2.1.1.

Interband absorption in BP

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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ point is allowed (forbidden) when incident light is polarized along AC (ZZ) direction. On the contrary, at $\mathrm{\Lambda }$ 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 $\mathrm{\Gamma }$ 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 $M_{cv}\propto \langle c|y|v\rangle$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.

Fig. 1

Light–matter interaction and atomic structures of anisotropic 2D semiconductors.

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 ($>8\mathrm{nm}$), BP thin film showed a bulk-like absorption spectrum, where an absorption edge was observed at $\sim 0.3\mathrm{eV}$, 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 $\sim 2.44$ 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 $\mathrm{\Gamma }$ 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 $v_1(c1),v_2(c_2),\dots ,v_n(c_n)$ in the Nth layer. Only interband transition from $v_n$ to $c_n$ (labeled as $E_{nn}$,) with the same index $n$ is allowed; thus, $N$-layer BP at $\mathrm{\Gamma }$ point possesses $N$ interband resonances. Consistent with theory predictions, absorption measurements showed all of $E_{nn}$ 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 $E_{nn}$ with layer number $N$ and sub-band index $n$, that is, $E_{nn}^N=E_{g0}-2\mathrm{\Delta }\gamma \cos (\frac{n}{N+1}\pi )$, where $E_{g0}$ is the monolayer bandgap, $\mathrm{\Delta }\gamma ={\gamma }_c-{\gamma }_v$, and ${\gamma }_c({\gamma }_v)$ is the nearest neighbor coupling between adjacent layers for conduction (valence) band. Through fitting experimental data, $E_{g0}=1.8\mathrm{eV}$ and $\mathrm{\Delta }\gamma =0.73\mathrm{eV}$ were obtained, with $E_{g0}$ comparable to the theory prediction ($\sim 2.0\mathrm{eV}$), 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 $E_{nn}^N$. 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 $E_{nn}$ 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 $E_{nn}$ exhibit the strongest absorption when light is polarized along AC. The relation between the absorption intensity $I$ and polarized angle $\theta$ (with respect to AC direction) follows the scaling $I\propto {\cos }^2\theta$.

Fig. 2

Interband absorption of anisotropic 2D materials. (a) Absorption spectra of 1L, 2L BP (left)^[34], bulk-like GeS (middle)^[64], and 1L–3L ${\mathrm{ReS}}_2$ (right)^[37]. The photoreflectance of GeS is also shown in the middle panel. (b) Polarization dependence of interband absorption of BP (left)^[35], GeS (middle)^[64], and ${\mathrm{ReS}}_2$ (right)^[37]. 0° corresponds to AC direction for BP and GeS, but for ${\mathrm{ReS}}_2$, it corresponds to $b$ axis (Re chain). (c) Illustration of interband transitions at $\mathrm{\Gamma }$ point of BP. (d) Photocurrent spectra of thin ${\mathrm{TiS}}_3$ ($\sim 15\mathrm{nm}$) (left)^[99] and absorption spectra of thin ${\mathrm{Nb}}_2{\mathrm{SiTe}}_4$ ($\sim 18\mathrm{nm}$) (right)^[102]. (e) Layer-dependent bandgap and interband transitions. Left panel: first three transition energies of BP and ${\mathrm{ReX}}_2$ versus layer number. Data are taken from Refs. [35,37,38]. Right panel: bandgap ranges of various anisotropic materials (BP, ${\mathrm{ReX}}_2$, MX, ${\mathrm{TiS}}_3$, NST) with thickness from bulk to monolayer. Data are taken from Refs. [35,37,38,60,97,102].

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 ${\mathrm{O}}_2$, forming ${\mathrm{P}}_x{\mathrm{O}}_y$ compounds on the BP surface. These ${\mathrm{P}}_x{\mathrm{O}}_y$ 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 $E_{nn}^N$ 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.

2.1.2.

Interband absorption in group IV monochalcogenides

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 $\sim 1.81(1.07)$ and $\sim 1.24(1.0)\mathrm{eV}$ in bulk to $\sim 2.32(1.54)$ 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 ($\sim 26\mathrm{meV}$). 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 $\mathrm{\Gamma }\text{-}\mathrm{X}$ direction (AC) and the other along the $\mathrm{\Gamma }\text{-}\mathrm{Y}$ 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 ($E_1$ and $E_2$) with a contrasted polarization dependence near the absorption edge were identified^[62–64]. As shown in Fig. 2, the interband transition $E_1$ at $\sim 1.6\mathrm{eV}$ is polarized along ZZ, whereas $E_2$ at $\sim 1.65\mathrm{eV}$ is polarized along AC. Due to the different polarization dependence of $E_1$ and $E_2$, the absorption edge ($E_g$) 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 $E_1$ and $E_2$ stir controversy. Hsueh et al. assigned multilayer GeS to a direct bandgap semiconductor, and $E_1$ corresponds to the transition from the direct bandgap at $\mathrm{\Gamma }$ point; $E_2$ is originated from the selection-rule forbidden transition near $\mathrm{\Gamma }$ point along $\mathrm{\Gamma }\text{-}\mathrm{Y}$ 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, $E_1$ and $E_2$ were assigned to transitions at $\mathrm{\Gamma }$ point and $\mathrm{\Delta }$ 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.

2.1.3.

