVD). The recent progress in 2-dimensional nanosheets offers an opportunity to fabricate SLs by using a low cost technique. a
Institute of Photonics & Photon-Technology and Department of Physics, National Key Laboratory of Photoelectric Technology and Functional Materials (Culture Base) in Shaanxi Province, National Photoelectric Technology and Functional Materials & Application of Science and Technology International Cooperation Base, Northwest University, Xi’an, 710069, China. E-mail: [email protected], [email protected]; Fax: +86-29-8830-2661; Tel: +86-29-8830-2661 b School of Physics and Engineering, Henan University of Science and Technology, Luoyang, 471023, China c National Supercomputer Center in Jinan, Shandong Computer Science Center, Jinan 250101, P. R. China † Electronic supplementary information (ESI) available: The partial density of states and the layer thickness dependence of MoS2/WSe2 superlattices, local density of states of strained MoS2/WSe2 and MoS2/WS2 superlattices. See DOI: 10.1039/c3cp54080d
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If two (or more) kinds of nanosheets can be alternately stacked layer-by-layer, a nanosheet SL can be obtained, as shown in Fig. 1(a). Recently, a self-assembled superlattice was successfully fabricated from two types of oxide nanosheets, and the ferromagnetism properties were investigated.2 This inspired us to find more nanosheet materials to build SLs and to evaluate the possibility of forming QWs. The most widely studied nanosheet has been graphene since its isolation in 2004.3–5 However, graphene is not suitable for building quantum wells due to its gapless nature. Alternatively, the MoS2 nanosheet with a certain band gap is a good choice. It can be easily obtained by chemical or mechanical exfoliation due to the weak van der Waals forces between the layers.6–8 MoS2 nanosheets have many advantages over graphene. On one hand, they all have band gaps, which makes them promising semiconductor materials for a range of applications, e.g. transistors.9 The band gap can also be tuned over a wide range by either the intrinsic mismatch strain10,11 or external electrical field.12,13 On the other hand, in contrast to sp2-bonded graphene, MoS2 with strongly-correlated d electrons could show distinctively new physics. For example, MoS2 was found to undergo a transition from indirect to direct band gap behavior as layering is reduced to a monolayer.14,15 Giant enhancement (104) in photoluminescence quantum yield emerges because of the direct band gap of MoS2 monolayers.16 All these prove that MoS2 nanosheets have great potential to act as optoelectronic materials. Furthermore, the ample compositional variation of transition-metal dichalcogenides
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Fig. 1 (a) Illustration of the nanosheet-superlattice structure. The cyan and rose colored layers represent MX2 and MoS2 nanosheets, respectively. (b) Side and (c) top views of the MoS2/MX2 superlattice. The big and small spheres denote the metal and dichalcogenide atoms, respectively. By periodic translational symmetry, the position of the grayer atoms can be anywhere within a red dashed irreducible parallelogram.
(TMDs) makes them a large family of materials. Actually, complementing graphene, two-dimensional TMDs have attracted growing attention and emerged as promising materials with potential applications in nanoelectronics and nanophotonics.16–19 More importantly, TMDs have similar in-plane lattice constants and different band gaps,14,20 so we have many choices when seeking for suitable building blocks to form nanosheet assembled SLs. Although the MoS2 nanosheets based composite or heterostrctures were theoretically or experimentally investigated,21–23 the formations of nanosheet SLs have been rarely reported so far. In the present work, MoS2, MoSe2, WS2 and WSe2 nanosheets are chosen as building blocks. As the quantum confinement effect in MoS2 has been proven to be effective in enhancing the optoelectric properties,15,16 we would like to form quantum wells in MoS2 layers and use three other materials as potential barriers. By using the first principles calculations, we will find out whether this is achievable by investigating the stablity of the SLs, the possibility of forming QWs, and the influence of strain on these SLs. Our theoretical results will guide the choice of future nanosheet SLs to work on and suggest a new route to facilitate the design of QW based optoelectronic devices.
