Since the discovery of graphene in 2004^[1], amazing quantum phenomena of 2D materials have been unraveled, such as the distinctive half-integral quantum Hall effect^[2,3], Hofstadter’s butterfly and fractal quantum Hall effect^[4], and unconventional superconductivity in magic-angle graphene superlattices^[5], which have made 2D materials fascinating ever since their birth. Furthermore, 2D materials exhibit strong light–matter interaction. For instance, single-layer graphene exhibits a considerable interband absorption of exactly πα≈2.3% in the infrared (IR) regime (α is the fine structure constant)^[6,7]. More surprisingly, due to the quantum confinement and reduction in dielectric screening, robust excitonic absorption at room temperature can be as large as 20% in monolayer transition metal dichalcogenides (TMDCs)^[8,9]. These make 2D materials a novel platform to explore light–matter interaction. Many unique optical properties in 2D materials have been revealed, including universal absorption regardless of the wavelength^[7], helicity of valley excitons^[10,11], near unity excitonic reflection and photoluminescence (PL) quantum yield^[12,13], single-photon emission^[14], Dirac-type plasmons^[15], and so on. In addition, the high tunability of 2D materials dramatically enriches optical properties. In particular, due to the non-bonding nature of the van der Waals (vdWs) interlayer interaction, 2D materials can be arbitrarily stacked layer by layer to form homo/heterostructures without a lattice matching limitation^[16,17], which in principle offers infinite freedom to manipulate optical properties. Well-known examples include intriguing interlayer excitons in heterostructures and moiré excitons in moiré superlattices^[18–21].

At present, hundreds of 2D materials with diverse band structures have been discovered experimentally, including metals, semimetals, semiconductors, insulators, magnetic materials, and superconductors^[22–25]. Among them, a special type of 2D materials with in-plane anisotropic properties attracts abundant research interest due to the in-plane low crystal structure symmetry. The most representative and extensively studied one is black phosphorus (BP). BP joined the 2D family in 2014, when Li et al. at Fudan University first reported few-layer BP field effect transistors (FETs) with superior properties^[26]. First, BP attracted attention due to its highly tunable properties and high carrier mobility, bridging graphene and TMDCs, such as a strongly thickness-dependent direct bandgap covering from the mid-IR (MIR) to the visible regime (0.3 to 2 eV from bulk to monolayer), and exactly complementing the bandgaps of graphene and TMDCs. Soon after, its anisotropic electrical, optical, mechanical, and thermal properties were discovered in both theory and experiment^[27–30], which evokes (not only in BP) much attention to anisotropic 2D materials. As for optical properties, the anisotropy not only induces well-known phenomena such as linear dichroism in absorption and birefringence, but also renders additional freedom to control and detect light. On the other hand, anisotropy gives rise to different optical conductivities along different in-plane axes σx≠σy. In particular, when σx,σy have different signs, the material can hold so-called hyperbolic polaritons (including phonon, plasmon, and exciton polaritons), whose iso-frequency contours (IFCs) are hyperbolas. These special polaritons with extremely large photonic densities of state (DOSs) and directional propagating rays are very fascinating in photonics^[31]. Thus, anisotropic 2D materials benefit from the common merits of 2D materials, such as strong light–matter interaction and high tunability, and at the same time, with additional anisotropy, are expected to show much richer physics, which is very promising for basic research and practical applications in photonics and optoelectronics. In fact, optical anisotropy can be found in in-plane isotropic 2D materials with anisotropic nanostructures as well^[15,32,33]. However, it should be noted there is an essential difference between low symmetry 2D materials and symmetric 2D materials with anisotropic nanostructures. The optical anisotropy in the former is due to the anisotropic band structure, which can be tuned by various electronic band structure engineering techniques. What is more, the working wavelength region can spread widely, ranging from terahertz (THz) to ultraviolet regions. The optical anisotropy of the latter is mainly due to the anisotropic effective dielectric constant in fabricated nanostructures, in which the tunability of optical properties and the working wavelength region are limited by the nanostructure fabrication.

To date, dozens of anisotropic 2D materials, including semiconductors and semimetals, have been discovered. According to the crystal structure, anisotropic 2D materials can be divided into orthorhombic systems [e.g., BP, black arsenic (B-As), group IV monochalcogenides MX (M = Ge,Sn; X = S, Se), etc.], monoclinic systems (e.g., 1T’ WTe2, TiS3), and triclinic systems (e.g., ReX2, X = S, Se). With different crystal structures, anisotropic 2D materials exhibit diverse optical properties. For instance, in BP, strong interlayer interaction gives rise to multiple interband transitions with polarization along the armchair (AC) direction^[34,35]; in group IV monochalcogenides, multi-valley band structure leads to multiple interband transitions as well but with different polarization dependences^[36]; in ReX2, excitonic polarization dependence exhibits layer dependence though with a weak interlayer coupling^[37,38]; in WTe2, anisotropic electronic structure gives rise to the prominent hyperbolic plasmon response^[39]; in α-MoO3, hyperbolic phonon polaritons (PhPs) with high confinement and long lifetimes have been discovered^[40]. In this paper, we thoroughly review the fundamental optical properties and polaritons of various anisotropic 2D materials, aiming to evoke more research interest in this field. The paper is organized as follows: after this introduction, we discuss the fundamental optical properties, including interband absorption, PL, exciton, non-linear optical properties, and band structure engineering in Section 2, where anisotropic materials are limited to semiconductors, including BP, group IV monochalcogenides MX (M = Ge, Sn; X = S, S), ReX2 (X = S, Se), and TiS3. In Section 3, first, the hyperbolic polaritons including phonon, plasmon, exciton polaritons are introduced; then the engineering for hyperbolic polaritons and their applications are discussed. Finally, conclusions and perspectives for future research in this field are given in Section 4.

Most optoelectronic devices work in the IR to ultraviolet regime, where the interband transition plays a critical role in light absorption. The interband transition is closely related to the material band structure, through Fermi’s golden rule: α∝g(hν)|Mcv|2,(1)where g(hν) (hν=Ec−Ev) is the joint DOSs (JDOSs) involving both conduction (c) and valence (v) bands, Mcv=〈c|Hop|v〉 is the transition matrix element, and Hop is the Hamiltonian for light–matter interaction. In general, in the dipole approximation, Hop can be expressed as −Pe·E0, where Pe=er is the dipole moment of the electron, and E0 is the electrical field of light. |Mcv|2 defines the transition probability from band v to band c^[41]. In anisotropic 2D materials, the band structure is anisotropic as well. Thus, the value of |Mcv| is polarization dependent, which results in anisotropic absorption. The band structure is layer dependent due to the interlayer interaction. Hence, we can gain insight into two main characteristics of anisotropic 2D materials from the absorption, namely, layer dependence and anisotropy. Next, we will particularly focus on these two characteristics of optical properties in anisotropic 2D materials.

BP belongs to the orthorhombic structure. As shown in Fig. 1, in-plane P atoms are covalently bonded, forming a hexagonal lattice with a puckered structure constituting two sublayers. According to atomic morphology, two main in-plane crystal axes are denoted as AC and zigzag (ZZ). During early studies of 2D BP, theoretical calculations predicted BP is always a direct bandgap semiconductor regardless of the thickness^[27,42–44]. The bandgap shifts from Z point (bulk) to Γ point (few or monolayer) in the Brillouin zone (BZ), with the energy ranging from 0.3 eV (bulk) to 2.0 eV (monolayer). Furthermore, theoretical studies also suggested the anisotropy in light absorption^[27,43,44]. For example, the interband transition at Γ point is allowed (forbidden) when incident light is polarized along AC (ZZ) direction. On the contrary, at Λ point, it is forbidden (allowed) under AC (ZZ) polarization^[43]. The polarization dependence of the interband transition can be understood by group theory or symmetry analysis^[43,45]. Briefly, due to a mirror symmetry with respect to the x–z plane, wave functions of the conduction band and valence band at Γ point both have even symmetry with respect to the x–z mirror plane. Consequently, when incident light is polarized along ZZ (y-axis) direction, the transition matrix element Mcv∝〈c|y|v〉is zero, giving rise to a forbidden transition. It should be noted, though, that selection rules will be relaxed with the interband transition away from the high symmetry point in BZ.

Experimentally, Xia et al. first measured the polarization resolved IR absorption of mechanically exfoliated BP film through a Fourier transform IR (FTIR) spectrometer^[28]. Due to the relatively large thickness (>8nm), BP thin film showed a bulk-like absorption spectrum, where an absorption edge was observed at ∼0.3eV, corresponding to the bulk bandgap. The layer-dependent absorption property of atomically thin BP was not discovered until Li et al. performed an optical absorption study of one- to five-layer BP^[34]. As shown in the reflection spectra of Fig. 2(a), a strong resonance peak, corresponding to the optical bandgap, was observed at 1.73 eV in monolayer BP. With increasing thickness, the optical bandgap redshifts to 1.15 eV in bilayer and 0.83 eV in tri-layer. In few-layer BP above the bandgap, additional absorption resonances were observed, such as peaks at ∼2.44 and 1.93 eV in bi- and tri-layer, respectively. In fact, the multiple resonance absorption peaks originate from interband transitions between different valence sub-bands and conduction sub-bands at Γ point as shown in Fig. 2(c). Due to the interlayer interaction, the original valence (conduction) band v(c) in monolayer splits into N sub-bands v1(c1),v2(c2), ... ,vn(cn) in the Nth layer. Only interband transition from vn to cn (labeled as Enn,) with the same index n is allowed; thus, N-layer BP at Γ point possesses N interband resonances. Consistent with theory predictions, absorption measurements showed all of Enn are strongly layer dependent, which redshifts with increasing layer numbers. Through the 1D tight-binding model, Li et al. gave a phenomenological description of the systematic evolution of Enn with layer number N and sub-band index n, that is, EnnN=Eg0−2Δγcos(nN+1π), where Eg0 is the monolayer bandgap, Δγ=γc−γv, and γc(γv) is the nearest neighbor coupling between adjacent layers for conduction (valence) band. Through fitting experimental data, Eg0=1.8eV and Δγ=0.73eV were obtained, with Eg0 comparable to the theory prediction (∼2.0eV), validating the model. It should be noted that, even without including an excitonic effect (as will be discussed in Section 2.3), the model can still well capture the evolution of the transition energy EnnN. In fact, this model is not only applicable to few-layer BP, but also to other 2D semiconductors, such as TMDCs. At the same time, Zhang et al. systematically studied IR absorption of the 2L–15L BP in MIR to near-IR (NIR) regime through FTIR, and multiple resonance absorption peaks were observed as well. Especially in thicker samples (e.g., 13-layer), at least four resonance absorption peaks were clearly discerned^[35]. Zhang et al. revealed that the transition energy of Enn can be well described by the quantum well based formula, suggesting that few-layer BP can be regarded as a natural vdWs quantum well, where sub-bands correspond to quantized bands in a quantum well due to quantum confinement. The anisotropy in absorption was revealed by absorption spectroscopy as well. As shown in Fig. 2(b), all Enn exhibit the strongest absorption when light is polarized along AC. The relation between the absorption intensity I and polarized angle θ (with respect to AC direction) follows the scaling I∝cos2θ.

Although atomically thin BP possesses superb absorption properties, it is very unstable under ambient conditions^[46–49]. Due to unpaired electrons on the surface, BP is reactive to O2, forming PxOy compounds on the BP surface. These PxOy compounds can be dissolved in air moisture and accelerate the degradation of BP. The degradation profoundly affects the BP optical properties. Wang et al. revealed that an optical bandgap of 3L has significant blueshifts even when exposed to air only within 5 min, accompanied by peak broadening and weakening^[49]. In addition, the transition energy of EnnN with small N and large n is more sensitive to degradation, which can be well explained by the quantum well model. The instability is a bottleneck that hinders BP from practical applications. Nevertheless, there are some ways to protect BP from degradation, such as encapsulation with boron nitride (BN) layers^[48,50,51], while there are still many limitations in practice.

With the same puckered crystal structure as that of BP (as shown in Fig. 1), B-As is regarded as a cousin of BP. B-As inherits superb properties of BP, such as strong in-plane anisotropy, large on–off current ratio, and highly tunable bandgap at the IR regime^[52–55]. What is more, B-As has better ambient stability. Zhong et al. showed that few-layer B-As based FET still works even after exposure to air for one month^[53]. Hence, B-As is a good candidate to complement BP. Unfortunately, to date, due to the limitation of synthesis of high quality B-As crystals, experimental study of optical properties of few-layer B-As is rather limited.

