InSe as a case between 3D and 2D layered crystals for excitons

InSe is a promising material in many aspects where the role of excitons is decisive. Here we report the sequential appearance in its luminescence of the exciton, the biexciton, and the P-band of the exciton-exciton scattering while the excitation power increases. The strict energy and momentum conservation rules of the P-band are used to reexamine the exciton binding energy. The new value ≥20 meV is markedly higher than the currently accepted one (14 meV), being however well consistent with the robustness of the excitons up to room temperature. A peak controlled by the Sommerfeld factor is found near the bandgap (~1.36 eV). Our findings supported by theoretical calculations taking into account the anisotropic material parameters question the pure three-dimensional character of the exciton in InSe, assumed up to now. The refined character and parameters of the exciton are of paramount importance for the successful application of InSe in nanophotonics.


SUPPLEMENTARY FIGURES
The band structure of bulk InSe was computed numerically many times using different theoretical methods 2,[9][10][11][12][13][14][15] . It was also investigated experimentally using photoluminescence (PL) spectroscopy under high pressure, as well as angle resolved photoemission spectroscopy (ARPES) 12,16,17 . InSe has three principal polytypes: the rhombohedral γ-polytype (C 5 3v space group), hexagonal β-polytype (D 4 6h ), and hexagonal -polytype (D 1 3h ) 18 . It is well established that γ-and β-polytypes have direct band structure, which is rather important for optoelectronic applications. They have nearly identical values of bandgaps at ambient conditions: 1.2829 and 1.2931 eV 16,19 . On the contrary, thepolytype is assumed to have indirect band structure with a higher bandgap value around 1.4 eV 20 . The excitonic PL at ∼1.3 eV reported in this work excludes the studied samples to be the polytype. The structural characterization described in Supplementary Methods 1 and 2 has demonstrated that they most likely belong to the β polytype, whose band structure has been analyzed in several papers 2,21 . We present in Supplementary Figure 1 the tetralayer arrangement and the schematic of the electronic band structure for the β-InSe in a vicinity of optical gap, where the characteristic type of states (s, p z , p xy ) is shown for each band.

SUPPLEMENTARY NOTE 2 : Selection rules of optical transitions
Following Ref. 22 , we can construct the direct dipole transitions in the β-InSe of the D 4 6h symmetry from the Bloch orbits Γ + 1 and Γ − 2 taking into account spin by going into the double group representation where D (1/2) is the two-dimensional representation of the Pauli spinors. Consistently, from Koster's tables 23 optically allowed representations. The Γ 2 states are dipole-forbidden. The optical transition Γ − 5 comprising x ± iy components occurs possible only when both spin and orbit are coupled. This coupling gives rise to a mixing of the highest valence band with the lower-lying bands E v2,v3 (see Supplementary Figure 1). Interband mixing between conduction bands can be neglected because their separation is much higher. Thus the transitions involving states of s-symmetry are only partly allowed for E ⊥ c, while they are fully allowed for light polarization E c axis.
The excitonic states near the optical gap in Γ point are mediated by these selection rules. In the weak spin-orbit coupling regime, like in our case when the E v2,v3 bands are distant, the singlet component of excitons originating from Γ 1 is strong in E c, while the triplet component Γ 5 interacts weakly with light when E ⊥ c. This situation in relevance to the optical absorption is illustrated by Fig. 5a in the main text. Thus, the exciton transitions at E ⊥ c in InSe cannot be considered neither allowed (bright) nor forbidden (dark) in the usual sense of these terms. We have to claim them as partly allowed (gray).
Direct confirmation of the impact of the selection rules on optical properties has been obtained by measuring the polarized PL from the cleaved edge of InSe crystal in two polarizations: E c and E ⊥ c. It turns out to be that their intensities differ more than by the order of magnitude (see Supplementary Figure 2). The possibility to observe such symmetry-related phenomenon also indicates the high quality of the studied samples.

