Bandgap engineering of two-dimensional semiconductor materials

Abstract

Semiconductors are the basis of many vital technologies such as electronics, computing, communications, optoelectronics, and sensing. Modern semiconductor technology can trace its origins to the invention of the point contact transistor in 1947. This demonstration paved the way for the development of discrete and integrated semiconductor devices and circuits that has helped to build a modern society where semiconductors are ubiquitous components of everyday life. A key property that determines the semiconductor electrical and optical properties is the bandgap. Beyond graphene, recently discovered two-dimensional (2D) materials possess semiconducting bandgaps ranging from the terahertz and mid-infrared in bilayer graphene and black phosphorus, visible in transition metal dichalcogenides, to the ultraviolet in hexagonal boron nitride. In particular, these 2D materials were demonstrated to exhibit highly tunable bandgaps, achieved via the control of layers number, heterostructuring, strain engineering, chemical doping, alloying, intercalation, substrate engineering, as well as an external electric field. We provide a review of the basic physical principles of these various techniques on the engineering of quasi-particle and optical bandgaps, their bandgap tunability, potentials and limitations in practical realization in future 2D device technologies.

Nearly all modern semiconductor devices employ some type of band-structure-engineered configuration, through the use of heterostructures¹, superlattices², strain³, alloying⁴, or other effects. Such band structure engineering can lead to profoundly different properties for the constituent materials. For instance, strain engineering in silicon has allowed transistors with up to fourfold higher hole mobilities as compared with unstrained devices⁵, and strain engineering in lasers enables vastly reduced threshold currents to be achieved, and allows precise control over the emission wavelength⁶. Multilayer heterostructures have been demonstrated to provide charge carrier and optical confinement to improve transistor performance⁷ and realize lasers and light-emitting devices⁸, while miniband formation in superlattice structures can enable groundbreaking new device concepts such as quantum cascade lasers⁹. Over the past few decades, heterostructures have also led directly to advances in fundamental physics, such as quantum tunneling phenomena¹⁰ and the fractional quantum Hall effect¹¹. In fact, it is now difficult to imagine a world before band-structure-engineered devices and materials.

In current electrical and optical technology, it is highly desirable to be able not only to tune the semiconductor properties, but also to miniaturize the corresponding devices toward nearly atomically thin dimensions. These considerations focused attention to natural or synthesized materials that are formed by stacking individual atomic layers commonly referred to as 2D materials. 2D materials are characterized by a layered crystal structure with strong in-plane bonds, where layers are coupled together by weak van der Waals (vdW) forces. Currently, hundreds of 2D materials are known^12,13. Many are natural semiconductors, along with metals and insulators. Because of the weak bonding between layers, numerous heterostructures between 2D materials can also be formed^14,15.

Unlike conventional quantum well semiconductors, vdW semiconductors allow for atomic control of their thicknesses. It is well known that thickness variations in semiconductor quantum wells lead to degradation in device performance¹⁶. We review the physics of bandgap scaling with layer number and how it differs from the usual effective mass model used in conventional quantum wells, direct-to-indirect bandgap transitions, and the influence of interlayer coupling. One of the most relevant ways of tuning bandgaps in 2D materials originates from the fact that they can be stacked in arbitrary fashion, largely unconstrained by the lattice-matching requirements as in conventional quantum well heterostructures. We review the large degrees of freedom that such an approach can offer in terms of bandgap and band alignment engineering, the influence of stacking, twist angle, interlayer coupling, and experimental growth of these heterostructures. Most importantly, because of the very large surface-to-volume ratios of the 2D layers, their band structure should be very sensitive to external interactions, suggesting that their electronic structure and bandgaps could be modified by external perturbations. We review the tuning of bandgap with the application of an electric field, such as the Franz–Keldysh and Stark effect, the influence of dielectric environment on optical and quasi-particle (QP) gaps, and strain engineering. In all these cases, the bandgap tunability through the above-mentioned external perturbations is about one order of magnitude stronger than in their bulk counterparts. Last, the layered nature of these materials also lends itself to unique chemical approaches to materials engineering. We review the most common schemes, which include alloying, chemical doping, and intercalation of chemical species. These approaches can be very effective, allowing the smooth interpolation of electronic properties, such as through ternary, quaternary, or quinary alloys; tuning of optical gaps and their photoluminescence (PL); and semiconductor-to-metal transitions.

This review is organized as follows: we first provide a survey of the bandgaps of several 2D materials, along with a brief discussion about their most interesting features. We then discuss the influence of the number of layers on the bandgap and the excitonic properties in van der Waals heterostructures (vdWH). Next, we review the latest developments on the use of external electric fields, dielectric environment, and strain engineering for the control of the energies of QP gaps and excitonic peaks in 2D materials. Further, we review the effects of molecular intercala- tion, chemical doping, and alloying on the bandgap. Finally, the bandgap closing and metal–insulator transition in transition-metal dichalcogenides (TMDCs) mediated by structural phase transitions is discussed, followed by our concluding remarks.

Bandgaps in 2D materials family

In Fig. 1, we survey the crystal structures and bandgaps of 2D materials. Currently, research on 2D materials beyond graphene has become very extensive, and is raising expectations for a wide range of applications¹³. Following similar mechanical exfoliation approaches, monolayers of TMDCs were isolated following graphene^12,17. The TMDCs have chemical composition MX₂ (M = transition metals, and X = S, Se, and Te) and exhibit various structural phases, such as the 2H, 1T, 1T’, and 1T_d. In particular, the semiconducting 2H TMDCs (e.g., MoS₂) have bandgaps in the 1–2-eV range and received significant attention due to their interesting circular valley dichroism and exciton physics¹⁸. On the other hand, the T phases of TMDCs (e.g., WTe₂) are generally metallic to semimetallic¹⁹, and are also interesting for their topological properties²⁰. The chalcogenide-based 2D materials also include the semiconducting transition-metal trichalcogenide (e.g., TiS₃) whose crystal structure consists of quasi-one-dimensional atomic chains of stacked triangular prism²¹. There are also group-III-elements 2D chalcogenides, such as GaSe²², which come in different polytypes per their layer-stacking configuration. The group-IV-elements 2D chalcogenides consist of the buckled and puckered monochalcogenides (e.g., SnS and GeSe)²³, while their 1T dichalcogenides (e.g., SnS₂)²⁴ are mostly semiconducting. The family of 2D transition-metal carbides, nitrides, and carbonitrides are collectively referred to as MXenes²⁵, which exhibits favorable properties of ceramics, such as structural stiffness and good thermal and electrical conductivity of metals. They can have the following chemical formulae: M₂X, M₃X₂, and M₄X₃, where M is an early-transition metal and X is carbon and/or nitrogen (e.g., Ti₃C₂). Wide-bandgap III–V 2D semiconductors, such as Ga₂N₃, were also recently synthesized²⁶.

Fig. 1: Selected family of 2D materials and their bandgaps.

2D materials are chosen for their experimental significance and demonstration, with depictions of a perspective view of their crystal structures. Arrangement is in accordance with their bandgap, guided by the bottom wavelength/bandgap scale, whereas the bar beneath each structure indicates bandgap range from bulk to monolayer. Typically, the bulk bandgap is smaller than that of its monolayer (black bars), but there are exceptions (red bars). 2D materials on the far left, indicated by a gray box, are zero or near-zero bandgap, metallic, or semimetallic.

Large-bandgap materials such as hexagonal boron nitride (h-BN)²⁷ play a critical role in 2D materials, as its inert and ultraflat nature allows it to serve as a substrate for high-mobility 2D devices. Other large-bandgap materials include the transition-metal oxides (TMO) such as the 2H phase MoO₂, 1T phase MnO₂, and, more recently, the octahedral α-MoO₃, which exhibit hyperbolic optical behavior²⁸, the chromium oxide (e.g., Cr₂O₃), known for its multiferroic properties²⁹, and mica³⁰.

Other examples of layered materials include perovskites (e.g., SrTiO₃), which have been intensively studied as solar cell materials^31,32, and topological insulators (e.g., Bi₂Se₃ and Sb₂Se₃) known for their topologically protected and spin-momentum-locked electronic transport³³. Recent discovery of 2D Cr-based (e.g., CrI₃)³⁴ and Fe-based (e.g., Fe₃GeTe₂)³⁵ ferromagnetic materials, as well as semiconducting ferroelectric materials (e.g., In₂Se₃)³⁶, introduced an arsenal of functional materials to the existing 2D family. Besides graphene, there are also several notable examples of elemental 2D materials, such as the various phases of phosphorus (e.g., black and blue phosphorus)³⁷, silicene³⁸, germanene³⁹, tellurene⁴⁰, gallenene⁴¹, antimonene⁴², and borophene⁴³, which can range from metallic to semiconducting. Figure 1 depicts the bandgap range of the above-mentioned 2D materials from monolayer to bulk, showing the span of energy gaps across the electromagnetic spectrum from terahertz, infrared, and visible to ultraviolet.

Bandgap in van der Waals multilayers

Let us start with the layer dependence of bandgaps in a widely studied elemental semiconductor, namely, black phosphorus (BP). This material is a stable allotrope of phosphorus that can be exfoliated into multilayer puckered honeycomb lattices of phosphorus³⁷, see Fig. 1 for illustration of its crystal structure. The dependence of the absorption spectra of BP on its number of layers N has been experimentally probed by reflectance measurements^44,45,46, showing optical gaps ranging from 1.66 eV in monolayers to 0.30 eV in bulk. Typical absorption curves of N-layer black phosphorus (N-BP) are shown in Fig. 2a, showing characteristic peaks that can be traced to transitions between the electron and hole subbands of the same index i⁴⁷. Figure 2b compiles the energies of these subband transition peaks labeled as E_ii in Fig. 2a as a function of the number of layers (symbols). As we will discuss in what follows, the peak features observed in the absorption spectrum of N-BP can be well described within a simple theoretical model, i.e., solid lines in Fig. 2b.

Fig. 2: Bandgap engineering by stacking BP layers.

a Experimentally observed absorption spectrum in N-BP, N = 4, 5, 7, 8⁴⁶, where peaks are identified as the E_ij transitions labeled in (d). b Positions of experimentally observed (symbols) peaks labeled as E₁₁ (black), E₂₂ (red), E₃₃ (blue), and E₄₄ (green), as a function of the number of layers N. Theoretical predictions (see text) are shown as curves. c Energy bands in monolayer, bilayer, and trilayer BP. d Conduction (E_e) and valence (E_h) band edges, separated by a ${E}_{gap}^{(1)}$ gap in the monolayer case. As the number of layers N increase, more bands appear around E_e(h), with energies E_e(h) ± t_e(h), for 2-BP, and ${E}_{e(h)}\pm \sqrt{2}{t}_{e(h)}$, for 3-BP. For N-BP, N bands appear around E_e(h), whose energies can be estimated by the eigenvalues of a Toeplitz matrix (see text). Panels a, b reused from Springer Nature/Zhang et al.⁴⁶, permissible under a CC-BY [4.0/3.0] license.

