Modeling the electronic structure of TMDCs

Important contributions to the development of the field of 2D crystals have come from theory. While theory offers an explanation of many fundamental experimental effects, many phenomena have been predicted by computer simulations. In the following, some of these examples are compiled, with a focus on our own contributions. It is important to note that calculations on 2D materials may require different approaches and computational details, so the interested reader is referred to our recent tutorial reviews of the topic.^{Reference Kuc and Heine1,Reference Heine36}

Theoretical modeling offers fast screening of many 2D crystals. We have compiled the structural and electronic properties of a large selection of 2D crystals, thus providing a database of bandgaps, effective electron and hole masses, lattice parameters, and band structures for further investigation by experiment and theory.^{Reference Miro, Audiffred and Heine37} Materials with spectacular properties invite further investigation; for example, in the case of PdS₂, a Group 10 TMDC with 1T symmetry is semiconducting as a ML, but becomes semimetallic as a bilayer (BL).^{Reference Miro, Ghorbani-Asl and Heine38}

When Group 6 semiconducting –MX₂ materials with the 2H (trigonal prismatic) structure type (Figure 1b) are confined from the bulk to the ML, their crystal symmetry lowers, and their electronic structure changes significantly. An important example of this quantum confinement is the observed enhanced photoluminescence signal from a MoS₂ ML, which has been rationalized by band structure calculations.^{Reference Splendiani, Sun, Zhang, Li, Kim, Chim, Galli and Wang19,Reference Mak, Lee, Hone, Shan and Heinz20} MoS₂, when thinned down to a ML, undergoes an indirect to direct bandgap transition, as illustrated in Figure 2. We have shown that this phenomenon is not restricted to MoS₂ but is also present in other Group 6 MX₂ materials.^{Reference Kuc, Zibouche and Heine21}

Figure 2.

(a) Structure of 2H MX₂ monolayers (MLs) with high-symmetry reflection planes R_x and R_z, and the Brillouin zone with its high-symmetry k points. (c) The same information for the bilayer (BL) system, with the inversion point I indicated. (b) Band structure of MoS₂ (top) and WSe₂ (bottom) for ML (a) and BL (c). The Fermi level (E _F) is indicated by a red dashed line. The valence band (blue) and conduction band (green) frontier orbitals are marked. The red arrow shows the lowest energy electronic excitation, indicating a direct bandgap for the MLs, but an indirect one for the BLs. Bandgaps (Δ) and spin–orbit splitting (Δ_SOC) are indicated. Adapted with permission from Reference 40.

Figure 2 also shows the importance of relativistic effects. Schwingenschlögl et al.^{Reference Zhu, Cheng and Schwingenschlögl39} predicted giant spin–orbit (SO) splitting for MoS₂ ML on the basis of density functional theory calculations. This strong quantum effect alters the electronic levels of the material, depending on their spin signature, due to the interaction of the intrinsic electron spin with its angular orbital momentum. This effect is also general for all 2H Group 6 MX₂ materials, with values reaching 480 meV for WTe₂.^{Reference Zibouche, Kuc, Musfeldt and Heine40}

It is intriguing to compare the electronic property changes of MX₂ MLs and BLs. While MLs show giant SO splittings, these are absent in the bulk and BLs, which is explained by the presence of an inversion center, resulting in the presence of two degenerate orbitals of opposite spin for any point in the band structure. Thus, it is possible to exploit the symmetry for tuning the SO splitting, offering interesting materials for spintronic applications. Any symmetry breaking will introduce SO splitting in the BL. One way to do this is by an external electric field normal to the basal plane, for example, provided by a gate voltage, as shown in Figure 3. This introduces a Stark effect––the applied electric field quasi-linearly splits the bands––and the SO splitting converges quickly to about the same values as found for the respective MLs.^{Reference Zibouche, Philipsen, Kuc and Heine41} After a field strength of ∼5 V/nm, a value that can be reached experimentally, the SO splitting is saturated. However, MX₂ MLs are insensitive to the presence of even strong gate voltages.^{Reference Zibouche, Philipsen, Heine and Kuc42}

Figure 3.