Interband absorption in ReX₂

Among anisotropic 2D materials, besides BP and its cousins, rhenium dichalcogenides (${\mathrm{ReX}}_2$, where X = S, Se) attract abundant research interest as well^[67–69]. ${\mathrm{ReX}}_2$ 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. ${\mathrm{MoS}}_2$, ${\mathrm{WS}}_2$), ${\mathrm{ReX}}_2$ belongs to the distorted 1T triclinic structure with a lower in-plane symmetry, which gives rise to anisotropic properties. ${\mathrm{RX}}_2$ is very air-stable. All of these make ${\mathrm{RX}}_2$ suitable for novel opto-electronic applications, although studies on 2D ${\mathrm{ReX}}_2$ from different groups show discrepancies, particularly with respect to the direct or indirect bandgap nature and strongly coupled or decoupled interlayer interaction. Take ${\mathrm{ReS}}_2$ as an example. Several theoretical calculations showed that ${\mathrm{ReS}}_2$ from bulk and monolayer has a direct bandgap located at $\mathrm{\Gamma }$ point of BZ^[70–72]. However, other studies showed that the bulk ${\mathrm{ReS}}_2$ 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 ${\mathrm{ReS}}_2$ 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 $\sim 55\%\text{–}90\%$ of those in ${\mathrm{MoS}}_2$^[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 ${\mathrm{ReS}}_2$^[73], also indicating considerable interlayer coupling. Similar controversies can be found in ${\mathrm{ReSe}}_2$ as well^[38,72,79]. Studies have suggested that in ${\mathrm{ReX}}_2$, there coexist different stacking orders with different optical properties^[77,78,80], which can interpret these contradictory results. To fully reconcile these discrepancies of ${\mathrm{ReX}}_2$, more studies are needed.

As for the absorption of ${\mathrm{ReX}}_2$, Ho el al. preliminarily performed a series of studies in bulk ${\mathrm{ReX}}_2$ through modulated optical spectroscopy before ${\mathrm{ReX}}_2$ joined the 2D family^[81–84]. In bulk ${\mathrm{ReX}}_2$, two resonance peaks (labeled as $E_1$ and $E_2$) 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 ${\mathrm{ReS}}_2$, $E_1$ ($E_2$) centered at $\sim 1.49$ (1.52) eV occurs with incident light polarized parallel (vertical) to the $b$ axis. $E_1$ ($E_2$) in ${\mathrm{ReSe}}_2$ has the same polarization dependence but with a lower transition energy of $\sim 1.31$ (1.33) eV. Previous optical spectroscopy studies also showed that both bulk ${\mathrm{ReS}}_2$ and ${\mathrm{ReSe}}_2$ have indirect bandgaps of $\sim 1.37$ and 1.19 eV, respectively, which are very close to direct bandgap values. The layer-dependent and anisotropic absorption properties of atomically thin ${\mathrm{ReS}}_2$ (1L–7L) were first experimentally studied by Aslan et al.^[37]. Three resonance peaks ($E_1$ to $E_3$) were discerned, among which $E_1$ and $E_2$ correspond to those previously reported in the bulk counterpart, while $E_3$ at higher energy could be due to the onset of a band-edge free-carrier transition or an excited excitonic state of $E_1$ or $E_2$. All three peaks exhibit obvious layer dependence, with $E_1$, $E_2$, and $E_3$ 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 $E_1$, $E_2$, and $E_3$ from bulk to monolayer in ${\mathrm{ReS}}_2$ are much larger than those observed for the direct transitions [at $\mathrm{K}({\mathrm{K}}^{\prime })$ point in BZ] in 2H-TMDCs (e.g., ${\mathrm{MoS}}_2$) but smaller than those for indirect transitions (valence band extrema at $\mathrm{\Gamma }$ point in BZ) in the same materials. In addition, Aslan et al. revealed a complex polarization dependence of $E_1$ and $E_2$ absorption in atomically thin ${\mathrm{ReS}}_2$. In monolayer, $E_1$ and $E_2$ exhibit a maximum for light polarized at $\sim 15°$ and $\sim 50°$ with respect to the $b$ axis, respectively, while in tri-layer, the maximum polarization angles change to $\sim 15°$ and $\sim 85°$ with respect to the $b$ axis, respectively, which are different from those reported in bulk by Ho et al. Similarly, in ${\mathrm{ReSe}}_2$, 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 $X_{1,2}$ and $X_{3,4}$) 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 $\mathrm{\Gamma }$ point, four absorption peaks originate from the direct transition at $\mathrm{\Gamma }$ point with different combinations of spin, with energy splitting from the exchange interaction. $X_{1,2}$ and $X_{3,4}$ also exhibit an obvious layer dependence, with transition energy blueshifting $\sim 120$ and 140 meV, respectively, when the thickness decreases from bulk to monolayer, comparable to those in ${\mathrm{ReS}}_2$. The polarization dependence of these four transitions is more complicated. In fact, there are controversies in the polarization dependence and interband transition assignment in ${\mathrm{ReX}}_2$^[85], and the mechanism underlying the complex polarization dependence in atomically thin ${\mathrm{ReX}}_2$ is still unclear. Further studies are necessary.

2.1.4.