The number of electron and hole bands in the band structure scales with N. Figure 2c shows the evolution of the band structure of few-layer BP with 1–3 layers, as obtained from tight-binding model and continuum approximation⁴⁸. At the Γ point, where the QP gap resides, the appearance of these new bands, with lower (higher) energy in the electron (hole) branch, allows the QP gap of black phosphorus to be tuned from 2.2 eV for monolayer to 0.3 eV in bulk. This is also consistent with predictions by density functional theory (DFT) calculations with GW corrections⁴⁹. The evolution of the bandgap with layer number can be drawn from a very simple model, which we will elaborate below.

If we assume each BP layer as a quantum well for electrons (holes), whose ground state is the band edge E_e(h), a coupled double-well model would predict the splitting of the conduction (valence) band in 2-BP, with energies E_e(h) ± t_e(h), as sketched in Fig. 2d, where t_e(h) is the nearest-neighbor interlayer hopping for electrons (holes). In fact, for N-BP, this simple model would consist of a tridiagonal matrix Hamiltonian, ${H}_{e(h)}={E}_{e(h)}\left|i\right\rangle \langle i| -{t}_{e(h)}| i\rangle \langle i+1| -{t}_{e(h)}^{* }| i+1\rangle \langle i| ,{\rm{where}}\ | i\rangle$ represents a state confined to the ith BP layer. Exact diagonalization of such a N × N matrix for N-BP leads to ${E}_{e(h)}^{n}={E}_{e(h)}-2{t}_{e(h)}\cos \left[n\pi /(N+1)\right]$ as the energies of the nth band edge for electrons (holes) at Γ. The peak features in the experimental absorption spectra in Fig. 2a are due to transitions between electron and hole bands with the same index, with energies

$${E}_{nn}^{(N)}={E}_{e}^{n}-{E}_{h}^{n}={E}_{gap}^{(1)}-2({t}_{e}-{t}_{h})\cos \left(\frac{n\pi }{N+1}\right),$$

(1)

where ${E}_{gap}^{(1)}={E}_{e}-{E}_{h}$ is the QP gap of 1-BP. The success of such simple theory is attested by the fair agreement between experimental data (symbols) and predicted transition energies (curves) with layer number N as shown in Fig. 2b.

In contrast to N-BP, the stacking of TMDC layers leads to systems with strikingly different electronic properties. Studies have found that the electronic bandgap, and its band structure in general, depends sensitively on the stacking order, which dictates its interlayer coupling. Indeed, stacking order significantly alters the crystal symmetry and electronic spectrum⁵⁰, from which second-order harmonics generation⁵¹, magnetism³⁵, and superconductivity⁵² might arise. Nevertheless, for the hexagonal 2H phase of TMDC, these multilayers show weak interlayer interactions at arbitrary fault angles, due to momentum mismatch between the electrons in neighboring layers. This lack of interlayer registry suppresses coherent interlayer motion of electrons, and the conduction and valence band-edge states of each layer remain quantum mechanically decoupled.

Here, we focus our discussion on AB-stacked TMDC, the energetically most favorable atomic configuration in bulk TMDC. As a representative of the TMDC materials, the band structure of bulk and monolayer WS₂ is shown in Fig. 3a. Spin–orbit splitting is disregarded here for the sake of simplicity. In the bulk, the valence (conduction) band edge is observed at the Γ (Q) point and the bandgap is therefore indirect. As the number of layers decreases down to bilayer TMDC, both band edges become comparable in energy with those at the K point. In monolayer WS₂, the band edges are located at K and the gap becomes direct. The direct transitions at K lead to prominent features in both absorption and PL spectra of monolayer TMDC^18,53,54. In the former, these peaks are usually labeled as A and B and assigned to excitonic transitions between different pairs of same-spin states at K. As for the latter, it exhibits an extra lower energy feature, due to the indirect gap transitions, usually labeled as I, as shown in Fig. 3b.

Fig. 3: Bandgap of bulk TMDC and dependence on the number of layers.

a Band structure of bulk (top) and monolayer (bottom) WS₂. High symmetry points are shown on the right side. Band structures of bulk MX₂ are qualitatively similar to this, where the lowest energy gap is indirect. This allows for photoluminescence originating from indirect gap transitions, labeled as I, as well as direct gap transitions, labeled as A and B. b Photoluminescence (PL) spectra of monolayer (1ML) to 8-layer (8ML) WS₂, measured at temperature T = 5 K, along with optical images of the samples (insets)⁵⁶. Labels A and I refer to the direct and indirect transitions sketched in (a), respectively. The B transition has higher energy, outside the range of energies in the figure, and is therefore not shown. c Dependence of the energy of the most prominent PL peak on the number of layers for different TMDC⁵⁸. Panel b used with permission from The Royal Society of Chemistry/Molas et al.⁵⁶; panel c reprinted by permission from Springer Nature Nature Communications⁵⁸, Copyright (2014), advance online publication, 06th February 2014 (https://doi.org/10.1038/ncomms4252 Nat. Commun.).

The decrease of the intensity of A and B PL peaks with the number of layers was observed in experiments^53,55,56,57, as a result of the transition from direct-to-indirect bandgap. The energies of these peaks, however, exhibit very weak dependence on the number of layers, being just slighly red-shifted (few tens of meV) from monolayer to bulk. This is reasonable, in light of the band structure shown in Fig. 3a, since the energy band is almost flat from K to H (out-of-plane crystallographic direction).

On the other hand, the I peaks observed in PL experiments from bilayer to bulk TMDC, where this peak represents the lowest energy transition, are highly sensitive to the number of MX₂ layers, as illustrated in Fig. 3c and reported in refs. ^53,55,56,57. Red shifts in this peak are observed in the range of 200–300 meV. Again, this can be understood from the band structure shown in Fig. 3a: in contrast to the energy bands along the K–H direction, the valence band in the Γ–A direction is not flat, which suggests much stronger interlayer hoppings for hole states. Equation (1) would predict the dependence of the I peak energy on the number of layers, as it has been done for WS₂ in ref. ⁵⁶, where the layer dependence of the I peak energy can be fitted with a quantum well-like expression as ${E}_{I}^{(N)}={E}_{0}+{\hslash }^{2}{\pi }^{2}/2{\mu }_{\perp }{L}^{2}$, where E₀ = 1.39 eV, μ_⊥ = 0.58m₀ is the reduced electron–hole effective mass along the vertical direction, m₀ is the free electron mass, and the system thickness is approximated as L = Na, where a = 6.16 Å is the thickness of monolayer WS₂. The correspondence between such quantum well approximation and the tight-binding model in Eq. (1) is revealed as one takes N → ∞ in this equation, and ${E}_{I}^{bulk}={E}_{gap}^{1}-2({t}_{e}-{t}_{h})\cos (ka)$. This cosine function is then an approximation of the energy dispersion along the vertical direction—its second derivative around Γ (Q) for holes (electrons) leads to electron (hole) effective masses μ_⊥ in ref. ⁵⁶, which are related to t_e(h) hopping parameters by ${m}_{e(h),\perp }={\hslash }^{2}/2{t}_{e}{a}^{2}$. As one can see in Fig. 1, the bandgap of layered materials is usually reduced as the number of layers increases, with only a few exceptions, such as mica³⁰, a wide-bandgap material with the unique property of exhibiting bandgap narrowing with the decrease in the number of layers.

The behavior of ReS₂, on the other hand, is particularly different from the other TMDC, as shown in Fig. 3c. Its lowest energy peak in PL is not strongly affected in either energy or intensity by the number of layers. The band structure of ReS₂ is very different from the one of MoS₂: as a consequence of a strong decoupling between ReS₂ layers, the band structure barely changes from monolayer to bulk⁵⁸. Similar behavior is also observed for ReSe₂⁵⁹. The octahedral 1T phase of TMDC, e.g., for ReS₂, undergoes a Peierls distortion resulting in buckled S layers and zigzag Re chains along one of the in-plane lattice vectors, which also lead to electronically and mechanically decoupled monolayers. These materials are thus not suitable for bandgap tuning via the number of layers.

Conversely, there is also TMDC that exhibits an exceptionally strong interlayer coupling energy. This is the case of the noble metal dichalcogenides, such as PdSe₂, PtSe₂, PdS₂, and PtS₂, whose bandgap is highly tunable by the number of layers, while their indirect nature is preserved from bulk all the way to the monolayer form. Bandgap in PdSe₂, for example, can be tuned from 1.37 eV (monolayer) to 0.5 eV (bulk)⁶⁰. As for PtSe₂, the monolayer gap 1.2 eV drops to only 0.3 eV already in the bilayer, and it becomes metallic for any higher number of layers⁶¹. These materials are thus promising candidates for future infrared and mid-infrared photonic and optoelectronic applications.

Bandgap and excitons in vdWH

Vertical stacking of 2D materials into heterostructures allows for the realization of a vastly expanded set of material combinations^15,62 that cannot be achieved in 3D materials. In type-I band alignment, conduction band minimum and valence band maximum are in the same material, thus maximizing the electron–hole overlap, which could enable new classes of light-emitting devices^63,64. On the other hand, type-II alignment, where conduction band minimum and valence band maximum are in different materials, as well as the semimetallic (zero-bandgap) type-III cases, has potential to create ultrafast photodetectors⁶⁵, monolayer solar cells⁶⁶, memory devices⁶⁷, and tunneling transistors⁶⁸.

An example of 2D vdWH is illustrated in Fig. 4a, which shows a microscope image of heterobilayer composing of WSe₂ monolayer deposited on top of a MoSe₂ monolayer⁶⁹. When different TMDCs are stacked to form vdWH, they typically do not have particular registry, leading to negligible interlayer coupling. Absorption peaks measured in the heterostructure reveal simply a superposition of exciton and trion peaks in each separate TMDC, except for small energy shifts. PL measurements, on the other hand, consistently exhibit an extra low-energy peak. This is associated with an interlayer exciton⁷⁰, i.e., an exciton where electrons and holes lie in different layers⁶⁹, as sketched in the inset of Fig. 4b. An example of such PL spectrum is shown in Fig. 4b, where neutral X${}_{Mo(W)}^{0}$ and negatively charged X${}_{Mo(W)}^{-}$ exciton peaks of Mo(W)Se₂ are seen at energies greater than ≈1.65 eV, along with an interlayer X_IL exciton peak at ≈1.38 eV⁶⁹. Indeed, the X_IL feature is present only in the PL of the vdWH region of the sample in Fig. 4a, whereas measurements at the monolayer Mo(W)Se₂ region exhibit only X${}_{Mo(W)}^{0}$ and X${}_{Mo(W)}^{-}$. Hence, vdWH figures as an efficient way to control optical bandgaps in TMDC.

