NANR Journal

1.       Introduction

Due to their physicochemical properties - strong covalent hybrid sp³-bonds inside crystal layers and the low van der Waals bonds between them - layered In₂Se₃ crystals and MX and ternary MX compounds (where M-In, Ga and X-S, Se) of the A³B⁶ Periodic Table group are low-dimensional semiconductors. As has been recently shown, MX crystals, and especially InSe crystals, are perspective materials for 2D electronic devices, e.g. field effect transistors [1,2], photodetectors [3-6], phototransistors [7,8], thermoelectric materials [9] and memory cells [10]. In addition, bulk layered MX crystals are good candidates for solid state hydrogen storage [11-13] and supercapacitors [14].

In contrast to the better-known InSe crystals, bulk In₂Se₃ crystals have been less often investigated due to technological growth difficulties; they are more inhomogeneous, do not have mirror-like cleavage surfaces and are characterized by different crystal structure modifications. Parts II and III of this article present the experimental data of our Scanning Electron-Microscopy (SEM), Energy-Dispersed (EDX) and optical (wide temperature photoluminescence and light absorption) investigations of Bridgeman-grown bulk In₂Se₃ crystals. Some aspects of physicochemical bonding in In₂Se₃ crystals are discussed in Part IV, together with their optical properties in the vicinity of the band-gap edge.

Part IV also presents a discussion of the determination of a band-gap edge at T = 5 K, the exciton binding energy, the effective mass of carriers, dielectric permeability, the character of electron- and exciton-phonon interactions in bulk In₂Se₃ crystals and the role of nano-sized interspersions of InSe phase in the optical properties of In₂Se₃ crystals. Separate effort is devoted in Part IV to the investigation of optical properties (band-gap tuning) and character of exciton-phonon interactions in 3D InSe nanocrystals with radii between 15 and 650 nm in the confinement of an In₂Se₃ matrix. Finally, Part V presents a summary of this work and our conclusions.

2.       Experimental Method

Bulk In₂Se₃ crystals as in [15] were grown using the Bridgeman method from a stoichiometric melt, with a temperature gradient of 15 K/cm at the crystallization front and a growth velocity of 1 mm/h. They have n-type   conductivity with concentration of carrier’s n = 2.0·10¹⁸ cm^-3. Characterization of In₂Se₃ crystals was carried out using a Vega3 SBU scanning electron microscope (Tescan) equipped with Second Electron (SE) and Back-Scattering Electron Emission (BSE) detectors, and an EDX microanalysis spectrometer (OXFORD Instruments) with an X-act (10 mm²) SDD detector and INCA EDS software.

Measurements of light absorption and Photoluminescence (PL) spectra were made using a 0.6 m optical spectrometer MDR-23 (LOMO) with a grating of 600 groves/mm. Investigations of light absorption and PL spectra for T = 5-300 K were made using a helium cryostat A-255 designed at the Institute of Physics National Academy of Sciences of Ukraine (IP NASU). This was equipped with a UTRECS K-43 system, allowing control of the sample temperature within the range 4.2 to 330 K with an accuracy of 0.1 K. Bulk samples of In₂Se₃ crystal were prepared for optical investigations and freely mounted on a large finger before being placed into the helium cryostat.

The excitation of PL spectra was performed using a current-wave semiconductor laser with a wavelength of 532 nm (P = 100 mW) equipped by a laser (SL-532-10 Thorlabs) and edge (LP-03-532-RS Semrock’s) filters. A stable power tungsten lamp (P = 60 W) and standard edge filter with transmission region λ > 600 nm were used for light absorption measurements. A liquid nitrogen mini-cryostat system MK-80 (IP NASU) was used to cool a near-infrared photomultiplier tube FEU-62 (400 < λ > 1200 nm), which served as a radiation detector. The geometry of light absorption near the band gap edge and PL investigations of In₂Se₃ crystals are presented in the insets of Figures 2a, c respectively.

3.       Results

3.1.  SEM and EDX investigations of In₂Se₃ crystals

Figures 1a, c show SEM images of In₂Se₃ crystal surfaces with 5,510× and 90,000× magnification. It can be seen from Figure 1a that the fresh cleavage surfaces contain spatial dislocations of 100-200 nm in radius, and the interspersion of foreign sub-micron phases of 400-600 nm in size can be observed in the vicinity of these dislocations. EDX investigations of these interspersions Figure 1b at 20,400× magnification) show that they are: i) phases consisting of 100% Se atoms (point 1); ii) phases of InSe (point 3), which contain 50.17% Se atoms and 49.83% In atoms; or iii) phases of In₆Se₇ (points 2, 4 and 5). Note that the analytical area for EDX investigation conducted at an electron beam acceleration of 15 kV was 0.53 mkm for In and 1.4 mkm for Se atoms.

With a magnification of 90,000× for the chosen BSE geometry (Figure 1c) and a low-current electron beam (the latter was compressed down to 25 nm), one can see dislocation veins of radius 20 to 30 nm and inclusions; these have an ellipsoidal shape with main axes a = b = 30 nm and c = 150 nm. The cleavage surfaces of the In₂Se₃ crystal, which do not contain dislocations, and sub-micron interspersions (see the selected rectangular area in Figure 1b may be attributed to pure In₂Se₃ crystals. For the example of GaSe crystals, it has been shown [12] that the interspersion of pure Se phase is a monoclinic modification inherent to the red β-Se (group 2.2/m). Moreover, according to [13], one can observe on the cleavage surfaces of bulk GaSe crystals after natural intercalation in distillated water (Tescan high resolution SEM Lira3 with immersed magnetic lenses) a shrub-like formation of dendrites, consisting of Se atoms grown from the spontaneous nucleation of red β-Se. The spatial size of these shrubs in GaSe crystals after half-year water natural intercalation was about 2-5 mkm, and their whiskers were about 25-50 nm in diameter, tapering to the end with a step of 15 ± 2 nm.