Interband absorption in other anisotropic 2D materials

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) (${\mathrm{TiS}}_3$, ${\mathrm{TiSe}}_3$, ${\mathrm{ZrS}}_3$, ${\mathrm{ZrSe}}_3$, ${\mathrm{HfS}}_3$, ${\mathrm{HfSe}}_3$), 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 ${\mathrm{TiS}}_3$. ${\mathrm{TiS}}_3$ 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 ${\mathrm{TiS}}_3$ is usually needle shaped. Monolayer ${\mathrm{TiS}}_3$ has been predicted in theory with a direct bandgap of $\sim 1.1\mathrm{eV}$ and with a very high electron mobility up to ∼10,000 cm²/(V s)^[95]. The photoresponsivity of devices based on ${\mathrm{TiS}}_3$ shows a strong polarization dependence with the highest value reaching up to 2500 A/W^[96]. All of these give ${\mathrm{TiS}}_3$ great potential for a new generation of optoelectronic device applications as well. Unlike other 2D materials, theory calculation showed that the bandgap of ${\mathrm{TiS}}_3$ is quite robust, exhibiting an $\sim 1\mathrm{eV}$ direct bandgap at $\mathrm{\Gamma }$ 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 ${\mathrm{TiS}}_3$, unaffected by interlayer coupling. Through theoretical calculations, Torun et al. suggested three interband transitions at $\mathrm{\Gamma }$ 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 $\sim 1$ to $\sim 1.8\mathrm{eV}$ in both thick ($\sim 200\mathrm{nm}$) and thin ($\sim 15\mathrm{nm}$) ${\mathrm{TiS}}_3$ [as shown in Fig. 2(d)]^[99]. Both thick and thin ${\mathrm{TiS}}_3$ exhibit resonance absorption peaks when light is polarized along the $b$ axis, in agreement with theoretical calculations. In thick ${\mathrm{TiS}}_3$, two absorption peaks $E_1$ and $E_2$ centered at 1.23 and 1.41 eV, respectively, were identified, while in thin ${\mathrm{TiS}}_3$, only $E_1$ centered at 1.34 eV was observed, indicating different band structures in thick and thin ${\mathrm{TiS}}_3$. According to their first-principles calculations, the CBM in thick ${\mathrm{TiS}}_3$ shifts to Z point, giving rise to an indirect bandgap, and $E_1$ is due to the transition from the valence band to a higher conduction band at $\mathrm{\Gamma }$ point; $E_2$ corresponds to two direct transitions near X and Z points. In thin ${\mathrm{TiS}}_3$, the band structure is similar to that of the monolayer, which has a direct bandgap at $\mathrm{\Gamma }$ point due to the lack of Z point; $E_1$ corresponds to the direct transition at $\mathrm{\Gamma }$ point, and $E_2$ 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 ${\mathrm{TiSe}}_3$ and ${\mathrm{Ta}}_2{\mathrm{NiS}}_2$ were reported with narrow bandgaps of $\sim 0.2$ and 0.3 eV in MIR, respectively^[93,100], while the optical properties of their 2D counterparts are hardly known. Recently, ${\mathrm{Nb}}_2{\mathrm{SiTe}}_4$ (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 ${\mathrm{MoS}}_2$ but with two types of atoms (Nb and Si) in the inner layer instead of one (Mo) in the ${\mathrm{MoS}}_2$ (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 $\sim 100{\mathrm{cm}}^2/(\mathrm{V}\mathrm{s})$ 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, $\mathrm{B}({\mathrm{B}}^{*})$ and C] centered at $\sim 0.49$, 0.66, and 1.1 eV was observed. Among them, A and B are allowed when light is polarized along the $b$ axis, whereas ${\mathrm{B}}^{*}$ 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 $v_1$ to different conduction bands $c_n$ [e.g., $c_{1(3)}$ for A (B) and $c_{3(5)}$ for ${\mathrm{B}}^{*}(\mathrm{C})$] at different k-points in BZ.

2.2.1.

PL in BP

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 $\sim 1.7$, 1.1, and 0.8 eV in monolayer, bilayer, and tri-layer respectively, with a small Stokes shift of $\sim 20\text{–}40\mathrm{meV}$, 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 $\sim 1.36\mathrm{eV}$ through artificially introducing oxygen defects by controllable etching with ${\mathrm{O}}_2$ plasma^[107]. By using a plasma-enhanced chemical vapor deposition (PECVD) oxide substrate, Xu et al. obtained defect state PL centered at energy of $\sim 1.35\mathrm{eV}$ 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 ($\text{thickness}>9\mathrm{L}$) 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 ${\cos }^2\theta$ 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].

2.2.2.

PL in group IV monochalcogenides

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 $\sim 1.62\mathrm{eV}$ in multilayer GeS ($\sim 40\mathrm{nm}$) at room temperature [as shown in Fig. 3(c)]^[63]. PL is due to the recombination of interband transition $E_A$. It should be noted that there are two interband transitions near the band edge, namely, $E_A$ and $E_B$, but only one PL peak assigned to $E_A$ is discernable. This is because the excited holes from transition $E_B$ 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)].

2.2.3.

