1.
Introduction

Besides the breakthrough fabrication of single-layered graphene^{[1, 2]}, the transition-metal dichalcogenides (TMDs) with MoS₂ as the prototype material become another attractive family of atomically thin two-dimensioned (2D) materials that were successfully exfoliated from three-dimensional bulk phases^[3]. The 2D TMDs show reasonably high carrier mobility (e.g., ~200 cm²V^?1s^?1 for MoS₂ at room temperature^{[4, 5]}) and direct optical band gap in the visible regime at monolayer limit (e.g., 1.8 eV for MoS₂^[6]). Additionally, it has been demonstrated that the flexibly tunable electronic and optical properties can be realized by controlling the structural parameters such as layer thickness^[6–9] and stacking pattern of multiple layers^[10–13], as well as applying strains^[14–19] and external fields^{[20, 21]}. Therefore, the family of 2D TMDs has attracted significant research interest for potential electronic^{[5, 22–24]} and optoelectronic applications^[25–27].



Carrier effective mass (m^*), defined as the second derivative of energy dispersion, is a physical quantity influencing almost all the transport properties. It is therefore a critical property of the 2D TMDs for advancing their electronic and optoelectronic applications. While the carrier effective masses of MoS₂ have been measured experimentally (for instance evaluated from the dispersion curve from the angle-resolved photoemission spectroscopy measurement^[8]) and calculated theoretically^{[19, 28]}, the systemic study of the effective masses of all the members in the 2D TMDs family is, to our best knowledge, not yet available. Moreover, the multiple homogenous layers of 2D TMDs and the heterostructures consisting of different sublayers have been used/considered for electronic and optoelectronic devices^{[5, 22–27]}. Studies of the layer dependence of carrier effective masses and how the effective masses change in the heterostructures are desired to purposely optimize carriers transport and improve the corresponding device performance.



In addition, the most remarkable feature of the 2D TMDs is the indirect to direct band-gap transition from multiple layers to monolayer^[6], which converts the system from an optically inactive material into a strong light absorber/emitter. The physical mechanism underlying this indirect to direct band-gap transition was attributed to the kinetic-energy controlled quantum confinement effect^{[6, 8, 9]}, or the weak interlayer coupling of extended electronic states^[29–31], which is still not yet conclusive. In fact, the mechanism could to some extent be revealed by simply evaluating the out-of-layer component of carrier effective mass. This is because in the purely quantum confinement dominated picture, the dependences of electron and hole energy levels on sample size can be described by the finite potential-well model, where the carrier effective mass with respect to the confinement direction enters as a constant (as shown in Fig. 1). The slower change of the curve (i.e., the red one) means the larger carrier effective mass. If the dependences of electron and hole energy levels do not comply with the finite potential-well model, the underlying physics is not the quantum confinement effect.



In such a context, we present here a theoretical study of the carrier effective masses of the family of 2D TMDs by using high-throughput density functional theory based calculations. We considered various TMD members in the formula of MX₂ with M = Mo, W and X = S, Se, Te and several heterostructures based on them. The in-layer and out-of-layer components of carrier effective mass, dependence of mass value on the layer thickness, and variation of mass value with different heterostructure configuration are extensively calculated. The results demonstrate chemical trends of carrier effective mass, insights on how the effective mass controls optoelectronic properties, and suggestion on optimizing the effective mass to improve carrier transport.

2.
Computational approaches

First-principles DFT calculations were carried out by using the plane-wave pseudopotential method as implemented in the VASP code^[32–35]. The electron–ion interaction was treated with the projector-augmented plane-wave pseudopotentials^{[36, 37]}. We used the generalized gradient approximation of the Perdew–Burke–Ernzerhof parameterization^[38] as exchange-correlation functional. The electronic integration within the Brillouin zone was done with the k-point meshes of 22 × 22 × 5, 22 × 22 × 2, and 22 × 22 × 1 for the bulk material, thinner (1 to 4 layers) and thicker (more than 4 layers) slabs, respectively. A vacuum thickness of more than 20 ? was adopted along the out-of-layer direction to isolate 2D TMD layers. We considered the interlayer van der Waals (vdWs) interactions essential for the 2D layered materials through the DFT-D2 scheme of Grimme^[39]. For all the TMDs, the lattice parameters and internal atomic positions were fully optimized with the total energy convergence threshold of 10^?4 eV/atom. The carrier effective masses are evaluated through the second derivative of the energy dispersion curve. We note that the standard DFT-GGA calculations are known to underestimate the band gap and overestimate the band dispersion of the semiconductors, so the calculated effective mass values are in principle overestimated as well. In spite of this problem, DFT-GGA calculations could give the reliable dependence of effective mass with thickness, and have also been adopted in previous publications^[40–42]. Therefore, the conclusion will not be affected.

