⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 088502 Normal Strain-Induced Tunneling Behavior Promotion in van der Waals Heterostructures Yi-Fan He (何旖凡), Lei-Xi Wang (王雷茜), Zhi-Xing Xiao (肖智兴), Ya-Wei Lv (吕亚威)*, Lei Liao (廖蕾), and Chang-Zhong Jiang (蒋昌忠) Affiliations Key Laboratory for Micro/Nano-Optoelectronic Devices of the Ministry of Education, School of Physics and Electronics, Hunan University, Changsha 410082, China Received 27 May 2020; accepted 19 June 2020; published online 28 July 2020 Supported by the National Key Research and Development Program of China (Grant Nos. 2018YFB0406603 and 2018YFA0703704), the National Natural Science Foundation of China (Grant Nos. 51991341, 61904052, 61851403 and 61704051), the Key Research and Development Plan of Hunan Province (Grant No. 2018GK2064), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB30000000).
*Corresponding author. Email: lvyawei@hnu.edu.cn Citation Text: He Y F, Wang L Q, Xiao Z X, Lv Y W and Liao L et al. 2020 Chin. Phys. Lett. 37 088502    Abstract Van der Waals heterostructures (vdWHs) realized by vertically stacking of different two-dimensional (2D) materials are a promising candidate for tunneling devices because of their atomically clean and lattice mismatch-free interfaces in which different layers are separated by the vdW gaps. The gaps can provide an ideal electric modulation environment on the vdWH band structures and, on the other hand, can also impede the electron tunneling behavior because of large tunneling widths. Here, through first-principles calculations, we find that the electrically modulated tunneling behavior is immune to the interlayer interaction, keeping a direct band-to-band tunneling manner even the vdWHs have been varied to the indirect semiconductor, which means that the tunneling probability can be promoted through the vdW gap shrinking. Using transition metal dichalcogenide heterostructures as examples and normal strains as the gap reducing strategy, a maximum shrinking of 33% is achieved without changing the direct tunneling manner, resulting in a tunneling probability promotion of more than 45 times. Furthermore, the enhanced interlayer interaction by the strains will boost the stability of the vdWHs at the lateral direction, preventing the interlayer displacement effectively. It is expected that our findings provide perspectives in improving the electric behaviors of the vdWH devices. DOI:10.1088/0256-307X/37/8/088502 PACS:85.35.Ds, 73.63.-b, 73.40.Gk, 73.40.-c © 2020 Chinese Physics Society Article Text The application-oriented material industry has stimulated tremendous efforts to explore materials with new physical or chemical properties. Among these efforts, the investigations aiming at rescuing the stagnant semiconductor technology and seeking the next-generation transistor channel material are a typical example. Not only the material innovation, the 60 mV/dec physical limit in traditional thermionic emission transistors also should be overcome. Therefore, tunnel field-effect transistors (TFETs) based on the band-to-band tunneling (BTBT) mechanism are proposed.[1,2] To realize this mechanism, heterostructure channels are usually needed to block and promote the tunneling at the off- and on-states simultaneously. However, restricted to the bulk material technology, dangling bond-induced junction interface quality problems have seriously hindered the on-state behavior of the TFETs.[3,4] After the successful exfoliation of graphene in 2004,[5] academics have been looking at the unique properties of two-dimensional (2D) materials concerning the electronic, photoelectric and mechanical aspects.[6–11] Among the properties, the extraordinary strain tolerance makes the combination of different 2D components through the lateral covalent bonding easier compared with the heterostructures composed of bulk materials,[12] resulting in clean and dangling-bond-free junction interfaces, showing huge potential for the tunneling application. However, the dangling-bond-free interface does not guarantee a steep band structure transition, since the junction intrinsic strains also cause interface states or traps, leading to the Fermi level pinning effect and impeding the electron tunneling.[13–15] One possible solution to the intrinsic strain is to design a 2D heterostructure with the same material but different shapes at the two sides of the junction. Typical structures include the width- and thickness-modulated graphene nanoribbon and phosphorene heterostructures which avoid the strains but inevitably induce new edge states.