In [16], γ-In₂Se₃ nanorods were directly grown on Si (111) substrates. Using high-resolution SEM JSM-7001F (JEOL), it was shown that In₂Se₃ nanorods are straight and are not tapered.  The average diameter and height of the In₂Se₃ Nano rods are about 64 and 460 nm, respectively. Thus, SEM and EDX investigations have shown that Bridgman-grown bulk In₂Se₃ crystals contain great numbers of small nanoparticles, which contain different concentrations of In and Se atoms.

3.2.  Edge Absorption and Photoluminescence of Bulk In₂Se₃ Crystals

Figure 2a shows the light absorption spectra of In₂Se₃ crystals at T = 5-300 K with geometry E ^ C, k || C (where E is the electromagnetic wave vector and C is the crystal optical axis), with the account of light reflection. In this case, the dielectric permeability of In₂Se₃ crystals was chosen as ε = 10.0, which is close to the value of ε = 10.5 for InSe crystals [17].  As the crystal has a number of nano-interspersions, a diameter of 2 mm was chosen for the investigated crystal surfaces and a crystal thickness of 1.2 mm for the light absorption measurement. We consider two independent series of absorption measurements. In the first case, the width of the spectrometer entrance slit was 0.1 mm, and in the second it was 0.8 mm. This allowed us to carry out practical absorption measurements of the In₂Se₃ crystal matrix without accounting for the absorption of light by foreign interspersions. For these cases, the spectral slit width during the experiment did not exceed 0.33 and 2.7 meV. We obtain identical absorption spectra in both cases.

At T = 5 K, light absorption in the bulk In₂Se₃ crystal rapidly increases at a wavelength of λ < 800 nm. At λ > 800 nm, a tail of electron density states of the conductive band was observed. This allows us to determine the band gap of In₂Se₃ crystal with sufficient accuracy at T = 5 K, at a value of E_g = 1.550 ± 0.001 eV. This value is in good agreement with that in [18], where E_g = 1.56 eV was calculated for an α-In₂Se₃ crystal at T = 4.2 K.

Moreover, in [19], PL investigations show that pure α-In₂Se₃ crystals exhibit two PL bands at liquid helium temperature: one at a higher energy of 1.523 eV, due to the radiative recombination of impurity-bound excitons, and the other at a lower energy of 1.326 eV, due to recombination on luminescence centers formed by intrinsic defects due to disorder in the cation sublattice. A temperature increase (Figure 2a) of up to 300 K leads to a long-wave shift in absorption edge, accompanied by a reduction in the absorption coefficient inclination corner caused by electron temperature redistribution (k_BT) in the tail of the conductive band (CB). Furthermore, with an increase in temperature, an increase of light scattering on crystal lattice vibrations was observed in the forbidden gap.

PL spectra were investigated from different points in the same In₂Se₃ crystal. The diameter of the laser beam on the crystal surface was 1.5 mm, and the size of the spectrometer entrance slit up to 80 K did not exceed 50 mkm with a spectral resolution of 0.17 meV. Figure 2b shows the PL spectra of bulk In₂Se₃ crystal for two different points, at T = 5 and 150 K. At T = 5 K, the PL spectra for the first point (black curve) consist of a wide band named the B band, with a peak at 1.3476 eV, and for the second point (grey curve) a B1 band with a peak at 1.3511 eV. In addition, on the high-energy side of the B1 band, a shoulder D at 1.3666 eV was observed. Figure 2c shows the PL spectra from the first point on the crystal surface of the bulk In₂Se₃ crystal at T = 5-300 K. It can be seen that at T > 200 K, the B band practically disappears and a new high-energy A band starts to prevail, which begins to be detected at T = 80 K (Figure 2c). The temperature shift of the band-gap width (E_g) and the energetic position maxima for the A, B, and B1 bands are presented in Table 1

4.       Discussion

In Figure 3, the open squares indicate the experimental dependence of E_g(T), obtained from absorption measurements of bulk In₂Se₃ crystals. The open circles and triangles show the energetic shift in the PL spectra of the A and B bands versus temperature. It can be seen that the temperature shifts of E_g and the A band are described by identical curves, 1 and 2, which are separated by 14 ± 1 meV. At the same time, the temperature shift of the B band essentially differs from the temperature shift of E_g and the A band. It is known [18,20] that single In₂Se₃ crystals grown by the Bridgman method possess four modifications (α, β, γ, and δ), which differ from each other in terms of their sequence of crystalline layer stacking. Each crystal layer consists of five monolayers of Se - In - Se - In - Se atomic sheets, bounded by hybrid sp³ covalent bonds. Interatomic distances between atomic sheets within a layer are as follows: i) In-Se: 0.287 nm (In situated in octahedral holes at 523 K) and 0.269 nm (In situated in tetrahedral holes at room temperature); and ii) Se-Se: 0.401 nm (Se between each layer) [18].