PL in ReX₂

Under room temperature, two prominent PL peaks with different polarization dependences in ${\mathrm{ReX}}_2$ 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 ${\mathrm{ReX}}_2$ also exhibit discrepancies. For example, in ${\mathrm{ReS}}_2$, Tongay et al. first reported that PL peak position slightly blueshifts from $\sim 1.5$ to $\sim 1.6\mathrm{eV}$ 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 ${\mathrm{MoS}}_2$. Calculations by Tongay et al. suggested that ${\mathrm{ReS}}_2$ is always a direct bandgap semiconductor regardless of thickness. Aslan et al. later reported that the lowest energy PL peak position in ${\mathrm{ReS}}_2$ 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 $\sim {10}^{-4}$ in 1L–7L ${\mathrm{ReS}}_2$ regardless of the layer number, which is at least one-order of magnitude smaller than those in monolayer 2H-TMDCs (e.g. ${\mathrm{MoS}}_2$) with direct bandgaps^[120]. These low PL quantum yields were ascribed to the indirect bandgap nature in ${\mathrm{ReS}}_2$. Similarly, in ${\mathrm{ReSe}}_2$, 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 $\sim 6\times {10}^{-4}$, comparable to that of ${\mathrm{ReS}}_2$^[85]. In addition, discrepancies can be found in the PL line shape as well. For example, Aslan et al. showed that in atomically thin ${\mathrm{ReS}}_2$, PL intensity due to transition $E_2$ is stronger than that due to $E_1$ ($E_2$ at a higher energy than $E_1$), 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 $E_1$ is obviously stronger than that due to $E_2$^[85,120]. The discrepancies in PL studies might be due to different crystal qualities or stacking orders of ${\mathrm{ReX}}_2$.

Under low temperature, the fine structure of PL in ${\mathrm{ReX}}_2$ 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 $E_3$, $E_{s1}$, and $E_{s2}$ respectively) in the few-layer ${\mathrm{ReS}}_2$ (${\mathrm{ReSe}}_2$) 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 $E_3$ ($E_{s1,2}$) was consistent with that of $E_2$ ($E_1$). In addition, Jadczak et al. observed two series of four PL peaks centered between 1.63 and 1.67 eV in bulk ${\mathrm{ReS}}_2$ 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 $E_1$ and $E_2$ (discussed in the next section).

2.2.4.

PL in other anisotropic 2D materials

Linearly polarized PL has also been studied in other anisotropic 2D materials. As a quasi-1D direct bandgap semiconductor, ${\mathrm{TiS}}_3$ is expected to have a strong anisotropic PL. Recently, Curto et al. reported an NIR PL centered at 0.91 eV in multilayer ${\mathrm{TiS}}_3$ (thickness $\sim 32\mathrm{nm}$)^[123], as shown in Fig. 3(e). PL emission in ${\mathrm{TiS}}_3$ 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 $\sim 4$ at room temperature and increasing to $\sim 5$ at 20 K. Due to anisotropic absorption, PL in ${\mathrm{TiS}}_3$ also exhibits excitation polarization dependence (laser wavelength $\sim 532\mathrm{nm}$). 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 ${\mathrm{TiS}}_3$, ${\mathrm{ZrS}}_3$ 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 ${\mathrm{ZrS}}_3$ 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 $\sim 5.7$ 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.

2.3.1.

