1. Introduction
In recent years, several two-dimensional layered semiconductors, including graphene, transition metal dichalcogenides (TMDCs), and III–VI semiconductors, have been investigated both experimentally and theoretically due to their unique optical, electronic and mechanical properties [
1,
2,
3,
4,
5]. Among these different layered semiconductors, the indium selenide (InSe) semiconductor consisted of stacked layers of Se-In-In-Se atoms with van der Waals bonding between the quadruplet layers. There are three common polytypes described as
β-,
ε- and
γ-InSe, which differ in the symmetry and structure of the crystal lattice, and display different optical properties [
1,
4]. Boukhvalov et al. and Shubina et al. have shown that, apart from
ε-InSe, which exhibited an indirect bandgap, both
β-InSe and
γ-InSe have a direct bandgap in the near infrared energy range [
1,
2]. Accordingly, only
β and
γ phases of InSe can be developed as an attractive material for versatile optoelectronic devices [
3,
5,
6].
To the best of our knowledge, there are few relevant studies on the temperature dependence of the absorption edge in InSe material [
7,
8]. Kundakçi et al. and B Gürbulak have investigated the optical properties of InSe single crystal by absorption measurement and observed that an Urbach’s tail appeared below the edge of the absorption spectrum.
However, no information about the actual phase of their samples was provided, and hence only a qualitative comparison can be made. Urbach’s tail or below-band edge optical absorption coefficient (
α) can be described by an exponentially varying function with respect to photon energy [
9,
10]. Cody et al. indicated that the absorption tail below the absorption edge most probably originates from the temperature-independent imperfections of the material and static disorder caused by the impurities, dislocations, or stacking faults [
9]. Abay et al. have experimentally and theoretically explained the origin of this tail as exciton–phonon interaction in their material [
10]. Therefore, the investigations on Urbach’s tail can provide important information on the localized states extended into the optical gap region, which are related to the structural and thermal disorder, and the electrons/excitons–phonons interactions in the layered semiconductors.
In this work, we reported the temperature dependence of the absorption edge of
γ-InSe by analyzing the
α derived from photoconductivity (PC) measurements [
11,
12]. Moddel et al. and Loveland et al. have shown that properly normalized PC data can indeed give an accurate measure of
α [
11,
12]. The energy gap estimated from the PC spectra of
γ-InSe as a function of temperatures was analyzed by Varshni’s relationship [
13] and the Bose–Einstein empirical equation [
14]. Additionally, the effective phonon energy was further obtained based on the temperature dependence of the Urbach energy extracted from PC spectra. The detailed discussion of experimental data can provide useful information for the optoelectronic materials and solar cell devices fabricated by chalcogenides-based layered semiconductors.
3. Results and Discussion
The XRD pattern and Raman spectrum of the as-grown bulk InSe crystals were measured at room temperature and are depicted in
Figure 1a,b, respectively. It is known that the five XRD peaks at about 10.51°, 21.31°, 32.20°, 43.96°, and 67.94°, as labelled and shown in
Figure 1a, can be assigned to define the crystal structure of the as-synthesized InSe. Referring to the reported literature [
3,
4,
5,
6], Grimaldi et al. [
4] Aitzhanov et al. [
5] have attributed these peaks to the (0, 0, 2
n) planes for the hexagonal
β- or
ε-polytype, but Wu et al. [
3] and Liu et al. [
6] have attributed these peaks to the (0, 0, 3
n) planes for the
γ-polytype. The present uncertainty on the structural phase of the as-grown InSe sample can be resolved by analyzing the two additional diffraction peaks at about 29.41° and 47.05° (marked by the red arrow in
Figure 1a).
By comparing the JCPDS (Joint Committee on Powder Diffraction Standards) standard database, the two diffraction peaks at about 29.41° and 47.05° are unable to be defined by the standard data file of PDF 34-1431 for
β- or
ε-polytype [
4,
15], but are matched to the standard data file of PDF 71-0447 for
γ-polytype and defined as (104) and (0111) plane, respectively [
6]. The calculated lattice parameters a = b = 4.32 Å and c = 24.96 Å are in good agreement with the JCPDS #71-0447 (a = b = 4 Å and c = 24.96 Å) [
6].