The space group of the β modification are rhombohedral D⁵_3d. The unit cell has dimensions a = 0.405 nm, C = 2.941 nm at 523 K, and Z = 3. The unit cell of the α modification (C⁵_3v - space group) has dimensions a = 0.405 nm, C = 2.877 nm at room temperature, and Z = 3. For bulk GaSe and InSe single crystals, the variation in E_g(T) is mainly attributed to a change in the electron-phonon interaction (the self-energy effect) due to the participation of a homopolar half-layer A¹_1g-phonon, which reduces the band-gap width when the temperature increases, as shown in [21-23]. In this case, the self-energy contribution obeys the equation

where the subscripts v and c refer to the valence band (VB) and conductive band (CB), n^* is the number of homopolar phonons in that particular mode, g_v and g_c are the coupling constants, m*_v and m*_c are the effective masses of carriers, and Q is the radius of the Brillouin zone. The energy of the homopolar phonon is taken to be constant over the whole Brillouin zone.

According to [21], using crystal parameters of a = 0.4001 nm and c = 2.5232 nm for g-InSe crystals (C⁵_3v space group) [24], values of effective mass for carriers m*_c = 0.135 m₀ and m*_v = 0.7 m₀ in CB and VB, and exciton binding energy ΔE_exc = 14.5  meV for InSe [17,25] an analytical expression for E_g(T) dependence can be obtained for bulk InSe crystals due to the self-energy effect with homopolar A¹_1g phonon (hw = 13.6 meV [26]) participation as follows:

Note that in [21], the temperature dependency of the E_n=1-ground exciton state shift was investigated for the case of exciton-phonon interaction, and an averages exciton-phonon coupling constant g = 0.5 (for electrons in CB and holes in VB) was found.         

Following [21,17], and taking into account the crystal parameters of bulk In₂Se₃ crystal [18] and our experimental data for E_g(T) dependence (Table 1), an analytical dependenc

was found for bulk In₂Se₃ crystals with g = 0.65 (g_c = g_v = 0.65), hw = 6.0 meV and m*_c = 0.12 m₀,  m*_v = 0.6 m_0.  

Note that the same analytical dependence

was found for the A-band shift versus temperature.

The experimental data for E_g(T) and A(T) dependences are in accordance with Eqs. (3) and (3^/), and differ one from another at a constant value of 14 meV. This allows us to make the assumption that the A band displays free exciton emission in bulk In₂Se₃ crystals, with an exciton binding energy ΔE_exc = 14 ± 1 meV. Note that the obtained value is very close to the exciton binding energy ΔE_exc = 14.5 meV in bulk InSe crystals. Now, using the obtained fitting values for m*_c and m*_v and the well-known equation for exciton binding energy ΔE_exc[meV] = 13605·μ^*/ε², where μ^* = (m*_c·m^*_v)/(m*_c+m*_v) is the exciton effective mass, a value for the dielectric permeability of ε = 9.9 ± 0.1 in bulk In₂Se₃ crystals can be obtained. This value is also close to ε = 10.5 [17] in bulk InSe crystal.

It can be seen that in the case of In₂Se₃ crystals, the temperature shift of E_g occurs with the participation of phonons whose energy (hω = 6.0 meV) and vibration frequency (ν₁ = 48 cm^-1) are significantly lower than the energy (hω = 13.6 meV) and the frequency (ν = 110 cm^-1) of a half-layer homopolar A¹_1g - phonon in the bulk InSe crystal. This reduction in frequency vibration for a half-layer homopolar phonon in an In₂Se₃ crystal is caused by: i) Аn increase in the atomic mass of the In₂Se₃ molecule, as compared to the In₂Se₂ molecule of InSe crystal. This increase in atomic mass in a simple model of one-dimensional linear chain corresponds with the equation ν₁ =ν/ [1 - (M - M1)/M]^1/2 (where ν_1, M1 are the frequency and atomic mass of the In₂Se₃ molecule, and ν, M are those of In₂Se₂) reducing the frequency for the half-layer homopolar  phonon in In₂Se₃ crystal to ν₁ = 96 cm^-1. ii) The difference in the distribution of valence electrons between the In and Se atoms in molecules of InSe and In₂Se₃ crystals. As can be seen from Figure 4, the transition from the InSe crystal to the In₂Se₃ crystal is accompanied by the appearance of an additional Se atom between two intra-layer In atoms, which takes up missing electrons from each of its six neighboring In atoms. This internal covalent sp³ bond between In→Ga←In atoms (six nearest-neighbor In atoms) in a layer of In₂Se₃ crystal is much weaker than the (2s) double electron In<=>In covalent bond in the InSe crystal. This leads, in accordance with the Hooks law, to reduction of the half-layer homopolar vibration frequency and increase layer thickness.

Note that according to [27], optical vibration at ν = 54 cm^-1 was observed in the Raman spectra of γ-In₂Se₃ crystals. In [28], a weak singularity at 41 cm^-1 with damping parameter γ = 97 cm^-1 for α-In₂Se₃ crystals was attributed to a plasmon contribution to the far infrared spectrum.

Based on the experimental data for E_g(T) dependency, and the calculations and discussion above, it is reasonable to suggest that the frequency of a half-layer homopolar A-phonon in Bridgeman-grown bulk In₂Se₃ crystal is ν₁ = 48 ± 3 cm^-1.