Excitons in BP

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 $\sim 800\mathrm{meV}$, and what is more interesting, they found that the wave functions of typical bright excitons (e.g., $1\mathrm{s},2\mathrm{s}\dots$) 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 ${\mathrm{SiO}}_2$ 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, $\sim 0.9\mathrm{eV}$, in monolayer BP on a ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate^[106]. They observed that a PL signal centered at $\sim 1.3\mathrm{eV}$ (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 $\sim 2.2\mathrm{eV}$. 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 ${\mathrm{SiO}}_2/\mathrm{Si}$ substrate with central wavelengths at $\sim 705\mathrm{nm}$ (1.76 eV) and $\sim 760\mathrm{nm}$ (1.6 eV)^[108]. The former was assigned to ground state exciton ($1\mathrm{s}$) PL, and with a comparison to the measured quasiparticle bandgap of $\sim 2.05\mathrm{eV}$ by Pan et al. using scanning tunneling spectroscopy (STS)^[139], Yang et al. determined the binding energy in supported monolayer of only $\sim 0.3\mathrm{eV}$, much smaller than that reported by Wang et al. but comparable to previous theoretical predictions. It should be noted that PL at $\sim 760\mathrm{nm}$ was assigned to the trion (charged exciton), leading to a large trion binding energy of $\sim 100\mathrm{meV}$, almost five times higher than that in the isotropic TMDC monolayer (e.g. ${\mathrm{MoS}}_2$), 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 $\sim 51\mathrm{meV}$, and decreases to $\sim 34\mathrm{meV}$ 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 ($2\mathrm{s}$ and $3\mathrm{s}$) in high quality monolayer BP on a polydimethylsiloxane (PDMS) substrate^[110]. Through extracting energy differences between $1\mathrm{s}$, $2\mathrm{s}$ (${\mathrm{\Delta }}_{12}\sim 230\mathrm{meV}$) and $2\mathrm{s}$, $3\mathrm{s}$ (${\mathrm{\Delta }}_{23}\sim 120\mathrm{meV}$) from the spectrum, binding energy of $\sim 452\mathrm{meV}$ 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 $n$, $2\mathrm{s}$ and $3\mathrm{s}$ states can be assumed to follow the ideal 2D hydrogenic model. Then the binding energy can be simply estimated as $E_b^{1\mathrm{s}}=(\frac{25}{16}){\mathrm{\Delta }}_{23}+{\mathrm{\Delta }}_{12}=418\mathrm{meV}$, consistent with previous predictions. In addition, it should be noted that the optical bandgap reported by Wang et al. was $\sim 1.69\mathrm{eV}$, and by comparing to the measured quasiparticle bandgap of $\sim 2.05\mathrm{eV}$ by STS, it suggests a comparable binding energy of $\sim 360\mathrm{meV}$ 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 $\sim 400\mathrm{meV}$, rather than $\sim 900\mathrm{meV}$, 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 $2\mathrm{s}$ 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 $1\mathrm{s}$ and $2\mathrm{s}$ 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 $1\mathrm{s}-2\mathrm{s}$ state is too close to discern. In the 2D hydrogenic model, it is well known that the energy separation of $1\mathrm{s}$ and $2\mathrm{s}$ 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 $E_b$ 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 $E_b\approx 3/(4\pi \alpha )$, where $\alpha$ is the 2D sheet polarizability of the 2D material, suggesting that $E_b$ is independent of the excitonic effective mass and depends only on $\alpha$. On this basis, Zhang et al. further proposed an effective polarizability as ${\alpha }_{\mathrm{eff}}(N)={\alpha }_0+N{\alpha }_1$, where ${\alpha }_0$ and $N{\alpha }_1$ account for the screening from the substrate and $N$-layer BP itself, respectively. $E_b$ can be expressed as $E_b=\frac{3}{4\pi ({\alpha }_0+N{\alpha }_1)}$^[141]. The model can well fit the layer-dependent $E_b$ in few-layer BP, with reasonable fitting values of ${\alpha }_0$ and ${\alpha }_1$, comparable to previous theorical calculations, which suggests that the model is feasible. In addition, this model predicts that $E_b$ 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 $L_z·{|{\varphi }_{\mathrm{ex}}(0)|}^2$, where $L_z$ is the thickness of the sample (layer number $N$ for BP), and ${|{\varphi }_{\mathrm{ex}}(0)|}^2$ is the modular square of the excitonic wave function at the origin with a $1/L_z^2$ 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 $1/({\alpha }_0+N{\alpha }_1)$. Equally interestingly, Zhang et al. found that absorption from the continuum transition close to the band edge is a constant $\pi \alpha \sqrt{{\mu }_y/{\mu }_x}$, where $\alpha$ is the fine structure constant with a value of 1/137 ($\pi \alpha$ is the universal absorption constant in graphene), and ${\mu }_{x(y)}$ 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 $\mathrm{\Gamma }$ 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.

2.3.2.

Excitons in group-IV monochalcogenides

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 ${\mathrm{SiO}}_2$ 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.

2.3.3.

Excitons in ReX₂

Zhong et al. first studied excitons in freestanding monolayer ${\mathrm{ReX}}_2$ in theory. Based on their results, the lowest-energy exciton in ${\mathrm{ReS}}_2$ at $\mathrm{\Gamma }$ point with a binding energy of 1.07 eV exhibits a perfect 8-shaped polarization dependence, while that of ${\mathrm{ReSe}}_2$ with binding energy of $\sim 0.87\mathrm{eV}$ exhibits only partial polarization dependence, due to the mixture of two bright excitons at $\mathrm{\Gamma }$ and $\mathrm{\Lambda }$ points with different polarization dependences^[132]. Theoretical calculations by Echeverry et al. suggested that in both monolayer ${\mathrm{ReS}}_2$ and ${\mathrm{ReSe}}_2$, due to the doubly degenerated VBMs and CBMs at $\mathrm{\Gamma }$ 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 ${\mathrm{ReS}}_2$ and ${\mathrm{ReSe}}_2$, respectively^[72].

In experiments, Arora et al. reported four excitonic absorption and PL peaks with different spin combinations at $\mathrm{\Gamma }$ point in ${\mathrm{ReSe}}_2$ [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 ${\mathrm{ReS}}_2$ under low temperatures [Fig. 4(d)]^[144]. Different from those of ${\mathrm{ReSe}}_2$, two prominent peaks were assigned as bright exciton absorption at Z and ${\mathrm{K}}_1$ 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 $E_3$, as reported by Aslan et al. in bulk ${\mathrm{ReS}}_2$, is composed of several excited excitonic states of ground state excitons $E_1$ and $E_2$^[122]. Through a 3D hydrogenic Rydberg model, the binding energies of excitons $E_1$ and $E_2$ were estimated as $\sim 0.12$ and 0.08 eV, respectively. It should be noted that the binding energy in bulk ${\mathrm{ReX}}_2$ is much larger than that in other bulk materials, which could be attributed to the weak interlayer coupling in ${\mathrm{ReX}}_2$, resulting in the exciton being confined in individual layers. The binding energy of monolayer ${\mathrm{ReSe}}_2$ 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) ${\mathrm{ReS}}_2$^[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.

2.3.4.