With orthorhombic structures, group IV monochalcogenides compounds MX (M = Ge, Sn and X = S, Se) have been extensively studied for their exceptional thermoelectric properties for decades. Now as a type of 2D vdWs material, they have regained extensive research interest^[56–58]. Orthorhombic MX compounds exhibit a puckered layer structure similar to that of BP (see Fig. 1), and thus MX is regarded as the cousin of BP as well. Different from BP, monolayer MX lacks inversion symmetry, giving rise to additional unique properties, e.g., giant piezoelectricity^[59], large spin–orbit splitting^[60], multiferroics^[61], and multi-valley band structure^[36]. The ambient stability and environmental compatibility make 2D MX particularly attractive in applications as semiconductor devices. Using density functional theory (DFT) with the Heyd-Scuseria-Ernzerhof (HSE06) hybrid function, Gomes et al. predicted that the bandgaps of GeS (GeSe) and SnS (SnSe) enlarge from ∼1.81(1.07) and ∼1.24(1.0)eV in bulk to ∼2.32(1.54) and 1.96 (1.33) eV in monolayers, respectively, covering the NIR to visible regime^[60]. Almost all MX have indirect bandgaps, except for 1L SnSe and 1L–3L GeSe, while other studies showed multilayer GeS could have a direct bandgap as well^[62]. In fact, theoretical calculations showed that the energy difference between direct and indirect bandgaps is tiny, comparable to room-temperature energy (∼26meV). In view of this, MX can be regarded as quasi-direct bandgap semiconductors, which could be responsible for the discrepancies of the direct versus indirect nature among different studies. In addition, theoretical calculations also showed that in monolayer MX, conduction and valence band valleys always appear in pairs at almost the same k-point in BZ, one pair along the Γ-X direction (AC) and the other along the Γ-Y direction (ZZ), giving rise to multiple interband transitions from different valleys with contrasted polarization dependence^[36].

In experiments, due to the fragility of MX, atomically thin MX with high crystal quality is not available yet; nevertheless, anisotropic interband transitions have been demonstrated in their bulk-like thin films. Taking multilayer GeS as an example, through modulated optical spectroscopy (a powerful technique for probing the direct interband transition), two direct interband transitions (E1 and E2) with a contrasted polarization dependence near the absorption edge were identified^[62–64]. As shown in Fig. 2, the interband transition E1 at ∼1.6eV is polarized along ZZ, whereas E2 at ∼1.65eV is polarized along AC. Due to the different polarization dependence of E1 and E2, the absorption edge (Eg) of multilayer GeS shifts from 1.6 to 1.65 eV when the light polarization changes from AC to ZZ direction. However, the origins of E1 and E2 stir controversy. Hsueh et al. assigned multilayer GeS to a direct bandgap semiconductor, and E1 corresponds to the transition from the direct bandgap at Γ point; E2 is originated from the selection-rule forbidden transition near Γ point along Γ-Y direction, evoked by the local structural symmetry breaking^[62], while Totooczko et al. showed multilayer GeS has an indirect bandgap by comparing absorption and photoreflectance, which was rationalized by their DFT calculations^[64]. According to their calculation results, E1 and E2 were assigned to transitions at Γ point and Δ point in ZZ direction, respectively. Similar to GeS, multiple interband transitions near the absorption edge with different polarization dependences of multilayer SnSe, SnS, and GeSe were revealed as well^[65,66]. Multiple interband transitions are a consequence of the multivalley band structure in MX, which can offer a new platform for valleytronics.

Among anisotropic 2D materials, besides BP and its cousins, rhenium dichalcogenides (ReX2, where X = S, Se) attract abundant research interest as well^[67–69]. ReX2 is a member of TMDCs, with its atomic structure shown in Fig. 1. In each layer, Re atoms are sandwiched between two S atom sheets, and four Re atoms are grouped into parallelograms, forming a Re chain along the b axis (ZZ). Different from hexagonal TMDCs (e.g. MoS2, WS2), ReX2 belongs to the distorted 1T triclinic structure with a lower in-plane symmetry, which gives rise to anisotropic properties. RX2 is very air-stable. All of these make RX2 suitable for novel opto-electronic applications, although studies on 2D ReX2 from different groups show discrepancies, particularly with respect to the direct or indirect bandgap nature and strongly coupled or decoupled interlayer interaction. Take ReS2 as an example. Several theoretical calculations showed that ReS2 from bulk and monolayer has a direct bandgap located at Γ point of BZ^[70–72]. However, other studies showed that the bulk ReS2 bandgap is marginally indirect^[73], or a bandgap with either direct or indirect nature depends on thickness^[74,75]. As for interlayer coupling, PL peak positions and Raman phonons showed little and no layer dependence, indicating electronic and vibrational decoupling due to the weak interlayer interaction^[70]. Consequently, bulk ReS2 behaves like monolayer. However, contrary results were reported by ultralow frequency Raman spectroscopy^[76–78], where force constants of interlayer shear modes were estimated as ∼55%–90% of those in MoS2^[77], suggesting strong interlayer coupling. By using k-space photoemission microscopy, Gehlmann et al. found that electrons in the valence band are significantly delocalized across the vdWs gap in ReS2^[73], also indicating considerable interlayer coupling. Similar controversies can be found in ReSe2 as well^[38,72,79]. Studies have suggested that in ReX2, there coexist different stacking orders with different optical properties^[77,78,80], which can interpret these contradictory results. To fully reconcile these discrepancies of ReX2, more studies are needed.

As for the absorption of ReX2, Ho el al. preliminarily performed a series of studies in bulk ReX2 through modulated optical spectroscopy before ReX2 joined the 2D family^[81–84]. In bulk ReX2, two resonance peaks (labeled as E1 and E2) near the band edge were unambiguously discernable, and were assigned to direct interband transitions from the valence to the conduction band at different k-points. In ReS2, E1 (E2) centered at ∼1.49 (1.52) eV occurs with incident light polarized parallel (vertical) to the b axis. E1 (E2) in ReSe2 has the same polarization dependence but with a lower transition energy of ∼1.31 (1.33) eV. Previous optical spectroscopy studies also showed that both bulk ReS2 and ReSe2 have indirect bandgaps of ∼1.37 and 1.19 eV, respectively, which are very close to direct bandgap values. The layer-dependent and anisotropic absorption properties of atomically thin ReS2 (1L–7L) were first experimentally studied by Aslan et al.^[37]. Three resonance peaks (E1 to E3) were discerned, among which E1 and E2 correspond to those previously reported in the bulk counterpart, while E3 at higher energy could be due to the onset of a band-edge free-carrier transition or an excited excitonic state of E1 or E2. All three peaks exhibit obvious layer dependence, with E1, E2, and E3 blueshifting from 1.47, 1.51, and 1.58 eV in bulk to 1.61,1.68, and 1.88 eV in monolayer, respectively. The blueshifts of E1, E2, and E3 from bulk to monolayer in ReS2 are much larger than those observed for the direct transitions [at K(K′) point in BZ] in 2H-TMDCs (e.g., MoS2) but smaller than those for indirect transitions (valence band extrema at Γ point in BZ) in the same materials. In addition, Aslan et al. revealed a complex polarization dependence of E1 and E2 absorption in atomically thin ReS2. In monolayer, E1 and E2 exhibit a maximum for light polarized at ∼15° and ∼50° with respect to the b axis, respectively, while in tri-layer, the maximum polarization angles change to ∼15° and ∼85° with respect to the b axis, respectively, which are different from those reported in bulk by Ho et al. Similarly, in ReSe2, two prominent absorption peaks near the band edge were observed as well. Through polarization-dependent extinction and PL spectroscopy in conjunction with DFT calculations, Arora et al. revealed that two prominent absorption peaks (labeled as X1,2 and X3,4) are actually superpositions of four different interband excitonic absorption peaks^[38]. Due to the two-fold degenerated valence band maximum (VBM) and conduction band minimum (CBM) at Γ point, four absorption peaks originate from the direct transition at Γ point with different combinations of spin, with energy splitting from the exchange interaction. X1,2 and X3,4 also exhibit an obvious layer dependence, with transition energy blueshifting ∼120 and 140 meV, respectively, when the thickness decreases from bulk to monolayer, comparable to those in ReS2. The polarization dependence of these four transitions is more complicated. In fact, there are controversies in the polarization dependence and interband transition assignment in ReX2^[85], and the mechanism underlying the complex polarization dependence in atomically thin ReX2 is still unclear. Further studies are necessary.

New types of low symmetry 2D materials with different crystal structures have been discovered, though their bulk counterparts were already studied many years ago. For example, group IV–group V compounds (GeP, GeAs, SiP, SiAs) and group IV transition metal tri-chalcogenides (TMTCs) (TiS3, TiSe3, ZrS3, ZrSe3, HfS3, HfSe3), which belong to the monoclinic system, have been recently rediscovered as emerging anisotropic 2D materials^[86–94]. Most of these materials are indirect bandgap semiconductors, except for TiS3. TiS3 is a TMTC with a unique quasi-1D Ti-chain in layers (as shown in Fig. 1), which can be regarded as a quasi-1D material. Due to the much weaker interaction between quasi-1D chains, the exfoliated TiS3 is usually needle shaped. Monolayer TiS3 has been predicted in theory with a direct bandgap of ∼1.1eV and with a very high electron mobility up to ∼10,000 cm²/(V s)^[95]. The photoresponsivity of devices based on TiS3 shows a strong polarization dependence with the highest value reaching up to 2500 A/W^[96]. All of these give TiS3 great potential for a new generation of optoelectronic device applications as well. Unlike other 2D materials, theory calculation showed that the bandgap of TiS3 is quite robust, exhibiting an ∼1eV direct bandgap at Γ point for all thicknesses^[97]. This is mainly attributed to the fact that the locations of CBM and VBM states are at the center atoms of TiS3, unaffected by interlayer coupling. Through theoretical calculations, Torun et al. suggested three interband transitions at Γ point, among which the interband transition from the bandgap (with lowest transition energy) is allowed only when light is polarized along the out-of-plane direction, while the other two are both allowed when light is polarized along the in-plane b axis (parallel to the Ti chain)^[98]. Recently, through photocurrent spectroscopy, Lian et al. revealed anisotropic absorption with photon energy ranging from ∼1 to ∼1.8eV in both thick (∼200nm) and thin (∼15nm) TiS3 [as shown in Fig. 2(d)]^[99]. Both thick and thin TiS3 exhibit resonance absorption peaks when light is polarized along the b axis, in agreement with theoretical calculations. In thick TiS3, two absorption peaks E1 and E2 centered at 1.23 and 1.41 eV, respectively, were identified, while in thin TiS3, only E1 centered at 1.34 eV was observed, indicating different band structures in thick and thin TiS3. According to their first-principles calculations, the CBM in thick TiS3 shifts to Z point, giving rise to an indirect bandgap, and E1 is due to the transition from the valence band to a higher conduction band at Γ point; E2 corresponds to two direct transitions near X and Z points. In thin TiS3, the band structure is similar to that of the monolayer, which has a direct bandgap at Γ point due to the lack of Z point; E1 corresponds to the direct transition at Γ point, and E2 is absent.

Although various anisotropic 2D materials have been discovered, those with a narrow bandgap located at MIR and far-IR (FIR) are really scarce, which could limit 2D materials for IR opto-electronics applications. Bulk layered TiSe3 and Ta2NiS2 were reported with narrow bandgaps of ∼0.2 and 0.3 eV in MIR, respectively^[93,100], while the optical properties of their 2D counterparts are hardly known. Recently, Nb2SiTe4 (NST) was discovered as an IR anisotropic 2D material^[101,102]. NST belongs to the monoclinic crystal system, possessing in-plane anisotropic atomic structure formed by a Te-(Nb, Si)-Te sandwich layer, which is similar to the 2H-TMDCs such as MoS2 but with two types of atoms (Nb and Si) in the inner layer instead of one (Mo) in the MoS2 (as shown in Fig. 1). Theoretical calculations suggest that NST has a narrow indirect bandgap ranging from 0.89 eV in the monolayer to 0.39 eV in the bulk, situating in the NIR to MIR regime^[102]. Devices based on few-layer NST show a relatively high carrier mobility of ∼100cm2/(Vs) and photon responsivity of 0.66 A/W. All of these suggest that NST is a good candidate for IR opto-electronic devices. Wang et al. studied the anisotropic absorption properties of atomic thin NST through FTIR spectrometry^[102]. As shown in Fig. 2(d), a series of polarization-dependent absorption resonances [labeled as A, B(B*) and C] centered at ∼0.49, 0.66, and 1.1 eV was observed. Among them, A and B are allowed when light is polarized along the b axis, whereas B* and C are allowed when light is polarized along the a axis. In addition, the absorption linear dichroism in NST is switchable through changing the wavelength, that is, NST absorbs more light polarized along the a axis in photon energy ranging from 0.49 to 0.66 eV, whereas it absorbs more light polarized along the b axis when above 0.66 eV. Through first-principle calculations and group theory, these interband transitions can be well assigned as transitions from the first valence band v1 to different conduction bands cn [e.g., c1(3) for A (B) and c3(5) for B*(C)] at different k-points in BZ.