SUPPLEMENTARY NOTE 3 : Anisotropic model for excitons in InSe
We treat an exciton in InSe in the framework of the effective mass approach with anisotropic mass and anisotropic dielectric function 24,25 . The respective Hamiltonian reads as The effective Rydberg Ry = µ xx e 4 2 2 κ xx κ zz and Bohr radius were used as energy and length units. Here x, y and z are Cartesian coordinates chosen along the crystal axes. The z-axis is directed along c-axis, µ ii and κ ii are the components of the reduced exciton mass and dielectric constant tensors, where m ii,e and m ii,h are the components of electron and hole effective masses tensors, respectively. The relations µ xx = µ yy = µ zz and κ xx = κ yy = κ zz are assumed.
In InSe, the charge carriers effective masses as well as the exciton reduced mass are smaller along the z-axis than in the perpendicular plane so the exciton could has somewhat 1D-like character. However, the parameter which actually control the exciton wave function is χ = κ ⊥ µ ⊥ /κ µ . In our calculations, we have used three parametrizations for effective masses as in Ref. 18 shown in Supplementary Table 1, which are two strongly different theoretical sets (I and II) in order to estimate the influence of parameters dispersion and one more (III) based on experimental data obtained by alternative methods, such as the study of acceptor-bound exciton states under pressure 4 and cyclotron-resonance studies of electron masses 3 . The latter data are close to those obtained by magnetoabsorption spectroscopy 26 . For all cases, the relative dielectric constants κ xx = 8, κ zz = 6.4 as in the Ref. 16 have been taken.
The binding energy E b of ground 1s exciton state (marked as R X in the main text) has been calculated by both variational approach and numerical calculations using the same Hamiltonian to check their consistency. Because we have obtained very good agreement between the obtained results, the developed numerical method was used to calculate the binding energies of excited 2s and 3s exciton states. In addition, we calculate the binding energy of the 2p state. It cannot be observed in the single-photon absorption; two-photons spectroscopy is needed for that. However, being situated closely to 3s, the 2p state can be mixed with it that effects on the possibility to resolve the 3s state. The results for magnetic quantum number m = 0 are shown in Supplementary Table 2.
We notice the good agreement between E 1S b = 21.7 meV, calculated using the experimental data obtained by the alternative methods (see Supplementary Table 2), and our result R X 22 meV, derived from the experiments on exciton scattering (see main text). The corresponding χ value varies from 2.68 (see Supplementary Table 1) up to 2.86 when the parameters from Ref. 26 was used. The dependence of E 1S b on the anisotropy parameter χ, calculated in the validity range of the effective mass approach (0.1 < χ < 5), is presented in Fig. 5b of the main text.

SUPPLEMENTARY NOTE 4 : Analysis of published data on exciton energies
We have analysed the published data on the exciton binding energy R X in InSe and GaSe. The data presented in Supplementary Table 3 show that in GaSe the values obtained from the differences ∆ 2,1 = E n=2 − E n=1 and ∆ 3,2 = E n=3 − E n=2 agree with each other. On the contrary, in InSe such an agreement is absent. The R X derived from the ∆ 3,2 value is around 21 meV that is markedly higher than the ∆ 2,1 one equal to 14 − 15 meV. We should note that this inconsistency does not allow one to declare the 3D character of excitons in InSe.
The outstandingly high R X magnitude obtained by Andriyashik et al. 7 may be related to the attribution of the Sommerfeld peak as the first excited state. That is why their bandgap value reasonably correlates with our value (1.36 eV), taking into account the temperature-induced shift (they performed the measurements at 80 K). This analysis is very important in the context of this paper, because the seeming applicability of the 3D model used was a crucial argument in favor of the 3D character of excitons in InSe.

SUPPLEMENTARY METHOD 1 : X-ray diffraction
Supplementary Figure 3 shows the θ/2θ scans of the undoped and Zn-doped InSe bulk samples. The periodic peaks indicate a perfect stacking of the layers along the z-axis for each crystal. The c lattice parameter is obtained from the linear dependence of the interplanar distance d vs inverse integer index (not shown). The values are c = 16.542Å and 16.650Å for the undoped and p-type crystals, respectively. This could correspond the β-InSe 14 because the rhombohedral InSe has the lattice parameter about c = 24.95Å 2,27 . In order to completely define the polytype the crystal has been oriented on purpose for the detection of an asymmetric line. The (107) plane specific to the hexagonal structure has indeed been observed for the expected crystal orientation (not shown here). This signature rules out other stacking orders.

SUPPLEMENTARY METHOD 2 : Raman studies
The Raman measurements were performed at room temperature using a T64000 (Horiba Jobin-Yvon) spectrometer equipped with a confocal microscope. The line at 532 nm (2.33 eV) of Nd:YAG laser (Torus, Laser Quantum, Inc.) was used as the excitation source. The laser power on the sample was as low as 25 µW in a spot size of ∼1 µm. Supplementary Figure 4 shows the typical polarized Raman spectra of InSe obtained in two backscattering geometries: z(xx)z and z(xy)z. Here, the z direction is parallel to the c axis, and x and y are mutually orthogonal and oriented in an arbitrary manner in the layer plane. Five narrow Raman lines have been observed at 19.0, 41.3, 115.4, 177.2, and 227.1 cm −1 . The most important finding is the observation of the low-frequency interlayer mode (vibration of tetralayer as a whole) at 19.0 cm −1 , very close to the laser excitation line. This is the main signature of the β-or -polytype of InSe. The primitive unit cell of γ-polytype contains only one InSe layer 18 . Therefore all optical modes are intralayer ones and the interlayer vibrations should not be observed in the first-order Raman spectra of γ-InSe. In our case, -InSe polytype should be also excluded because of the registered PL energy which is characteristic for the β-InSe. Small FWHM of Raman lines and a very good agreement with the Raman selection rules 18,28 clearly demonstrate that the studied samples can be classified as the high-quality β-InSe. This conclusion is fully consistent with the X-ray data and the observation of polarized PL satisfying the exciton selection rules.