Excitons in other anisotropic 2D materials

With quasi-1D structure, ${\mathrm{TiS}}_3$ is expected to have robust excitons with large binding energy. Theoretical studies showed that in monolayer ${\mathrm{TiS}}_3$, there are three interband transitions at $\mathrm{\Gamma }$ point, accompanied by three bright excitons $E_x^{1,2,3}$ with binding energies of 0.59, 0.68, and 0.32 eV, respectively^[98]. The lowest energy exciton $E_x^1$ located at $\sim 1.07\mathrm{eV}$ is bright only for out-of-plane polarization, but with a quasi-1D nature, theoretical calculations for ${\mathrm{TiS}}_3$ suggested that the wave function of an exciton in ${\mathrm{TiS}}_3$ 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 $\sim 0.07\mathrm{eV}$ in bulk to $\sim 0.8\mathrm{eV}$ in monolayer^[99]. The large binding energy in monolayer ${\mathrm{TiS}}_3$ results in the lowest energy excitonic absorption ($E_x^1$) at $\sim 0.58\mathrm{eV}$. This value is inconsistent with the aforementioned $\sim 1.07\mathrm{eV}$, 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 ${\mathrm{TiS}}_3$ thin film ($>25\mathrm{nm}$) of $\sim 1.2$ and 1.07 eV, respectively^[136]. The binding energy was determined as $\sim 0.13\mathrm{eV}$, 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 ${\mathrm{SiP}}_2$, where the electrons constituting the excitons are confined within 1D phosphorus–phosphorus chains, while the holes are extended over the 2D ${\mathrm{SiP}}_2$ plane^[146]. This unique exciton with binding energy of $\sim 140\mathrm{meV}$ (in ${\mathrm{SiP}}_2$ with $\text{thickness}>5\mathrm{nm}$) 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 $\sim 50\mathrm{meV}$, 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 ${\mathrm{NiPS}}_3$, 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 $10{\text{exciton-A}}_{1\mathrm{g}}\text{-}\text{phonon bond}$ 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 $m_X/m_Y$ can be as large as $\sim 50$^[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 ($E_{\mathrm{laser}}\text{–}{E}_{1\mathrm{s}}$) corresponding to ${\mathrm{A}}_{\mathrm{g}}^2$ and ${\mathrm{A}}_{\mathrm{g}}^3$ phonons in the PLE spectrum. In this quasi-1D electronic system combined with magnetic order, the magneto-exciton physics calls for further exploration.

2.4.1.

Saturable absorption

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

Fig. 5

Nonlinear optics of anisotropic 2D materials. (a) Saturable absorption. Left panel: schematic illustration of saturable absorption. Middle left panel: power-dependent transmittance with incident polarization angle changing from $E\parallel x$ axis (AC) to $E\parallel y$ axis (ZZ) in the exfoliated thick BP flake (thickness $\sim 300\mathrm{nm}$)^[155]. Middle right panel: typical open aperture Z-scan curves of SnSe nanosheet dispersion under 230 fs excitation pulses with wavelength of 1030 nm^[160]. Right panel: polarization-dependent absorption of ${\mathrm{ReS}}_2$ (thickness $\sim 320\mathrm{nm}$) at low incident power of $1.55\mathrm{GW}/{\mathrm{cm}}^2$ and at high power of $72.8\mathrm{GW}/{\mathrm{cm}}^2$ (left top), and polarization-dependent ratio of absorption between that under 1.55 and $72.8\mathrm{GW}/{\mathrm{cm}}^2$ incident power (left bottom)^[156]. The right one is a schematic of excited-state absorption in ${\mathrm{ReS}}_2$. (b) Third harmonic generation (THG). Left panel: schematic illustration of THG. Middle panel: typical THG spectrum of multilayer BP flake (thickness $\sim 30\mathrm{nm}$) under excitation wavelength of $\sim 1560\mathrm{nm}$ (left), and polar plot of polarization dependence of THG from BP flake (thickness $\sim 26\mathrm{nm}$) on glass substrate (right), where red and purple arrows indicate AC and ZZ directions, respectively^[163]. Right panel: THG power of BP in transmission (orange) and in reflection (blue) versus thickness. The resonance THG occurred at thickness $\sim 3\mathrm{nm}$ as indicated by pink area^[161]. (c) Second harmonic generation (SHG). Left panel: schematic illustration of SHG. Middle left panel: SHG spectra from bilayer ${\mathrm{ReS}}_2$ and monolayer ${\mathrm{WS}}_2$ pumped by a laser at 1558 nm. Insets are polar plots of SHG parallel components from bilayer ${\mathrm{ReS}}_2$ and monolayer ${\mathrm{WS}}_2$^[165]. Middle right panel: SHG intensity in 1L–8 L ${\mathrm{ReS}}_2$^[165]. Right panel: SHG of SnS film (thickness $\sim 30\mathrm{nm}$) at different excitation wavelengths^[166].

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 $\sim 300\mathrm{nm}$) 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 ${\mathrm{ReS}}_2$ 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 $\sim 100°$ 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 ${\mathrm{ReS}}_2$, 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 ${\mathrm{ReS}}_2$ 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 $\sim 175\mathrm{nm}$) at dual wavelengths of 800 and 850 nm^[159]. The anisotropic ratio of the modulation depth $\delta T$ 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 ${\alpha }_s$ and modulation depth $\delta T$ 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 $\sim 44\%$ 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 $\sim 6\mathrm{nm}$), 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 $\sim 9.5\mathrm{nm}$) dispersion was observed as well but with higher incident power, which is related to the layer-dependent bandgap in SnSe.

2.4.2.