PL is the re-emission of light from material after absorbing a photon of higher energy, which is another fundamental optical process in materials. Materials with high quantum yield (ratio of the radiative photon number and generated electron–hole pairs) PL have great advantages for luminescent device applications such as lasers and light emitting diodes (LEDs). PL results from the radiative recombination of electrons in the conduction band and holes in the valence band, termed as interband PL. The selection rules of interband PL and interband absorption are consistent since they are governed by the same transition matrix elements. Different from absorption, the PL process is prone to emit photons with the lowest energy, and thus the recombination of electrons and holes usually occurs at the bandgap. Even when carriers (electron or hole) are excited at a higher energy level, they will soon relax back to the CBM (or VBM) before recombination, as shown in Fig. 3(b). If there are defect states located within the bandgap, PL due to defect states emitting longer wavelengths could be prominent, especially under low temperatures. It should be noted that interband absorption is insensitive to defect states due to the low density of defect states compared to that at band extrema. Hence, it will give rise to the difference between PL and absorption peak positions, which is termed as a Stokes shift. The Stokes shift can be a gauge for crystal quality. Due to defect states, PL is not a good method to determine the intrinsic bandgap; nevertheless, it offers vitally important emission properties of the material.

As a direct bandgap semiconductor, BP is expected to have a strong PL signal. In early studies of BP, several groups observed pronounced PL in atomically thin BP but with huge discrepancies in PL peak positions^{[34,49,103–111]}. For example, PL in monolayer BP was reported by several groups with emission energy ranging from 1.3 to 2 eV, and ranging from 1.1 to 1.6 eV in bilayer and 0.8 to 1.0 eV in tri-layer [as shown in Fig. 3(a)]. Such huge discrepancies in PL peak positions are most likely due to the vastly different sample quality, since BP degrades under ambient conditions and gives rise to a blueshift of PL peak position. On the other hand, degradation also introduces defect states, rendering PL emission at a lower energy^[49]. To suppress degradation, Li et al. encapsulated BP by BN, and observed PL at ∼1.7, 1.1, and 0.8 eV in monolayer, bilayer, and tri-layer respectively, with a small Stokes shift of ∼20–40meV, attributed to defect states^[34]. Recently, Yan’s group reported PL at 1.69, 0.98, and 0.86 eV in ultrahigh quality 1L–3L BP (no Stokes shifts)^[49,110], respectively.

Since PL is sensitive to defect states, one can manipulate PL properties through defect engineering. Defect state PL with emission energy lower than the bandgap can widen the working wavelength range of devices. In addition, it has been demonstrated as a good candidate for single-photon emitters as well. Pei et al. reported a bright defect state PL in monolayer BP at ∼1.36eV through artificially introducing oxygen defects by controllable etching with O2 plasma^[107]. By using a plasma-enhanced chemical vapor deposition (PECVD) oxide substrate, Xu et al. obtained defect state PL centered at energy of ∼1.35eV as well, whose intensity was 33.6 times stronger than the intrinsic one^[112]. This defect state PL was further confirmed by the sublinear relation between emission intensity and excitation intensity in a power dependence study. PL due to defect states was also observed in few-layer and bulk BP^[113–115].

Only thin BP, with a layer number less than five, was investigated in early PL studies, with PL situated in the visible to NIR range, readily detectable by a grating spectrometer. Chen et al. investigated MIR PL emission in thicker BP (thickness>9L) by FTIR spectroscopy^[116]. They showed that PL emission in multilayer BP is quite bright, comparable to InAs multiple quantum well samples grown by molecular beam epitaxy (MBE). PL peak positions in the IR regime can also be well fitted through the 1D tight-binding model (as applied to absorption peaks) with consistent fitting parameters, further confirming the direct bandgap nature in multilayer BP. Inspired by superb PL properties in BP, anisotropic MIR LEDs and lasers based on BP were achieved in experiments as well^[117,118], suggesting the great potential of BP for IR luminescent device applications.

As mentioned above, the polarization dependence of interband PL and absorption is consistent. As shown in Fig. 3(a), the PL emission of BP is linearly polarized along AC, with intensity following a cos2θ scaling^[34]. Interestingly, defect state PL has the same polarization dependence as well^[112]. PL intensity also exhibits excitation polarization dependence. When the excitation light (wavelength of 532 nm) is polarized along AC, PL intensity is maximal, which is mainly due to the anisotropic absorption of the excitation light^[106].

As quasi-direct bandgap semiconductors, group IV monochalcogenides MX (M = Ge, Sn and X = S, Se) are expected to have prominent PL under room temperature in the NIR to visible range. Take GeS as an example. Ho et al. reported bright PL centered at ∼1.62eV in multilayer GeS (∼40nm) at room temperature [as shown in Fig. 3(c)]^[63]. PL is due to the recombination of interband transition EA. It should be noted that there are two interband transitions near the band edge, namely, EA and EB, but only one PL peak assigned to EA is discernable. This is because the excited holes from transition EB at a lower energy level will soon relax to VBM before radiative recombination. A similar situation can be found in few-layer BP, where only one PL peak with the lowest energy can be observed among multiple interband transitions. Surprisingly, Zhao et al. showed that laser thinned monolayer GeSe exhibits eight PL peaks ranging from the visible to IR range^[119], and the intensities and positions of these PL peaks are abnormally independent of temperature, since in general, PL intensity can be significantly enhanced under low temperatures due to the suppressed nonradiative recombination. These results need further experimental verification. Due to the atomic structure similar to BP, the polarization dependence of PL in MX is consistent with that in BP. For instance, GeS PL emission is linearly polarized along AC direction. When the excitation light (e.g., wavelength of 633 nm) is polarized along the AC direction, the maximal PL intensity can be achieved due to anisotropic absorption [as shown in Fig. 3(c)].

Under room temperature, two prominent PL peaks with different polarization dependences in ReX2 can be observed^[37,38], which are consistent with those observed in the absorption spectrum. Due to controversial findings of the band structure, studies related to PL in ReX2 also exhibit discrepancies. For example, in ReS2, Tongay et al. first reported that PL peak position slightly blueshifts from ∼1.5 to ∼1.6eV when the thickness decreases from bulk to monolayer^[70]. PL intensity decreases with the decrease in sample thickness, indicating no indirect-to-direct bandgap transition similar to that in MoS2. Calculations by Tongay et al. suggested that ReS2 is always a direct bandgap semiconductor regardless of thickness. Aslan et al. later reported that the lowest energy PL peak position in ReS2 exhibits a larger blueshift when thickness is reduced from bulk (PL at 1.40 eV) to monolayer (PL at 1.65 eV), and through a comparison between PL and absorption spectra, they assigned PL at 1.40 eV in bulk to the indirect bandgap emission, slightly below the direct bandgap at 1.47 eV^[37]. Mohamed et al. quantitatively obtained a low PL quantum yield of ∼10−4 in 1L–7L ReS2 regardless of the layer number, which is at least one-order of magnitude smaller than those in monolayer 2H-TMDCs (e.g. MoS2) with direct bandgaps^[120]. These low PL quantum yields were ascribed to the indirect bandgap nature in ReS2. Similarly, in ReSe2, Zhao et al. reported that the PL position slightly shifts from 1.26 eV in bulk to 1.32 eV in monolayer, and the PL intensity decreases monotonically when decreasing the layer number, suggesting a much weaker interlayer interaction and no indirect bandgap to direct bandgap transition^[121]. Ho et al. also evaluated PL quantum yields of ∼6×10−4, comparable to that of ReS2^[85]. In addition, discrepancies can be found in the PL line shape as well. For example, Aslan et al. showed that in atomically thin ReS2, PL intensity due to transition E2 is stronger than that due to E1 (E2 at a higher energy than E1), indicating the possible presence of a lower lying indirect transition or of rapid nonradiative relaxation associated with defects^[37]. In other reports, PL intensity due to E1 is obviously stronger than that due to E2^[85,120]. The discrepancies in PL studies might be due to different crystal qualities or stacking orders of ReX2.

Under low temperature, the fine structure of PL in ReX2 can be clearly distinguished. As shown in Fig. 3(d), Ho et al. found three additional PL peaks centered at 1.63 (1.45), 1.64 (1.47), and 1.65 (1.48) eV (labeled as E3, Es1, and Es2 respectively) in the few-layer ReS2 (ReSe2) at 4 K, which were assigned to interband excitonic PL at different k-points in BZ with higher energy^[85]. Among them, the polarization of E3 (Es1,2) was consistent with that of E2 (E1). In addition, Jadczak et al. observed two series of four PL peaks centered between 1.63 and 1.67 eV in bulk ReS2 at 7 K^[122]. One series of four PL emission peaks was polarized along the b axis, while the other series was perpendicular. These two series of PL are ascribed to excited states of excitons E1 and E2 (discussed in the next section).

Linearly polarized PL has also been studied in other anisotropic 2D materials. As a quasi-1D direct bandgap semiconductor, TiS3 is expected to have a strong anisotropic PL. Recently, Curto et al. reported an NIR PL centered at 0.91 eV in multilayer TiS3 (thickness ∼32nm)^[123], as shown in Fig. 3(e). PL emission in TiS3 is partially linearly polarized with maximal (minimal) intensity along the b(a) axis, parallel (perpendicular) to the quasi-1D chain, with a polarization anisotropy ratio of ∼4 at room temperature and increasing to ∼5 at 20 K. Due to anisotropic absorption, PL in TiS3 also exhibits excitation polarization dependence (laser wavelength ∼532nm). At low temperatures, defect state PL with lower emission photon energy is prominent. Interestingly, such PL exhibits a contrary polarization dependence compared to that of interband PL, whose maximal emission intensity is along the a axis, as shown in Fig. 3(e).

Although a cousin of TiS3, ZrS3 has an indirect bandgap, due to the in-plane quasi-1D chain atomic structure, it is expected to exhibit strong PL as well. Pant et al. reported polarized PL centered at 1.45 and 1.8 eV in ZrS3 thin film. PL at 1.8 eV is due to the indirect bandgap, whose polarization dependence mainly results from the geometry anisotropy of the quasi-1D chain structure^[124]. For the width of the quasi-1D chain much smaller than the excitation light wavelength, absorption will be stronger when the excitation light is polarized along the chain. However, the PL anisotropy ratio is only ∼5.7 to 10.8, smaller than that in true 1D nanowires or carbon nanotubes. The discrepancy might be attributed to the large effective permittivity or interactions between quasi-1D chains.

Above, the absorption and PL were discussed without considering Coulomb interactions between electrons and holes. In fact, due to the enhanced quantum confinement and reduced dielectric screening, Coulomb interactions in 2D materials between electrons and holes are prominent even at room temperature, giving rise to electron–hole bound states termed as excitons. In 2D materials, excitons play a dominating role in optical properties, significantly enhancing light–matter interaction. For example, without excitons involved, the absorption of monolayer graphene is only 2.3%^[7], while in monolayer TMDCs at exciton resonance, the absorption can be as large as ∼20%^[8,9]. There are many excellent reviews of excitons in 2D materials^[125–127]. Here we mainly focus on excitons in low symmetry 2D materials. Before a detailed discussion on representative materials, basic properties of excitons will be briefly reviewed.

As mentioned above, an exciton is formed by a pair of electrons and holes due to Coulomb interactions, which can be simply conceived as an analogy to a hydrogen atom. Analogous to hydrogenic states, there is a series of excitonic states as illustrated in Fig. 4(a), which can be labeled by principal and magnetic quantum numbers as (n, m), where n=1,2,3 ... and |m|<n. With the same notation as in the hydrogen atom, m=0,1,2 ... can be denoted as s,p,d ... states. Due to selection rules determined by the symmetry of the exciton wave function, in general, only s states can be observed in a single-photon absorption process. As shown in Fig. 4(a), the resonance energy at the exciton ground state (1s), typically called optical bandgap, is lower than the true bandgap (quasiparticle bandgap) Eg, with the energy difference termed as exciton binding energy Eb. The binding energy is a critical parameter of the exciton, which measures the energy needed to separate the bounded electron–hole pairs, and gauges the strength of Coulomb interactions. In bulk semiconductors, due to the strong screening from the material itself, the binding energy is usually comparable to or smaller than room-temperature energy (∼26meV); thus, the exciton can be observed only at low temperatures, while in atomically thin 2D materials, screening from the material is reduced significantly. In addition, quantum confinement restricts electrons and holes in-plane, and hence, the binding energy can acquire a surprisingly large value, e.g., ∼500meV in monolayer TMDCs^[126]. Interestingly, in anisotropic 2D materials, anisotropic effective mass makes carriers more mobile along one crystal axis than the other (see Table 1). It is plausible to anticipate excitons with quasi-1D properties, such as larger binding energy.

Tran et al. first theoretically studied the anisotropic exciton in 1L–3L BP^[44]. They predicted that suspended monolayer BP has a large binding energy of ∼800meV, and what is more interesting, they found that the wave functions of typical bright excitons (e.g., 1s,2s ...) are anisotropic, specifically, more extended along the AC direction to form striped patterns, similar to those found in nanowires [as shown in Fig. 4(a)]. They ascribed this quasi-1D like exciton wave function to the anisotropic mobility of electrons. Subsequently, Gomez et al. predicted that the binding energy of monolayer BP on a SiO2 substrate would be significantly reduced to only 380 meV, suggesting a strong screening effect from the substrate^[104]. Later on, comparable binding energies of freestanding and supported BP were predicted in theory through different calculation methods^[43,137,138].