(a) The electric field (E _field) normal to the lattice plane is indicated. (b) The electric field breaks the degeneracy of the opposite spin orbitals at different valleys K and K′, thus introducing a spin–orbit (SO) splitting. σ– (σ+) denotes left (right)-handed circularly polarized light that couples to the band-edge transitions at K (K′) point. (c) SO splitting (Δ_SOC) in MX₂ bilayers (BLs) as a function of an external electric field normal to the lattice plane. Electric field already induces SO splitting in the MX₂ BLs for weak fields and quickly saturates to values typically observed in their respective monolayers.^{Reference Zibouche, Philipsen, Kuc and Heine41} Note: SOC, spin–orbit coupling.

External fields have another important effect on the electronic structure of MX₂ materials and can be used to tune the bandgap. Again, MLs are almost unaffected by external electric fields, with bandgap and effective electron and hole masses remaining stable.^{Reference Zibouche, Philipsen, Heine and Kuc42} Only at very high fields of about 50 V/nm do the bandgaps get smaller and eventually vanish, but at such a high field strength in the laboratory, MLs would not remain intact. BLs show a quasi-linear decrease in the bandgap in the presence of an external field (Figure 4).

Figure 4.

(a) Bandgap change (Δ) of Group 6 2H MX₂ materials as a function of a perpendicular electric field (E _field).^{Reference Zibouche, Philipsen, Kuc and Heine41} Semiconductor–metal transition is achieved for a field strength of at least 1.2 V Å^–1. (b) Bandgap change as a function of lattice parameter a.^{Reference Ghorbani-Asl, Borini, Kuc and Heine44} The equilibrium value is marked with a red circle. The range of direct bandgap transition is marked with dotted vertical lines and labeled Δ_D. Outside this small region around the equilibrium distance, the bandgap is indirect (Δ_I). The bandgap reduces almost linearly for compression (to the left) and expansion (to the right), where a semiconductor–metal transformation is reached at ∼10% strain. The figure also includes the reciprocal lattice points at which the smallest energy transition from valence to conduction band is possible.

Mechanical deformation is another way to change the electronic structure in MX₂ materials. Group 6 2H MX₂ materials readily form nanotubes and nano-onions, which have been investigated for their tribological properties. WS₂ nanotubes resist strain for elongations of 10% and more, thereby showing a linear strain–stress relation up to rupture.^{Reference Kaplan-Ashiri, Cohen, Gartsman, Ivanovskaya, Heine, Seifert, Wiesel, Wagner and Tenne43} In the 2D crystal, deformation is accompanied by strong electronic effects (straintronics). Under isotropic and unidirectional strain, the bandgap is almost linearly lowered until a semiconductor–metal transition is observed at about 10% isotropic strain (Figure 4).^{Reference Ghorbani-Asl, Borini, Kuc and Heine44} Note that strained MoS₂ ML, as well as related Group 6 MX₂ MLs, are no longer direct bandgap semiconductors.

The metallic state has been confirmed by explicit calculations of the quantum conductance, indicating that transport channels are opened. Unidirectional strain gives similar results but with a smaller magnitude. Also, unidirectional strain can be employed to introduce anisotropy to the electronic properties.^{Reference Ghorbani-Asl, Zibouche, Wahiduzzaman, Oliveira, Kuc and Heine45} However, it is difficult to introduce strain experimentally. One way is to employ an atomic force microscope (AFM) tip; computer simulations of an indentation experiment showed that the electronic structure of MoS₂ remains remarkably stable up to the point of piercing by the tip.^{Reference Lorenz, Ghorbani-Asl, Joswig, Heine and Seifert46} Another method is bending, or thermal expansion, which could be carried out on a substrate with a strong thermal expansion coefficient. However, due to the low friction coefficient of Group 6 MX₂, only a small amount of isotropic strain or compression can be introduced into a MoS₂ ML.^{Reference Plechinger, Castellanos-Gomez, Buscema, van der Zant, Steele, Kuc, Heine, Schüller and Korn47} Nevertheless, the first experiments with strain up to 2% confirm that the bandgap of MoS₂ ML can be tuned by strain with an accompanying bandgap decrease of ∼70 meV per % strain.^{Reference He, Poole, Mak and Shan48,Reference Conley, Wang, Ziegler, Haglund, Pantelides and Bolotin49}