3.
Results and discussions

We begin with the evaluation of the out-of-layer component of effective mass to probe the mechanism underlying the indirect to direct band-gap transition upon dimensional reduction in TMDs. For the semiconductor nanostructures where the electronic states are dominated by the quantum confinement effect, the dependences of carriers’ energy levels (E_n) on the nanostructure size (L) can be depicted within the finite potential-well model, ${E_{{n}}} = frac{{{n^2}{h^2}}}{{8{m^*}{L^{
m{alpha }}}}} + {C_0}$, where the integer n represents ground (n = 1) or excited states (n > 1); h is Planck’s constant, m^* is the carrier effective mass, and α and C₀ are two material-specific parameters. The evolutions of electron and hole energy levels with L are demonstrated in the schematic Fig. 1 (where only the ground-state levels are shown). One sees that the larger m^* (i.e., the β case) corresponds to a more gradual change in the evolutions of energy levels.






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Figure1.
(Color online) Schematic plot of the evolutions of electron and hole energy levels with the sample size L for the semiconductor nanostructures dominated by quantum confinement effect. The evolution of energy can be fitted into the finite potential-well model, ${E_{n}} = frac{{{n^2}{h^2}}}{{8{m^*}{L^{
m{alpha }}}}} + {C_0}$ (see the main text). The larger m^* (i.e., the β case) corresponds to a more gradual change in the energy evolution.




Fig. 2(a) shows the calculated band structure of bulk MoS₂, where the band-edge states determining the indirect to direct band-gap transition are indicated. These involve T_C, K_C, Γ_V, and K_V states. The direct band gap occurs when the band edges changed from (T_C, Γ_V) to (K_C, K_V) with decreasing layer thickness^[43–46]. It is known that while the T_C and Γ_V states are delocalized states, the K_C and K_V are rather localized ones^[44]. Figs. 2(b)–2(e) show the energy dispersion of these states along the out-of-layer direction, based on which the out-of-layer components of effective mass ($m_{
m{out}}^*$) are calculated. The T_C and Γ_V states have small $m_{
m{out}}^*$ of 0.18m₀ and 0.29m₀, respectively, whereas the K_V state shows the larger $m_{
m{out}}^*$ of 0.63m₀, and the K_C state shows an extremely large $m_{
m{out}}^*$ (with numerical value above 10m₀). Within the pure quantum confinement picture, one expects that the T_C state would show the most dramatic change in the dependence of band energies on the layer thickness L.






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Figure2.
(Color online) (a) Calculated band structure of bulk MoS₂, where the band-edge states forming the indirect and direct band-gap transitions are indicated. (b–e) The energy dispersion of the states marked in (a) along the out-of-layer direction for each band-edge state, based on which the out-of-layer components of effective mass ($scriptstyle{m_{
m{out}}^*}$) are calculated.




Fig. 3 shows the evolution of calculated band energies of the aforementioned band-edge states with the layer thickness L for MoS₂. The absolute band energies were obtained by aligning the band structures at different L with respect to the averaged electrostatic potential in the vacuum region. The layer thickness L is defined by the distance between two S atoms terminating the layer plus a 1 ? region for electronic wave-function extension. As seen the Γ_V state shows the most dramatic change in the evolution of band energies (green squares), though the corresponding $m_{
m{out}}^*$ (0.29m₀) is larger than that of the T_C state (0.18m₀). This contradicts with the quantum confinement dominated picture. The solid lines represent the fitting results based on the finite potential-well model (i.e., the equation mentioned above with n = 1); for all the band-edge states, the uniform α value is adopted by considering the common confinement condition. The fittings give the α of 1.798, reasonably falling in the region of quantum wells embedded in finite barriers. For the localized states of K_C and K_V, the band energies show quite gradual (or invisible) evolution with L and the agreement between the fitting curve and the calculated data is good. However, this is not the case for the delocalized T_C and Γ_V states, where one clearly observes significant deviation of the fitting from the actual calculation (particularly in the thin layers region).






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Figure3.
(Color online) Evolution of calculated band energies of the band-edge states determining the indirect to direct band-gap transition with the thickness L for MoS₂. The solid lines represent the fitting results based on the finite potential-well model (i.e., the equation mentioned above with n = 1). The dash lines are drawn as a guide to the eye.




We further extend the above calculation and analysis to other TMD materials. Table 1 lists the calculated $m_{
m{out}}^*$ for the relevant band-edge states. Fig. 4 shows the evolution of band energies of the corresponding states with the layer thickness L. Fitting into the finite potential-well model is performed for all the band-edge states. For all the investigated TMDs, there appears significant deviation of the fitting curve from the actual calculated data in the thin layers region for the delocalized T_C and Γ_V states. These results indicate that the quantum confinement effect on its own can never be responsible for the indirect to direct band-gap transition upon dimensional reduction. This is consistent with our previous studies^{[31, 44]}. The underlying physical mechanism has to involve the other effect, i.e., the interlayer coupling of extended electronic states.