In experiments, through PL excitation (PLE) spectroscopy, Wang et al. first obtained the exciton binding energy of a surprisingly large value, ∼0.9eV, in monolayer BP on a SiO2/Si substrate^[106]. They observed that a PL signal centered at ∼1.3eV (corresponding to the optical bandgap) was resonantly enhanced with excitation energy at 2.45 eV, which was attributed to the resonance absorption of the quasiparticle bandgap edge extracted as ∼2.2eV. This unusually large binding energy of supported monolayer BP was attributed to the quasi-1D excitonic feature by Wang et al. Nevertheless, as mentioned above, there are huge discrepancies in the monolayer BP PL peak position, which could give rise to different binding energies. For example, later studies by Yang et al. reported two PL peaks in monolayer BP on a SiO2/Si substrate with central wavelengths at ∼705nm (1.76 eV) and ∼760nm (1.6 eV)^[108]. The former was assigned to ground state exciton (1s) PL, and with a comparison to the measured quasiparticle bandgap of ∼2.05eV by Pan et al. using scanning tunneling spectroscopy (STS)^[139], Yang et al. determined the binding energy in supported monolayer of only ∼0.3eV, much smaller than that reported by Wang et al. but comparable to previous theoretical predictions. It should be noted that PL at ∼760nm was assigned to the trion (charged exciton), leading to a large trion binding energy of ∼100meV, almost five times higher than that in the isotropic TMDC monolayer (e.g. MoS2), which was rationalized by their quasi-1D charge complex model. However, such a claim was challenged by later theoretical studies^[138,140]. Calculation results by Chaves et al. showed that, although larger than the usual trion binding energy in TMDCs, even the trion binding energy in suspended monolayer BP is only ∼51meV, and decreases to ∼34meV in supported BP, which is much smaller than the claim by Yang et al.^[138]. Due to the discrepancy, the validity of the 1D model of the trion binding energy in monolayer BP was questioned by Chaves et al.

Very recently, through reflection spectroscopy, Wang et al. reported excited exciton states (2s and 3s) in high quality monolayer BP on a polydimethylsiloxane (PDMS) substrate^[110]. Through extracting energy differences between 1s, 2s (Δ12∼230meV) and 2s, 3s (Δ23∼120meV) from the spectrum, binding energy of ∼452meV in monolayer BP on PDMS was obtained, slightly larger than previous predictions. On the other hand, since states are expected to be more Rydberg-like (as discussed later) with an increasing exciton quantum number n, 2s and 3s states can be assumed to follow the ideal 2D hydrogenic model. Then the binding energy can be simply estimated as Eb1s=(2516)Δ23+Δ12=418meV, consistent with previous predictions. In addition, it should be noted that the optical bandgap reported by Wang et al. was ∼1.69eV, and by comparing to the measured quasiparticle bandgap of ∼2.05eV by STS, it suggests a comparable binding energy of ∼360meV in supported monolayer BP. All of these results indicate that, regardless of the quasi-1D excitonic feature in BP, the binding energy in supported monolayer BP is roughly ∼400meV, rather than ∼900meV, as claimed by Wang et al.^[106]

Due to the direct bandgap nature of BP, regardless of thickness, the layer dependence of excitons can be studied in experiment. Through absorption spectroscopy, Zhang et al. observed the 2s states in few-layer BP (2L–6L) with high crystal quality on PMDS substrates at room temperature [as shown in Fig. 4(b)]^[141]. The energy difference between 1s and 2s states was directly extracted from the spectra as 122, 87, 76, 62, and 58 meV for 2L to 6L BP, which decreases with increasing thickness. As for 2s states in thicker BP (>6L), the energy separation of the 1s−2s state is too close to discern. In the 2D hydrogenic model, it is well known that the energy separation of 1s and 2s states accounts for 8/9 of total binding energy. Through numerical calculations, Zhang et al. showed that the binding energies of 2L–6L BP on PDMS are 213, 167, 139, 120, and 106 meV, respectively, suggesting that the energy separation between 1s and 2s states accounts for only about half of the total binding energy. This is mainly due to the nonlocal dielectric screening of Coulomb interactions in 2D materials. In fact, this nonlocal screening induced deviation between excitonic Rydberg series and 2D hydrogenic Rydberg series was earlier studied in monolayer TMDCs^[128]. Therein, Olsen et al. proposed a generalized hydrogen model for nonlocal screened excitons in 2D materials, where the media dielectric constant ɛ in the Eb expression in the hydrogen model was substituted by an effective 2D dielectric constant through averaging the screening over the extent of the exciton^[130]. Remarkably, in this generalized hydrogen model, the binding energy can be generally simplified as Eb≈3/(4πα), where α is the 2D sheet polarizability of the 2D material, suggesting that Eb is independent of the excitonic effective mass and depends only on α. On this basis, Zhang et al. further proposed an effective polarizability as αeff(N)=α0+Nα1, where α0 and Nα1 account for the screening from the substrate and N-layer BP itself, respectively. Eb can be expressed as Eb=34π(α0+Nα1)^[141]. The model can well fit the layer-dependent Eb in few-layer BP, with reasonable fitting values of α0 and α1, comparable to previous theorical calculations, which suggests that the model is feasible. In addition, this model predicts that Eb of suspended monolayer BP and that supported on PDMS are 762 and 316 meV, respectively, in good agreement with previous theoretical predictions.

Besides the binding energy, Zhang et al. further investigated the layer-dependent absorption strength of excitons in 2-7 L BP on a PDMS substrate^[129]. In common sense, thicker materials induce stronger absorption strength. Surprisingly, for exciton absorption in 2D BP, less is more. Zhang et al. found that frequency-integrated exciton absorption was enhanced in thinner BP [as shown in Fig. 4(b)]. This interesting phenomenon can be quantitatively explained by the quantum well model, i.e., the integrated excitonic absorption is mainly determined by Lz·|ϕex(0)|2, where Lz is the thickness of the sample (layer number N for BP), and |ϕex(0)|2 is the modular square of the excitonic wave function at the origin with a 1/Lz2 scaling law. This gives rise to a 1/N relation for the excitonic absorption strength, hence, the thinner the sample, the larger the excitonic absorption. After taking into account the substrate screening effect, this relation was modified as 1/(α0+Nα1). Equally interestingly, Zhang et al. found that absorption from the continuum transition close to the band edge is a constant παμy/μx, where α is the fine structure constant with a value of 1/137 (πα is the universal absorption constant in graphene), and μx(y) is the reduced effective mass in the AC (ZZ) direction. This universal continuum absorption in BP is due to the cancellation mechanism between the transition matrix element and 2D JDOS, i.e., the band-parameter-dependent terms are perfectly canceled out. It should be noted that in N-layer graphene, N pairs of sub-bands are involved in the optical absorption, giving rise to N-fold absorption compared to the single layer. On the contrary, due to the bandgap at Γ point in BZ, only one pair of sub-bands in BP is involved for continuum absorption near the bandgap; hence, the continuum absorption near the bandgap is constant regardless of thickness.

As for group-IV monochalcogenide compounds MX, there are pairs of excitons from different valleys with different polarization dependences due to the multi-valley band structure. Xu et al. systematically investigated the exciton in monolayer MX in theory, as shown in Fig. 4(c)^[36]. The lowest energy exciton A is polarized along AC direction in all MX, with binding energies of 0.98, 0.38, 0.5, and 0.28 eV in GeS, GeSe, SnS, and SnSe, respectively, in good agreement with earlier calculation results^{[131,142,143]}. Binding energies of the lowest energy exciton B in ZZ direction were calculated as 0.77, 0.36, 0.5, and 0.28 eV in GeS, GeSe, SnS, and SnSe, respectively. In the MX bulk counterpart, the binding energy is relatively small, less than 0.1 eV. The substrate screening effect also significantly affects the exciton binding energy, e.g., A exciton binding energy in monolayer GeSe significantly shrinks from 0.38 eV in a suspended one to 0.19 eV in one supported on a SiO2 substrate. It should be noted that among four monolayer MX, exciton in GeS is the most localized due to the largest binding energy, while exciton in SnSe is the most delocalized. Binding energy is closely related to the screening length of the material itself but weakly depends on the reduced mass. The exciton spatial distribution anisotropy in GeX is larger than that in SnX. The radii of A exciton in GeS along AC and ZZ are 6.74 and 4.46 Å, respectively, while in SnS, the A exciton radius is 15 Å along both directions.

Zhong et al. first studied excitons in freestanding monolayer ReX2 in theory. Based on their results, the lowest-energy exciton in ReS2 at Γ point with a binding energy of 1.07 eV exhibits a perfect 8-shaped polarization dependence, while that of ReSe2 with binding energy of ∼0.87eV exhibits only partial polarization dependence, due to the mixture of two bright excitons at Γ and Λ points with different polarization dependences^[132]. Theoretical calculations by Echeverry et al. suggested that in both monolayer ReS2 and ReSe2, due to the doubly degenerated VBMs and CBMs at Γ point, the lowest energy exciton with different spin combinations splits into four due to the exchange interaction, with average binding energies of 0.74 and 0.57 eV in ReS2 and ReSe2, respectively^[72].

In experiments, Arora et al. reported four excitonic absorption and PL peaks with different spin combinations at Γ point in ReSe2 [as shown in Fig. 4(d)^[38]], discussed in Section 2.1. Dhara et al. reported four excitonic absorption peaks near the bandgap in few-layer ReS2 under low temperatures [Fig. 4(d)]^[144]. Different from those of ReSe2, two prominent peaks were assigned as bright exciton absorption at Z and K1 points in BZ according to previous DFT results^[133], and two additional weaker peaks were ascribed to the splitting of singlet and triplet states of excitons. Dhara et al. also revealed that the existence of low lying dark excitons causes an anomalous PL intensity variation at 30 K in a temperature dependence study, which was rationalized by the rate equation model. In addition, through low temperature PL spectroscopy, Jadczak et al. revealed that resonance E3, as reported by Aslan et al. in bulk ReS2, is composed of several excited excitonic states of ground state excitons E1 and E2^[122]. Through a 3D hydrogenic Rydberg model, the binding energies of excitons E1 and E2 were estimated as ∼0.12 and 0.08 eV, respectively. It should be noted that the binding energy in bulk ReX2 is much larger than that in other bulk materials, which could be attributed to the weak interlayer coupling in ReX2, resulting in the exciton being confined in individual layers. The binding energy of monolayer ReSe2 supported on graphene was directly determined as 0.52 eV by Qiu et al. through measuring the quasiparticle bandgap by STS and the optical bandgap by reflectance spectroscopy, which could also be continuously tuned from 0.46 to 0.68 eV by electrostatic gating^[134]. In addition, through gate-dependent low temperature PL spectroscopy, Wang et al. reported negative trions in few-layer (1L–5L) ReS2^[135]. The trion binding energy is strongly layer dependent, and can be as large as 60 meV in monolayer, almost two times larger than that in isotropic TMDC monolayers.

With quasi-1D structure, TiS3 is expected to have robust excitons with large binding energy. Theoretical studies showed that in monolayer TiS3, there are three interband transitions at Γ point, accompanied by three bright excitons Ex1,2,3 with binding energies of 0.59, 0.68, and 0.32 eV, respectively^[98]. The lowest energy exciton Ex1 located at ∼1.07eV is bright only for out-of-plane polarization, but with a quasi-1D nature, theoretical calculations for TiS3 suggested that the wave function of an exciton in TiS3 is almost isotropic^[145]. In addition, in spite of the robust nature of a quasiparticle bandgap with little layer dependence, Lian et al. found that binding energy exhibits strong layer dependence, which is significantly enhanced from ∼0.07eV in bulk to ∼0.8eV in monolayer^[99]. The large binding energy in monolayer TiS3 results in the lowest energy excitonic absorption (Ex1) at ∼0.58eV. This value is inconsistent with the aforementioned ∼1.07eV, and needs further study to resolve the discrepancy. In experiments, by using STS and photoelectrochemical measurements, Mendoza et al. obtained a quasiparticle bandgap and optical bandgap in TiS3 thin film (>25nm) of ∼1.2 and 1.07 eV, respectively^[136]. The binding energy was determined as ∼0.13eV, in agreement with the predicted result of 0.1 eV by their calculations.

Very recently, through PL and reflectance spectroscopy, Zhou et al. reported an unconventional exciton with hybrid dimensionality in 2D SiP2, where the electrons constituting the excitons are confined within 1D phosphorus–phosphorus chains, while the holes are extended over the 2D SiP2 plane^[146]. This unique exciton with binding energy of ∼140meV (in SiP2 with thickness>5nm) can strongly couple with phonons, resulting in a broad sideband PL at the low energy side of the prominent PL [as shown in Fig. 4(e)]. The exciton–phonon coupling induced energy shift of the sideband PL was ∼50meV, in good agreement with their calculation results. Both exciton emission and an emergent sideband feature show linear dichroism, and due to the electron–phonon coupling, both dramatically redshift with increasing temperature, in contrast to the slight change of the quasiparticle bandgap. This unique hybrid low dimensional system is very promising for enriching excitonic physics and further understanding exciton–phonon coupling.