4.1.  B Band Emission

As can be seen in Figure 2c, the PL intensity of the B band prevails at T=5 K and decreases with temperature, while the PL intensity of the A band at T > 80 K increases with temperature. At T=5 K, the B band for some crystal points has a structure consisting of a B1 band and a shoulder D, whose energetic position slightly differs from the B band (Figure 2b). In Figure 3, the energy shift of the B band versus temperature is shown by open triangles. It can be seen that the energy position of the B band at T = 5 K (E_B = 1.3476 eV) and the energy shift of the B band with temperature significantly differ from the energy position and shift of the A band and E_g. The energy shift with temperature for the B band is well approximated by curve 3, based on the following analytical dependence:

Curve 3 satisfactorily describes the temperature shift of the B band with participation of phonons possessing energy hw = 5.1 meV (n = 41 cm^‑1). This half-layer transversal vibration of E-symmetry with frequency n  = 40 cm^‑1 takes place in InSe crystals and is pronounced in IR and Raman spectra [29,30].

In the following discussion, we assume that the B band:

i) Is caused by radiative recombination of excitons in InSe nanocrystals;

ii) Has an emission peak at T = 5 K which is shifted by 11 meV to the short-wave side relative to the emission line of the ground exciton state (E_n=1 = 1336.7 meV) in bulk crystals γ-InSe [31] and

iii) Has a temperature shift which occurs with the participation of polar half-layer phonons of E-symmetry, which has a frequency n  = 40 cm^‑1 similar to that in bulk crystals, in the electron-phonon interaction process.

In this case, the half-layer E-vibration in InSe nanocrystals with frequency n  = 40 cm^‑1 is almost in resonance with the fully symmetric half-layer A-vibration of the In₂Se₃ matrix of n₁ = 48 cm^‑1 (Figure 4). It was shown in [31] that the availability of layer Stacking Faults (SF) - a mixture of γ- and ε-modifications - results in the appearance of additional bands at T = 4.5 K in the spectra of exciton photoluminescence which are 1-1.5 meV higher than the energy of the ground exciton state E_n=1 = 1336.7 meV in γ-InSe single crystals. In ε-GaSe crystals, the availability of SF leads to the appearance of a thin structure related to free excitons localized at SF and which is below the energy of the ground exciton state in ε-GaSe single crystals [32-34].

However, the increase in free exciton recombination energy of 11 meV observed in InSe nanocrystals is problematic to explain based on the availability of stacking faults, since for the InSe nanocrystal sizes shown in Figure 1c these stacking faults should, as a rule, extend from the nanocrystal bulk to its surface. A more likely reason for the high-energy shift of the exciton band is therefore the so-called “Blue” shift, caused by a decrease in the geometrical sizes of bulk crystals to those of nanocrystalline particles. Based on this assumption, we perform some calculations. For γ-InSe, the volume of a unit cell to a good approximation is V₀ ≈ 0.4 nm³, which corresponds to the volume of a sphere with radius r_s = 0.457 nm. Then, from the inequality defining the ratio of a nanocrystal surfaces to its volume

N³/2 > 6N²,            (5)

where N is the number of unit cells along one of the faces for a nanocrystal of the cubic shape, it follows that Eq. (5) is valid for an InSe spherical particle with radius R^QD > 5.5 nm. Note that the Rydberg radius of a Wannier-Mott exciton in InSe bulk crystals is r_exc = 4.5 nm. Hence, for shown in Figure 1c (SEM images) InSe nanocrystals with radius R^QD = 30 nm, the number of unit cells is (R^QD/r_s)³ ≈ 2.8×10⁵.

For the bulk InSe crystals, the average values of the anisotropic effective masses of electrons and holes are m_e = 0.14m₀ and m_h = 0.72m₀ [35,36], while their Rydberg radii are r_e = 4.0 nm and r_h = 0.8 nm, respectively. We then suppose that the following condition is valid for InSe nanocrystals at R^QD > 5.5 nm:

R^QD > r_exc > r_e > r_h  ,    (6)  

which allows a description of phonon branches, exciton behavior and exciton binding energy as in a classic case of bulk InSe crystals, based on quantum-dimensional corrections in the first approximation, while the behavior of charge carriers should be considered within the framework of quasi-3D conduction and valent minibands. In the case where R^QD < r_exc, radiative recombination will be observed of an electron-hole pair non-bound by Coulomb interaction. Here, it will be important to take into account the ratio of the radii of charge carriers and the quantum dot.

For the set i = 1, ... n of nanoparticles possessing various radii , the following relationship is valid:

That is obtained using the hyperbolic model for the blue shift of the forbidden gap inherent to the crystal when its geometrical dimensions are decreased [37], where  is the forbidden gap width corresponding to the nanocrystal of the radius . In accordance with Eq. (7), Figures 5a, b shows the fitting of calculated and experimental data obtained at Т = 5 K from the measured PL spectra of the В and B1 bands. The fitted spectra well approximate the В and B1 bands by the set of six Lorentz lines L1-L6 of different intensities (which describe the emission of Wannier-Mott free excitons in InSe nanocrystals with radii within the range 16.2 to 627 nm) and Gauss lines G1, G2 (describing the emission of excitons bound by defects at the boundary between InSe nanocrystals and the crystalline matrix In₂Se₃). It should be noted that in the case of the B1 band, the weakly intensive G2 line is practically absent.