Additionally, Raman spectroscopy can be employed to further clarify the structural phase of the as-synthesized InSe. As shown in
Figure 1b, the Raman spectrum exhibits three significant peaks at 117, 178, and 229 cm
−1, as well as a weak feature at 200 cm
−1. These observed features have been attributed to the
,
,
, and
modes, respectively [
3,
4]. It should be noted that the
mode has been identified as a significant characteristic for the non-centrosymmetric structure of either
ε-polytype or
γ-polytype [
3,
4]. The observation of the
mode signal provides additional evidence to exclude the occurrence of
β-polytype. Therefore, by integrating the data from both XRD and Raman analysis, it can be inferred that the structural phase of the as-prepared bulk InSe sample is
γ-polytype [
6].
Figure 2 shows the normalized PC spectra of
γ-InSe in the energy range ~1.0–1.4 eV at selected temperatures of 15, 100, 200, 260, and 300 K.
It is observed that the lower-temperature PC spectra at 15, 100, and 200 K exhibit an initial decrease in PC intensity in the energy range 1.1–1.2 eV. A minimum is observed in the range 1.2–1.25 eV, which increases to a maximum in the range 1.25–1.3 eV, and then finally decreases monotonically with any further increase in photon energy within the investigated energy range. The higher-temperature PC spectra at 260 and 300 K show an almost constant below-band edge PC intensity and increase to a maximum as the photon energy sweeps across the band edge, and then decrease with any further increase in the photon energy. It is noted that the existence of impurity or defect states may play an important role in the PC spectra at low temperature, leading to a larger PC intensity in the vicinity of 1.1 eV. Thermal ionization of the defect states is evident in the observed PC spectra at higher temperature. The temperature dependence of the change in PC intensity can provide information about the fundamental absorption properties of
γ-InSe. As stated earlier, correlation of our PC intensity measurements to the relevant absorption coefficient can be found in references [
11,
12] and will not be elaborated on here. From here on, we use the term absorption coefficient (
α) freely in place of normalized PC intensity. It is known that
γ-InSe is a direct bandgap material [
1,
3]. Quereda et al. and Patilhave et al. have used the PC spectroscopy for extraction and estimation of bandgap by using Tauc-relation [
16,
17], which is given as follows:
where
hv is the photon energy,
is the absorption coefficient, A is a constant, and
Eg is the bandgap for the
γ-InSe sample. The optical bandgap of the
γ-InSe at 15 and 300 K can be extracted by extrapolating the straight line of the plot (
αhv)
2 versus energy (
hv) to the energy axis, as shown in
Figure 3a,b. In this way, the bandgap of
γ-InSe at 15 and 300 K is estimated to be about 1.220 ± 0.002 and 1.152 ± 0.003 eV, respectively. It is observed that the obtained bandgap value is less than the reported results [
1,
3], which may be caused by the band tailing effect frequently observed in absorption measurement methods used to estimate bandgap energy [
16].
The extracted bandgaps of
γ-InSe at different temperatures are displayed as an open circle in
Figure 4. The temperature-dependent
Eg of
γ-InSe are further analyzed by the empirical Varshni’s relationship and Bose–Einstein empirical equation. The Varshni empirical relationship is written as follows [
13]:
where
Eg (0) is the transition energy at 0 K and
a and
β are constants referred to as Varshni coefficients. The constant
is related to the electron/exciton–phonon interaction and
is approximately equal to the Debye temperature (Θ
D) [
18].
The Bose–Einstein empirical equation is written as shown in [
14]:
where
represents the strength of the electron/exciton–phonon interaction and Θ corresponds to the average phonon temperature. The theoretical curve, shown as a red solid line in
Figure 4a,b, from the empirical Varshni’s relationship and Bose–Einstein empirical equation, respectively, can accurately describe the variations in the temperature-dependent
Eg of
γ-InSe. The parameters fitted to the empirical Varshni’s relationship and Bose–Einstein empirical equation are obtained and listed in
Table 1. The obtained
is about 279 K for Varshni’s equation and the average phonon temperature Θ is about 205 K for the Bose–Einstein relation. It is known that the Θ is closely related to the Debye temperature (Θ
D) by the relation
[
18]. Hence, the estimated Debye temperature in this work is about 273 K, which is in close agreement with the fitted parameter
. The good agreement indicates consistency of our fitted parameters using both Varshni’s and Bose–Einstein’s empirical equation.