The recent emergence of 2D vdWs magnets provides a new platform to study excitonic physics with intrinsic magnetic order. Hwangbo et al. reported the observation of excitons coupled to ZZ antiferromagnetic order in layered antiferromagnetic semiconductor NiPS3, as manifested by the near-unity linear polarization of exciton PL and absorption under low temperatures and almost vanishing above the Neel temperature^[147]. In addition, the strongly coupled exciton–phonon bound states can be observed even in atomically thin samples (e.g., 3L–5L) below the Neel temperature, and more surprisingly, over 10exciton-A1g-phonon bond states on the high energy side of exciton resonance were clearly resolved as shown in Fig. 4(e), which were ascribed to the strong modulation of the ligand-to-metal charge-transfer energy by electron–lattice interactions. Klein et al. reported an extremely anisotropic electronic band structure of layered A-type antiferromagnetic semiconductor CrSBr, in which the anisotropic ratio of electron effective mass mX/mY can be as large as ∼50^[148]. Such a strong quasi-1D electronic character gives rise to strongly anisotropic excitons with large binding energies of 0.88 and 0.9 eV for the two lowest energy excitons in the monolayer limit, and strong exciton–phonon coupling as revealed by resonances of the 1s state exciton at detuning energy (Elaser–E1s) corresponding to Ag2 and Ag3 phonons in the PLE spectrum. In this quasi-1D electronic system combined with magnetic order, the magneto-exciton physics calls for further exploration.

The above discussions are all in the realm of linear optics, where the polarizability P of the material depends linearly on the electric field E of light. This is valid only when incident light intensity is low. With a high intensity of incident light, P depends nonlinearly on E, that is, P=ɛ0(χ(1)E+χ(2)E2+χ(3)E3+…),(2)where ɛ0 is the vacuum permittivity, and χ(n) is the nth-order susceptibility. χ(1) is the normal linear susceptibility, responsible for linear absorption, reflectance, and transmittance; χ(2) is responsible for second-order nonlinear effects, such as second-harmonic generation (SHG), optical rectification, and Pockels effect; χ(3) is responsible for third-order nonlinear effects, such as third-harmonic generation (THG), saturation absorption, two-photon absorption (TPA), and optical Kerr effect. It should be noted that second- or even-order nonlinear effects exist only in non-centrosymmetric materials, while third- or odd-order nonlinear effects can take place in all materials. Due to the quantum confinement effect and anisotropy for anisotropic 2D materials, nonlinear optical properties are unique. Recent studies have been performed to investigate nonlinear optical effects in anisotropic 2D materials, and optoelectronic devices based on nonlinear optical effects, such as anisotropic saturable absorbers, ultrafast pulse generation, optical switches, and optical diodes in anisotropic 2D materials, have been demonstrated. There are dedicated review papers for nonlinear optical properties and applications in anisotropic 2D materials^[149,150]. In this section, nonlinear optical properties including saturable absorption, harmonic generation (SHG and THG) of low symmetry 2D materials will be briefly discussed.

Saturable absorption is a phenomenon in which absorption of light decreases with increasing incident light intensity, which results from the Pauli-blocking principle. Under strong incident power, CBM and VBM are filled with excited electrons and holes, respectively, impeding the interband transition and giving rise to a reduction in absorption [as shown in Fig. 5(a)]. The saturable absorption effect is very useful for ultrafast pulsed lasers, optical modulators, and so on; thus, it is an important research field in nonlinear optics and has been extensively studied. The saturable absorption coefficient can be expressed as α=αs1+I/Is,(3)where α and αs are the saturable and non-saturable absorption coefficients, respectively, and I and Is are incident light intensity and saturable intensity, respectively. Based on the Beer–Lambert law, the saturable absorption induced transmittance can be expressed as T=exp(−δT1+I/Is),(4)where δT=αsL is modulation depth. In experiments, through open (or closed) aperture Z-scan and I-scan methods, the normalized transmittance spectrum can be obtained.

The saturable absorption in anisotropic 2D materials was first studied in few-layer BP dispersions obtained by liquid phase exfoliation with a broad bandwidth ranging from visible (400 nm) to MIR (at least 1930 nm)^[151–153]. The saturable absorption response in BP dispersions is much stronger than that in graphene dispersions. Afterwards, Li et al. revealed the salient layer dependence of saturable absorption in BP flakes^[154]. The saturable absorption induced transmittance increase in thicker exfoliated BP flakes (1100 nm) was around eight times larger than that in thinner BP (25 nm). Such thickness dependence is useful for pulse generation since nonlinear saturable absorbers with different parameters are needed.

Saturable absorption also exhibits polarization dependence. Sotor et al. found that BP flakes (thickness ∼300nm) exhibit saturable absorption but with different modulation depths when light (wavelength of 1560 nm) is polarized along AC and ZZ directions. The highest transmittance is obtained under ZZ polarization, consistent with linear absorption polarization dependence^[155]. With a further increase in light intensity, a roll-off effect (reverse saturable absorption) in the transmittance spectrum with ZZ polarization was observed, ascribed to the TPA effect, whereas it retains saturable absorption with AC polarization [as shown in Fig. 5(a)]. Anisotropic saturable absorption was also reported in bulk ReS2 by Liu et al., which can be transformed to reverse saturable absorption by rotating incidental light polarization^[156]. For instance, the normalized absorption increases under a polarized angle of 0° (with respect to the b axis) due to saturable absorption, while it decreases under an ∼100° polarized angle due to reverse saturable absorption [as shown in Fig. 5(a)]. Since reverse saturable absorption occurs at relatively low intensity power, and due to the lack of free carriers in undoped ReS2, the finding was ascribed to excited state absorption, consistent with angle-resolved photoemission spectroscopy (ARPES) results^[157]. The same group found that nonlinear optical absorption depends on the stacking order. Saturable absorption occurs only in AB stacking ReS2 at polarized angles from 30° to 105° (with respect to the b axis), and is absent in AA stacking. The reverse saturable absorption due to excited state absorption occurs in both stackings, but with a larger anisotropy in AB stacking^[158]. This peculiar polarization and stacking order dependence were attributed to an additional absorption channel of exciton–exciton annihilation in AB stacking.

In fact, the saturable absorption effect is closely related to linear absorption. According to Fermi’s golden rule, linear absorption at different wavelengths with different transition matrix elements (polarization dependent) and JDOSs, changes the final excited states differently, giving rise to different saturable absorption effects. Zhang et al. investigated saturable absorption effects in SnSe flakes (thickness ∼175nm) at dual wavelengths of 800 and 850 nm^[159]. The anisotropic ratio of the modulation depth δT between AC and ZZ directions was 6.7 at 800 nm wavelength, much larger than that of 1.4 at 850 nm. In addition, the saturable absorption coefficient αs and modulation depth δT at 800 nm were found to be smaller than those at 850 nm, indicating weaker saturable absorption. On this basis, novel all-optical switching with a higher on–off ratio of ∼44% was proposed. Ye et al. studied the saturable absorption effect at a wavelength of 1030 nm in few-layer SnSe dispersion^[160]. Since incident photon energy is lower than the bandgap of few-layer SnSe (average thickness of ∼6nm), the saturable absorption effect was attributed to the depletion of defect states located in the bandgap. With increasing incident light intensity, reverse saturable absorption due to TPA was observed. The transition from saturable absorption to reverse saturable absorption in thicker SnSe (average thickness of ∼9.5nm) dispersion was observed as well but with higher incident power, which is related to the layer-dependent bandgap in SnSe.

Harmonic generation is a process converting n incident photons at frequency ω into a higher frequency nω photon as shown in Figs. 5(b) and 5(c), which can be employed to produce a broadband wavelength-tunable coherent light source. SHG (n=2) and THG (n=3) are most frequently used for frequency conversion applications; hence, here we mainly focus on SHG and THG processes in anisotropic 2D materials.

As a centrosymmetric material, BP permits only THG. THG in BP has been studied by several groups^[161–163]. A typical THG signal is shown in Fig. 5(b), with the peak centered at ∼520nm, exactly three times the frequency of the pump light (wavelength 1560 nm)^[163]. As anticipated, THG is polarization dependent, and the intensity of THG I3ω in BP can be expressed as I3ω=Iω3[(χxx(3)cos3θ+3χxy(3)cosθsin2θ)2+(χyy(3)sin3θ+3χyx(3)sinθcos2θ)2],where Iω is the intensity of the pump light, χ(3) is the third-order susceptibility tensor of BP, subscripts x and y refer to AC and ZZ directions, respectively, and θ is the polarized angle of pump light with respect to the AC axis. Different from the two-fold polarization dependence of linear absorption, THG exhibits a four-fold polarization dependence pattern, which is mainly due to the nonzero χyx(3) component. It should be noted that maximal THG intensity is obtained not along the AC direction as shown in Fig. 5(b). In fact, due to the higher tensor ranks of χ(2) and χ(3), the polarization dependence of SHG or THG could be very different from linear absorption. Even in isotropic material, SHG or THG could be anisotropic. Hence, they are usually used to determine the crystal orientation of materials.

The intensity of THG in BP also exhibits strong layer dependence. Rodrigues et al. found that THG in atomically thin BP can be significantly enhanced due to excitonic resonance [as shown in Fig. 5(b)]^[161]. With the pump light wavelength at 1560 nm, which is close to the excitonic resonance absorption of 3L BP, the effective third-order susceptibility |χeff(3)| in 3L BP can be enlarged to ∼10–20 times that in graphene. In a non-resonance regime, THG also has layer dependence in BP with thickness ranging from 5 to 50 nm. The maximal THG signal was obtained in BP with ∼10 to 20 nm thickness, which was attributed to the competition between different absorptions of pump light and THG in materials. THG has also been studied in other anisotropic 2D materials. In particular, monolayer ReS2 exhibits strong and anisotropic THG, in which the third-order susceptibility |χ(3)| is of 10−18m2/V2, almost one order of magnitude larger than that in 2H TMDCs such as MoS2^[164].

SHG exists only in non-centrosymmetric materials, such as 2H-TMDCs with odd-layer thickness. Interestingly, Song et al. found that anisotropic SHG in atomically thin ReS2 exhibits an opposite dependence on thickness^[165]. Strong SHG exists only in even-layer ReS2, whereas it is negligible in odd-layer ReS2 [as shown in Fig. 5(c)]. This unusual layer-dependent SHG is due to the unique distorted 1T structure of ReS2, i.e., inversion symmetry exists in mono (odd-) layer but breaks in bi- (even-) layer. In group IV monochalcogenides MX, prominent and anisotropic SHG was observed as well [see Fig. 5(c)]^[166]. Theoretical studies predicted that effective second-order susceptibility |χ(2)| in monolayer MX can be as large as 10 nm/V within a broadband regime (photon energy from 0.8 to 4 eV)^[167], which is remarkably high among those reported in other 2D materials, such as 0.16 nm/V in MoS2 and 0.001 nm/V in BN. The large anisotropy of SHG in monolayer MX could be also closely related to the intrinsic ferroelastic and ferroelectric orders^[168].

2D materials composed of single or several atomic layers, are very sensitive to external perturbations, such as the dielectric environment, electrical field, and strain. Thus, the band structure can be easily modulated artificially, that is, band structure engineering. Band structure engineering has been extensively studied in 2D materials, through various tuning methods, including electrical gating, applying strain or high pressure, chemical doping or intercalation, alloying, stacking vdWs homo or heterostructures, and so on^[169]. Due to band structure engineering, novel properties beyond natural properties of materials have been revealed; one of the most famous examples is the unconventional superconducting in magic angle twisted bilayer graphene. Thus, band structure engineering is vitally important, not only for fully exploring the novel properties of materials in fundamental research, but also offering more possibilities for device design and optimization in practical applications. Anisotropic 2D materials with polarization dependence can offer additional freedom for band structure engineering. In this section, several types of band structure engineering to manipulate optical properties of anisotropic 2D materials will be discussed, particularly for BP, since band structure engineering of BP has been most extensively studied, and serves as a good representative.