[16–18] Clearly, interface states also seem to be unavoidable in 2D heterostructures made of the lateral covalent bonding. The above interface state induced mechanisms, such as the intrinsic strain and material edge, can be avoided naturally by van der Waals heterostructure (vdWH) structures,[19–23] in which different 2D materials are stacked together layer by layer and interact with each other through weak vdW forces, maintaining their independent atomic and electronic properties to the maximum. Therefore, the electronic junction interfaces are believed to be the sharpest among different heterostructures, very suitable for the electron interlayer tunneling transport.[24,25] Using highly doped bulk Ge and monolayer MoS$_{2}$ as the source and channel, Sarkar et al. synthesized a TFET based on the Ge/MoS$_{2}$ vdWH for the first time and demonstrated a minimum subthreshold swing of 3.9 mV/dec.[26] Later, other fascinating performance produced by the vdWH tunneling transistors such as the negative differential conductance behavior with large current peak-to-valley ratios and high current on-to-off ratios were reported.[27–30] However, the on-state behaviors of the TFETs are still unsatisfactory.[2] The largest on-state current is 40 µA/µm reported by Qiu et al., still lower than the values of the traditional Si devices.[31] Therefore, theoretical works on the electronic property of vdWHs' interfaces are needed before further experiments. In this work and using the first-principles method, we theoretically investigate the electronic properties of vdWHs composed of transition metal dichalcogenides (TMDs) MX$_{2}$, where M = Mo and W, X = S, Se, and Te. We find that the band structures of the vdWHs response differently and independently to the normal strain induced vdW gap shrinking and the electric field. The energy states at the $\varGamma$ point of the $k$-space are more sensitive to the strains, whereas the state movement at the $K$ point by the electric field is almost immune to the strains. Therefore, to a certain degree, the strain can improve the tunneling probability through the narrowing of the vdW gap when the band structures of the vdWHs are changed from type II to type III, instead of degrading the tunneling performance by aggravating the interlayer orbital hybridization and causing the Fermi level pinning effect. Moreover, the stability of the heterostructures is also enhanced. We hope the results could provide a new orientation to improve the device performance based on vdWHs. Each vdWH is composed by two different $2H$ phase TMDs. Considering different element combinations and excluding the two materials with too large lattice mismatches (such as the MoS$_{2}$/MoTe$_{2}$ vdWH), 11 TMD vdWHs are computed, among which the results of the MoS$_{2}$/MoSe$_{2}$ and MoS$_{2}$/WS$_{2}$ vdWHs are shown in detail. The first-principles calculations are carried out using the Quantum Espresso plane-wave density functional theory (DFT) package.[32,33] The Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional is adopted with ultrasoft pseudopotentials.[34–36] The kinetic energy cut-offs for the plane-wave basis and charge density are 51 and 379 Ry, respectively.[34,37,38] The in-plane parallelogram cells are built and a vacuum layer of more than 40 Å is added in the thickness direction in each supercell to avoid image interactions. The value is much larger than the common treatment,[39] since the dipole correction and finite electric field require additional vacuum spaces.[40,41] The Brillouin zone $k$-point sampling in ground-state calculations is $10 \times 10 \times 1$, where the $10 \times 10$ is the in-plane sampling.[42,43] Before the band structure calculation, the crystal structures and atomic positions are fully optimized using the variable-cell method until the force on each atom and the total energy variations are smaller than $1 \times 10^{-4}$ Ry/bohr and $1 \times 10^{-4}$ Ry.[35,44] As a calibration, the relaxed monolayer MoS$_{2}$ and WS$_{2}$ cell parameters are 3.16 Å, very similar with the previous works.[45] After the first optimization step, the normal strains are initially exerted through the manual vdW gap reducing. Then, another crystal optimization procedure is adopted and the forces in the thickness direction are neglected. To realize external potential differences, homogeneous finite electric fields according to the modern theory of the polarization are applied in the thickness direction, inducing voltage drops between the two material layers. The vdW corrections are carried out by the semiempirical DFT-D3 method.[46] Note that the PBE functional tends to underestimate the bandgaps, but the underestimation is acceptable (0.27 eV compared with the HSE result for the MoS$_{2}$/MoSe$_{2}$ vdWH) and will not affect the band structure behaviors under finite electric fields here. We have also examined the results using different vdW functional (optB88-vdW functional) and found negligible differences.[47] The dipole correction is also employed to restrain the potential interaction between adjacent cells.