Calculation details and models All geometry optimizations and electronic properties calculations were performed by using the density functional theory (DFT) as implemented in the Vienna ab-initio simulation package (VASP).24,25 Projector-augmented-wave (PAW) potentials26 were used to account for electron–ion interactions, while the electron exchange–correlation interactions were treated by using local density approximation (LDA).27 The structural relaxations were performed by using a conjugate gradient algorithm with a force tolerance of 0.01 eV Å1. Electronic minimization was performed with a tolerance of 106 eV and electronic convergence was accelerated with a Gaussian smearing of the Fermi surface by 0.05 eV. The kinetic energy cutoff was set at 400 eV for all the calculations. A k-point Monkhorst–Pack sampling of 11 11 5 was used for the relaxation calculations, and a dense 27 27 7 mesh was used for generating accurate charge densities and density of states (DOS).
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Table 1 The calculated lattice constants and band gaps of bulk materials and the SLs. Experimental data and previous theoretical values are also shown as reference
LDA results a (Å) MoS2 WS2 MoSe2 WSe2 MoS2/WS2 SL MoS2/MoSe2 SL MoS2/WSe2 SL a
3.135 3.130 3.262 3.254 3.132 3.196 3.194
c (Å) 12.052 12.092 12.690 12.816 12.039 12.605 12.474
Experimental data Eg (eV) a (Å) 0.825 1.022 0.798 0.990 0.765 0.747 0.680
3.160 3.153 3.299 3.286
Eg (eV)
c (Å) a
12.294 1.290a 12.323a 12.938a 1.100a 12.976a 1.200a
Experimental lattice constants and Eg data of bulk MX2.28–32 Eg data of reference.20
b
0.875b 0.984b 0.851b 0.940b
Calculated
The calculated lattice constants and energy band gaps of the bulk materials are listed in Table 1. The experimental and theoretical values reported in the previous literature20,28–32 are also shown as references. The agreement is very good and shows the reliability of our calculation method. The optimized values of lattice constants are all smaller than corresponding experimental data within 2% due to the shortcoming of semilocal functionals in DFT.27 All these materials have similar in-plane lattice constants as can be seen from Table 1. As the interaction between the layers is the weak van der Waals force, we also introduce the vdWs interaction by adding a semiempirical dispersion potential (through a pairwise force field following Grimme’s DFT-D2 method33) as implemented in VASP. The comparison between the LDA and vdWs results is shown in Table S1, Fig. S1 and S2 (ESI†). It can be seen that LDA well reproduces all the results from vdWs, including the lattice parameters, band gaps, the positions of band edges and the depth of the QW. This also agrees with a previous work where LDA and vdWs gave the same calculation results.22 It should also be noted that although the band gaps were underestimated in our calculations due to the well known shortcoming of the semi-local functional, and the GW approach gives more accurate band gap value by considering the many body effect, it has been reported that the semilocal function and GW give the same trends and similar values when calculating the relative positions of the band edges in TMDs.34
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We have three types of SLs, i.e. MoS2/MoSe2, MoS2/WS2, and MoS2/WSe2. The structures of the SLs are shown in Fig. 1(b). Hereafter these SLs consisting of one monolayer of MoS2 and one monolayer of another TMD nanosheet will be investigated unless otherwise stated. Then the most energy favorable configuration of these SLs, i.e. the relative position of the two layers, should be determined. This was done by using the following methods: fix the MoS2 layer and move the MX2 layer both in the in-plane and out-of-plane direcitons, and calculate the total energy at each step. The results are shown in Fig. 2. It should be noted that due to the periodic conditions, in the in-plane direction only the irreducible region35 is considered, as indicated by red lines in Fig. 1(c). Fig. 2(a) shows the energy landscape in the irreducible region of the MoS2/WSe2 SL. There are 3 special points, as indicated by A, B and C, which are the configurations of X atoms located right above Mo atoms, M atoms located right above Mo atoms, and M atoms located right above S atoms, respectively. As shown in Fig. 2(a), A has the highest total energy and is the most unstable, while C is the most stable configuration. For the other two SLs, the C configuration is also the most stable one. Therefore in the following calculations, only C configuration is considered. The total energy of MoS2/ WSe2 as functions of the a and c lattice constants is shown in Fig. 2(b). Then the optimum values can be obtained as listed in Table 1. Those for the other two SLs are calculated by using the same method and are also listed in Table 1.
Results and discussion 1. Stability of the superlattices The stability of the SLs relates to three types of energy: the strain energy, the adhesive energy and the binding energy. The last one is the sum of the former two energies, and determines the energy cost when forming the nanosheet SLs. The strain energy Estrain and adhesive energy Ead can be calculated by Estrain = (EMX2 + EMoS2) (E 0 MX2 + E 0 MoS2) Ead = ESL (EMX2 + EMoS2) where ESL, EMX2 and EMoS2 are energies of the SL, MoS2 and MX2 nanosheets with the same in-plane lattice constant as the SLs, respectively; E 0 MoS2 and E 0 MX2 are the energies of the freestanding MoS2 and MX2 nanosheets, respectively. The calculated energies are all listed in Table 2. From Table 2, it can be seen that all the strain energies are positive, but the values are all very small. This indicates that all the nanosheets are in the strained state in the SLs, but the lattice mismatch between different nanosheets is small. The adhesive energies are all negative, indicating that the nanosheets tend to assemble to form SLs. Although the strain energies are positvie, they can be easily compensated for by the negative adhesive energies,19 therefore the binding energies are all negative. This means that the nanosheet SLs are all stable.
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Fig. 2 (a) The total energy landscape of the MoS2/WSe2 SL within the irreducible region shown in Fig. 1(c). (b) The total energy as a functions of a and c lattice constants of the MoS2/WSe2 SL.
Table 2
The adhesive, strain and binding energies of SLs (eV per unit-cell)
Strain energy Adhesive energy Binding energy
MoS2/MoSe2
MoS2/WS2
MoS2/WSe2
0.068 0.241 0.173
0.001 0.218 0.217
0.025 0.240 0.215
2. Formation of the quantum wells The electronic structures of the SLs are calculated to find out whether QWs can be formed. Band offset should exsit in order to form QWs. The valence band offset (VBO) or the conduction band offset (CBO) is defined by the difference between the positions of the valence band maximum (VBM) or the conduction band minimum (CBM) of the two materials, respectively. There are two methods to calculate the band offset: planaraveraged electrostatic potential and the local density of states (LDOS). These two methods give nearly the same results as reported in the literature.36 Here we use the LDOS method, and the results of the three SLs are shown in Fig. 3(a). It is clear that band offsets at the CBM in MoS2/WSe2 and MoS2/MoSe2 SLs are 0.17 eV and 0.02 eV, respectively, and in both cases the QWs are formed in MoS2 layers. Therefore electrons will be confined in MoS2 layers. Compared to MoS2 bulk material, stronger optoelectric effects will emerge in these SLs due to the quantum confinement effect in the QWs.16 However, in the MoS2/WS2 SL there is no band offset, i.e. no QW can be formed. This indicates that a large lattice mismatch and different transition metals are both needed to form a deep QW, which will be further discussed below. As the band offset of the MoS2/WSe2 SL is the largest, in the following calculations we only focus on the electronic properties of this SL. The formation of QWs in the conduction band is illustrated in Fig. 3(b). For the MoS2/WSe2 SL, the QWs are formed in the MoS2 layers, and the WSe2 layers act as potential barriers. So by alternately stacking MoS2 and WSe2 nanosheets, as shown in Fig. 3(c), QWs can be formed due to the CBO of the two materials, or equally, due to the CBM difference of the two layers. The CBO is mainly contributed by the Mo 4d orbitals, as shown in Fig. S3 in the ESI.†
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Fig. 3 (a) Local density of states of MoS2/MX2 SLs. (b) Illustration of the multiple quantum wells formation at the conduction band bottom. (c) The optimal stacking pattern of MoS2 and WSe2. MoS2 and WSe2 are potential well and barrier materials, respectively.
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To verify this, the influence of the barrier thickness on the type of band gap was examined, and the results are shown in Fig. S4 in the ESI.† It can be seen that the valence band maximum at the K point does approach the Fermi level in MoS2/2WSe2, compared with that in MoS2/WSe2. This supports our scenario that increasing the barrier thickness will lead to a direct band gap in the SL. However, as the barrier thickness further increases, the SL still shows an indirect band gap. This is because the band structures are calculated at the equilibrium lattice constant of the SLs, in which MoS2 is in tensile strain due to the larger in-plane lattice constant of WSe2, and the strain can turn MoS2 monolayers into an indirect band gap seimconductor.11,20 Even by fixing the in-plane lattice parameter of the SL equal to that of the free standing MoS2 (i.e. no strain for the MoS2 layer but a compressive strain for the SL), we still cannot obtain the direct band gap. This is because the depth of the QW is highly sensitive to the strain of the SL, as shown in Fig. 6(a) and discussed in detail below. When the SL is under compressive strain, the QW vanishes and the conduction band, which makes the MoS2 layer bulk like: the CBM leaves the K point, as shown in Fig. S5 in the ESI.† Based on the analysis above, we ´ structure can be formed, such as that in propose that if a Morie MoS2/MoTe2,22 and when the barrier is thick enough, the direct band gap can be obtained in the SLs. However, this cannot be done in our calculation because a large supercell is needed, and the Brillouin zone folding in such a large supercell will make it difficult to find whether the material is direct band gap or not.37 So this still needs an experimental verification. When two nanosheets are assembled to form a SL, it is important to know whether there is charge transfer between the two constituents. It can be calculated as Dr = rSL rMoS2 rWse2
Fig. 4 Band structures of (a) the MoS2 monolayer, (b) the MoS2/WSe2 SL, and (c) the WSe2 monolayer. The Fermi energy is set to 0 eV.
The band structure of the MoS2/WSe2 SL is shown in Fig. 4. The band structures of the MoS2 and WSe2 monolayer nanosheets are also shown as references. It can be seen that MoS2 and WSe2 monolayers both have a direct band gap with the VBM and CBM located at the same high symmetry point K, consistent with previously reported results.15 While MoS2/WSe2 SL has an indirect band gap, with the CBM locating at K and the VBM locating at G. This is because the thickness of potential barriers, i.e. one layer of WSe2, is not enough to block off the interaction between the neighbouring MoS2 layers. The direct band gap of MoS2 monolayers originates from the quantized out-of-plane electronic momenta k.16 In a SL when QW is formed, the energy levels are quantized. But if the interaction between two quantum wells is strong enough, the electronic monenta cannot be quantized and the direct band gap cannot be achieved.16
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where rSL, rMoS2 and rWse2 are the charge densities of the MoS2/ WSe2 SL, the free standing MoS2 and WSe2 nanosheets with the same in-plane lattice constant as the SL, respectively. The results are shown in Fig. 5(a) and (b). The interaction between the two layers is rather weak, as evidenced by the small amount of electron transfer. From a Bader population analysis, we find that the S atom gains 0.008 e, while the Se atom losses 0.018 e. These results indicate that the interlayer bonding of MoS2 and WSe2 should be rather weak, which comprises long-range van der Waals forces. 3. Adjustable quantum well depth through strain engineering Strain engineering is a powerful approach to tune the electronic properties of monolayer or bilayer nanosheets.10,11 For example,
Fig. 5 Charge transfer of the MoS2/WSe2 SL, where the isosurface value is 5 104 e Å3. (a) positive charge and (b) negative charge.
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QW depth in the SLs can be tuned by strain, which is a universial behaviour of the nanosheets and SLs. To illustrate this, Fig. S7 in the ESI† shows that a CBO of 0.16 eV can be formed in the MoS2/WS2 SL under a 2% tensile strain, while this SL shows no band offset at its equilibrium lattice constant.
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Conclusions
Fig. 6 (a) Band offset of the MoS2/WSe2 SL as a function of the strain e, which varies from 4% to 5%. The negative and positive values of e represent compressive and tensile strains, respectively. (b) The band gap of strained MoS2 and WSe2 layers in the SL as a function of strain e.
the strain can induce a semiconductor-to-metal transition,10,11 and can induce a direct-to-indirect band gap transition in MoS2 monolayers.11,20 Therefore in these SLs, the electronic properties can also be tuned by mechanical strain. Strain is introduced by adjusting the in-plane lattice constants of the SLs, and the out-of-plane lattice constant and atomic positions are fully relaxed by using the same criteria as mentioned above. After the geomety optimization, the electronic structures of the strained SLs are calculated. Fig. 6(a) shows the band offset of the MoS2/WSe2 SL as a function of the mechanical strain e, which varies from 4% to 5%. The negative and positive values of e correspond to the compressive and tensile strain, respectively. It can be seen at the equilibrium state (e = 0), there is a CBO of 0.17 eV. The value of CBO increases with the increasing tensile strain, and a linear relationship is clear. Therefore the depth of the QWs also increases with the increasing tensile strain. This means that the confinement of electrons in the QWs becomes stronger, which is in favor of the quantum confinement effect. For compressive strain, the band offset vanishes at e = 1%. However, as the compressive strain further increases, a VBO forms. The value of the VBO increases with the increasing compressive strain. The right and left insets in Fig. 6(a) illustrate the formation of CBO and VBO, respectively. If the strain exceeds a certain value, the SL becomes metallic. This is similar to the findings in monolayer and bilayer nanosheets.11 The LDOS values of the MoS2/ WSe2 SL at e = 1%, 1%, 3% are shown in Fig. S6 in the ESI.† From Fig. S6 (ESI†), it can be seen that the QWs locate in MoS2 layers when a CBO is formed, while the QWs locate in WSe2 layers when a VBO is formed. To facilitate a closer inspection of such a behavior of band offsets as a function of strain, the band gaps of the two layers are calcuated by using the LDOS, and the results are shown in Fig. 6(b). It can be seen that the band gap dependence on the strain of MoS2 and WSe2 layers is different, similar to the trends in the corresponding freestanding nanosheets.11 Therefore the band gap difference varies as a function of the biaxial strain. In other words, the change of the band offset and the QW depth can be attributed to the different behaviours of the MoS2 and WSe2 layers under strain. That’s why the band offset and the
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We investigated the geometric and electronic structures of MoS2/MX2 (MX2 = MoSe2, WS2, and WSe2) nanosheet assembled superlattices (SLs). These SLs can be easily formed due to the same crystal structure and similar in-plane lattice constants of the TMD nanosheets. The calculated binding energies are all negative, suggesting that the SLs are stable. The most stable stacking pattern is the configuration where M atoms are located right above S atoms. MoS2/WSe2 has the largest band offset among these SLs, and a deep QW of 0.17 eV is formed in the MoS2 layer. The depth of the QW can be tuned over a wide range by the mechanical strains, which can be attributed to the different behavior of the different layers under strain. It should be noted that MoS2 nanosheets were investigated in this paper mainly because MoS2 nanosheets have been well studied and the number of the experimental findings increase dramatically, which makes the experimental verification of our theoretical predictions easier. We believe that there should be many other types of nanosheet assembled SLs, which still need a systemic exploration in the further theoretical and experimental work.
Acknowledgements This work was supported by the high-level talent project of Northwest University, National Natural Science Foundation of China (No. 11104220, 11274251), Natural Science Foundation of Shaanxi Province (No. 2011JQ1012), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (excellent), and Foundation of Key Laboratory of Photoelectric Technology in Shaanxi Province (12JS094). We thank the National Super Computing Center in Jinan for making some of the computations possible.
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