In practice, applying an electrical field is the most feasible way to achieve tunable optoelectronic devices. When 2D materials are under a vertical (along out-of-plane direction) electrical field, the band structure distorts in real space with the wave functions of electrons and holes moving to opposite directions and partially tunneling into a forbidden bandgap, giving rise to a redshift of interband absorption, termed the Franz–Keldysh effect. Due to the strong quantum confinement effect in 2D materials, it is also called the quantum-confined Franz–Keldysh effect. As mentioned above, excitons usually dominate the optical properties in 2D materials. Exciton resonance energy redshifts with an external electrical field, which is also known as the quantum-confined Stark effect. In addition, since atomically thin materials can be approximated as natural infinite quantum wells, interband transitions are allowed only between valence and conduction sub-bands with the same quantum number n (e.g., Enn in BP). In the presence of a vertical electrical field, the additional otherwise selection-rule forbidden interband transitions (e.g., Enm, m≠n) could occur due to broken inversion symmetry. In experiments, the vertical electrical field is usually applied through the gate voltage based on FETs. Gate voltage not only supplies a vertical electrical field, but also changes the carrier populations (doping effect), which can blueshift the interband transition energy due to Pauli-blocking. In such a case, the interband transition energy shift will be nonlinear due to the competition between the electrical field effect and doping effect. Theoretical and experimental studies have shown that interband absorption peaks in few-layer BP redshift first at a low gate voltage and then blueshift at a higher gate voltage due to the quantum-confined Franz–Keldysh or Stark effect and Pauli-blocking, respectively^[170–172]. Hybrid transitions such as E21 and E12 were observed as well. In fact, the electrical field and doping effect in FET can be separated by applying a dual gate voltage. By stacking heterostructures, materials can float in a vertical electrical field to avoid the doping effect. Through BN encapsulated BP devices with BN serving as the dielectric layer, Chen et al. revealed the pure electrical field effect on PL emission of few-layer BP (∼20L), as shown in Fig. 6(a)^[173]. With a moderate displacement field up to 0.48 V/nm, PL emission with linear polarization can be continuously tuned from 3.7 to 7.7 μm, and it is expected to redshift to longer than 7.7 μm with a further increased field, which suggests the widely tunable IR light emission in BP. However, under the electrical field, PL intensity is reduced due to the spatial separation of electron and hole wave functions. Through in situ surface doping of potassium (K) on few-layer BP, where the carrier distribution is different between top and bottom BP surfaces, Kim et al. introduced a vertical electrical field in BP flakes^[174]. Due to the giant Stark effect, the semiconducting BP was converted into a semimetal as a consequence of the bandgap closing, which was monitored by ARPES. What is more interesting, at the critical semiconductor to semimetal transition point, BP becomes an anisotropic Dirac semimetal, in which the electronic band disperses linearly (quadratically) along AC (ZZ) direction. In addition, under an ambient condition, the unintentional doping induced electrical field effect on few-layer BP was observed, as manifested by the redshift of the optical bandgap and appearance of hybrid transitions, which was further confirmed by the intentional chemical doping through HNO3 vapor^[35].

Strain and pressure affect band structure in the same way in essence, i.e., modification of the lattice constants to tune the band structure, which can be easily understood in the tight-binding model. For example, the monolayer BP bandgap Eg can be approximately expressed as Eg≍4t1∥+2t2∥, where t1∥ and t2∥ are hopping parameters between different atoms^[175]. In general, the modulus of hopping parameter |t| is proportional to 1/r2, where r is the distance between atoms connected by hopping parameter t. Changes in lattice constants induce changes in hopping parameters, hence the bandgap. It should be noted that hoping parameter t can be a positive or negative value, e.g., in monolayer BP t1∥>0 and t2∥<0. The sign of t determines the blueshift or redshift of the bandgap with increasing (or decreasing) lattice constants.

In-plane uniaxial strain can be applied by bending or stretching the substrate, or by introducing wrinkles in the material [as shown in Fig. 6(b)]^[176,177]. Experimental studies have shown that uniaxial strain can efficiently tune the bandgap size. For example, in few-layer BP with moderate uniaxial strain, Kim et al. reported that PL wavelength can change from 5.5 μm (0.66% compressive strain) to 3.2 μm (1.21% tensile strain), corresponding to the bandgap change from 0.22 to 0.53 eV [see Fig. 6(b)]^[177]. For anisotropic 2D materials, it is natural to anticipate that a uniaxial strain effect on band structure will be different when strain is applied along different directions^[178,179], while experimental studies found that changes in bandgap and resonance absorption peaks above the bandgap are almost the same with strain along AC and ZZ directions^[35,180]. This counterintuitive result was ascribed to different Poisson ratios along AC and ZZ directions. Previous studies showed that the Poisson ratio along AC direction is positive, which means that the puckered height d of monolayer BP will decrease (increase) under the tensile (compressive) strain along AC. On the contrary, the Poisson ratio along ZZ direction is negative, which means that the puckered height d changes in the opposite way^[29]. The different changes in monolayer puckered height d cancel the difference between the strain effect along AC and ZZ directions. Recently, Wang et al. reported that the strain effect for interband absorption in few-layer Nb2SiTe4 exhibits strong orientation dependence^[102], in particular, for transitions B and B*, which blueshift (redshift) with tensile strain along the a axis (b axis). Other interband absorption peaks exhibit anisotropic strain effects as well, manifested by the different shift rates of the absorption peak with strain along different axes.

In-plane biaxial strain can be commonly applied through the thermal expansion of the substrate underneath the 2D material, whose thermal expansion coefficient should be much larger than that of the 2D material [as shown in Fig. 6(b)]^[176,181]. Different from uniaxial strain, the biaxial strain effect is isotropic and believed to tune the band structure more efficiently. In addition, Huang et al. revealed that in-plane biaxial strain can efficiently modulate the interlayer coupling in few-layer BP, as manifested by the strongly layer-dependent strain effect^[181]. According to their results, 1% in-plane biaxial compressive (tensile) strain can induce ∼10% increase (decrease) in interlayer coupling, which is remarkable. More surprisingly, they found that the biaxial compressive strain in few-layer BP strengthens interlayer coupling, which is abnormal since the interlayer distance is anticipated to increase with in-plane compressive strain due to the Poisson effect. This unusual result is actually due to the unique puckered atomic structure of BP. More specifically, under biaxial compressive strain, the interlayer distance D+d increases, but the “effective distance” D determining interlayer coupling decreases, resulting in the enhancement of interlayer coupling.

To tune the interlayer interaction, applying pressure is the most direct method. Generally, hydrostatic pressure up to several to dozens of GPa can be applied through a diamond anvil cell (DAC) [as shown in Fig. 6(b)]^[182,183]. Under such high pressure, the material properties will undergo prominent changes, such as bandgap closing and crystal structure phase transition. For atomically thin 2D materials supported on a substrate, due to the adhesiveness between the material and the substrate, it is prone to induce non-hydrostatic pressure. In fact, if the substrate is stiff enough, the 2D materials undergo uniaxial strain along the out-of-plane direction^[184,185]. Pressure can directly reduce the interlayer distance and increase interlayer coupling, which can shrink the bandgap size. For example, under 1.8 GPa, the bandgap of bulk BP is closed^[186,187]. Recently, Huang et al. revealed that the pressure effect on band structure of BP exhibits strong layer dependence, e.g., under 1.8 GPa, the bandgap of bilayer BP does not shrink but enlarges^[188]. This layer-dependent pressure effect results from the competition between the strengthened interlayer coupling and enlarged bandgap of the monolayer (the puckered height in monolayer decreases, inducing a larger bandgap). Their results show that a moderate pressure (∼1.9GPa) induces ∼18% relative change in interlayer coupling.

Alloying is a very traditional strategy to tune the band structure. Due to the weak interlayer coupling in ReX2, the bandgap is hardly tuned by thickness, while it can be efficiently tuned by alloying. In ReS2(1−x)Se2x alloys, the optical bandgap can be continuously tuned from 1.62 (1.52) to 1.31 (1.26) eV in monolayer (bulk) by changing the constituent ratio x from zero to one^[189]. In B-As–phosphorus alloys (b-AsxP1−x), which keeps the orthorhombic structure with a puckered honeycomb lattice like that of BP or b-As but with two elements [as shown in Fig. 6(c)], the bandgap can be continuously tuned from 0.3 to 0.15 eV by changing the constituent ratio x from 0 to 0.83^[190,191]. At the same time, the anisotropy is always retained. This gives b-AsxP1−x great potential for long-wavelength optoelectronic applications.

Besides alloying, due to their interlayer non-bonding nature, layered 2D materials possess a physical vdWs gap, where guest species can be intercalated^[192]. Usually, due to the small atomic size, the alkali ion is chosen as the guest species for intercalation, which can significantly change the carrier density and even induce a crystal structure transition, hence affecting the material properties. Studies showed that interaction is orientation dependent for anisotropic materials, e.g., intercalated ions have a more favorable channel along ZZ direction in BP^[193,194]. While due to the high reactivity of an alkali ion, the intercalated materials are unstable under an ambient condition. In addition to alkali ions, a stable organic molecule [such as cetyl-trimethylammonium bromide (CTAB)] can also be chosen for intercalation. Due to the large size of the molecule, the intercalation can significantly increase the interlayer distance, giving rise to weakened interlayer coupling. Taking BP as a model system, Wang et al. intercalated CTAB into bulk BP through the electrochemical method, as shown in Fig. 6(c)^[195]. Their atomic force microscope (AFM) and X-ray diffraction (XRD) results suggested that the interlayer distance increases more than 115% after intercalation, effectively isolating the single layer to form a monolayer superlattice. Through leveraging the electrochemical current, bilayer, tri-layer, or few-layer (even four to 10) BP superlattices can be engineered as well. PL was observed in mono-, bi-, and tri-layer superlattices at energies of 2.26, 1.75, and 1.38 eV, respectively. These PL energies are slightly higher than those of intrinsic BP, due to the strain induced bandgap expansion or orbital symmetry breaking caused by intercalation. Since monolayer BP can be obtained only by mechanical exfoliation with low yield and small lateral size at present, organic molecular intercalation offers an avenue to overcome these bottlenecks. The intercalated monolayer BP superlattice is very stable under ambient conditions. Thus, this technique is vitally important for 2D BP in both research and applications. More broadly, intercalation is a promising engineering technique, since intercalating different functional species into layered 2D materials could achieve a versatile material platform for both studies and applications.

Due to the non-bonding vdWs interlayer interaction, 2D materials can be stacked arbitrarily with the same or different materials to form homo- or heterostructures without a lattice matching requirement. Based on the band alignment, heterostructures are divided into three types. In type I, both VBM and CBM of one material are located inside the bandgap of the other material; in type II, VBM and CBM of two stacking materials are staggered; and in type III, the CBM of one material is lower than the VBM of the other material. Due to the staggered band alignment in type II heterostructures, the excited electrons and holes can separately reside in different materials and form interlayer excitons. With the merits of a long lifetime and strong Stark effect, interlayer excitons have been extensively studied in type II heterostructures based on isotropic TMDCs^[196]. It is expected that anisotropic materials could bring in anisotropy for interlayer excitons. Unfortunately, as far as we know, there is no experimental study on interlayer excitons in anisotropic heterostructures yet, even though a few studies revealed a superb anisotropic photo-response based on BP heterostructures^[197–199]. In addition, in type I heterostructures, excited electrons and holes can transfer from material with a wider bandgap to another material, giving rise to an increase in light emission intensity. Recently, Zong et al. revealed that few-layer BP and monolayer WS2 form a type I heterostructure [as shown in Fig. 6(d)]^[200]. Due to the larger bandgap in monolayer WS2, photo-excited electrons and holes are transferred from WS2 to BP, giving rise to ∼192% enhancement in MIR PL intensity at a wavelength of 2.79 μm [PL of 4 nm (8L) BP]. Although the enhancement of PL decreases with increasing thickness of BP, this enhancement effect can persist up to the wavelength of 3.89 μm. It should be noted that in their heterostructure, BP and WS2 were stacked with random interlayer angles. The aligned stacking angle could affect PL efficiency, which deserves further study.

The twisting interlayer angle also dramatically enriches the phase diagram of homo/heterostructures. In particular, with a moiré superlattice, new properties emerge, such as unconventional superconductivity in magic angle twisted graphene, moiré excitons, and strongly correlated states in TMDC moiré superlattices^{[127,201,202]}. In homo/heterostructures based on anisotropic materials, due to the lower symmetry, interlayer twisting is expected to be unique. For example, in homo/heterostructures based on graphene or TMDCs with hexagonal structure, the prominent moiré effect can be observed only within a tiny interlayer twisted angle (<2°). Recently, Zhao et al. revealed that a strong moiré effect persists even at a large angle of ∼19° in mono- and bilayer BP homostructures, as manifested by a new PL emission peak at a lower photon energy with polarization different from PL in mono- and bilayer BP [as shown in Fig. 6(d)]^[203]. Their DFT calculation suggested that such strong moiré effects in BP are not only related to the rectangular lattice of phosphorene but also due to the bandgap at Γ point. Cao et al. theoretically investigated the 90° twisted bilayer BP earlier^[204]. They showed that interlayer hybridization still exists for the conduction band but is absent for the valence band, and as a consequence, the transport and optical anisotropy can be switched by gate voltage. Subsequently, Srivastava et al. experimentally demonstrated resonance tunneling diodes based on 90° twisted BP homostructure^[205]. At present, the study of homo/heterostructure involving BP is still in its infancy, and calls for more systematic studies in the future.

The concept of polaritons was first proposed by Kun Huang in 1951, describing the hybrid modes of a radiation field and lattice vibration in ionic crystals^[206]. The solutions reveal two dispersive branches—the so-called PhPs. “Polariton” was further coined by Hopfield in 1958 based on the implication of polarization and photons, when applied to the interaction of incident light with excitons, namely, exciton polaritons^[207]. In principle, polaritons are quasiparticles that contain both photons and collective oscillations of polarization charges (electric dipoles) under the illumination of light. Such electric dipoles can originate from the collective oscillation of free carriers due to the Drude response, or phonons in polar crystals and excitons in semiconductors, forming so-called plasmon polaritons, PhPs, and exciton polaritons, respectively. Note that these electric dipoles provide the negative real part of permittivity (positive imaginary part of conductivity), which is key for sustaining polaritons at the interface between dielectrics and the host material.