Second- and third-harmonic generation

Harmonic generation is a process converting $n$ incident photons at frequency $\omega$ into a higher frequency $n\omega$ 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 $\sim 520\mathrm{nm}$, exactly three times the frequency of the pump light (wavelength 1560 nm)^[163]. As anticipated, THG is polarization dependent, and the intensity of THG $I_{3\omega }$ in BP can be expressed as

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 $|{\chi }_{\mathrm{eff}}^{(3)}|$ in 3L BP can be enlarged to $\sim 10\text{–}20$ 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 $\sim 10$ 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 ${\mathrm{ReS}}_2$ exhibits strong and anisotropic THG, in which the third-order susceptibility $|{\chi }^{(3)}|$ is of ${10}^{-18}{\mathrm{m}}^2/{\mathrm{V}}^2$, almost one order of magnitude larger than that in 2H TMDCs such as ${\mathrm{MoS}}_2$^[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 ${\mathrm{ReS}}_2$ exhibits an opposite dependence on thickness^[165]. Strong SHG exists only in even-layer ${\mathrm{ReS}}_2$, whereas it is negligible in odd-layer ${\mathrm{ReS}}_2$ [as shown in Fig. 5(c)]. This unusual layer-dependent SHG is due to the unique distorted 1T structure of ${\mathrm{ReS}}_2$, 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 $|{\chi }^{(2)}|$ 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 ${\mathrm{MoS}}_2$ 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].

2.5.1.

Electrical field effect

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., $E_{nn}$ in BP). In the presence of a vertical electrical field, the additional otherwise selection-rule forbidden interband transitions (e.g., $E_{nm}$, $m\ne n$) 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 $E_{21}$ and $E_{12}$ 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 ($\sim 20\mathrm{L}$), 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 ${\mathrm{HNO}}_3$ vapor^[35].

Fig. 6

Band structure engineering of anisotropic 2D materials. (a) Electrical field effect on BP. Top left panel: schematic illustration of the BN-sandwiched device with graphene as top gate and Si substrate as bottom gate for applying vertical electrical field^[173]. Top right panel: PL spectra of the 20-layer BP under different displacement fields ($D$) ranging from 0 to 0.48 V/nm^[173]. Middle left panel: bandgap of the 20-layer BP extracted from PL spectra (dots) and first-principles calculations (curve) versus $D$^[173]. Bottom left panel: schematic illustration (top) and ARPES spectra (bottom) of BP band structure under different surface K-dopings^[174]. Bottom right panel: absorption spectrum of a 9L BP before and after chemical doping through ${\mathrm{HNO}}_3$ vapor (top) and enlarged view of the hybrid peaks between $E_{11}$ and $E_{22}$ (bottom)^[35]. (b) Strain and pressure effect. Left panel: schematic illustration of application of uniaxial strain (top)^[177], biaxial strain (middle)^[181], and pressure (bottom)^[188]. Right top panel: normalized PL spectra of multilayer BP ($\sim 20\mathrm{nm}$) under zero strain, compressive strain ($\varepsilon =0.66\%$ along the zigzag direction), and tensile strain ($\varepsilon =1.21\%$ along the zigzag direction)^[177]. Right middle panel: linear shift rates of interband transition ($E_{11}$, $E_{22}$, and $E_{33}$) peaks induced by biaxial strain versus layer number in 2L–10L BP^[181]. Dots are experimental data, and solid curves are fittings through the tight-binding model. Right bottom panel: relative change of the optical bandgap of BP versus pressure^[188]. Dots are experimental data, and solid curves are the fittings. (c) Alloying and intercalation. Top panel: crystal structure of $\mathrm{b}\text{-}{\mathrm{As}}_x{\mathrm{P}}_{1-x}$^[195] (left) and absorption spectra of $\mathrm{b}\text{-}{\mathrm{As}}_x{\mathrm{P}}_{1-x}$ with different constituent ratios x from 0 to 0.83 (right)^[190]. Middle panel: schematic illustration of intercalation process by CTAB to form the BP superlattice (left) and high resolution TEM images of BP before and after intercalation (right)^[195]. After intercalation, in-plane lattice parameter negligibly changes but interlayer distance has nearly two-fold enlargement. Bottom panel: electrochemical gate current as a function of the applied electrochemical potential (left), and PL signal observed at different stages of intercalation (right)^[195]. (d) Homo and heterostructures based on BP. Top panel: schematic illustration of the crystal structure and band diagram of few-layer BP and monolayer ${\mathrm{WSe}}_2$ heterostructure (left), and PL spectra of different thickness BP and $\mathrm{BP}\text{-}{\mathrm{WSe}}_2$ heterostructures at 80 K^[200]. Bottom panel: illustration of BN encapsulated 1L and 2L twisted BP homostructure and TEM image of a 5.7° twisted 1L/2L BP structure (left), scale bar 2 nm. PL spectra of the 1L, 2L BP and 19° twisted BP homostructure (right)^[203]. An emerging moiré optical transition at 0.83 eV is observed in twisted homostructure.

2.5.2.

Strain and pressure

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 $E_g$ can be approximately expressed as $E_g\asymp 4t_1^{\parallel }+2t_2^{\parallel }$, where $t_1^{\parallel }$ and $t_2^{\parallel }$ are hopping parameters between different atoms^[175]. In general, the modulus of hopping parameter $|t|$ is proportional to $1/r^2$, 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 $t_1^{\parallel }>0$ and $t_2^{\parallel }<0$. 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 ${\mathrm{Nb}}_2{\mathrm{SiTe}}_4$ exhibits strong orientation dependence^[102], in particular, for transitions B and ${\mathrm{B}}^{*}$, 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 $\sim 10\%$ 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 ($\sim 1.9\mathrm{GPa}$) induces $\sim 18\%$ relative change in interlayer coupling.