cpl-37-8-088502-fig1.png
Fig. 1. (a) Schematic illustration of the normal strain inducing using the MoS$_{2}$/MoSe$_{2}$ vdWH as an example. The band structures of the MoS$_{2}$/MoSe$_{2}$ and MoS$_{2}$/WS$_{2}$ heterostructures under strains are shown in (b) and (c). The exhibited maximum strains for the two structures are 18% and 8%, since they are the maximum values to maintain the direct tunneling manner as summarized in Table 1. All the band energies are referenced by the common zero-vacuum energy. Every calculated energy state is projected onto different layers. The MoS$_{2}$ and MoSe$_{2}$ (WS$_{2}$) contributed energy states are shown by the red and blue dots in (b) and (c), and the colors between them denote hybrid states.
Most of the TMDs are the direct-gap semiconductor with the valence band maximum (VBM) and conduction band minimum (CBM) located at the $K$ point of the hexagonal Brillouin zone, so do their heterostructures.[42,45] For the heterostructures, the interlayer atomic orbital hybridization tends to move up the valence band edges at the $\varGamma$ point. In some cases, such as the MoS$_{2}$/MoSe$_{2}$ vdWH shown in Fig. 1(b), this tendency can be well restrained without strain. However, in the MoS$_{2}$/WS$_{2}$ vdWH shown in Fig. 1(c), it has already taken the VBM away from the $K$ point without strain, making the heterostructure an indirect-gap semiconductor. The hybridization can be clearly observed by the state colors (colored dots), in which the red and blue represent that they are contributed by the MoS$_{2}$ and WS$_{2}$, and the colors between them indicate that they are hybrid states. In the two cases, the up moving of the valence band edges at the $\varGamma$ point can be further accelerated by the normal strain. In addition, the conduction band edges at the $Q$ point are also moved down. Comparatively, the band edges at the $K$ point are insensitive to the strain. Therefore, all the TMD vdWHs will be changed into the indirect semiconductor under strains.
cpl-37-8-088502-fig2.png
Fig. 2. (a) Schematic illustration of the applied voltage using the homogeneous finite electric fields. The band structures of the 18% and 8% strained MoS$_{2}$/MoSe$_{2}$ and MoS$_{2}$/WS$_{2}$ heterostructures along voltages are shown in (b) and (c). All the band energies are referenced by the common zero-vacuum energy. Every calculated energy state is projected onto different layers. The MoS$_{2}$ and MoSe$_{2}$ (WS$_{2}$) contributed energy states are represented by the red and blue dots in (b) and (c).