Fig. 4: Optical gap from interlayer excitons in TMDC vdWH.

a Microscope image of a MoSe₂/WSe₂ van der Waals heterostructure. b PL spectrum (at T = 20 K) of the sample shown in (a)⁶⁹. The inset sketches intralayer (red and green) excitons in the Mo(W)Se₂ layer, associated with the features labeled as X${}_{Mo(W)}^{0}$ in the PL spectrum, and the interlayer (yellow) exciton, associated with the X_IL peak. X${}_{Mo(W)}^{-}$ refers to intralayer trions. c Top panels: band structures of single-layer MoS₂ (green) and WS₂ (red) superimposed, and band structure of a MoS₂/WS₂ heterostructure. Colors in the latter represent probability-density projections in each material. Bottom panels: the same as top, but for GeS (green) and SiTe (red) layers in the buckled form. d Survey of intralayer (triangles) and IL (circles) exciton peaks, experimentally observed in PL measurements of different combinations of TMDC as vdWH. Background colors delimit the range of energies observed for exciton absorption in single-layer experiments with MoS₂ (green), MoSe₂ (red), WS₂ (blue), and WSe₂ (yellow)^{69,75,76,77,78,79,80,81,82,83,84,85,86,87}. Colors of triangles are chosen as to match those corresponding to each material involved in the observed intralayer exciton peak. Panels a, b reprinted by permission from Springer Nature Nature Communications⁶⁹, Copyright (2015), advance online publication, 24th February 2015 (https://doi.org/10.1038/ncomms7242 Nat.Commun).

For rotation angles where the crystal lattices of the stacked TMDC layers are commensurate, namely, 0^∘ (AA stacking) and 60° (AA’ stacking), the conduction (valence) band minimum (maximum) is found at the K point of the Brillouin zone. The band-edge states are mostly composed of d orbitals of the transition-metal atoms, which are buried in-between the chalcogens. These orbitals from transition metals of different layers do not interact strongly, i.e., its bandgap at K should not differ much from a simple superposition of the bandgaps of its constituent materials⁷¹, as one can verify in the top panels of Fig. 4c, obtained from first-principles DFT calculations for a WS₂/MoS₂ heterostructure. In the case of separate monolayers, the band structure of Mo(W)S₂ is shown as green (red) lines, whereas in the vdWH, green (red) refers to the projection of the states on the Mo(W)S₂ layer. At K, the conduction (valence) band-edge state of WS₂/MoS₂ vdWH is practically fully confined within the Mo(W)S₂ layer, leading to negligible electron–hole overlap. As a consequence, despite the direct gap nature of the K-point electron–hole state, light absorption via valence-to-conduction transition at K point is much suppressed, since the optical susceptibility, as given, e.g., by Elliot formula⁷², depends not only on the dipole-matrix element of the bands states, but also on the electron–hole overlap of the exciton state. This explains the lack of an IL exciton peak in the absorption spectra of TMDC vdWH. In PL measurements, however, light emission from IL states is made possible as electrons and holes excited by light in the same material may drift toward different layers seeking for lower energy band edges, thus producing a charge-separated IL exciton.

This relaxation process is shown to be faster than recombination of intralayer excitons^73,74. Recent observations of IL excitons in PL spectra of MoSe₂/WSe₂ vdWH^75,76,77 suggest that these states may be indirect not only in real space, but also in reciprocal space. In refs. ^75,76, two IL exciton peaks are observed at ≈1.4 and ≈1.42 eV and interpreted as reciprocal space indirect excitons where the hole is at the K point, while the electron lies in a spin-split Q valley (see Fig. 4c, top). In ref. ⁷⁷, an IL exciton peak is observed at 1.33 eV and assigned to a similar reciprocal space indirect exciton, but the second IL exciton peak, at 1.38 eV, is attributed to a K–K direct transition. In a MoS₂/WSe₂ vdWH, evidence from the dependence of the lowest energy PL peak on the twisting angle between layers points toward an interpretation of this peak as being due to a reciprocal space indirect Γ-to-K IL exciton⁷⁸. These Γ–K and K–Q indirect excitons share the characteristic of having one of the quasi-particles spread among both TMDC layers, i.e., the particles are only partially separate in real space, which helps to enhance the light emission efficiency of these states.

Figure 4d surveys energies of experimentally observed IL exciton peaks (black circles) in vdWH: MoS₂/MoSe₂^79,80, MoS₂/WSe₂^78,81, MoSe₂/WSe₂^{69,75,76,77,82,83}, MoS₂/h-BN/MoSe₂⁸⁴, and MoSe₂/WS₂^85,86,87. A simple theoretical model can be made to predict the energy of such IL excitons: (i) once the band edges involved in the IL transition are identified, the effective masses of electrons and holes in those band edges are obtained, (ii) a tight-binding model for the stacked layers similar to the one proposed earlier is constructed for electron and hole states, from where IL hopping parameters t_e and t_h are obtained, (iii) electron–hole binding energies E_ij are calculated assuming an electron in layer i and a hole in layer j, using an appropriate interaction potential^88,89, and (iv) the system Hamiltonian is approximated to be ${H}_{\rm{exc}}={E}_{ij}\left|ij\right\rangle \langle ij| -[{t}_{e}| ij\rangle \langle i+1\,j| -{t}_{h}| ij\rangle \left\langle i\,j+1\right|+c.c.]$ and properly diagonalized, leading to the exciton energy states. Similar recipes were followed, e.g., in refs. ^71,90, as well as in ref. ⁷⁸ for a MoS₂/WSe₂ vdWH, which helped to understand the apparently high IL exciton-binding energy and peak intensity as being due to a partially charge-separated Γ–K IL exciton. K–K IL exciton transitions have also been observed in PL⁹¹ and electroluminescence⁹² experiments with MoS₂/WSe₂ vdWH. In this case, the IL peak is observed with lower energy (≈1.1–1.3 eV) as compared with the Γ–K one (≈1.58 eV, see Fig. 4d), as a consequence of the smaller IL gap at K point, as compared with the one at Γ (see, e.g., Supplemental Material of ref. ⁷⁸).

Intralayer (direct) exciton transitions are also featured in PL measurements of vdWH. Figure 4d shows the energies of these transitions as triangles. The vdWH surveyed in Fig. 4d are those that were experimentally realized to date, and where IL excitons were observed, but possible combinations of TMDC into vdWH are potentially vast. In order to investigate this further, Fig. 5a surveys the bandgaps theoretically obtained for stacked TMDC in the limit of zero interlayer coupling for different TMDC. In this limit, the bandgap is obtained by the difference between the minimum conduction and maximum valence band-edge energies of the constituent monolayers (Anderson model). These TMDC heterostructures exhibit bandgaps in the terahertz-to-infrared range, with different band alignments, namely, of types I–III.

Fig. 5: Bandgaps in TMDC and MMC vdWH.

a Quasi-particle bandgap values and types for various 2D vertical heterostructures composed of TMDC, obtained using DFT–PBE calculations and displayed as a heatmap. Monolayer bandgap values are given in the left column, while heterostructure bandgap values, as obtained by the Anderson rule, are shown in the upper triangular half. Type-I, -II, and -III heterostructures are represented by green-, red-, and blue-checkered boxes, respectively, in the bottom triangular region. Top and side views of crystal structure for the different TMDC phases, namely 2H, 1T, and 1T_d, are displayed on the far left. b The same as a, but for MMC heterostructures. Bandgap values for monolayer IV chalcogenide in puckered (buckled) phase are shown on the left (right) panel. In general, they do not obey the Anderson’s rule, due to strong interlayer coupling. Monolayer values are shown in the first column of each panel. Top and side views of crystal structures for puckered and buckled crystal groups are displayed as the inset. For crystal structure labels, refer to Fig. 1a.

The metal (group-IV) monochalcogenides (MMC) are also layered semiconductors that have recently been synthesized, and are theoretically predicted to form mechanically rigid heterostructures^{23,93,94,95,96,97,98}. Their monolayers are stable buckled and puckered structures, as depicted in the inset of Fig. 5b. The interlayer coupling for the puckered phase is generally stronger than the buckled counterpart. As a result, the energy gaps for the puckered heterostructures are smaller, mostly in the infrared. On the other hand, the gap in buckled heterostructures mostly spans from near-infrared to red. Figure 5b summarizes the bandgaps of group-IV-chalcogenide heterostructures, obtained from DFT calculations. Due to the strong interlayer coupling, the electronic wavefunctions are highly delocalized across the two layers⁹³, in contrast to the case of TMDC vdWH previously discussed. Hence, their bandgaps do not follow Anderson rule⁹⁹, as one can verify by comparing the band structures of, e.g., monolayer GeS (green) and SiTe (red) with those of a SiTe/GeS vdWH, which are shown in the bottom panels of Fig. 4c. Thus, from the point of view of electronic structure, a MMC stack should be viewed as a new material, instead of a heterostructure.

The vast majority of reports on 2D material-based heterostructures and devices have used flakes exfoliated and transferred from naturally occuring crystals or crystals grown by chemical vapor transport (CVT). However, those flakes have high levels of extrinsic impurities and intrinsic point defects, which lead to degraded material and device performance with high variability^100,101. Furthermore, it is well established that heterostructures that are created through exfoliation and even the “cleanest” subsequent transfer processes¹⁰², still result in residual contamination at interfaces and surfaces, which again degrades the heterostructure properties¹⁰³.

Chemical vapor deposition (CVD) of TMDC has been shown to be quite promising^104,105. In particular, CVD films have been demonstrated with very large grains¹⁰⁶ and reduced impurity¹⁰⁷ and structural defect levels. Transport in CVD-grown TMDC is also quite competitive with reported field-effect mobilities of 20–90 cm²/Vs^{106,108,109,110,111} for MoS₂ and 30 cm²/Vs (true MOCVD)^8,112 and 95 cm²/Vs (powder vaporization)^113,114 for WSe₂ that is approaching the values reported for flakes exfoliated from geologic crystals. Heterostructures using MOCVD have also been reported. Successful heterostructures include MoS₂/WSe₂^115,116, WSe₂/MoSe₂¹¹⁷, WTe₂/WSe₂¹¹⁸, GaSe/MoSe₂¹¹⁹, and SnS₂/WSe₂¹²⁰. However, CVD typically requires growth temperatures in excess of 700 °C that is often much too high to be compatible in heterostructures with underlying chalcogenide layers¹²¹. The elevated growth temperatures can result in point defect and vacancy formation and undesired reactions at interfaces¹¹⁶.