The electronic properties of MX₂ materials are strongly affected by defects that often form due to the synthesis procedure. For instance, point defects lead to new photoemission peaks and enhanced photoluminescence intensity in MoS₂ MLs.^{Reference Tongay, Suh, Ataca, Fan, Luce, Kang, Liu, Ko, Raghunathanan, Zhou, Ogletree, Li, Grossman and Wu50} These effects can be attributed to the trapping potential of free charge carriers and to localized excitons. Simulations of the electronic structure of defective MoS₂ ML show that defects introduce mid-gap states that act as scattering centers.^{Reference Ghorbani-Asl, Enyashin, Kuc, Seifert and Heine51} Our non-equilibrium Green’s function transport simulations, where the current due to a potential drop between two leads in the presence of a bias voltage is calculated, show that single atomic vacancies can significantly reduce the average conductance and lead to significant anisotropy of electron transfer in MLs with grain boundaries. These results indicate that defects are important limiting factors in electronic transport,^{Reference Qiu, Xu, Wang, Ren, Nan, Ni, Chen, Yuan, Miao, Song, Long, Shi, Sun, Wang and Wang52} thus explaining a large variation of conductivity in different samples.

Similar to freestanding graphene, MX₂ MLs are not completely flat, which was shown from the microscopic structure of thin MoS₂ layers using high-resolution transmission electron microscopy and atomic force microscopy.^{Reference Brivio, Alexander and Kis53} MoS₂ MLs form ripples, which may reach heights of 6–10 Å for a lateral flake length of 6–10 nm. This was supported by Born–Oppenheimer molecular dynamics simulations. Here, forces on the atoms are calculated from first principles, while the dynamics of the atoms is calculated using classical theory. These simulations confirmed that the inherent dynamics of MoS₂ MLs lead to spontaneous ripple formation even at temperatures as low as 77K.^{Reference Miro, Ghorbani-Asl and Heine54} The ripples converged to the heights observed experimentally for supercells with lateral lengths of about 9 nm.

Field-effect devices

One of the most direct ways of accessing the electrical properties of a material is to incorporate it in an electronic device, for example in a FET (see Figure 5). In this multi-terminal device, current is injected by applying a voltage between source and drain electrodes, while the charge density can be modulated and the Fermi level varied using the gate electrode. FETs are not just convenient devices for probing the electrical properties of materials, but they are also the main building blocks of modern electronics, so demonstrating that a well-behaved FET can be made using a new semiconducting material is an important step in highlighting their technological relevance.

Figure 5.

(a) Schematic representation of the first top-gated field-effect transistor with a 2D transition-metal dichalcogenide (MoS₂) channel. (b) Device current as a function of gate voltage. The transistor could be completely turned OFF for gate voltages <–2 V, and it showed an ON/OFF current ratio of 10⁸. Inset: current as a function of bias voltage, showing linear I _ds–V _ds curves for small bias voltages.^{Reference Radisavljevic, Radenovic, Brivio, Giacometti and Kis7}

Radisavljevic et al. reported the first FET based on a 2D TMDC material in 2011.^{Reference Radisavljevic, Radenovic, Brivio, Giacometti and Kis7} It included a HfO₂ top-gate dielectric and showed a current ON/OFF ratio exceeding 10⁸ at room temperature and OFF-state currents smaller than 25 fA/µm. For comparison, an ON/OFF ratio of 10⁶ and an OFF-state current on the order of 1 nA/µm is considered acceptable for digitial electronic circuits. This showed for the first time that electronic devices with interesting performance levels could be realized based on 2D materials other than graphene. It resulted in the inclusion of MoS₂ and semiconducting TMDCs in the International Technology Roadmap for Semiconductors⁵⁵ in 2011.