When the thickness of the host material is much smaller than the polariton wavelength, only the in-plane optical response is prominent. Such polaritons in thin films of layered 2D materials have gained much attention since the discovery of plasmon polaritons in graphene^[15,208,209], and promise high field confinement and good tunability. The dispersion of polaritons in such a thin film limit is intrinsically determined by the sheet optical conductivity of the film itself: ω=−iσ(ω)2ɛ0ɛenvkp,(5)where ω and kp are the frequency and in-plane wave vector of polaritons, respectively, ɛ0 is the vacuum permittivity, ɛenv is the dielectric constant of the surrounding environment, and σ(ω) is the sheet optical conductivity, which follows σ(ω)=σ3D(ω)·t, with σ3D(ω) the optical conductivity of the bulk and t the film thickness.

For 2D materials with in-plane anisotropy, the sheet optical conductivity takes the complex tensor form σ↔, written as [σaa00σbb], where σaa, σbb represent the principal components of the tensor (these components are dispersive with frequency ω, and we omit them for simplicity). Note that in this review, we mainly consider the tensor with zero nondiagonal elements. Assume that the material is lossless, so that there are only imaginary parts of conductivities with σaa=iσaa′′ and σbb=iσbb′′, and the wavelength of the plasmon is much smaller than that of light. The dispersion relation of surface waves can be simplified as^[210]qa2σbb′′+qb2σaa′′=2pω(ɛ0σaa′′σbb′′−μ04),(6)where p=qa2+qb2−k02, k0 is the vacuum wave vector, and qa,qb are in-plane wave vector components of polaritons. By fixing the frequency ω, the in-plane IFC can be plotted in momentum space, which governs the optical dynamics of polariton propagation. For example, in isotropic surfaces such as graphene, where σaa′′=σbb′′>0 all the time, Eq. (6) indicates that the geometry of the contour is a circle, and the polariton wavefront propagates isotropically. In anisotropic surfaces where σa′′≠σbb′′ and σaa′′>0, σbb′′>0, the contour is quasi-elliptic with the long axis along the direction of the smaller conductivity component, which implies that the same wave vector of polaritons along different crystal axes will lead to splitting of the resonance frequency. However, at the frequency regime where the conductivities along the two principal axes have opposite signs, that is, σaa′′·σbb′′<0, the IFC appears as a hyperbola [as shown in Fig. 7(a)]. This topological transition of the IFC resembles the Lifshitz transition, where the deformation of the Fermi surface from a closed to an open geometry significantly affects the electronic properties^[211].

Polaritons with hyperbolic contours (σaa′′·σbb′′<0) are well known as hyperbolic polaritons, exhibiting great advantages compared to those in isotropic materials. For example, the open geometry of the contour in momentum space leads to theoretically infinite wave vectors and consequently extremely high DOSs. It should be noted that the wavefront and energy flow direction in anisotropic surfaces are not necessarily the same. It is the group velocity vg=∇qω(q) that dictates the energy propagation, which is also orthogonal to the IFC. The asymptotes of the hyperbolic curve are expressed as qb=±qa|σaa′′/σbb′′|, and the perpendicular direction, namely, the direction of the polariton beam, is always along y=±x|σbb′′/σaa′′|. In other words, a polariton propagates directionally on the hyperbolic surface. This peculiarity can be used to realize abnormal manipulation of polariton beams, such as focusing, negative refraction, and reflection^[212–218]. In extreme cases, the asymptotes are parallel to one of the coordinate axes, while the energy flow is then parallel to the other, giving rise to highly ray-like polaritons, which is also known as the canalization effect^[219,220].

The hyperbolic characteristic in layered material is attributed to the sign-changing conductivity along different crystal axes (σaa′′·σbb′′<0). To be more precise, one of the axes is conductive (inductive) while the other is dielectric (capacitive), the mechanisms of which are diverse for different types of polaritons. Take plasmon polaritons as an example. The electromagnetic response of the material originates from free carrier intraband transitions and bound carrier interband transitions, which can be described by the Drude–Lorentz model: σjj(ω)=iπDjjω+iΓ+iπωSjjω2−ωr2+iωη,(7)where the index j=a,b, D is the Drude weight, S and ωr denote the spectral weight and frequency of the interband resonance, respectively, and Γ and η represent the full width at half maximum (FWHM) of intraband and interband transitions, respectively. The first term on the right-hand side of Eq. (7) describes the Drude response of free carriers, which leads to a positive imaginary part of the conductivity, and the second term accounts for the interband transition, which provides a negative imaginary part of conductivity below the resonance frequency. The interplay between them could result in a critical point where σjj′′(ω)=0. Apparently, in anisotropic 2D materials, the effective mass, spectral weight, and frequency of interband transitions can be different along two principal axes, leading to splitting of the critical points and formation of the hyperbolic regime as in Fig. 7(c), which can be further manipulated by strain, intercalation, chemical or electrostatic doping, and so on.

Now let us discuss PhPs in polar dielectric crystals. A high reflectance is observed in the reststrahlung band (RB), which is between the transverse optical (TO) and longitudinal optical (LO) phonon frequencies. It is analogous to the high reflectance below the plasma frequency in plasmonic systems, such as metals. The coherent lattice vibrations (phonons) in polar crystals play a role similar to free carriers in metals to sustain polaritons. The difference is that phonons are bound states. The optical conductivity induced by phonon resonance can be expressed by the Lorentz model as well, with ωTO as the resonance frequency, which serves as the critical point. As a result, the anisotropic phonons along two crystal axes generate a hyperbolic regime as shown in Fig. 7(b). Such hyperbolic PhPs intrinsically possess ultra-long lifetimes (narrow scattering width) and high figures of merit compared to plasmon polaritons, due to the damping only from the scattering of optical phonons^[40]. On the other band, the hyperbolic characteristic is intrinsically fixed because of the stubborn phonon frequency, though it has been successfully tuned extrinsically by twisted stacking^[221–227] and substrate engineering^[228,229].

As for exciton polaritons in 2D semiconductors, excitons as bound states are also depicted by the Lorentz model, which enables σ′′(ω)>0 at a certain frequency regime. As a result, they are able to sustain propagating exciton polaritons. In particular, in anisotropic 2D materials such as BP, exciton absorption is prohibited when incident light is polarized along the ZZ direction but allowed in the AC direction, leading to a dielectric response in ZZ direction with σ′′(ω)<0. Then, the hyperbolic characteristic can emerge at the region where σaa′′·σbb′′<0, as shown in Fig. 7(d). The exciton polariton is actually a bosonic quasiparticle constituted of an exciton and a photon, which is promising for high temperature Bose–Einstein condensation (BEC) due to the exceedingly light effective mass^[230].

In fact, in-plane hyperbolic dispersion from visible to microwave frequency ranges has already been realized in artificial metasurfaces. However, metasurfaces demand complicated design and fabrication of sophisticated subwavelength structures, which could hinder more extensive applications. More importantly, due to the energy dissipation and limitations on the fabricated geometric size of structure elements, the wave vector of the hyperbolic polariton in metasurfaces cannot be large. Therefore, natural hyperbolic 2D materials, with merits of natural atomic scale size and high tunability, provide a promising platform for hyperbolic polaritons.

Unlike other optical measurements (e.g., absorption, PL), direct illumination of light from free space on materials is unable to excite polaritons due to the momentum mismatch of photons and polaritons. Some methods have been explored to compensate for the momentum mismatch, including both near- and far-field techniques.

Far-field techniques usually detect the reflection, transmission, absorption, or scattering of a specimen at macroscopic distance with the aid of spectrometers. To excite polaritons, incident light can couple to a prism to obtain an extra momentum as k′=nprismsinθk0 (nprism is the refractive index of the prism, and θ is the angle of incidence). Thus, a prism with a high refractive index is needed, either in touch with (Kretschmann configurations) or above (Otto configurations) the surface, as shown in Fig. 8(a). On the other hand, a periodic grating array fabricated by the side of the material can also provide additional momentum to excite polaritons, with the momentum expressed as k′=k0sinθ+2πnd (d is the grating period, and n is an integer). The above methods are to probe propagating polaritons. Localized non-propagating resonant modes of polaritons in artificial nanostructures (e.g., cavities, particles, disks, and ribbons) can be directly probed by incident light, such as the localized surface plasmon resonance (LSPR) of graphene nanoribbons shown in Fig. 8(b). Far-field techniques with merits of convenience, multifunctionality, and full-wave detection are powerful tools to investigate polaritons in momentum space.

One of the near-field techniques is known as scattering-type scanning near-field optical microscopy (s-SNOM), which combines an AFM with frequency-tunable coherent light sources, typically a quantum cascade laser (QCL) or interband cascade laser (ICL). It is able to probe spatially resolved light–matter interactions as well as the topographical information of sample surfaces beyond the diffraction limit, through the focusing and collimating of the laser and the nanometer-scale proximity of the AFM tip. More importantly, s-SNOM can be used to both launch and probe polaritons. In detail, a metal-coated nanoscale AFM tip scatters the incident IR laser, which induces an out-of-plane evanescent wave carrying high wave vectors so as to compensate for the momentum mismatch to excite polaritons. These excited polaritons propagate forwardly along the smooth surface of the specimen and interfere with the backward wave of polaritons reflected by edges or defects, which is also referred to as a round-trip component^[233]. Then the interference fringes, with a period of half the polaritonic wavelength, can be detected by the tip as the near-field scattering amplitude, shown in Fig. 8(c). Note that a physical edge of the 2D material or an artificial metallic nanostructure (such as a hole, rod, or disk) is also able to scatter incident beams to excite propagating polaritons (referred to as the direct component), which interfere with the illumination itself (or exactly the local field of material polarizability) to form interference fringes. The mechanism and launching efficiency of different components can be found in recent studies^[233,234]. In addition, s-SNOM can be used to merely probe the electromagnetic response of fabricated polaritonic nanostructures such as resonant antennas, where a dielectric (such as Si) tip should be adopted to avoid electromagnetic interaction between the antennas and metallic tip^[235]. It should be noted that the aforementioned operating mode in s-SNOM can detect only one polaritonic wavelength at a certain frequency through one measurement. s-SNOM integrated with a frequency-tunable laser and Michelson interferometer, that is, nano-FTIR, can acquire the dispersion relation of polaritons through a single measurement, which is achieved by collecting a series of nano-FTIR spectra at every pixel of a line scan. In conclusion, s-SNOM has the advantage of high spatial resolution and is capable of imaging propagating polaritons, compensating for the weakness of far-field techniques. However, s-SNOM is restricted by the available frequency bands of laser sources (e.g., very few commercial lasers in the FIR regime), although a synchrotron offers an option. In addition, the spatial imaging requires an intrinsically long lifetime of polaritons, which limits the applicability of s-SNOM.

In addition to the above two optical techniques, which make use of photons as the probe, energetic electrons can do the job as well. A scanning transmission electron microscope (STEM) is able to probe polaritons in the form of electron energy loss spectroscopy (EELS). For an EELS measurement, accelerated keV relativistic (also referred to as fast or swift) electrons are mono-chromated and focused by an electromagnetic lens on the specimen. The scattered electrons, both elastically and inelastically, are then collected by detectors placed at different emergent angles [Fig. 8(d)]. High angle scattered electrons are employed to investigate the atomic structure by a high angle annular dark-field (HAADF) detector, while the energy loss of low angle inelastic scattered electrons reflects the electromagnetic response of materials. The process of elastic scattering without energy exchange leads to the typical zero-loss peak (ZLP) in EELS at 0 meV, the FWHM of which indicates the monochromaticity of incident electrons or the energy resolution of the measurement, as shown in Fig. 8(e). In fact, EELS is proven to be a powerful tool for polariton research, due to swift electrons serving as localized, ultrafast, evanescent broadband (white) sources to excite polaritons beyond the optical selection rules, which can be focused within the sub-angstrom scale with advances in aberration correctors, achieving spatial resolution at the atomic level^[236]. At present, LSPR in metallic nanoparticles has been widely investigated by EELS, where a full set of resonance modes can be captured spatially^[237]. Such LSPR via Mie scattering contains both optical bright and dark modes, where the dark mode with a net zero dipole moment due to the selection rule is inaccessible through far-field techniques in the non-retarding regime (quasi-static limit).

In this section, we will introduce state-of-the-art experiments on observations of natural hyperbolic PhPs in anisotropic 2D materials. Efforts paid to manipulate topological transitions and some applications will be discussed in detail in Sections 3.5 and 3.6.