Molecular beam epitaxy (MBE) can oftentimes solve these challenges enabling the growth of 2D materials and heterostructures with ultraclean interfaces, as illustrated in Fig. 6. Enhanced quality films and interfaces with very low-impurity concentrations are enabled in MBE through the utilization of elemental sources of high purity coupled with the cleanliness of ultrahigh vacuum. Moreover, the lower growth temperature capability of MBE minimizes issues associated with vertical heterostructure growth, including vacancy formation, layer intermixing, and interface chemical reactions. MBE, therefore, enables a variety of novel materials, including 2D-layered oxides, nitrides, arsenides, and the wide variety of 2D chalcogenides^{122,123,124,125,126,127,128}. TMDC materials, such as WSe₂¹²⁹, MnSe₂¹³⁰, MnTe₂ and VSe₂¹³¹, VTe₂ and HfSe₂¹³², SnSe₂¹²⁵, MoSe₂¹³³, MoTe₂¹³⁴, and WTe₂¹³⁵, have all been grown by MBE. In addition, TMDC alloys with mixed metals or mixed chalcogens have been advanced lately, further expanding the library of potential TMDC and heterostructures^{136,137,138,139,140,141}.

Fig. 6: TEM images of a variety of van der Waals heterostructures grown by molecular beam epitaxy.

a HfSe₂ grown on MoS₂¹³², b WSe₂ grown on Bi₂Se₃²⁷⁷, c WTe₂ grown on MoS₂¹³⁵, and d helical Te grown on graphite²⁷⁸. In each case, the interfaces between the two materials are atomically sharp with no misfit dislocations due to the lack of strain in the covalent-free interface interactions. Heterostructures can therefore be designed and synthesized with minimal regard for lattice matching. Panels reproduced with permission from a reprinted with permission from¹³² ACS Nano 2015, 9, 1, 474–480 Publication Date: December 11, (2014) https://doi.org/10.1021/nn5056496 Copyright (2014) American Chemical Society; panel b reused from IOP Publishing/Yue et al.²⁷⁷ permissible under a CC-BY [4.0/3.0] licence; panel d used with permission from John Wiley and Sons/Zhou et al.²⁷⁸.

In TMDC and other van der Waals materials, the layer-to-layer or layer-to-substrate interactions are much weaker than the intralayer bond strength. Coupled with the surface termination of van der Waals crystals, layer-by-layer growth without islanding can be achieved, although determining the growth parameter window that enables that this can be a challenge¹⁴². Experimental results of epitaxially grown TMDC have demonstrated that TMDC grown on other inert, hexagonal substrates (other TMDC, graphite, etc.) grows unstrained, with its own lattice constant, and does not form any misfit dislocations¹⁴³. Utilizing this so-called van der Waals epitaxy enables¹⁴⁴ vertical heterostructures with constituent materials chosen almost exclusively for their electronic properties with signifcantly relaxed criteria for crystal lattice matching. Recent findings have also indicated that through controlled growth conditions, the stacking sequence of thin films of TMDC can be altered¹⁵. In a high chalcogen background environment in conjunction with a relatively fast growth rate, for example, one can promote differences in edge dimerization and growth rates that result in AA stacking in TMDC that normally has AB stacking in equilibrium (like WSe₂ or MoS₂). Atomic heterostructuring is an exciting approach whose full potential is yet to be explored.

External electric field effect

In the presence of an external electric field F, energy band edges in a semiconductor are distorted in real space, so that electrons and holes are pushed toward opposite directions. As a consequence, their QP envelope wave functions are no longer sine waves, but rather Airy functions. These functions, however, still exhibit a tail that makes the electron–hole overlap nonnegligible, thus allowing for a finite probability of interband transition mediated by light absorption. As illustrated in Fig. 7a, neglecting electron–hole interactions and excitonic effects, the QP-state transition has energy E_FK, which is smaller than the zero-field gap E_g. This phenomenon, also known as the Franz–Keldysh effect, is responsible for a tail in the absorption spectrum of semiconductors at energies below the zero-field QP gap of the material, which suggests the use of an electric field to actively tune optical gap properties of semiconductors.

Fig. 7: Tuning the bandgap by an applied electric field.

a Sketch of the quantum-confined Franz–Keldysh effect (top): in the presence of an applied field (blue, F ≠ 0), the otherwise constant bands (black, F = 0) are tilted, which pushes electrons and holes toward opposite sides of the system. Electron and hole wave functions are given by Airy functions (red) with tails that slightly overlap, thus allowing for a light absorption-induced transition with energy E_FK lower than the original quasi-particle gap E_g of the system. Bottom panels show an experimental observation of the Franz–Keldysh effect in light absorption for few-layer (9-nm-thick) BP, with several values of applied bias V_g (in volts) and light polarized along the armchair (AC) and zigzag (ZZ) directions of the material¹⁴⁵. b Experimental observation of Franz–Keldysh effect as a subgap shoulder (FK) in the absorption spectrum, which emerges among A exciton peaks as the gate voltage V_B increases in monolayer WSe₂ at temperature T = 30 K¹⁵¹. c Band structure of 2-BP, as obtained with tight-binding (red dashed) and continuum (blue solid) models, in the presence of 1.5-eV (top) and 2-eV (bottom) bias¹⁵⁸. d Experimental verification of the bandgap closing and consequent formation of a Dirac cone in few-layer BP due to an effective bias across the sample¹⁵⁵. Experimental data and theoretical simulations are shown together, for comparison. e Stark shift of the PL peak (at T = 4.2 K) due to IL excitons in a MoSe₂/WSe₂ vdWH, as a function of bias induced by top and bottom gates, set at potentials V_TG and V_BG, respectively. Two energy peaks, labeled as IX₁ and IX₂ are consistently observed⁷⁶. Panel a reprinted with permission from¹⁴⁵ Nano Lett. 2017, 17, 10, 6315–6320 Publication Date: September 27, 2017 https://doi.org/10.1021/acs.nanolett.7b03050, Copyright (2017) American Chemical Society; panel b reused from Springer Nature/Massicotte et al.¹⁵¹, permissible under a CC-BY [4.0/3.0] license; panel d used with permission from AAAS/Kim et al.¹⁵⁵; panel e reprinted by permission from Springer Nature Nature Photonics⁷⁶, Ciarrocchi, A., Unuchek, D., Avsar, A. et al. Polarization switching and electrical control of interlayer excitons in two-dimensional van der Waals heterostructures. Nature Photon. 13, 131–136 (2019), Copyright (2019) advance online publication, 31st December 2018 (https://doi.org/10.1038/s41566-018-0325-y Nature Photon.).

The 2D nature adds another feature to this scenario: due to the small thickness of the material, electrons and holes pushed toward the top and bottom layers become strongly confined, in what is called the quantum-confined Franz–Keldysh (QCFK) effect. The consequence of this effect is clearly seen in the modulation of the absorption spectrum of 9-nm-thick few-layer BP in the bottom panels of Fig. 7a¹⁴⁵. When light is polarized along the zigzag (ZZ) direction of the N-BP crystal lattice, no feature is observed, as expected, due to particular selection rules in BP¹³. For polarization along the armchair (AC) direction, however, very clear features are observed, such as the presence of absorption peaks and a modulation of the absorption coefficient. Although the Franz–Keldysh effect alone tunes the bandgap only within the range of a few tens of meV, much broader bandgap tuning is achieved as stronger electric fields are applied perpendicularly to multilayer BP^146,147 and TMDC¹⁴⁸.

For monolayer TMDC, the excitonic Stark shifts are predicted to be small, of the order of a few meV^149,150. This has been confirmed in a recent experiment with monolayer WSe₂¹⁵¹. On the other hand, a clear subbandgap shoulder related to the Franz–Keldysh effect in this case is observed ≈70 meV below the QP bandgap of this material, amid excitonic state peaks. This is illustrated in Fig. 7b, where the shoulder is seen to emerge as the gate potential V_B increases from −0.2 V up to 2.5 V, while the few meV Stark shift of the A exciton peak is not visible in the plot.

Theoretical works^{146,152,153,154} have suggested that applying an electric field perpendicular to N-BP lowers its QP gap and may close it for high enough fields¹⁵⁵. In fact, a variation of the bandgap of few-layer BP as large as 200 meV has been observed for a 2 V/nm displacement field, due to quantum-confined Stark effect, in a SiO₂/BP/BN-sandwiched structure¹⁵⁶. Such gate-tunable giant Stark effect in few-layer BP was confirmed by a direct measurement, using low-temperature scanning tunneling microscopy¹⁵⁷. Further increase in the electric field then produces a linear band crossing along the AC direction, as illustrated in Fig. 7c¹⁵⁸. This perculiar band structure was subsequently experimentally observed in ARPES measurements¹⁵⁵, as illustrated in Fig. 7d, where experimental data and theoretical simulations are shown side by side for comparison. In this experiment, the strong electric field accross the N-BP layers is obtained by doping the uppermost layer with potassium atoms. Similar surface-doping technique has also been used to tune the bandgap of multilayer TMDC up to hundreds of meV¹⁵⁹.

Bandgap modulation induced by a vertical electric field has also been observed in electrical experiments^{156,160,161,162,163}, with significant impact on the transport properties of 2D materials as well. For example, a room-temperature hole Hall mobility of few-layer BP as high as 5000 cm²/Vs is reported in BN/BP/BN van der Waals quantum well¹⁶⁰, which exceeds the theoretical limit¹⁶¹ without consideration of QCFK^156,162,163.

Electric field tuning of electronic properties of multilayer TMDC is also relevant in the context of vdWH composed of these materials⁷¹. As previously discussed, vdWH exhibits a low-energy PL peak related to the IL exciton state. Due to the intrinsic electric dipole across the layers, application of a perpendicular electric field yields a predominantly linear Stark shift⁶⁹. Figure 7e shows an example of such linear Stark effect, experimentally observed in a MoSe₂/WSe₂ vdWH, where two interlayer exciton states, labeled IX₁ and IX₂, are consistently observed and interpreted as being formed by indirect and direct transitions in the reciprocal space, respectively⁷⁶. In this case, a 138-meV tuning of this lowest-transition energy peak is obtained. Similar tuning is also obtained for IL exciton peaks in MoS₂/WSe₂ vdWH in ref. ⁹².