SUPPLEMENTARY METHOD 3 : Optical measurements
The nonlinear optical processes can be studied in two different experimental geometries: i) incident excitation to the sample surface and detection of the emission from the surface ("backscattering geometry") 29,30 ; ii) excitation to the surface but detection from a cleaved facet 31 . On one hand, the waveguiding effect (and symmetry-enhanced PL intensity), characteristic for the later geometry, can promote the stimulated emission. On the other hand, it leads to an additional red-shift and a broadening of the emission lines. We used the first geometry in our experiments.
The low-intensity photoluminescence (PL) was measured using a continuous 15 mW red laser (650 nm). High pump power was obtained with a Q-switched green laser (532 nm) with a pulse duration of 15 ns and a repetition rate of 1 kHz, which corresponds to a maximal power of 100 W per laser peak. The pump intensity was varied using the combinations of neutral filters. The PL was detected using a Hamamatsu InGaAs photomultiplier cooled at 77 K. The integration of pulsed response over time was performed by a lock-in amplifier.
A reasonable estimate of the laser spot diameter of ∼200 µm leads to a maximal power density of ∼ 0.3 MWcm −2 . Note that such a density is significantly lower than the 6 MWcm −2 used in the previous experiments on stimulated emission in InSe 32 .
The intensity of emission peaks measured at T = 10 K and for maximal laser power is strongly dependent on the position of the laser spot on the surface of the InSe crystal (see Supplementary Figure 5). The strong dependence of the peak intensities on the laser spot location results either from the softness of the InSe crystal, which can be easily damaged during cleavage, or from interlayer defects 33 . Both sources lead to spatially inhomogeneous intensity of optical response 34 . At the same time, the energy positions of the peaks are constant within a ±0.5 meV limit.
The transmission measurements presented here were performed on a thinner (∼ 40 µm) InSe sample with back illumination with a halogen lamp through a 1 mm diameter pinhole. All samples were mounted on the cold finger of a closed cycle helium cryostat with the possibility of temperature variation in the 10 − 350 K range.

SUPPLEMENTARY METHOD 4 : Defect-related emission
The defect related lines below the P-band exhibit intensity saturating with the power increase. Besides, they quench fast with the temperature rise due to the carrier delocalization from bounding centers. First line marked by (*) was reported in many previous studies as a donor-related transition 34,35 . It was proposed that this transition could be related to an indirect exciton 36 . However the currently accepted band structure of InSe (see Supplementary Figure  1) rules out this suggestion definitely. In our studies, this assignment is supported by its arising in the naturally n-doped InSe and quenching in the p-doped structures. This line is not resolved in spectra in Supplementary Figure  5 measured with high excitation power due to its saturation.
Excitons bound to very shallow neutral donors have been reported in InSe around 1.321 − 1.334 eV but are not detected in our samples [37][38][39] . The peaks detected at 971.5 nm, 991 nm, and 1013 nm have been assigned previously to donor-acceptor transition 34 , to donor-to-band transition in lithium intercalated InSe 38 or excitons bound to donor levels such as V Se 40 . Ferrer-Roca et al. interpreted these peaks as a band-to-acceptor recombination and LO-phonon replicas 4 . According to this analysis and assuming the 971.5-peak is the zero-phonon recombination we evaluate the acceptor binding energy to 76 meV. The energy separation between adjacent peaks is almost constant and is of the order of 25 − 27 meV. This value is in agreement with the 201 and 227 cm −1 optical phonons (25 and 28 meV respectively) 20,41,42 . Finally, a broad peak occurs at larger wavelength centered at 1084 nm (1.144 eV) (not shown in the presented PL plots) was previously observed in p-type InSe and attributed to the Zn acceptor level 43,44 . Indeed this PL peak is even more pronounced in the PL spectrum of the Zn-doped InSe sample.

SUPPLEMENTARY METHOD 5 : Temperature dependent photoluminescence and absorption
The temperature dependence of the free exciton energy shown in the inset of Fig. 3 in the main text can be modeled by E(T ) = A − B coth( ω ph /2k B T ). The fit of our data leads to a phonon energy of 13.7 meV, which is in agreement with the 14 meV value reported in Ref. 5 and the 13 meV estimate extracted from our transmission data. At high temperature the FX peak shifts linearly with temperature at a rate of −2k B B/ ω ph ≈ −0.41 meV/K.
The absorption spectra exhibit the pronounced Urbach tails (see Supplementary Figure 6). Their temperature dependence is given by where E U (T ) = k B T /σ is the Urbach energy equal to the energy width of the absorption edge. E 0 and α 0 are the coordinates of the convergence point of the Urbach "bundle", which is the crossing point of the fitting lines. It occurs close to 1.36 eV in our samples. The steepness parameter σ depends on the temperature and the energy of phonons involved in the absorption process: where ω ph is the energy of the phonon involved in the broadening, likely an LO phonon. We refrain from defining the exact value of E g based on the absorption data, because the 1s absorption peak is smoothed and we do not observe the features of excited levels.