Table 2 gives the calculated values of the fitted half-widths at half-heights Γ_Li,Gi, the energy positions of peaks E_Li,Gi and the integrated intensities S_Li,Gi of the L and G lines of emission inherent to free and bound excitons, respectively, for InSe nanocrystals of various radii at T = 5 K. For comparison, Figure 5c shows the PL spectrum (T = 5 K) of an especially non-doped InSe single crystal and the fitting spectra of the Lorentz L6 and Gauss G1, G2 components of the emission B band observed in InSe nanocrystals. It should be noted that the fitted G1 line is close in energy to the PL band observed in [19] at 1.326 eV; this is related to recombination on luminescence centers, formed by intrinsic defects caused by disorder in the cation sub lattice of In₂Se₃. In these calculations, we assume that, even for small sizes of InSe nanoparticles, the behavior of electrons and holes inside energy minibands and their Coulomb interaction can be described as for the case of bulk crystals. The radii of InSe nanocrystals (Table 2) were therefore calculated using Eq. (7) and the expression

At the same time, it should be noted that in the layered InSe and GaSe crystals, intercalated with hydrogen at concentrations х > 1.0 for InSe [12] and х > 2.0 for GaSe [38], non-linear behavior can be observed in value of the exciton binding energy, which is related to the filling of the van der Waals gap by hydrogen molecules and the increase in dielectric permittivity in the interlayer space.

In accordance with [12,38], this leads from the very beginning to an increase in the exciton radius and, with the subsequent increase in hydrogen molecules concentration in the interlayer space, to the localization of exciton motion within the planes of the crystalline layers. Since in InSe, excitons comprise 12 crystalline layers and in GaSe only seven layers, the 2D localization of exciton motion in GaSe crystals can be observed only at higher concentrations of hydrogen. It should also be noted that in studies of the optical properties of InSe nanoflakes of thickness d = 0.8-4.0 nm, which comprises one to five crystalline layers, beside the observed tuning of the forbidden gap [39-41], Coulomb interaction is available between 2D localized electrons and holes in these layers with the creation of 2D excitons. This should also result in an increase in the binding energy of 2D excitons.

In this approximation, it can be seen from the fitting spectra for the В and В1 bands at T = 5 K, shown in Figure 5, and from the parameters given in Table 2, that:

- The values of Г for the Lorentz L lines are correlated and increase, with a decrease in the nanoparticle radius, from Г = 15 meV for R^QD < 150 nm, to Г = 16-20 meV for R^QD < 30 nm;

- The obtained values for InSe nanoparticle radii correlate with the experimental values for the observed nano-crystalline InSe inclusions in the In₂Se₃ matrix, as shown in Figure 1c;

- For R^QD < 20 nm, the intensity of emission lines due to inclusions of InSe nanocrystals in the matrix of In₂Se₃ (at T = 5 K) drops sharply; for R^QD ≤ 7 nm, this intensity is equal to zero, in accordance with Eq. (7), since the energy of fundamental transition in the InSe nanocrystal E_g = 1.564 eV exceeds the forbidden gap width in the In₂Se₃ crystal.

Figure 6 shows the dependences of the integrated intensities S_Li for the В and В1 bands on the radius R_i^QD of the InSe nanocrystal at T = 5 K. It can be seen that the highest contribution to the emission of В and В1 bands is provided by free excitons in InSe nanocrystals of radii 28.8 to 136 nm; this correlates well with the results of SEM investigations. The insert in Figure 6 shows the dependence of S_Li on R_i^QD for InSe nanocrystals in the In₂Se₃ matrix; due to their sizes, this corresponds to the range 3D QD.

The above discussion allows us to determine the value of the radius cutoff in InSe nanocrystals, at which the emission of free excitons in the In₂Se₃ matrix takes place, as R^QD_min ≈ 15 ± 1 nm. The maximum value of the forbidden miniband width in these nanocrystals, at which radiative recombination of free excitons in InSe nanocrystals embedded into the In₂Se₃ matrix is observed, can be determined as E^QD_max < 1.450 ± 0.001 eV.

4.2.  Scheme of Energy Positions for Minibands in InSe Nanocrystals Embedded into Bulk In₂Se₃ Crystals

Figures 7a) and 7b) illustrate the scheme of energy positions for valence (MVB) and conductive (MCB) minibands in InSe nanocrystals relative to the valence (VB) and conductive bands (CB) of the In₂Se₃ matrix at T = 5 K and T = 300 K. The scheme is plotted in accordance with the analytical dependences obtained in this work for the energy shifts of the forbidden gap E_g and the А, В, В1 PL bands on temperature, and for the redistribution of intensities between the A, В and В1 bands, observed experimentally at T > 80 K, with an increase in temperature of the In₂Se₃ crystal. We assume here that in bulk In₂Se₃ crystals, in a similar way to InSe crystals, the main contribution to the change in the forbidden gap width with an increase in temperature is provided by the conduction band. It therefore follows that at T = 5 K in InSe nanocrystals of large radius (> 627 nm), the MCB minimum is localized near 100 ± 1 meV, below the conduction band of bulk In₂Se₃; the MVB maximum for InSe nanocrystals is 99 ± 1 meV higher than the maximum in the valence band in bulk In₂Se₃. Presented in the same place is the scheme of the energy positions for VB and CB of bulk InSe, the extremes of which practically coincide in terms of energy with those of InSe nanocrystals of large radii (> 627 nm).