The encapsulation of MoS₂ using insulating HfO₂ was crucial in achieving high performance levels in early devices. Subsequent temperature-dependent measurements of electrical transport in various geometries^{Reference Radisavljevic and Kis56} showed that encapsulation in HfO₂ can effectively increase the mobility of MoS₂ through a combination of homopolar phonon mode quenching^{Reference Kaasbjerg, Thygesen and Jacobsen57,Reference Kaasbjerg, Thygesen and Jauho58} as well as charged impurity screening by the dielectric and the top gate.^{Reference Ong and Fischetti59} A complementary approach to achieving high mobility involves vacuum annealing, resulting in the removal of adsorbates,^{Reference Jariwala, Sangwan, Late, Johns, Dravid, Marks, Lauhon and Hersam60,Reference Baugher, Churchill, Yang and Jarillo-Herrero61} lower noise,^{Reference Xie, Sarkar, Liu, Kang, Marinov, Deen and Banerjee62} and low-temperature mobilities in ML MoS₂ of ∼250 cm²/Vs extracted from Hall effect measurements and field-effect effective mobilities of ∼1000 cm²/Vs.^{Reference Baugher, Churchill, Yang and Jarillo-Herrero61} These values could be significantly underestimated and lower than the intrinsic limit^{Reference Kaasbjerg, Thygesen and Jacobsen57} due to the presence of electronic trap states^{Reference Zhu, Low, Lee, Wang, Farmer, Kong, Xia and Avouris63} and the formation of a band tail, as seen in Figure 6a. The smearing of the band edge due to various types of inhomogeneities could include sulfur vacancies, charges trapped at the TMDC/dielectric interface, or other structural defects. The extent of the band tail in MoS₂ was characterized by Zhu et al., who found a ∼100 meV characteristic energy width of the band tail using frequency-dependent capacitance–voltage measurements^{Reference Zhu, Low, Lee, Wang, Farmer, Kong, Xia and Avouris63} (Figure 6b).

Figure 6.

(a) Density of states (DOS) for MoS₂, including both localized band-tail and mid-gap localized states. E _mid is the energy level in the middle of the bandgap and E _M is the energy of the mobility edge. Inset shows a simplified band diagram of the device. (b) Multi-frequency capacitance of a MoS₂ test structure as a function of gate voltage (V _tg). C _ox is the oxide capacitance. The symbols are experimental results, and the lines indicate modeling results.^{Reference Zhu, Low, Lee, Wang, Farmer, Kong, Xia and Avouris63} Data and the model show reasonable agreement. Adapted with permission from Reference 63. © 2014 Nature Publishing Group. Note: HK, high-κ (dielectric constant) dielectric; E _F, Fermi energy.

While further increases in the mobility of MoS₂ are to be expected, it is already large enough to allow operation of MoS₂-based transistors in the GHz range, where current, voltage, and power gains were demonstrated.^{Reference Krasnozhon, Lembke, Nyffeler, Leblebici and Kis64} These performance figures are mostly limited by the contact resistance.

Reducing the contact resistance due to the Schottky barrier between the 2D semiconductor and the metal and realizing an ohmic contact is one of the most important challenges for engineering a high-quality device. This is, for example, crucial in achieving highly responsive photodetectors.^{Reference Lopez-Sanchez, Lembke, Kayci, Radenovic and Kis65} One way to achieve this is by using low work-function metals as contacts, such as Ni⁶⁶ or Sc.^{Reference Popov, Seifert and Tománek67,Reference Das, Chen, Penumatcha and Appenzeller68} In addition to choosing the metal with an advantageous work function, other factors such as the orbital overlap between the metals and TMDCs and the contact geometry (top versus edge contacts) also play important roles, as shown by Kang et al. first in their theoretical study^{Reference Kang, Liu, Sarkar, Jena and Banerjee69} and later in the experimental demonstration of low-resistance Mo contacts to MoS₂.^{Reference Kang, Liu and Banerjee70}

A complementary approach is to reduce the Schottky barrier height by doping the semiconducting region under the contact (e.g., by p-doping WSe₂ devices to NO₂^{Reference Fang, Chuang, Chang, Takei, Takahashi and Javey31} in order to improve Pd contacts, n-type doping using the chemical compound benzyl viologen to improve Ni contacts to MoS₂,^{Reference Kiriya, Tosun, Zhao, Kang and Javey71} or chlorine doping in the case of Ni on WS₂ and MoS₂).^{Reference Yang, Majumdar, Liu, Du, Wu, Hatzistergos, Hung, Tieckelmann, Tsai, Hobbs and Ye72} The biggest reduction in contact resistance reported so far was by using the radical approach of converting MoS₂ in the contact region from the semiconducting 2H to the metallic 1T phase by locally increasing the charge doping level using lithiation,^{Reference Kappera, Voiry, Yalcin, Branch, Gupta, Mohite and Chhowalla73} resulting in a contact resistance of less than 300 Ω·µm. Previous typical contact resistance values were above 1 kΩ·µm. Not only can MoS₂ be semiconducting or metallic, but at sufficiently high doping levels, it can also become superconducting, with a gate tunable transition temperature showing a maximum at 11 K.^{Reference Ye, Zhang, Akashi, Bahramy, Arita and Iwasa74}

Because 2D TMDCs are direct bandgap semiconductors, FETs based on these materials can also be used as simple and efficient photodetectors. Early phototransistors based on MoS₂ had a photoresponsivity of 7.5 mA/W.^{Reference Yin, Li, Li, Jiang, Shi, Sun, Lu, Zhang, Chen and Zhang75} Using layers with different thicknesses, the responsivity of the photodetector can be tuned to different wavelength ranges.^{Reference Lee, Min, Chang, Park, Nam, Kim, Kim, Ryu and Im76} Lopez-Sanchez et al.^{Reference Lopez-Sanchez, Lembke, Kayci, Radenovic and Kis65} reported ML MoS₂ phototransistors with a high photoresponsivity of 880 A/W.

Experimental work on other semiconducting TMDCs has also started. While MoS₂ is usually n-type, p-type FET devices based on WSe₂ have been realized showing a room-temperature hole mobility of 250 cm²/Vs³¹ as well as n-WSe₂ transistors with a room-temperature mobility of 142 cm²/Vs.^{Reference Liu, Kang, Sarkar, Khatami, Jena and Banerjee32} Using liquid gating, in which an ionic liquid is used in place of a solid dielectric, it was possible to achieve ambipolar operation in ML WSe₂^{Reference Allain and Kis77} as well as in WS₂.^{Reference Jo, Ubrig, Berger, Kuzmenko and Morpurgo78}

While most of the work in 2D TMDC-based transistors has focused on using a 2D material as the semiconducting channel, the wide range of electrical properties represented by 2D materials, from insulating such as boron nitride (BN) to superconducting, implies that, in principle, all parts of an FET structure can be replaced by 2D materials. The lack of dangling bonds in 2D materials also weakens the usual constraints of lattice matching that limit the possible combinations for “classical” three-dimensional materials. This allows for the realization of a host of new types of devices, such as vertical FETs with several BN¹² or TMDC layers^{Reference Georgiou, Jalil, Belle, Britnell, Gorbachev, Morozov, Kim, Gholinia, Haigh, Makarovsky, Eaves, Ponomarenko, Geim, Novoselov and Mishchenko79} sandwiched between graphene electrodes.

Being a good conductor, graphene can be used as an electrical contact for lateral electrical transport through MoS₂ and other 2D semiconductors. Bertolazzi et al.^{Reference Bertolazzi, Krasnozhon and Kis80} demonstrated flash memory cells based on ML MoS₂ contacted by low-resistance ohmic-like graphene contacts (see Figure 7). By using BN as the gate insulator, the entire FET structure could be implemented using 2D materials as the contact, semiconducting channel, and gate insulator.^{Reference Roy, Tosun, Kang, Sachid, Desai, Hettick, Hu and Javey81}

Figure 7.

(a) MoS₂/graphene heterostructure memory based on a MoS₂ transistor with graphene contacts. (b) Optical image of MoS₂ contacted by graphene (upper panel) and multilayer graphene floating gates deposited on top of the semiconducting channel (lower panel).^{Reference Bertolazzi, Krasnozhon and Kis80} Note: ML, monolayer.

Electronic circuits

Electronic circuits based on 2D TMDC semiconductors have already been demonstrated. Radisavljevic et al.^{Reference Radisavljevic, Whitwick and Kis82} realized the first such circuit, which consisted of two transistors on the same piece of ML MoS₂ and operating as an inverter. The gain of the inverter was higher than 4, indicating that such logic gates could be integrated into more complicated circuits without attenuation of voltage levels. The same device can also be operated as an analogue amplifier^{Reference Radisavljevic, Whitwick and Kis83} for small signals. Logic circuits based on BL MoS₂^{Reference Wang, Yu, Lee, Shi, Hsu, Chin, Li, Dubey, Kong and Palacios84} and complementary inverters based on WSe₂^{Reference Tosun, Chuang, Fang, Sachid, Hettick, Lin, Zeng and Javey85} were also demonstrated, as were ring oscillators and random access memory circuits.^{Reference Wang, Yu, Lee, Shi, Hsu, Chin, Li, Dubey, Kong and Palacios84} Bertolazzi et al.^{Reference Bertolazzi, Krasnozhon and Kis80} demonstrated flash memory cells, with operating principles similar to the ones used in consumer electronic devices such as cell phones and cameras, in a heterostructure geometry. Such devices could offer reduced crosstalk between neighboring memory cells and could be scaled to smaller sizes due to the high sensitivity of the ML MoS₂ semiconducting channel to the charge stored on the multilayer graphene floating gate.

Flexible devices

In addition to its interesting electronic applications, TMDC semiconductors have favorable mechanical properties. The first mechanical measurements of the stiffness and breaking strength of single and BL MoS₂ were carried out by suspending MoS₂ membranes on perforated substrates and deforming them using an AFM tip.^{Reference Bertolazzi, Brivio and Kis86} These experiments showed that MoS₂ has a Young’s modulus of ∼270 GPa, higher than that of steel (∼205 GPa). The breaking strength of single and BL MoS₂ was between 6% and 11% of their respective Young’s modulus, which represents the upper theoretical limit of a material’s breaking strength^{Reference Griffith87} and reflects the intrinsic properties of its interatomic bonds. This makes ML MoS₂ the strongest semiconducting material and suggests that MoS₂ is suitable for integration with flexible plastic substrates, such as polyimide, whose maximum strain before failure is ∼7%. Similar mechanical properties are expected in all TMDC materials because of the similar nature of interatomic bonds in these materials. Therefore, MoS₂ is an interesting material for flexible electronics and nanoelectromechanical systems. Castellanos-Gomez et al.^{Reference Castellanos-Gomez, van Leeuwen, Buscema, van der Zant, Steele and Venstra88} realized the first devices of this kind when they demonstrated nanomechanical resonators based on ML MoS₂ membranes with resonant frequencies of up to 30 MHz. A number of flexible FETs based on MoS₂ have been realized using liquid gels,^{Reference Pu, Yomogida, Liu, Li, Iwasa and Takenobu89} high-κ (where κ is the dielectric constant),^{Reference Chang, Yang, Lee, Tao, Hwang, Jena, Lu and Akinwande90} or BN^{Reference Lee, Yu, Cui, Petrone, Lee, Choi, Lee, Lee, Yoo, Watanabe, Taniguchi, Nuckolls, Kim and Hone91} as dielectrics, and show robust electrical operation against mechanical deformation.

Summary

We have summarized the most recent work on MoS₂ and other Group 6 TMDC materials, focusing on their electronic properties and their potential to enter the field of next-generation 2D electronics, including applications in low-power electronics (due to reduction of leakage currents and contact resistance), flexible electronics, optoelectronics, and straintronics. The enormous progress of the field, ranging from fundamental science up to device implementation, is reflected by the large number of scientists working in the field, even though the first transistor on a MoS₂ ML was reported only recently. MoS₂ and other TMDCs are being developed, in synergy with graphene and hexagonal boron nitride, as important material components for 2D electronics. The work has just begun, and a large number of TMDCs have not even been synthesized or studied.