Excitons trapped in defects in monolayer TMDC have been observed as intense features in the PL spectra with energy slightly lower than the optical gap. These optically active defects have attracted interest as possible quantum emitters at low temperature. The energy of these peaks can also be tuned by an external applied field, via quantum-confined Stark effect¹⁶⁴. In this case, trapped exciton states can exhibit either linear or quadratic Stark shifts, depending on the character of the defect. The linear contribution is dominant, provided there is an intrinsic electric dipole moment, as a result of a defect that lacks of a center of symmetry. Linear Stark shifts of a few tens of meV were experimentally observed for optically active defects in monolayer WSe₂ encapsulated in h-BN¹⁶⁴.

Dielectric screening and many-body effects

It is clear that in a 2D semiconductor, the combination of dielectric screening by the material layer itself and that provided by its surrounding environment plays a major role in defining not only exciton-binding energies, but also the QP gaps^18,165,166. However, it is known that the dielectric screening-induced shifts of these two effects are opposite; hence, their isolated effects are difficult to discern experimentally. Indeed, early attempts to understand optical gaps under the influence of the surrounding dielectric environment were inconclusive^{165,167,168,169,170,171,172,173,174,175,176}. In these reports, either fused silica or oxides, such as SiO₂, MgO, and sapphire, are used as substrates and/or capping layers. Only recently, experimental studies demonstrated that encapsulation of 2D semiconductors with different layered materials, such as h-BN and graphene^166,167, produced significant effects in the optical bandgap consistent with theory. This ushered in a new approach of optical gap engineering via encapsulation techniques and dielectric engineering.

A clear experimental evidence of this effect has been reported in ref. ¹⁶⁶, where the excitonic ground and excited states in a WS₂ monolayer in the presence of a few-layer graphene capping are carefully investigated by differential reflectivity measurements. On the one hand, additional screening from the graphene layers should reduce electron–hole interactions and lead to a blueshift of the excitonic peak. On the other hand, it should also reduce electron–electron interactions, which leads to a redshift in the QP gap renormalization^177,178. Overall, the latter was found to dominate, and a red-shift in the optical gap was observed.

Figure 8a surveys experimentally obtained optical gaps under different dielectric environment for various TMDC, with optical gap tuning in the range of a few hundred of meV^{165,166,167,176,179,180}. As for BP, it is commonly covered by capping layers of different materials, such as h-BN and sapphire. A recent theoretical study shows that such encapsulation causes significant changes (≈120 meV for 1-BP) on the optical gap¹⁷³. This has been experimentally verified in 2-BP and 3-BP^173,181, by comparing the results for N-BP supported on SiO₂, PDMS^45,46, and h-BN substrates⁴⁴, with and without further coating with sapphire.

Fig. 8: Bandgap engineering by dielectric environment.

a Survey of recent experimentally observed optical gaps of 2D materials in different dielectric environments. For TMDC, lowest energies correspond to samples encapsulated in h-BN. In general, energies increase as the substrate is changed from h-BN to SiO₂, CYTOP, MgO, Al₂O₃, and LaAlO₃, respectively¹⁶⁷. For N-BP, lower energies correspond to samples on sapphire covered by h-BN¹⁷³, whereas uncovered samples on Si/SiO₂ substrate¹⁸¹ exhibit higher energy. b Difference between optical bandgaps (top) of the suspended monolayer (ε = 1) and those in a medium with an effective dielectric constant ${\varepsilon }_{m}=\frac{{\varepsilon }_{1}+{\varepsilon }_{2}}{2}$ for four TMDC, as calculated by solving SBE for the gapped Dirac Hamiltonian⁷². The parameter r₀ was fitted for each TMDC as to obtain the experimental optical bandgap: 1.91 eV for suspended MoS₂²⁷⁹ using r₀ = 108 Å, 1.66 eV for MoSe₂ deposited on SiO₂²⁸⁰ with r₀ = 90 Å, 1.74 eV for WSe₂ deposited on SiO₂¹⁷⁸ with r₀ = 47 Å, and 2.04 eV for WS₂ deposited on SiO₂²⁸¹ with r₀ = 23 Å. The other parameters were obtained from the GGA calculations of ref. ²⁸². The effect on the QP bandgap, as calculated by the method explained in the Supplemental Material, is shown in the bottom panel.

Understanding the physics behind the dielectric screening effect on the optical gap is fundamental to provide control of its optoelectronic properties. Let us provide a glimpse of how such a model could be constructed. The simplest way to see the dependence of effective electron–electron interaction in 2D materials on the screening by the environment is to consider a thin slab with width d cladded between two different dielectrics with permittivities ε₁ and ε₂. In the d → 0 limit, the eletrostatic potential due to a charge bound to the slab is given by the Rytova–Keldysh (RK) potential^182,183

$${V}_{{\rm{R}}K}(q)=\frac{{e}^{2}}{2{\varepsilon }_{0}{\varepsilon }_{m}}\frac{1}{q({r}_{0}q+1)},$$

(2)

where r₀/2π is the 2D polarizability of the thin slab and ε_m = (ε₁ + ε₂)/2 is the mean value of the dielectric constants of the environment. The corresponding static dielectric function is linear in q, given by ε_2D(q) = V_RK(q)/V₀(q) = ε_m(1 + r₀q), with V₀(q) = e²/(2εq) being the 2D Fourier transform of the vacuum Coulomb potential. Therefore, in contrast to bulk materials, where the dielectric constant is an intrinsic property, for a 2D material, the dielectric constant depends on extrinsic properties (ε_m) and intrinsic properties (r₀). The inverse Fourier transform of the potential Eq. (2) is

$${v}_{{\rm{R}}K}(r)=\frac{2\pi {e}^{2}}{{\varepsilon }_{0}{\varepsilon }_{m}{r}_{0}}\left[{H}_{0}\left(\frac{r}{{r}_{0}}\right)-{Y}_{0}\left(\frac{r}{{r}_{0}}\right)\right],$$

(3)

where H₀ and Y₀ are the Struve function and the Bessel function of the second kind, respectively. This potential can be further generalized to include finite thickness⁸⁹, anisotropy¹⁸⁴, and a spacer between the 2D material and the substrate¹⁶⁸.

Equation (2) can also be obtained from a microscopic viewpoint, as the long-wavelength limit of the dielectric function in the random-phase approximation⁸⁸. This provides an ab initio way to calculate the parameter r₀ from the dielectric constant¹⁸⁵. In computational codes for ab initio calculations, it is necessary to put an interlayer distance L_c large enough to avoid interaction between layers. It can be shown that the effective dielectric constant scales with the interlayer distance L_c as⁸⁸

$$\varepsilon ({L}_{c})=1+\frac{2{r}_{0}}{{L}_{c}}+O\left(\frac{1}{{L}_{c}^{2}}\right).$$

(4)

The dependence of the screening of the carrier–carrier interaction on the environment, as shown by Eq. (2), opens avenues to engineer the desired electronic and optical properties of 2D materials^167,169,186.

The optical bandgap is effectively defined as the QP bandgap corrected by the exciton-binding energy. The latter can be obtained by solving the Wannier equation, namely, a hydrogen-like equation describing an exciton¹⁸⁷

$$\left(-\frac{{\nabla }_{{\bf{r}}}^{2}}{2\mu }+v(r)-{E}_{g}-{E}_{n\ell }\right){\psi }_{n\ell }({\bf{r}})=0,$$

(5)

where r is the relative coordinate between the electron and hole, E_g is the QP bandgap, μ = m_em_h/(m_e + m_h) is the reduced mass, ψ_nℓ(r) is the exciton envelope wave function, and n (ℓ) is the principal (angular) quantum number. When the RK potential Eq. (3) is used, this equation only has, as input, the external dielectric constant ε_m, the 2D polarizability r₀, and the effective masses. Eq. (5) can be straightforwardly generalized to account for the anisotropic mass and screened potential in, e.g., N-BP^184,188. This provides a good aproximation for the first energy level in the exciton Rydberg series¹⁸⁷.

While Eq. (5) accounts for the effect of dielectric environment on the exciton-binding energy, it does not account for the QP bandgap renormalization that results from it. On the other hand, both effects are naturally included in the semiconductor Bloch equations (SBE)¹⁸⁹, which has been already succesfully applied for 2D materials^72,190. Here, we use the SBE to study the effects of the environment on the exciton properties and the QP bandgap. Details of these calculations are shown in the Supplemental Material.

Since the results for N-BP were already previously discussed¹⁷³, let us now focus on TMDC. Exciton energies and QP bandgaps for TMDC, as obtained by the above-mentioned theoretical procedure, are presented as a function of the effective dielectric constant of the environment in Fig. 8b, which depicts the changes both in the optical ΔE_opt = E₀(ε_m) − E₀(ε_m = 1) and QP $\Delta {E}_{{\rm{Q}}P}={\Sigma }_{{\bf{0}},+}^{{\rm{xc}}}({\varepsilon }_{m})-{\Sigma }_{{\bf{0}},+}^{{\rm{xc}}}({\varepsilon }_{m}=1)$ gaps. Here, E₀ is the ground-state exciton energy, and ${\Sigma }_{{\bf{0}},+}^{{\rm{xc}}}$ accounts for the exchange-correlation function¹⁹¹. More details are presented in the Supplemental Material. One clearly sees that the optical bandgap has a weak dependence on the substrate, in accordance with previous calculations¹⁹² and experiments^179,193. However, the QP bandgap alone exhibits stronger dependence on the environment, also in accordance with experiments¹⁷⁰ and theory^172,175. It is also clear from Fig. 8b that tungsten-based materials are more susceptible to optical gap modulation by the dielectric environment, as compared with the molybdenum-based ones, which is in accordance with the experimental data shown in Fig. 8a. The results thus far suggest that more is yet to be done in this area in order to exploit the full capability of optical bandgap modulation through dielectric engineering.

Strain engineering effects

Strain engineering aims at modifying the bandgap or the electronic properties of materials by the application of strain. Traditionally, this technique has been developed in the context of conventional semiconductors (such as silicon) toward the enhancement of electronic transport properties¹⁹⁴. Its potential use to break the inversion symmetry of the crystal lattice in graphene, thus opening a bandgap in its otherwise gapless band structure, is among the first applications of strain in the context of 2D materials^195,196,197. According to theoretical calculations, in order to open a technologically relevant bandgap in graphene of a few hundred meV, a large strain ε ≈ 15–30% is needed¹⁹⁸.

Contrary to graphene, TMDC and N-BP are 2D semiconductors with bandgap energies in the range between 0.3 and 2.0 eV, which can be further engineered with strain. Theoretical calculations based on DFT^199,200 suggest that the direct QP gap energy in MoS₂ decreases by ≈0.8 eV with 9% of biaxial strain. The indirect (K–Q) gap energy undergoes an even stronger decrease; consequently, a direct-to-indirect bandgap transition is observed as biaxial strain is applied in MoS₂ and the same is predicted for other TMDCs. Conversely, in monolayer BP, an increase of ≈0.65 (0.4) eV is observed in the optical gap for 4.5% uniaxial strain applied in the AC (ZZ) direction²⁰¹. Such substantial modifications of the bandgap with the application of moderate strain levels in both cases demonstrate the feasibility of strain-based bandgap engineering of 2D semiconductors.

The tuning of the bandgap in TMDCs by the application of uniaxial and biaxial strain has been experimentally demonstrated in various works^{202,203,204,205,206}. Contrary to dielectric environment engineering, uniaxial strain is demonstrated to lead to large QP gap modulation, but with very weak effect on the exciton-binding energy²⁰⁶. The latter results from slight modifications of the effective masses with strain.

Typically, uniaxial strain is applied by bending or stretching the substrate underneath the 2D material, or by inducing ripples in the material itself. For example, uniaxially-strained monolayer MoS₂ can be achieved by bending its polycarbonate substrate in a four-point bending apparatus²⁰⁷. The top panel of Fig. 9a shows the PL of the unstrained and strained (up to a value of 1.8%) monolayer MoS₂. The A exciton PL peak, related to the direct bandgap of MoS₂, shifts to lower energies when a tensile uniaxial strain is applied, indicating a closing of the bandgap^204,207,208. Uniaxial strain tuning of the bandgap has been also demonstrated for other TMDCs, like MoSe₂²⁰⁹ and WSe₂²¹⁰, and it has been shown to have a strong impact on the exciton–phonon coupling²¹¹. The bottom panel of Fig 9a shows a different experiment in which biaxial strain of monolayer MoS₂ was achieved by exploiting the thermal expansion or contraction of the polypropylene substrate carrying the flake²¹². Alternatively to the use of thermal expansion, biaxial strain has also been demonstrated by transferring MoS₂ onto a piezoelectric substrate²⁰³, or by applying a pressure difference across suspended MoS₂ membranes²⁰⁵. The A exciton peak in the differential reflectance spectra shifts to lower energies when applying biaxial tensile strain (up to 0.9%) and to higher energies for a compressive strain (up to 0.1%). The reported bandgap tunabilities with strain (or gauge factors) are −45 meV/% and −25 meV/%, respectively, for uniaxial and biaxial strain, in good agreement with theoretical predictions²⁰⁶. A 100- (125) meV redshift of the optical (quasi-particle) gap has also been observed in monolayer WSe₂ under 2.1% uniaxial strain²¹³. Strain engineering also has the potential to modify the interlayer interaction in multilayered 2D materials. This has been recently employed to modify the interlayer interaction in few-layer black phosphorus²¹⁴, and to tune the energy of the interlayer excitons in bilayer MoS₂^215,216 flakes.

Fig. 9: Strain engineering in 2D materials.

a Bandgap tuning in uniaxially²⁰⁷ (top) and biaxially (bottom)²¹² strained monolayer MoS₂. b Superlattice built by introducing a local and periodic strain in a monolayer MoS₂ by patterning nanocones in the substrates²²¹. c Exciton funneling and inverse funneling effect in 2D semiconductors²⁰¹. The bottom panels show an experimental realization of inverse funneling in a locally strained HfS₂ photodetector²²⁵. Panel a reprinted with permission from²⁰⁷ Nano Lett. 2013, 13, 8, 3626–3630. Publication Date: July 2, 2013 https://doi.org/10.1021/nl4014748 Copyright (2013) American Chemical Society; b reused from Springer Nature/Li et al.²²¹, permissible under a CC-BY [4.0/3.0] license; c reused with permission from APS/San Jose et al.²⁰¹, and Springer Nature/Sanctis et al.²²⁵, permissible under a CC-BY [4.0/3.0] license.

Strain engineering has been recently used to fabricate straintronic devices based on 2D semiconductors. For example, the thermal expansion-based biaxial straining has been recently employed to tune photodetectors based on single-layer MoS₂. By controlling the level of strain, it was shown that one can tune the photoresponsivity (by 2–3 orders of magnitude), the response time (from <80 ms to 1.5 s), and the spectral bandwidth (with a gauge factor of 135 meV/% or 58 nm/%) of the device²¹⁷. Uniaxial strain engineering has also been exploited to modify the in-plane anisotropy of MoS₂-based photodetectors. Photodetectors with strong linear polarization-sensitive response from visible to near-infrared range have been shown for monolayer MoS₂ devices fabricated onto polydimethylsiloxane, and the polarization anisotropic ratio can reach over 2.0 under 4.5% stretching of the polymeric substrate²¹⁸.

While the experiments discussed above deal with global and homogeneous strain, many interesting physical effects can be achieved with local strain distributions, such as pseudomagnetic fields¹⁹⁷, the creation of wrinkles²¹⁹, or even “artificial atoms” single-photon emitters, by the exciton-funneling effect^220,221. The latter has been proposed as a way to modulate the intrinsic bandgap of monolayer MoS₂ and thus improve its photovoltaic and photodetecting properties¹⁹⁹. The diagram in the top panel of Fig. 9b sketches an example of local strain distributions induced by patterning nanocones in the substrate, where the bandgap of MoS₂ is modulated in such a way that the locations of maximum strain mimic the Coulomb potential around ions in a crystal. A similar approach, exploiting the local induced strain reduction of the bandgap, has been employed to deterministically define single-photon emitters^222,223. In the bottom panel of Fig. 9b, a tilted false-color scanning electron microscopy (SEM) image of the strained MoS₂ crystal defined by such nanocones array is shown.

A local strain center in a 2D semiconductor leads to exciton funneling, which is a way of controlling exciton motion by means of inhomogeneous strains. The top panel of Fig. 9c shows a representation of exciton funneling in MoS₂ (left)²²⁴ and inverse exciton funneling in black phosphorous (right)²⁰¹ together with a schematic band diagram of the process. An experimental realization of the inverse funneling effect was recently demonstrated²²⁵, where the photoresponse of a HfS₂-based device is tuned by introducing a local strain center through laser-assisted oxidation of HfS₂ to HfO₂. Other alternative local straining approaches are based on indentation of the 2D material with an AFM tip, that produces strong strain localization right under itself^226,227,228.

Strain also plays an important role in lateral heterojunctions between 2D materials with different lattice parameters. The fabrication of seamless lateral heterojunctions has been pursued over the past few years as a path toward monolayer diodes and transistors, see refs. ^{231,232,233,234}. for reviews on their synthesis, optoelectronic properties, and device applications. In this case, a spatial modulation of both the energy and intensity of PL peaks has been experimentally observed²²⁹ in the MoS₂ region nearby a MoS₂–WSe₂ lateral heterojunction, where energies were found between 1.79 and 1.91 eV. This modulation has been addressed to the effects on the band structure due to the strain distribution caused by the lattice mismatch between these two materials. Strain engineering in such monolayer lateral junctions has been proven a challenging task, but it would allow for control of the alignment between energy bands of the materials in the heterojunction. Theoretical predictions also suggest that the band structure modification due to tensile strain across a monolayer MoS₂/WS₂ lateral heterojuction would lead to charge carrier confinement at the interface, thus forming a one-dimensional channel that brings prospects for applications in high-electron-mobility transistors and low-power switching devices^230,231,233.

Molecular intercalation and chemical doping effects

Intercalation of 2D materials by organic molecules leads to organic–inorganic hybrid structures, such as p–n junctions²³⁵, heterojunctions, and superlattices^236,237. This opens a new avenue to modulate and manage materials electronic and optical properties, thermoelectric performances, and doping effects by tailoring their molecular structures, sizes, and symmetries. Molecular intercalation techniques thus build a bridge between traditional integrated circuits and atomically layered materials. In contrast to previous organic–inorganic hybrid systems, organic molecule-intercalated hybrid structures exhibit unique features, such as intrinsic interfaces, atomically seamless junctions, high yield, and reproducibility^235,236,237. Recent experiments involving intercalation of layered semiconductors, such as bulk TMDCs and BP with organic molecules²³⁷, forming stable superlattices of alternating monolayer atomic crystals with molecular layers, could effectively remove the interlayer coupling and thus allow access to tune the bandgap of bulk layered materials. In situ measurement of PL during the process that intercalates organic molecules (such as cetyl-trimethylammonium bromide, CTAB) into bulk BP by electrochemical approach reveals the gradual modulation of PL peak position from no apparent emission in bulk to remarkable emission at ≈1.38, ≈1.75, and ≈2.26 eV as the number of layers in the intercalation process increases (Fig. 10a). These three values roughly correspond to band-edge transition of trilayer, bilayer, and monolayer BP, respectively, but the latter is slightly more than that of ideal monolayer BP. This may be due to the bandgap expansion and orbital symmetry breaking due to the intercalation.

Fig. 10: Bandgap modulation by intercalation and absorption.

a PL signals of different stages of intercalation of monolayer BP molecular superlattices (MPMS)²³⁷. The simulated atomic structure of MPMS is shown in the side panel. b Band structure of intrinsic bilayer CVD SnS₂ and Cu-intercalated SnS₂. The inset in the left panel shows the lattice structure of bilayer SnS₂ with AA stacking. The inset in the right panel shows seamless integration of n-type SnS₂, p-type Cu–SnS₂, and metallic Co–SnS₂ within a single flake. Valence band V₁ and conduction bands C₁ and C₂ are marked by yellow, orange, and blue, respectively. The calculated Fermi level is set to be zero²³⁵. c PL spectra for undoped and F4TCNQ-doped monolayer MoS₂ (bottom-left panel). The atomic structure of the latter is shown in the top-left panel. The right panel illustrates relative potentials of monolayer MoS₂ and n- and p-type dopants. d PL images of monolayer MoS₂ before and after TFSI treatment. Insets show optical micrographs. Panel a reprinted by permission from Springer Nature²³⁷ Wang, C., He, Q., Halim, U. et al. Monolayer atomic crystal molecular superlattices. Nature 555, 231–236 (2018), Copyright (2018), advance online publication, 8th March 2018 (doi: 10.1038/nature25774 Nature); panel b reprinted by permission from Springer Nature Nature Nanotechnology²³⁵, Gong, Y., Yuan, H., Wu, C. et al. Spatially controlled doping of two-dimensional SnS2 through intercalation for electronics. Nature Nanotech 13, 294–299 (2018), Copyright (2018), advance online publication, 26th February 2018 (https://doi.org/10.1038/s41565-018-0069-3 Nature Nanotech.).

Furthermore, intercalating metal atoms into CVD SnS₂ flakes was found to change its charge-conductive states²³⁵. Naturally grown SnS₂ flakes have n-type charge transport behavior, resulting from S vacancy. However, after those flakes are intercalated by copper and cobalt atoms, they display p-type semiconductor properties, with record hole mobility, and highly conductive metal properties. DFT calculations provide an understanding of these changes. In Fig. 10b, we label the valence and conduction bands as V₁, C₁, and C₂, respectively. For intrinsic bilayer SnS₂ [left panel in Fig. 10b], the bandgap arises from the difference between V₁ bands and C₁ bands, ≈1.5 eV. When intercalating Cu atoms into SnS₂, the s-orbital electrons of Cu atoms extend and partly fill into C₁ bands due to the coupling among intercalated atoms and SnS₂ layers. The bandgap that at this point comes from the difference C₁–C₂ (Δ₁₂) decreases by 0.9 eV due to the increase in the C₁ bandwidth [right panel in Fig. 10b], compared with that of pristine SnS₂. Therefore, the Cu-intercalated SnS₂ behaves like a p-type semiconductor. For Co-intercalated SnS₂, the strong coupling hybridization among d orbitals of Co atoms, s orbitals of Sn atoms, and p orbitals of S atoms imports plenty of hybridized states around the Fermi level, leading to excellent metal conductivity. The intercalation technique has been used to demonstrate the seamless integration of n-type SnS₂, p-type Cu–SnS₂, and metallic Co–SnS₂ within a single flake. Therefore, such intercalation strategy can be regarded as an effective way to tailor the electro-optical properties of parent materials.

The traditional dopant diffusion or ion-injection scheme is not suitable for doping 2D-layered materials, due to their ultrathin nature. Therefore, many efforts have been devoted to develop new chemical doping techniques to modify their electrical and optical properties, such as absorbing NO₂ molecules²³⁸ or potassium (K) on their surface^239,240,241, covering them with organic molecular films^239,240,241, soaking them into organic solutions²⁴², and treating their surface with organic superacids²⁴³. Although these chemical doping methods do not usually modify the intrinsic band-edge transition energy, they can effectively dope 2D-layered materials and manipulate electron and exciton transport to enhance their electrical²⁴⁴ and optical^240,243 performances. Organic molecules, such as 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (F4TCNQ) and 7,7,8,8-tetracyanoquinodimethane (TCNQ), have been used as p-type dopants, while nicotinamide adenine dinucleotide (NADH) is used as n-type dopant to dope monolayer MoS₂²⁴⁰. With increasing F4TCNQ doping step, the PL intensity (peak A) continuously increases until saturation, and its peak energy undergoes a blueshift to 1.88 eV (left panel in Fig. 10c). It has been claimed²⁴⁰ that such shift in PL spectra does not originate from the modification of band-edge energy, but rather derives from the suppression of negative trion (X⁻) exciton recombination. For undoped monolayer MoS₂, X⁻ is still formed due to unintentional electron doping, and its PL peak (≈1.84 eV) is stronger than that of the intrinsic exciton (X) peak (at ≈1.88 eV). Hence, the low PL-intensity peak at 1.84 eV is observed. For p-type doping samples, monolayer MoS₂ samples, F4TCNQ, or TCNQ, electron acceptors consume excess electrons and drastically inhibit the X⁻ trion formation [right panel in Fig. 10c]. These flakes present intrinsic exciton properties and, for n-type doping of a MoS₂ flake using NADH as electron donor, excess electrons and more negative trions are formed, which decreases the PL intensity [right panel in Fig. 10c]. Similarly, the defects or unintentional doping can give rise to nonradiative recombination, thus resulting in low PL quantum yield (QY) in monolayer MoS₂ flake. A method of chemical treatment by organic superacid to remove the above defect-mediated nonradiative recombination was also reported²⁴³. After bis(trifluoro methane) sulfonimide (TFSI) treatment, the luminescence QY of monolayer MoS₂ flake can reach near-unity, and PL intensity enhances by two orders of magnitude (Fig. 10d).

Alloying effects

Historically, the alloying strategy allows one to achieve continuously tunable bandgap and electronic structures for various applications in optoelectronics. Due to their low dimensionality, 2D alloys are comparatively more difficult to realize. Theoretical calculations indicate that monolayer TMDC alloys can exhibit continuously tunable gaps²⁴⁵ spanning across those of its constituents. So far, mechanical exfoliation^139,246,247, as well as physical and CVDs (PVD¹³⁸ and CVD^136,137,248) have been explored to synthesize 2D- layered TMDC alloys. Although mechanically exfoliated 2D-layered alloys^139,246,247 exhibit long stability²⁴⁷ without phase separation^139,247, they have small size, poor thickness control, and low yield. This, along with the ultralong growth time of their parent bulk crystals, limits them to fundamental researches, in which the bulk crystals can be directly synthesized by CVT technique^247,249. Their PVD counterparts, on the other hand, seem to have a great potential for producing large-area continuous films; however, 2D PVD alloys to date are hard to synthesize at high dopant content, due to the component decomposition resulting from high temperature, so that only 2D MoS_2(1−x)Se_2x alloys with 20-μm domain size have been achieved²⁵⁰. Conversely, the CVD approach shows some outstanding advantages, so that it enables not only the mass production of single-crystal domains, but also the fabrication of large-area continuous films with a lower growth temperature than the PVD approach [see left panel in Fig. 11a], thus being preferred for practical applications. So far, numerous 2D TMDC alloys, including ternary (such as MoS_2(1−x)Se_2x, WS_2xSe_2(1−x), and WSe_2(1−x)Te_2x), quaternary (Mo_xNb_(1−x)S_2ySe_2(1−y)), and quinary (V_xW_yMo_1−x−yS_2zSe_2(1−z)), have successfully been synthesized²⁵¹ via sulfurization, selenization, and tellurization of transition metals or transition- metal compounds by CVD technique.

Fig. 11: Bandgap modulation by alloying.

a Typical optical images of monolayer and single-crystal domains and continuous MoS_2(1−x)Se_2x (x = 0.1) films (left panel), scanning transmission electron microscope (STEM) image of Se-doped MoS₂ with ≈12% local Se concentration (middle panel), and composition-dependent PL emission peak position of 0% Se-doped MoS₂ (monolayer pristine MoS₂,blue), 30% Se-doped MoS₂ (green), 50% Se-doped MoS₂ (purple), 75% Se-doped MoS₂ (orange), and 100% Se-doped MoS₂ (monolayer pristine MoSe₂, red), respectively¹³⁷. b Composition-dependent optical bandgap of 2D-layered alloys. c STEM image of quinary V_xW_yMo_1−x−yS_2zSe_2(1−z) alloy (left panel)²⁵¹, quaternary Mo_xW_1−xS_2ySe_2(1−y) alloy (middle panel), and DFT-calculated bandgap as a function of composition parameters (right panel). In the latter, experimental points are indicated as white (650 °C), blue (700 °C), and green (750 °C) dots²⁵⁵. Panel a reprinted with permission from¹³⁷ Nano Lett. 2014, 14, 2, 442–449. Publication Date: December 24, 2013 https://doi.org/10.1021/nl4032296, Copyright (2013) American Chemical Society; panel c reprinted by permission from Springer Nature²⁵¹ Zhou, J., Lin, J., Huang, X. et al. A library of atomically thin metal chalcogenides. Nature 556, 355–359 (2018), advance online publication, 18th April 2018 (https://doi.org/10.1038/s41565-018-0069-310.1038/s41586-018-0008-3 Nature) and from John Wiley and Sons/Susarla et al.²⁵⁵.

Alloying of 2D monolayer TMDCs, such as MoS_2(1−x)Se_2x and Mo_1−xW_xSe₂, has robust potential to continuously modulate the band-edge transitions, by changing their constituent stoichiometries, while maintaining direct gap character and thermodynamical stability. A MoS_2(1−x)Se_2x monolayer film with well-controlled Se concentration from 0 to ≈75% by CVD enables its bandgap modulation over 10% and features one dominant peak in PL spectroscopy, as demonstrated in ref. ¹³⁷. These observations suggest that the distributions of S and Se atoms in MoS_2(1−x)Se_2x alloy are random. This is consistent with the result of scanning transmission electron microscope (STEM) in Fig. 11a, which shows clear separation between doped- and undoped domains in Se-doped MoS₂ with ≈12% Se concentration, where each type of atom can be easily distinguished by quantitative image-intensity analysis. Noticeably, a redshift of peak position in PL spectrum with increasing Se concentration (Fig. 11a) suggests that the band-edge transition energy of monolayer MoS_2(1−x)Se_2x alloy can be continuously fine-tuned from 1.85 eV (for pure monolayer MoS₂) to 1.54 eV (for pure monolayer MoSe₂), where the band-edge transition energy exhibits a linear dependence on the Se concentration [see cyan open right triangles in Fig. 11b]. Such arrangement of constituent atoms and composition-dependent band-edge transition energy is usually a common feature in other TMDC alloys. Furthermore, monolayer WSe_2(1−x)Te_2x alloy exhibits a phase transition [magenta open circles in Fig. 11b], so that the optical bandgap is continuously red-shifted from 1.67 to 1.44 eV with increasing Te atom concentration from 0 to 60%, and suddenly jumps to 0 eV when the dopant concentration is over 60%¹⁴⁰. This indicates that the monolayer WSe_2(1−x)Te_2x alloy transforms from semiconducting 2H phase to metallic 1T_d phase. This transition will be discussed in more detail in the next section.

In general, the bandgap of 2D ternary TMDC alloys MX_2(1−x)Y_2x, can be modeled with²⁵²

$${E}_{\rm{PL,MXY}}={E}_{\rm{PL,MY}}x+{E}_{\rm{PL,MX}}(1-x)-bx(1-x),$$

(6)

where E_PL,MXY, E_PL,MX, E_PL,MY, and b are, respectively, the optical bandgap of the alloy, pure MX₂, pure MY₂, and bowing parameter. The latter characterizes the bowing effect of the bandgap of 2D TMDC alloys, which is a joint effect of alloy volume deformation, strengthened structure relaxation by low dimensionality, and chemical difference between two constituent compounds. Usually, a larger-size mismatch and chemical potential difference will lead to a higher bowing parameter such as b(S,Te) > b(Se,Te) > b(S,Se), and then result in an obvious bandgap bowing effect as shown for Mo_1−xW_xS₂ and Mo_1−xW_xSe₂ in Fig. 11b. Taking Mo_1−xW_xS₂ as an example²⁴⁶, the large bowing effect of the bandgap (b ≈ 0.28 eV) comes from the bowing of the LUMO energy level. Notice that the bowing parameter can also be negligible; in this case, the optical bandgap shows a linear modulation trend as a function of dopant concentrations (see WS_2xSe_2(1−x) and MoS_2xSe_2(1−x) in Fig. 11b).