Indirect semiconductors do not mean an indirect BTBT manner. For example, the 18% strained MoS$_{2}$/MoSe$_{2}$ vdWH in Fig. 1(b) is an indirect semiconductor. When the external voltage is applied in Fig. 2(b), the hybrid electron atmospheres between the two material layers make the energy states at the $\varGamma$ and $Q$ points response inertly to the voltage, compared with the band edges at the $K$ point. The up shifts of the valence band edges at the $\varGamma$ and $K$ points are 0.04 and 0.36 eV when the external voltage is 1.57 V, referenced by the zero-vacuum energy. Thus, the direct electron tunneling from the valence band edge of MoSe$_{2}$ to the conduction band edge of MoS$_{2}$ at the $K$ point is reserved. Furthermore, the required voltage exerted on the two material layers to turn on the direct tunneling is almost invariable whether the strain is applied.
cpl-37-8-088502-fig3.png
Fig. 3. PDOS of the MoS$_{2}$/MoSe$_{2}$ heterostructure at specific energy states. The valence band edges at the $\varGamma$ and $K$ points and the conduction band edges at the $K$ and $Q$ points are shown in (a)–(d), respectively. The normal strain and external voltage values are 18% and 1.57 V.
To further investigate the different response mechanisms of the TMD vdWHs on the strain and electric field, the projected density of states (PDOS) of the MoS$_{2}$/MoSe$_{2}$ heterostructure are analyzed in Fig. 3. The four subplots correspond to the energy states of the valence band edges at the $\varGamma$ and $K$ points and the conduction band edges at the $K$ and $Q$ points, respectively. As shown in Fig. 3(a), the valence band edge at the $\varGamma$ point is composed of the $d_{z}$ and $p_{z}$ orbitals of the MoS$_{2}$ and MoSe$_{2}$ layers and the contributions of the two layers are almost equal. The hybrid orbitals at the thickness direction are more sensitive to the normal strain and thus can be moved up easily. However, when the electric field is applied, the interlayer potential difference can be well balanced by the hybrid electron atmospheres, keeping the state energy unchanged and only causing the layer contribution variation. In contrast, the energies and layer contributions of the valence and conduction band edges at the $K$ point are slightly changed by the normal strain in Figs. 3(b) and 3(c), since the two band edges are composed of in-plane orbitals. When the electric field is applied, enough potential difference can be built between these orbitals in the two layers, effectively reducing the energy difference of the two band edges. Therefore, the heterostructures remain direct tunneling under the electric field modulation, even they have already been changed into the indirect semiconductor by the normal strain. The normal strain not only can maintain the tunneling manner of the vdWHs within a certain strain range, but also can promote the tunneling probability through the tunneling width shrinking. As shown in Fig. 4(a), the colored lines denote the electric potentials of the MoS$_{2}$/MoSe$_{2}$ heterostructure under difference strain and the same external voltage which has turned it into the type-III semiconductor. The CBM and VBM locate near the black dashed line energetically. The electrons of the MoSe$_{2}$ layer should tunnel through a barrier of several eV to the WS$_{2}$ layer. Thus, the tunneling probability can be calculated following the Wentzel–Kramer–Brillouin (WKB) approximation as[48] $$ T \propto \exp \Big(-2\sqrt \frac{2m^{\ast }(U-E)}{\hbar^{2}} a\Big), $$ where $m^{\ast}$ is the electron effective mass, $U$ is the barrier height, $E$ is the kinetic energy of the electron, $a$ is the tunneling width, and $\hbar$ is the reduced Planck constant. Here we use electrons' vacuum mass as their effective mass and their kinetic energy is zero. With the increasing normal strain, both the width and height of the barrier are reduced, promoting the tunneling probability significantly as shown in Fig. 4(b). The tunneling probability increases of the MoS$_{2}$/MoSe$_{2}$ and MoS$_{2}$/WS$_{2}$ heterostructures are more than 6 and 2 times under the 18% and 8% normal strains. The normal strain is also favorable for the stability of the vdWHs. Owing to the weak vdW force, the material layers in the vdWHs are easy to move laterally, leading to different interlayer atom combinations and causing the instability of the tunneling behavior.[19,49] As shown in Fig. 5(a), without strain, the material layer displacement barrier is only 0.04 eV/cell in the MoS$_{2}$/MoSe$_{2}$ heterostructure. When the 18% strain is applied, the barrier increases to 0.29 eV/cell, much larger than the value without strain. The similar phenomenon is also observed in the MoS$_{2}$/WS$_{2}$ heterostructure as shown in Fig. 5(b).