Additionally, all PC spectra of the bulk
γ-InSe sample at different temperatures shown in
Figure 2 displayed a significant absorption tail below the absorption coefficient, which is known as Urbach’s tail. Numerous theoretical and experimental studies about the origin of this tail have been presented. Zhu et al. have indicated that the width of the absorption tail is related to localized states in the optical gap of material, which are associated with structural and thermal disorder or a low-quality crystalline structure [
19]. Abay et al. have attributed the absorption tail to the electron–exciton interaction with phonons in the layered semiconductors [
10]. To understand the effects of defect states or electronic/exciton–phonon interaction in the bulk
γ-InSe sample, the absorption coefficient near the absorption edge is further investigated by the Urbach’s tail model as an exponential relation form, as shown in [
19,
20]:
where
is a constant, and
is the Urbach energy that has the dimensions of energy to describe the distribution of tail states extending into the forbidden gap. As shown in
Figure 5, the Urbach’s tail in the absorption edge for the
γ-InSe at 15 K and 300 K can be well described by the red solid line calculated using Equation (4). According to the fits, the Urbach energy
as a function of temperature is extracted and plotted in
Figure 6a.
Furthermore, the Urbach energy can be expressed by an inverse logarithmic slope relation
, where
is the Boltzmann constant and the
is called the steepness parameter [
19,
20].
Figure 6b shows the temperature-dependent steepness parameter, which is derived from the Urbach energy as plotted in
Figure 6a. The relationship between the steepness parameter
, temperature, and phonon energy is discussed in detail by [
19,
20] and is written as follows:
where
is a temperature-independent but material-dependent constant. The
is attributed to the effective phonon energy in the single-oscillator model describing certain information about the interaction of electrons and/or excitons with phonons in semiconductors. The red solid line displayed in
Figure 6b represents the best fit to the derived
data and the estimated value of
is about 52 meV, which is the effective average energy of phonons contributing to the Urbach tails. It has to be noted that the obtained value of
is almost double that of the actual phonon energy of about 28 meV, which is calculated using the observed highest optical mode (
at 229 cm
−1) in the Raman spectrum shown in
Figure 1b. The larger
indicates that in addition to the electron/exciton–phonon interaction, extra effects also contribute significantly to the effective phonon energy. Similar results have been investigated in previous reports [
10,
20,
21]. Abay et al. have attributed the origin in the higher value of
observed in mixed chalcogenides to the structural disorders caused by cation–cation disorder, cation vacancies, and interstitials [
10]. Rincón et al. and Hwang et al. indicated that larger
values could result from enhanced electronic distortion originating from the ordered defects and structural disorder caused by compositional deviation from the ideal stoichiometry [
20,
21].
The contributions from both thermal-induced and structural disorders for non-ideal single crystals should be incorporated into the consideration of the increased
[
21,
22]. The temperature dependence of
can be described by an Einstein oscillator model, which take into account contributions of dynamic thermal and static structural disorders. The relation is expressed as [
10,
23]:
where
A and
B are adjustable parameters related to the modification of thermal phonon distribution and structural disorders, respectively, and
is the Einstein temperature [
18]. Pejova et al. further indicated that the first term in Equation (6) represents the contribution of electron/exciton–phonon interaction, and the second term is due to the mean-square deviation of atomic positions from a perfectly ordered lattice, caused by the structural disorder [
23]. It is noted that structural disorder is independent of temperature, and the structural disorders would form due to the existence of multiple structural defects associated with the two-dimensional dislocation or complex stacking arrangements of the constituent elements in the layered lattice for the layered materials [
24]. As shown in
Figure 6a, the red solid line represented by Equation (6) can accurately describe the temperature dependence of
. The fitted values of
A,
B, and
are 17.7 meV, 18.6 meV and 198 ± 10 K, respectively. By computing with the fitted value of
A, the contribution of the thermally induced disorder at low temperature of 15 K shows a negligibly small thermal disorder. The room temperature thermal disorder is evaluated to be about 18.9 meV and is slightly larger than that of the structural disorder. Our analysis indicates that the contribution of
in the InSe-layered materials is dominated by the structural disorder at low temperature and thermal disorder becomes significant as the temperature increases.