2D-layered materials of group-V elements can tune their bandgap to mid-infrared regime. A clear example is BP (As), whose bandgap goes from 0.3 (0.3) eV in bulk to 2.2 (1.0) eV in monolayer form^253,254. So, black arsenic–phosphorus ternary alloy (b-As_xP_(1−x)) can be regarded as a promising candidate for application in industrial thermal imaging, infrared tracking, and free-space optical communication. Unlike 2D TMDC alloys, direct synthesis of 2D As_xP_(1−x) alloys has not been reported. Most of such alloys have been obtained by mechanical exfoliation from their parent crystals. These can be a mixture of red phosphorus and gray arsenic with a desired ratio, which are then placed, along mineralizer additives, into evacuated silica grass ampoules to perform phase formation and crystal growth via the vapor transport method^137,254. Different composition alloys may require different mineralizer additives and growth temperatures and times. Dark-yellow diamonds in Fig. 11b show the composition-dependent bandgap of b-As_xP_(1−x) alloy with flake thickness over 30 nm, in which the bandgap can be fully modulated from 0.3 to 0.15 eV with increasing As concentration²⁵⁴. The observed fluctuation of bandgap at each arsenic concentration originates from the polarization-sensitive optical properties of these materials⁴⁷.

Besides the binary and ternary alloys, multielement alloys have drawn intensive attention as they can also widely modify the bandgap and greatly enrich the physical properties. Recently, several groups^251,255 have successfully synthesized multiple monolayer, quaternary, and quinary alloys using the CVD method. For instance, a molten-salt-assisted CVD process, which can grow a wide variety of TMD alloys, was developed²⁵¹. This includes 11 ternary, one quaternary, and one quinary (such as MoS_xTe_2−x, Mo_xNb_1−xS_2ySe_2(1−y), and V_xW_yMo_1−x−yS_2zSe_2(1−z)), where mixing NaCl or KI together with metal or metal oxides served as precursors to decrease the melting point of the reactants and facilitate the formation of intermediate products. A facile CVD method to successfully synthesize monolayer quaternary Mo_xW_1−xS_2ySe_2(1−y) alloy was also developed²⁵⁵, where the content of metals and chalcogens can be tuned by controlling the growth temperature during the growth process. For these quaternary and quinary alloys, STEM image characterization (Fig. 11c) shows that each atom was mostly uniformly distributed throughout the alloy samples. CVD-synthesized monolayer quaternary Mo_xW_1−xS_2ySe_2(1−y) alloy exhibits a wide tunable range of bandgap, from 1.61 to 1.85 eV²⁵⁵. The DFT-calculated gap of such quaternary alloy presents good variability with values ranging from 1.60 to 2.03 eV. The right panel in Fig. 11c shows such variation of bandgap with composition. In summary, these alloys show the possibility to modulate the conductivity type (such as n-type, p-type, and semimetal), and provide a feasibile scheme for tunable material bandgap and electronic structure.

Bandgap closing: metal–insulator transition

An important topic on its own, the metal–insulator transition (MIT) of 2D semiconductor materials can also be considered as a rather extreme version of bandgap engineering. While the structural phase transition of bulk MoS₂ under alkali metal intercalation^256,257 and α- to β-phase transition of MoTe₂^258,259 under heating had been reported since the 1970s, the isolation of atomically thin layers of TMDC has rekindled the research interest. Owing to its polymorphism, the MIT in 2D TMDC may be either a pure electronic phase transition, or also accompanies a structural phase transition.

In a nutshell, the general driving force of the structural phase transition is attributed to the change in the oxidation number of the central transition-metal cation and its electron configuration. In trigonal prismatic (2H) phase, the typical oxidation number of the transition-metal cation is +4. Under 2H coordination, the d-orbital splitting has one lowest state at d${}_{{z}^{2}}$, which can host two d-orbital electrons. However, the reduction of central transition-metal atom requires an additional electron to occupy a high-energy state. Thus, it becomes energetically favorable for the crystal to assume the octahedral (1T/1T’) phase, with three degenerate low-energy states. Hence, alkali metal intercalation has shown to be an effective method to trigger a phase transition in TMDC^19,256,257 (for a more detailed discussion, refer to ref. ²⁶⁰). In this regard, triangular prism phase (H) MoTe₂ has been the model case of phase transition owing to the small energy difference between semiconducting trigonal prismatic (2H) phase and metallic distorted octahedral phase (1T’). Theory predicts that the smallest stimulus for triggering structural phase transition is in 2H-MoTe₂²⁶¹. To understand what makes MoTe₂ more susceptible to phase transition compared with its counterpart TMDC, an intuitive picture is presented to link the ionic radii of atoms and the preferred phases of different TMDCs^262,263. First, because of the aforementioned d-orbital splitting, the preferred coordination is trigonal prismatic. However, if we view the crystal as a close-packed layer of a large spherical anion (r⁻) and metal cation (r⁺), the close-packed spheres cannot be arranged in a trigonal prismatic phase when the radius ratio between r⁺/r⁻ is too small. From such a simple geometrical picture, the ratio limit for 2H phase is 0.527. This simple picture can lead to a clean separation of phases by the ratio of the cation radius and the anion radius r⁻ by a straight line with a slope of 0.49. In addition to this simple picture, the ionic nature of the bond between metal and chalcogen plays an important role: if the bonding is ionic, the ionic radii of chacogen anions are larger, resulting in a stronger anion–anion repulsion. Taking the bond ionicity and the ratio r⁺/r⁻ together, a separation of phases in TMDC is possible as illustrated in Fig. 12a (reproduced from ref. ²⁶³). As one can see, MoTe₂ is right on the boundary where a transition to the octahedral phase is feasible. According to this intuitive picture, any stimuli that increase the anion–anion repulsion (whether by reduction of anions or by tensile strain) would increase the repulsion between the Te atoms. This becomes an additional driving force for phase transition, as the repulsion can be relaxed by adopting the octahedral phase, where the distance between anions is greater²⁶³ (see Fig. 12b). Consequently, the structural phase transition in MoTe₂ can be triggered by a relatively small change in the crystal geometry, which is not the case for other TMDCs. Evidently, such structural phase transition (accompanied by the semiconductor-to-metal transition) of MoTe₂ has been demonstrated to be triggered by heating²⁶⁴, tensile strain²⁶⁵, electron doping²⁶⁶, and electrostatic gating²⁶⁷ (see Fig. 12c).

Fig. 12: Bandgap closing: metal–insulator transition of TMDC.

a Ratio of the single-bond covalent radii of the metal atom and chalcogen plotted against Paulings resonating bond ionicity (reproduced from ref. ²⁶³). b Intuitive picture of phase transition in MoTe₂ with the anion–anion repulsion. c Structural phase transition of monolayer 2H-MoTe₂ with ionic gating²⁶⁷. d Asymmetric critical exponent T₀ versus δ_n for monolayer and multilayer MoS₂, showing the co-existence of percolation-like and quantum-phase transition in the same sample²⁶⁹. Panel a used with permission from Elsevier B.V./Madhukar²⁶³; c reprinted by permission from Springer Nature²⁶⁷ Wang, Y., Xiao, J., Zhu, H. et al. Structural phase transition in monolayer MoTe₂ driven by electrostatic doping. Nature 550, 487–491 (2017), Copyright (2017), advance online publication, 11th October 2017 (https://doi.org/10.1038/nature24043 Nature); d reused from Springer Nature/Moon et al.²⁶⁹, permissible under a CC-BY [4.0/3.0] license.

MIT of 2D materials, especially in TMDC, that do not involve structural phase transition, has been an active research field as well. Both percolation-like transition through disorder^268,269 and quantum-phase transition^270,271 have been attributed as the underlying mechanism. It should be noted that although the device geometry employed maybe similar to that typical of bandgap engineering through an external electric field, the electrostatic gate-induced MIT in TMDC essentially differs from the excitonic Stark effect discussed in previous sections. The distinction between the disorder and quantum mechanisms is based on the different scaling behaviors: $\sigma =A{(n-{n}_{c})}^{\delta }$ for percolation-like transition, while the quantum-phase transition follows the conductivity scaling of σ(δ_n, T) = σ_c(T)F[T/T₀(δ_n)]. Naturally, the quality of the sample appears to play a critical role in determining the origin of the MIT, as both the percolation-like transition and quantum-phase transition were observed in the same sample in multilayer MoS₂²⁶⁹. With the recent progress in obtaining high-quality samples, more intricate physical phenomena are being revealed. It is intriguing that the critical carrier density is typically in the order of few 10¹² cm⁻², which is easily achievable for a monolayer (and even for few-layer TMDC) through electrostatic gating. From a technological point of view, the MIT offers complementary advantages to the bandgap engineering, whether it originates from structural phase transition or from other mechanisms, as it provides a route to improve the contacts as heterophased devices^19,264. Moreover, MIT enables optoelectric applications, such as optoelectrical switches, infrared detection, and more²⁷¹.

Conclusions

The emergence of 2D semiconductors offers a new platform to explore band structure engineering effects. Here, we reviewed the state-of-the-art bandgap engineering approaches, but their applications in materials science, large-scale growth, and devices are still in early stages. Outstanding issues include the realization of heterostructures with tailored band alignment for devices, such as tunnel field effect transistors, light-emitting diodes, solar cells, electrocatalysis, and exciton condensate devices, among many others. Another area where 2D materials depart from most conventional semiconductors is in their polymorphic nature, which allows them to display vastly different band structure and electronic properties in different phases. Such phase engineering can be used to improve transistor Ohmic contacts¹⁹ for the study of exotic quantum behavior, such as spin-valley and Weyl fermion physics, and realize memristive devices with potential applications in reconfigurable RF²⁷² and neuromorphic computing applications^273,274. In addition, 2D materials can also exhibit ferromagnetism^34,35, ferroelectricity²⁷⁵, topological effects²⁷⁶, as well as superconductivity⁵², which offers a multitude of other opportunities for a wide range of novel functional devices, made possible by the ability to manipulate the band structure in entirely new ways compared with conventional 3D semiconductors.