As can be seen in Fig. 7a), both CB electrons and VB holes in In₂Se₃ matrix can sufficiently quickly, under optical excitation of electrons into the conduction band, overcome the quantum barriers separating them from InSe nanocrystal minibands located at lower energies. Therefore, at T = 5 K, recombination of hot carriers in In₂Se₃ matrix is realized via InSe nanocrystal minibands with the emission of free and bound excitons. The reverse process of carrier transition from the InSe nanocrystal minibands into the VB and CB of the In₂Se₃ matrix is inhibited due to cooling of the carriers and the availability of quantum walls (QW) between the MCV and CV, and MVB and VB, respectively. When the temperature increases to above T = 150 to 200 K, the CB of In₂Se₃ crystal is reduced to below the MCB_i of InSe nanocrystal of the radius R_i^QD. This results in closing the channel for radiative recombination of carriers via MCB_i of InSe nanocrystal of the radius R_i^QD, and, as shown in Figure 7b, the wide A band inherent to emission of free excitons in In₂Se₃ matrix can be detected in the PL spectra at T > 80 K.

It is worth noting that the QW shown in Figure 7 are, in the simplest case, a set of defects in the crystalline lattice of InSe nanocrystals and the In₂Se₃ matrix. These regions provide binding of free excitons, which can be seen as the G1 and G2 emission lines in Figure 5. The size of the QW is commensurate with the thickness of several crystalline layers (1-2 nm). The simplest defects in bulk InSe and GaSe, responsible for the creation of these QW (donors and acceptors), are shown in Figure 8.

Note that in [42] it was reported on the basis of SEM, EDX and EBSD investigations that Bridgman-grown bulk GaSe single crystals contain numerous inclusions of residual monoclinic β‐Se (Laue group 2.2/m) in the interlayer space. At the same time, the average proportions of Ga atoms (53%) and Se (47%) essentially differ from the stoichiometric composition. In addition, GaSe crystals possess ideally mirrored split surfaces, are high‐Ohmic, and the concentration of free carriers at room temperature lies within the  range  p = 10¹³ to 10¹⁴ cm^‐3. It therefore follows that the presence of only point defects is obviously insufficient to describe both the disturbances in stoichiometric composition and the existence of a rather perfect crystalline structure. In view of these conditions, and considering the evolution of the simplest defect of the acceptor type, calculations were performed in [42] for the types of defects that can disturb crystal stoichiometry and cause the aggregation of simple point defects into complex plane and bulk spatial defects during the process of synthesizing the GaSe crystal.

Following [42], the layered crystals of the А³В⁶ group ordering the atoms both inside crystal layers and between them have the appearance …[X‐M‐M‐X]…WB…[X‐M‐M‐X]…, where WB is the van der Waals bond between [X‐M‐M‐X] crystal layers. Therefore, the loss of one of the metal atoms (M) in the course of synthesis can cause splitting of the [X‐M‐M‐X] layer and give rise to layer bifurcation (or creation of two layers) or, vice versa, the “Agglutination” of two layers into one. These areas are characterized by the creation of residual Se atoms “pocket” phases and a considerable amount of Frenkel and Schottky defects, both point and extended, along the perimeter of splitting/agglutination. Thus, in GaSe crystals of the р‐type, agglutination takes place when three Ga and seven Se atoms are lost in every adjacent crystalline layer in the course of growing.

High concentrations of agglutinations lead to disturbances in stoichiometric composition as a whole; however, in the remaining areas, the crystal remains perfect from a crystallographic viewpoint, since deformations decay quickly owing to the weakness of van der Waals bonds [42]. This splicing/splitting of layers also plays an essential role in the synthesis process of n‑In₂Se₃ crystals and, as seen in Figure 1, results in fallout of the nanocrystalline “Pocket” phases of InSe, In₆Se₇ and red β-Se between the In₂Se₃ layers. In addition, it leads to energy band bending and the appearance of local and gap vibrations in the crystalline lattice.

4.3.  Exciton-Phonon Interaction in InSe 3D Quantum Dots

Exciton-phonon interaction in InSe and GaSe crystals has been sufficiently well studied, both within the framework of the classical model by Elliott-Toyozawa [21,43,44] and based on polariton mechanisms of energy dissipation [17]. Since the dimensions of nanocrystals are five to seven orders of magnitude smaller than those of bulk crystals, the manifestation of polariton effects in the absorption/emission of free excitons is usually neglected (with account of essential decay of excitons), and the classical model is used. According to [21], in the case of weak exciton-phonon interaction (Wannier-Mott excitons), the half-width Γ of the absorption line for the ground n = 1 exciton state with an increase of crystal temperature is described by the expression

where Γ^inh is the inhomogeneous widening caused by crystal defects; =g²[hω(ΔE_exc - hω)]^1/2 is homogeneous widening  caused by optical phonons at T = 0; β = 1+[(ΔE_exc +hω) / | ΔE_exc - hω |]^1/2; g is the constant of exciton-phonon interaction; ΔE_exc and hω are the exciton binding energy and phonon energy, respectively; and n^*(T) = [exp(hω/kT) -1]^-1 is a value which characterizes the filling of the phonon branch with temperature.  In accordance with [21], the homogeneous widening of the exciton absorption band in bulk InSe single crystals and the shift of the forbidden gap with the temperature growth occur with the participation of the homopolar half-layer A¹_1g -phonon (n = 110 cm^-1), and the homogeneous half-width  for n = 1 exciton absorption line at T = 0 is  = 1.3 meV [17]. In the case of InSe nanocrystals, when an exciton is scattered by a polar half-layer E-phonon (n = 41 cm^‑1), the value  reaches 1.8 ± 0.2 meV in accordance with Eq. (11).