2.5.3.

Alloying and intercalation

Alloying is a very traditional strategy to tune the band structure. Due to the weak interlayer coupling in ${\mathrm{ReX}}_2$, the bandgap is hardly tuned by thickness, while it can be efficiently tuned by alloying. In ${\mathrm{ReS}}_{2(1-x)}{\mathrm{Se}}_{2x}$ 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 ($\mathrm{b}\text{-}{\mathrm{As}}_x{\mathrm{P}}_{1-x}$), 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 $\mathrm{b}\text{-}{\mathrm{As}}_x{\mathrm{P}}_{1-x}$ 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.

2.5.4.

Van der Waals homo/heterostructures

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 ${\mathrm{WS}}_2$ form a type I heterostructure [as shown in Fig. 6(d)]^[200]. Due to the larger bandgap in monolayer ${\mathrm{WS}}_2$, photo-excited electrons and holes are transferred from ${\mathrm{WS}}_2$ to BP, giving rise to $\sim 192\%$ 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 ${\mathrm{WS}}_2$ 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 $\sim 19°$ 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 $\mathrm{\Gamma }$ 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.

3.1.1.

Fundamentals of polaritons

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:

For 2D materials with in-plane anisotropy, the sheet optical conductivity takes the complex tensor form $\overleftrightarrow{\sigma }$, written as $\left[\begin{array}{cc}{\sigma }_{aa}& 0\\ 0& {\sigma }_{bb}\end{array}\right]$, where ${\sigma }_{aa}$, ${\sigma }_{bb}$ represent the principal components of the tensor (these components are dispersive with frequency $\omega$, 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 ${\sigma }_{aa}=i{\sigma }_{aa}^{\prime \prime }$ and ${\sigma }_{bb}=i{\sigma }_{bb}^{\prime \prime }$, and the wavelength of the plasmon is much smaller than that of light. The dispersion relation of surface waves can be simplified as^[210]

Fig. 7

Schematic of polariton topological transitions. (a) IFCs in the momentum space and the $z$ component of the electric field in real space of surface plasmons supported by BP in purely anisotropic, quasi-circular, and hyperbolic regimes^[231]. (b), (c), (d) Hyperbolic regimes induced by anisotropic phonons, plasmons, and excitons for phonon^[232], plasmon^[39], and exciton polaritons^[110], respectively.

Polaritons with hyperbolic contours (${\sigma }_{aa}^{\prime \prime }·{\sigma }_{bb}^{\prime \prime }<0$) 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 $v_g={\nabla }_q\omega (\mathit{q})$ that dictates the energy propagation, which is also orthogonal to the IFC. The asymptotes of the hyperbolic curve are expressed as $q_b=\pm q_a\sqrt{|{\sigma }_{aa}^{\prime \prime }/{\sigma }_{bb}^{\prime \prime }|}$, and the perpendicular direction, namely, the direction of the polariton beam, is always along $y=\pm x\sqrt{|{\sigma }_{bb}^{\prime \prime }/{\sigma }_{aa}^{\prime \prime }|}$. 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 (${\sigma }_{aa}^{\prime \prime }·{\sigma }_{bb}^{\prime \prime }<0$). 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:

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 ${\omega }_{\mathrm{TO}}$ 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 ${\sigma }^{\prime \prime }(\omega )>0$ 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 ${\sigma }^{\prime \prime }(\omega )<0$. Then, the hyperbolic characteristic can emerge at the region where ${\sigma }_{aa}^{\prime \prime }·{\sigma }_{bb}^{\prime \prime }<0$, 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.

3.1.2.

Probe of 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 $k^{\prime }=n_{\mathrm{prism}}\sin \theta k_0$ ($n_{\mathrm{prism}}$ is the refractive index of the prism, and $\theta$ 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 $k^{\prime }=k_0\sin \theta +\frac{2\pi n}{d}$ ($d$ 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.

Fig. 8

Methods to probe polaritons. (a) Schematic to stimulate polaritons in the far field. Due to the momentum mismatch, light in free space is unable to excite polaritons directly. Additional momentum can be supplied by Kretschmann or Otto configuration, fabricating gratings by the side of material or resonant nanostructures^[238]. (b) Resonant modes of LSPR in graphene nanoribbons with different widths^[15]. [(c)-a] Schematic illustration of the near-field technique to image propagating plasmon polaritons at the surface of graphene on ${\mathrm{SiO}}_2$ with s-SNOM. [(c)-b, c, d, e] Near-field amplitudes of interference pattern close to graphene edges (blue dashed lines), defects (green dashed lines and dot), and at the boundary between single and bilayer graphene (white dashed line)^[208]. (d) Diagram of STEM-EELS experimental setup. (e) Typical EEL spectrum of $\alpha \text{-}{\mathrm{MoO}}_3$ in aloof geometry. Blue: sharp ZLP with an FWHM of about 8 meV after spectrum normalization. Green: energy loss area. Yellow: fitted background. Red: extracted five major vibrational signals^[232].

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).