cpl-37-8-088502-fig4.png
Fig. 4. (a) Electrostatic potentials of the MoS$_{2}$/MoSe$_{2}$ heterostructure under different normal strains and a 1.57 V external voltage. The VBM and CBM locate near the black dashed line. The tunneling probability promotions in the MoS$_{2}$/MoSe$_{2}$ and MoS$_{2}$/WS$_{2}$ heterostructures along the applied strains according to the WKB approximation are shown in (b).
cpl-37-8-088502-fig5.png
Fig. 5. Relative energy variations of the MoS$_{2}$/MoSe$_{2}$ and MoS$_{2}$/WS$_{2}$ heterostructures during the interlayer lateral displacements. The structures with high symmetry are also schematically shown.
Table 1. The calculated switch voltages from the type-II–III semiconductors, maximum strain which can maintain the direct BTBT mechanism, and the final tunneling probability promotion for the 11 TMD vdWHs.
Heterostructure name Switch voltage (V) Maximum strain Tunneling promotion
MoS$_{2}$/MoSe$_{2}$ 1.50 18% 636%
MoS$_{2}$/WS$_{2}$ 3.04 8% 209%
MoS$_{2}$/WSe$_{2}$ 1.10 19% 694%
${}^*$MoSe$_{2}$/MoTe$_{2}$ 1.11 26% 2554%
MoSe$_{2}$/WS$_{2}$ 2.14 20% 765%
MoSe$_{2}$/WSe$_{2}$ 2.68 22% 1057%
MoSe$_{2}$/WTe$_{2}$ 0.79 32% 4140%
${}^*$MoTe$_{2}$/WSe$_{2}$ 1.67 29% 3680%
MoTe$_{2}$/WTe$_{2}$ 1.99 25% 1515%
WS$_{2}$/WSe$_{2}$ 1.78 21% 852%
WSe$_{2}$/WTe$_{2}$ 1.35 33% 4519%
$^*$Since the electronic property of the MoTe$_{2}$ is very sensitive to the in-plane strain, the two cases are calculated using the unstrained MoTe$_{2}$ cell parameters.
Although the strain has been approved to be beneficial to the tunneling behavior of the vdWHs, the maximum value for every single case should be noticed since the direct tunneling manner will be destroyed if the strain exceeds these criteria. Table 1 shows the maximum strain values for the 11 heterostructures to maintain a direct tunneling manner. In addition, the voltages needed to turn on the tunneling are calculated. Note that in-plane strains are inevitable in the calculations even we have avoided the heterostructures with too much lattice constant mismatches. In most cases, the in-plane strains do not affect the electronic properties. However, in the MoSe$_{2}$/MoTe$_{2}$ and MoTe$_{2}$/WSe$_{2}$ heterostructures, the in-plane strains cause non-negligible valence band discrepancies because of the unique property of the MoTe$_{2}$ layers. Therefore, the unstrained MoTe$_{2}$ lattice constant is used in the calculations of these two cases, instead of the values coming from the overall relaxed heterostructure cell. From the table, the contribution of normal strains on the tunneling promotion is proved to be universal for TMD vdWHs. In conclusion, electronic properties of the TMD vdWHs have been studied under the combined effect of the normal strain and electric field. The effect of the interlayer orbital hybridization on the tunneling behavior of the heterostructures is limited within certain normal strain ranges as the direct BTBT mechanism can always been maintained when the heterostructures are changed from the type-II–III semiconductors under external electric fields. Furthermore, the normal strain induced tunneling width shrinking and lateral stability enhancement promote the tunneling probability and quality efficiently. The promotion is universal for TMD vdWHs. Therefore, we believe that the normal strain modulation should be an effective strategy to improve the performance of vdWH-based tunneling devices. 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