For comparison, Figure 5c shows the PL spectrum of the bulk InSe single crystal containing an FE line of free exciton emission and a BE band corresponding to the set of lines for emission of bound excitons, the fitted G1 and G2 lines of bound exciton emission, and the L6 line of free exciton emission for InSe nanocrystals of radius R^QD = 627 nm along with the total L6+G1+G2 band. The energy position of the L6 line maximum E_L6 = 1.3375 eV practically coincides with the emission maximum of free excitons in the bulk InSe crystal, E_n=1 = 1.3371 eV. As can be seen from Figure 5c, emission related to excitons bound at the boundary of InSe nanocrystals results in a spectrum which is qualitatively close to the emission spectrum of the InSe bulk crystal. Figure 5c also shows that the total intensity of radiative recombination in InSe nanocrystals, the concentration of which does not exceed 0.1% in the In₂Se₃ matrix, is comparable in magnitude with that in bulk InSe crystals.

It should be noted that the fitted half-width for the L6 line (Table 2), Γ_L6 = 7 meV, is four to five times higher than the calculated value of for the homogeneously widened ground state of bulk InSe crystal at T = 0. In crystals of radius < 150 nm, the fitted values Γ_Li are already one order higher than . At the same time, for bulk InSe crystals containing only a small number of lattice defects, the experimentally observed half-width of the emission line (Figure 5c) and the absorption line [17] for free excitons at T = 5 K does not exceed 2-3 meV, which is only 1.5-2 times higher than .

This widening of the emission line of free excitons is indicative that this exciton in InSe nanocrystals is subjected to an essential additional scattering, which results in non-homogeneous widening оf the emission line. For L3-L5 lines, this non-homogeneous widening reaches Γ_Li^inh = 14.9 meV in accordance with Eq. (11) and provides the main contribution to the widening of the emission lines of free excitons at T = 0. Note that the sharp increase in the contribution from non-homogeneous widening to the emission band of free excitons in InSe nanocrystals of radii less than 150 nm, as compared with bulk crystals, cannot be explained by scattering from defects available inside nanocrystals, since defects extend to the surface at these dimensions.

4.3.1.         Photoluminescence of InSe QD in In₂Se₃ Confinement at T = 5 K:

Analytical calculations are presented in this and the following sections, where the radii of quantum dots are fixed values obtained in accordance with Eq. (7) when fitting the emission spectra of B and B1 bands by the set of emission lines of free and bound excitons. It is assumed that when considering the secondary emission of light by a crystal (PL process) in the case of R^QD → r_exc of the bulk crystal, it is important to take into account the polariton model offered by Hopfield [45] and developed in [46-48] for the propagation of light in crystals, within the ranges of excitonic resonances. In accordance with [45], the light wave propagating along the crystal within the range of excitonic resonances in the infinite ideal crystal is alternately either in the photonic or the excitonic state. In the general case, the plasma frequency of exciton-photon transition is as follows, according to [49]:

where w_exc is the frequency for n=1 excitonic transition, and w_Ω is the frequency of crystalline vibration which takes part in exciton scattering. According to Eq. (12), the plasma frequency for the InSe crystal is w_p = 2.05 x 10¹⁴ s^-1; from this, the polariton free path in the excitonic state is L_e-p = c/n_ow_p = 450 nm. From the other viewpoint, it was reported in [50] that light propagation in the vicinity of excitonic resonances in GaN essentially slows down, and the effective group velocity v_gr drops to 2.1 x 10⁶ m/s, which in the case of an InSe crystal gives the free path of excitonic polariton as L_e-p = hv_gr/2 = 506 nm, where  = 1.3 meV [17].

It is known that in InSe crystals, so-called “Thickness Effects” (related to the decay of excitonic polaritons at Т = 0, which leads to an increase in integrated intensity of the absorption excitonic band) begin to develop at a thickness of less than 40-50 mkm [17,51]. It therefore follows that, in the ideal infinite crystal, exciton decay takes place approximately once per 150 periods of a light wave in the excitonic state. Increases in the crystal temperature and number of defects, and a decrease in the geometrical sizes of crystals, result in a widening of the excitonic state. Figure 9 presents the data (for T = 5 K) from Table 2 (shown in circles), illustrating the dependence of Γ(D), the half-width at the half-height for the emission excitonic line, on a reduction in the linear dimensions of InSe crystals from L_bulk = 45 mkm to L = 2R^QD = 32.2 nm. For convenience, in the following analysis the linear dimension of the crystals L is expressed in the units of exciton diameter D = L/d_exc, where d_exc = 9.0 nm is the exciton diameter in bulk InSe. As can be seen from Figure 9, the data in Table 2 are well approximated at T = 0 by the analytical dependence of exciton line half-width on the crystal size, expressed in D units

where Λ is the analytical value of 23 for InSe, and Γ₀(D) is dependent on D the half-width of the exciton line for absorption/emission in pure crystal at T = 0. The analytical dependence expressed in Eq. (13) may be reduced to the well-known Davydov equation obtained for light propagation in the vicinity of exciton resonance [48], and can be rewritten as

Combining Eqs (13) and (14), the fitting parameter Λ in Eq. (13) can be expressed as

The second component in Eqs. (13) and (14) represents a correction to the exciton half-width that is associated with a finite crystal size.  In limiting cases, i.e. for D → ¥, the value Γ₀ → , and for D → 1 = d_exc, Γ₀ → ¥.