Natural hyperbolic materials were first discovered in bulk crystals such as hexagonal BN (h-BN)^[239,240], quartz^[241], and graphite^[242], where hyperbolic dispersion involves the out-of-plane direction. The first natural in-plane hyperbolic PhPs were observed in α-MoO3 by Ma et al. in 2018^[3]. α-MoO3 is a layered vdWs material in orthorhombic crystal structure with three different lattice constants, as shown in Fig. 9(a). Each Mo atom is surrounded by six O atoms to form a distorted MoO6 octahedron, and a unit layer contains two layers of such octahedrons. The strong in-plane anisotropy originates from different interlayer spacings between the (100) and (001) facets, leading to anisotropic optical phonons between 820 and 1010cm−1. The propagating PhP was observed by s-SNOM with a metallic AFM tip as a round-trip component. Figure 9(b) shows the interference fringes of an α-MoO3 slab with thickness of 250 nm at 990 cm⁻¹ (left panel) and 900cm−1 (right panel) laser frequencies. Under 990cm−1 (10.1 µm) laser excitation, the fringe period shows anisotropic polariton wavelengths of 950 and 1200 nm along the [100] and [001] crystal axes, respectively, much smaller than the free space wavelength of 10.1 µm, manifesting a deep subwavelength characteristic. Under 900cm−1 (11.1 µm) laser excitation, only the [100] direction sustains polaritons. The lifetime is 1.9±0.3ps based on the fitting of the s-SNOM amplitude line profiles, implying the intrinsically ultralow loss due to the high crystal quality. By nano-FTIR imaging, the dispersion was revealed at both upper and lower RBs, as shown in Fig. 9(c). In the upper RB, the fringe period increases with increasing frequency, indicating negative phase velocity. In the lower RB, there are fringes only along the [100] axis with positive phase velocity, which is compelling evidence for hyperbolic dispersion (fringes parallel to the [001] direction in the right panel are polaritons propagating along the [100] direction scattered by the [001] sample edge). The small group velocity [vg,i=(∂ki/∂ω)−1] of hyperbolic polaritons, extracted from Fig. 9(c), is about 0.7×10−2c at 893cm−1, due to the very small slope of its dispersion curve. To further investigate the propagation of PhPs, α-MoO3 was fabricated as a geometrically isotropic nanodisk for the interference fringes by s-SNOM in Fig. 9(d). In the upper RB, the pattern is elliptic with the largest wavelength along the [001] direction, with an almond shape in the lower RB. The IFCs were obtained directly by Fourier transform of corresponding spatial images, exhibiting a hyperbola at 893cm−1.

A similar study by Zheng et al. revealed that there are three RBs of α-MoO3 in the MIR range^[239]: band 1 (also referred to RBc) along the [001] axis from 545 to 851cm−1, band 2 (RBa) along the [100] axis from 820 to 972cm−1, and band 3 (RBb1) along the [010] axis (out of plane) from 958 to 1010cm−1. Therefore, PhPs in α-MoO3 have both in-plane hyperbolic dispersion with different opening directions below 958cm−1 and out-of-plane type I hyperbolic dispersion above 972cm−1. They also predicted the novel subwavelength focusing enabled by the propagation of in-plane hyperbolic polaritons via simulation, compared to the cases in uniaxial material h-BN. Dong et al. performed EELS in a STEM to probe PhPs in α-MoO3, which further expands the hyperbolic dispersion to the FIR regime^[232]. The dielectric functions along different crystal axes were fitted by multiple Lorentz oscillators. A new RB along the [010] axis tagged as “RBb2” was observed from 55 to 62 meV (444 to 500cm−1). They obtained the typical EEL spectrum of the sample placed in an arbitrary direction to capture all vibration modes. Five major features between 50 and 130 meV were observed, in good agreement with different RBs in α-MoO3. The sharp ZLP exhibits an FWHM of about 8 meV, indicating excellent energy resolution. When the electron beam is collimated parallel to a certain axis such as the [010] axis [out-of-plane b axis in Fig. 9(e)], the in-plane PhP can be selectively excited, which is spatially captured by EELS maps taken at the dashed box in Fig. 9(e). Analogous to a round-trip component in s-SNOM, PhP propagates and reflects at the edge along the [100] (a) axis, leading to the interference fringes shown in Fig. 9(f). From 71 to 92 meV, where the hyperbolic PhP resides in RBc, the highlights move towards the edge because the wavelength decreases as energy increases. PhP propagating along the [100] axis at 111 meV (in RBa) gives rise to homogeneous fringes parallel to the [001] (c) axis. No excitation is observed at 122 meV. They further verified the dispersion relation of hyperbolic PhPs in RBc by determining the polariton wave vector, since the distance between the interference maximum and minimum corresponds to 7π/8kp.

Subsequently, another in-plane hyperbolic PhP material—α-V2O5—was reported by Taboada-Gutiérrez et al.^[243]. α-V2O5 is a layered vdWs material with orthorhombic structure, and its crystal unit cell is shown in Fig. 10(a). The three inequivalent oxygen positions (labeled O1−3) relative to the vanadium atom lead to asymmetric V-O bonds and anisotropic optical phonons along different crystal axes. Very similar to α-MoO3, there are three RBs (RB1, RB2, RB3) along three crystal axes in the MIR regime, as shown in Fig. 10(c). They performed nano-FTIR line scans along the in-plane [100] and [001] axes. In RB1, both directions show interference fringes with different spacings (corresponding to polariton wavelength λp), which increase with frequency, suggesting elliptic topology and negative phase velocity. The group velocity obtained from the dispersion curves is also a small value (0.0009c). On the other hand, the fringes appear only along the [100] axis in RB2, and λp decreases with laser frequency, which implies in-plane hyperbolic topology and positive phase velocity. Finally, in RB3, there are fringes merely along the [001] axis, indicating hyperbolic dispersion with different opening directions. Similar to α-MoO3, the lifetime of the polariton is 6±1ps at 1010cm−1, revealing the low loss nature of PhPs in α-V2O5.

More similar materials have been discovered. Nörenberg et al. reported that α-GeS sustains low loss hyperbolic PhPs in the THz regime^[244]. As mentioned above, α-GeS is of a layered orthorhombic crystal lattice [Fig. 10(b)] with an AC structure along the [100] (x) direction and ZZ structure along the [010] (y) direction, similar to BP. The ratio of lattice constant is 1.18, 2.5 times larger than that in α-MoO3, implying stronger in-plane anisotropy. There are two in-plane RBs, RBy (ωTO,y=6.06THz and ωLO,y=9.47THz) and RBx (ωTO,x=7.74THz and ωLO,x=9.65THz), and two out-of-plane (z) TO phonons at ωTO,z1=7.1THz and ωTO,z2=8.4THz. To visualize the propagation of hyperbolic PhPs, they combined a free-electron laser (FEL) as a narrowband source with s-SNOM to detect the round-trip near-field component. At 7.33 THz, PhPs propagate only along the [010] direction with wavelength λp=2.6μm, indicating the hyperbolic characteristic. At 8.57 THz, the interference fringes exhibit wavelengths of 2.0 and 0.75 µm along [100] and [010] directions, respectively, and the IFC is propeller shaped and elongated along the [010] axis, which is demonstrated to host the natural canalization effect in the [100] direction. The dispersion relation was obtained from 6.0 to 8.7 THz by analyzing near-field images at various illumination frequencies, which agrees well with the calculated reflectivity RP(ω,k), as shown in Fig. 10(d). The topological transition from elliptic to hyperbolic regions is evident at ωTO,x=7.74THz. Intriguingly, two consecutive back bending effects emerge at frequencies above 7.9 THz, due to the coupling between in-plane elliptic PhPs and out-of-plane intrinsic phonons at 7.1 and 8.4 THz. Full-wave electromagnetic simulations illustrate the complete picture of PhPs in three dimensions [Fig. 10(d)-c, e, g], where the in-plane higher order l=1 modes at different frequencies are tuned by the sign-change of the permittivity along the [001] direction [Fig. 10(d)-b, d, f]. The quality factor and light confinement of PhPs in α-GeS were also extracted, which are comparable to α-MoO3 and α-V2O5.

Plasmon polaritons are the building blocks of plasmonic nanophotonics. However, experimental observation and manipulation of hyperbolic plasmons are rare. In this section, we will provide an overview on state-of-the-art experiments and theory about hyperbolic plasmons. Predictions of potential candidates are also summarized for further investigations.

To the best of our knowledge, naturally existing in-plane hyperbolic plasmon polaritons have been demonstrated only in WTe2^[39]. WTe2 is a layered TMDC with a distorted version of the usual MoS2 lattice structure [Fig. 11(a)]. In typical TMDCs, the dichalcogenide lattice is formed by metal layers sandwiched between adjacent chalcogenide layers, with stacks along the c axis of the hexagonal structure. However, in WTe2, there is an extra structural distortion as tungsten atoms form ZZ chains along the a axis of the orthorhombic unit cell, leading to in-plane anisotropy. Homes et al.^[245] and Frenzel et al.^[246] reported the band structure and temperature-dependent dielectric function along both in-plane and out-of-plane directions by means of ab initio calculation and polarized reflectance, implying an in-plane hyperbolic regime in the FIR regime at 10 K. The in-plane anisotropic mass ratio was extracted as ηab=me,b/me,a=ωp,a2/ωp,b2∼2.2 (ωp is the plasma frequency) at 10 K and almost constant below 100 K. They claimed that interband conductivity is fully attributed to transitions between topological trivial bands due to the Weyl points far above the Fermi level.

Hyperbolic plasmons in WTe2 were first observed experimentally by Wang et al. in 2020^[39]. WTe2 thin films were obtained by mechanical exfoliation [Fig. 11(c)] and characterized by the far-field extinction spectrum 1−T/T0 [Fig. 11(b)], where T and T0 refer to the transmission of light through the sample and bare substrate, respectively. The extinction spectra are determined by the sheet conductivity σ(ω) as 1−TT0=1−1|1+Z0σ(ω)/(1+ns)|2,(8)where Z0 is the vacuum impedance, and ns is the refractive index of the substrate. They measured the anisotropic optical absorption of bare films at different temperatures [Fig. 11(d)]. The extracted optical conductivity at 10 K shows σaa′′·σbb′′<0 from 427 to 623cm−1, which suggests that there are hyperbolic polaritons. Similar to previous results, the effective mass ratio is 2.05 and nearly unchanged below 100 K. The LSPR was studied by fabricating microdisks as shown in Fig. 11(g). The anisotropic effective mass leads to splitting of resonance frequencies (ωp), and the effective mass ratio can be determined as ηab=ωp,a2/ωp,b2=1.91. The plasmon linewidth is about 50cm−1 with a lifetime of about 0.1 ps at 10 K, comparable to graphene plasmons (0.05–0.1 ps). The detailed in-plane plasmon dispersions along two principal axes were determined through the plasmon peak positions of ribbons with various widths, as shown in Fig. 11(h), which are fitted by the loss function defined as −Im(1/ɛ). Note that the direction of the equivalent plasmon wave vector is perpendicular to the ribbon edge. Therefore, the IFCs were acquired by patterning ribbon arrays with a skew angle θ with respect to the a axis, and the topological transition from the elliptic to hyperbolic regime is clearly demonstrated in Fig. 11(i).

The physical properties of 2D materials change significantly from bulk to monolayer, and WTe2 is no exception. Monolayer WTe2 is a quantum spin-Hall insulator^[247] and possesses gate induced superconductivity^[248], while bilayer WTe2 is ferroelectric^[249]. Torbatian et al. predicted WTe2 single-layer hosts hyperbolic plasmons according to density-functional and many-body perturbation theories^[250]. The hyperbolic behavior could be tuned from the FIR to NIR region by chemical doping or strain. Jing et al. spatially imaged the temperature-dependent THz response of WTe2 down to monolayer using s-SNOM^[251]. Monolayer with a bandgap larger than 60 meV showed no near-field signal, while bilayer exhibited relatively higher signal with a small positive gap (<10meV). The signal of the WTe2 tri-layer with a semimetallic bandgap of ∼20meV was evident, which was proven to be plasmon polaritons. However, the in-plane anisotropic response was indistinguishable in such near-field studies.

More hyperbolic plasmonic 2D material candidates have been investigated theoretically. Several studies have been devoted to investigating the anisotropic plasmons in BP via theoretical calculations^{[210,231,252–254]}. Nemilentsau et al. proposed hyperbolic plasmon polaritons in BP in 2016 for the first time^[210]. They revealed the physical mechanism as we mentioned in Section 3.1. The minimal model of the optical conductivity for BP was constructed as σjj=ie2ω+iηnmj+sj[Θ(ω−ωj)+iπln|ω−ωjω+ωj|],(9)with j=a,b, where n is the concentration of electrons, mj and ωj are the electron effective mass and frequency of the onset of interband transitions along the j direction, respectively, and sj is the transition intensity. Here, as the second term on the right-hand side, the interband conductivity is expressed phenomenologically as a step function rather than the Lorentz model. By setting up a particular set of parameters (n,mj,sj,ωj), they studied the evolution of the electromagnetic response of BP when one or a few parameters change [Figs. 12(m) and 12(n)]. The optical conductivity manifesting the purely anisotropic (elliptic dispersion) regime at low frequency and the hyperbolic regime at higher frequency can both be identified. By solving Maxwell’s equations using a commercial finite-difference time-domain method from Lumerical, they obtained the corresponding field intensity and IFCs of surface plasmons in Fig. 12(o). At low frequency or high electron concentrations, plasmon beams tend to propagate along the a axis due to larger values of σaa′′ (conductive) than σbb′′. What is more, they confirmed that the plasmon is apt for the lower doped medium, when investigating the reflection and refraction at the interface between two sheets with different electron concentrations. The plasmon ray is able to bend by gradually varying the doping along the propagation direction as an analog to a graded-index plasmonic material.