Parameter	$m_{{ m{out}}}^*left( {{T_{ m{C}}}} ight)$	$m_{{ m{out}}}^*left( {{K_{ m{C}}}} ight)$	$m_{{ m{out}}}^*left( {{varGamma _{ m{V}}}} ight)$	$m_{{ m{out}}}^*left( {{K_{ m{V}}}} ight)$
MoS2	0.18	> 10.00	0.29	0.63
MoSe2	0.16	> 10.00	0.42	0.41
MoTe2	0.13	> 10.00	0.26	0.25
WS2	0.16	> 10.00	0.22	0.89
WSe2	0.30	> 10.00	0.33	1.98
WTe2	0.16	> 10.00	0.17	0.45

Table1.
Calculated out-of-layer components of effective mass ($m_{
m{out}}^*$) of the relevant band-edge states of T_C, K_C, Γ_V, and K_V for the bulk phase of MX₂ (M = Mo, W and X = S, Se, Te). The unit of $m_{
m{out}}^*$ is m₀.



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Parameter	$m_{{ m{out}}}^*left( {{T_{ m{C}}}} ight)$	$m_{{ m{out}}}^*left( {{K_{ m{C}}}} ight)$	$m_{{ m{out}}}^*left( {{varGamma _{ m{V}}}} ight)$	$m_{{ m{out}}}^*left( {{K_{ m{V}}}} ight)$
MoS2	0.18	> 10.00	0.29	0.63
MoSe2	0.16	> 10.00	0.42	0.41
MoTe2	0.13	> 10.00	0.26	0.25
WS2	0.16	> 10.00	0.22	0.89
WSe2	0.30	> 10.00	0.33	1.98
WTe2	0.16	> 10.00	0.17	0.45

We next calculated the in-layer component of effective mass ($m_{
m{in}}^*$) that is crucial for the usage of 2D TMDs in the electronic and optoelectronic applications. Table 2 lists the $m_{
m{in}}^*$ at the monolayer limit and the thick enough layers of 10 monolayers for the aforementioned band-edge states in all the six TMDs. The results of the layer dependence of $m_{
m{in}}^*$ are shown in Fig. 5. We have the following observations on the general variation trends: (i) At the monolayer limit, most of the states show rather low $m_{
m{in}}^*$ of 0.5m₀–0.75m₀. The exception is the Γ_V state, which shows large $m_{
m{in}}^*$ above 2.0m₀, consistent with the experimental observation^[8]. (ii) At the thick layers, the K_C and Γ_V states have the heavier $m_{
m{in}}^*$, whereas the T_C and K_V states show the lighter $m_{
m{in}}^*$. (iii) While the most of states show decreasing $m_{
m{in}}^*$ with the increasing L, the K_C state shows abnormally increasing $m_{
m{in}}^*$ with L. This effect is particularly strong in MoSe₂ and MoTe₂. (iv) For the Γ_V state, there is an especially strong negative dependence of $m_{
m{in}}^*$ on L. (v) Focusing on the CBM and VBM states, the VBM has the heavier $m_{
m{in}}^*$ than the CBM at the thick layers, and they become comparable at the monolayer limit.






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Figure5.
(Color online) Evolution of the in-layer component of effective mass ($m_{
m{in}}^*$) of the relevant band-edge states with the layer thickness L for all the investigated group-VIB TMDs MX₂ (M = Mo, W and X = S, Se, Te). The actual CBM and VBM states are depicted by the solid black and red lines, respectively.

Parameter	Layers	$m_{{ m{in}}}^*left( {{T_{ m{C}}}} ight)$	$m_{{ m{in}}}^*left( {{K_{ m{C}}}} ight)$	$m_{{ m{in}}}^*left( {{varGamma _{ m{V}}}} ight)$	$m_{{ m{in}}}^*left( {{K_{ m{V}}}} ight)$
MoS2	1	0.63	0.65	2.49	0.69
	10	0.61	0.87	0.67	0.65
MoSe2	1	0.60	0.74	3.63	0.77
	10	0.54	1.34	0.81	0.73
MoTe2	1	0.49	0.75	10.93	0.86
	10	0.41	1.34	1.15	0.83
WS2	1	0.59	0.75	2.18	0.57
	10	0.59	2.08	0.75	0.54
WSe2	1	0.54	0.48	3.43	0.62
	10	0.54	0.65	0.85	0.57
WTe2	1	0.43	0.53	8.74	0.67
	10	0.42	0.84	0.98	0.59

Table2.
Calculated in-layer components of effective mass ($m_{
m{in}}^*$) of the relevant band-edge states of T_C, K_C, Γ_V, and K_V at the monolayer limit and the thick enough layers of 10 monolayers. All the synthesized TMD materials of MX₂ (M = Mo, W and X = S, Se, Te) are calculated.



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Parameter	Layers	$m_{{ m{in}}}^*left( {{T_{ m{C}}}} ight)$	$m_{{ m{in}}}^*left( {{K_{ m{C}}}} ight)$	$m_{{ m{in}}}^*left( {{varGamma _{ m{V}}}} ight)$	$m_{{ m{in}}}^*left( {{K_{ m{V}}}} ight)$
MoS2	1	0.63	0.65	2.49	0.69
	10	0.61	0.87	0.67	0.65
MoSe2	1	0.60	0.74	3.63	0.77
	10	0.54	1.34	0.81	0.73
MoTe2	1	0.49	0.75	10.93	0.86
	10	0.41	1.34	1.15	0.83
WS2	1	0.59	0.75	2.18	0.57
	10	0.59	2.08	0.75	0.54
WSe2	1	0.54	0.48	3.43	0.62
	10	0.54	0.65	0.85	0.57
WTe2	1	0.43	0.53	8.74	0.67
	10	0.42	0.84	0.98	0.59

The interlayer coupling resulted in the dependence between $m_{
m{in}}^*$ and thickness. As shown in Fig. 4, the energy levels of the Γ_V state are much more sensitive to thickness than the others, due to the Γ_V states being mainly contributed by interlayer orbitals (Mo-d_z2 orbital and S-p_z orbital), especially by the out-of-plane extension of S-p_z orbital. This effect is much more significant from monolayer to bilayer. The energy levels of Γ_V state increase accompanied by the sharp decreasing of $m_{
m{in}}^*$ for all the six TMDs. On the other hand, the $m_{
m{in}}^*$ of K_C states slowly increase with the thickness. In contrast, the $m_{
m{in}}^*$ of T_C, K_C and K_V states are quite insensitive to thickness.






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Figure4.
(Color online) Evolution of band energies of the relevant band-edge states with the thickness L for all the investigated group-VIB TMDs MX₂ (M = Mo, W and X = S, Se, Te). Similar to Fig. 3, the solid lines represent the fitting results based on the finite potential-well model and the dash lines are drawn as a guide to the eye.




Finally, we construct the van der Waals heterostructures by combining together two TMD materials and calculate their carrier effective masses. Two types of heterostructures were considered: one consists of MoS₂ and MoSe₂ with common cations and the other is composed of MoS₂ and WS₂ with common anions. The heterostructures are in the 2H symmetry and along the out-of-layer direction have the period of 10 monolayers, in which two TMDs with different compositons are randomly arranged. All possible stacking orders for each composition are considered. The heterostructures (including structural parameters and internal atomic coordinates) are fully optimized for strain relaxation. Fig. 6 shows their calculated in-layer component of effective mass ($m_{
m{in}}^*$). We can see that except for the VBM state of MoS₂/WS₂ that shows a linear variation of $m_{
m{in}}^*$ between the two end-point materials, the other cases show complicated bowing behaviors. Meanwhile the magnitude of $m_{
m{in}}^*$ exhibits strong dependence on the orders of two TMDs arrangement along the out-of-layer direction at the same compositons; the discrepancy can be as large as ~0.3m₀ (in the case of the VBM state at MoS₂/MoSe₂). The scheme of the stacking orders for each structure with the lowest $m_{
m{in}}^*$ are displayed in Fig. S1. These imply that the van der Waals heterostructuring offers a possible degree of freedom to tune the carrier effective mass, and thus affect carrier transport behavior in 2D materials based devices.






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FigureS1.
(Color online) the scheme of the stacking orders for each structures with the lowest $m_{
m{in}}^*$ for CBMs and VBMs in MoS₂/MoSe₂ and MoS₂/WS₂ heterostructures.






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Figure6.
(Color online) Calculated in-layer component of effective mass ($m_{
m{in}}^*$) for the van der Waals heterostructures formed by two TMD materials (see the main text) as the function of the content of constituted TMD. Each data point represents the specific ordering of two TMDs arrangement along the out-of-layer direction at some compositon.

4.
Conclusions

In summary, through first-principle density functional calculations, we present a comprehensive study of carrier effective masses for the family of two-dimensional layered group-VIB transition metal dichalcogenides (i.e., MX₂ with M = Mo, W and X = S, Se, Te). Particularly we calculate the in-layer component of carrier effective masses for the single, multiple layers and also the van der Waals heterostructures, as well as the out-of-layer component for the bulk phase. Analysis of the out-of-layer effective masses within the framework of the finite potential-well model provides insight into the mechanism underlying the indirect to direct band-gap transition upon dimensional reduction. The ab initio calculated data of the in-layer effective masses give useful guidance for tuning the carrier effective masses in the two-dimensional transition metal dichalcogenides to meet the requirement of electronic and optoelectronic applications.