In Figure 9, the arrow indicates the free path corresponding to one period of the light wave in the excitonic state L_e-p = 450 nm, expressed in dimensionless unit’s D = L/d_exc.

In accordance with Eq. (13), when Γ = 100 meV the diameter of the InSe nanocrystal is 2.7% higher than that for exciton diameter in the bulk InSe crystal, and its lifetime τ = 3.29 x 10^‑15 s should indicate recombination of a non-bound electron-hole pair.  

                4.3.2.         Photoluminescence of the InSe QD Ensemble in In₂Se₃ Confinement at T = 5 K:

Earlier electron-microscopic and optical investigations of GaSe crystals intercalated in a RbNO₃ melt [14] showed that the concentration of RbNO₃ nanocrystalline inclusions can reach 10%, while the segmented polycrystal GaSe matrix retains its optical properties. When comparing the light absorption coefficients at T = 5 K in the range < 1.45 eV Figure 2a indicates the range of light absorption in InSe nanocrystals) and for absorption near the band-to-band transitions in In₂Se₃ matrix, it can be assumed (by analogy with [14]) that the concentration of InSe nanocrystals in In₂Se₃ matrix is less than 0.1%.

At the same time, the juxtaposition of the PL spectra shown in Figure 5c for the bulk InSe crystal and the calculated L6, G1 and G2 emission lines obtained for InSe nanocrystals under practically identical experimental conditions confirms the high efficiency of radiative recombination for InSe nanocrystals confined in the In₂Se₃ matrix. In view of the above, the resultant half-width of the B band for the different intensity l = 6 lines of free exciton emission and m = 2 emission lines of bound excitons at T = 0 is equal to

where M_i, N_j and Γ_Li (0), Γ_Gj are weighting coefficients defining the intensities and half-widths of fitted emission lines. As can be seen from Figure 6 and Table 2, the main contribution at T = 5 K (up to 95%) to the formation of B and B1 bands is made by the L2-L5 and G1 fitted emission lines.

Therefore, neglecting the contributions of the L1, L6 and G2 lines, according to Eq. (16) and given in Table 2 values of half-widths for L2-L5 and G1 fitting lines (when the weighting coefficients M₂-M₅ and N₁ are equal to unity), the following calculated values of half-widths can be obtained for the B and B1 bands: Γ_B = 44.6 meV and Γ_B1 = 42.7 meV. These coincide with good accuracy with the experimental data at T = 5 K: Γ_B = 45.0 meV and Γ_B1 = 40.5 meV, as given in Table 1.

Based on the corrections to the half-widths of the L2-L5 and G1 lines, obtained according to Eqs. (13) and (14), half-width values of Γ_B = 43.1 meV and Γ_B1 = 40.9 meV can be deduced. Taking into account that the G2 line also gives a non-zero contribution (approximately 15% relative to G1-line) to the emission intensity of the B band, there is practically full correspondence between the fitted L1-L6, G1and G2 lines and the emission spectra of the В and В1 bands at T = 5 K.

                4.3.3.         Photoluminescence of the InSe QD Ensemble in In₂Se₃ Confinement at T > 5 K:

It is notable that essential growth of free exciton emission line half-width occurs when the sizes of the InSe crystal become smaller than the free path of the polariton in the exciton state L_e-p, for the case of the ideal infinite crystal. To investigate the influence of temperature on widening the exciton lines of absorption/emission in nanocrystals, we separate the half-width using two components, homogeneous and non- homogeneous, as for the case of bulk crystals. As an initial homogeneous component, one can take the homogeneous half-width = 1.3 meV of the absorption exciton line in the bulk InSe crystal. Then, based on the half-width values given in Table 2 for the L3-L5 lines, we can obtain the value of the non-homogeneous component as Γ_Li^inh = 14.9 meV.

Figure 10 shows the experimental dependences of the half-width Γ on temperature for the В and В1 bands (shown with squares and circles), taken from Table 1. The same figure shows curve 1, plotted in accordance with Eqs. (11) and (16) and parameters M_1,6 = 0; N₂ = 0; M_2,3,4,5 = 1; N₁ = 1;  = 14.9 meV;  = 35.0 meV. It is obvious that curve 1 only qualitatively describes the widening of band B with increasing temperature. An additional component providing a contribution to the widening of the B band with temperature can be explained by taking into account the fact that the kinetic energy of excitons h²k²/2M (where M = m_e +m_h), being closed inside a sphere of radius less than the free path of polaritons in the exciton state, is increased; this results in an increase in the frequency of exciton scattering on the walls of a sphere with radius R^QD.

In our case, a 3D exciton gas, the exciton kinetic energy (similarly to that in an ideal gas) is proportional to (3/2) kT. Then, Eq. (16), based on Eqs (11) and (13), can be transformed to a general form: