The Franz–Keldysh effect in the optical absorption spectrum of a TlGaSe2 layered semiconductor caused by charged native defects

The Franz–Keldysh effect in the optical absorption edge of a bulk TlGaSe2 layered semiconductor poled under an external electric field was investigated in the present work. The Franz–Keldysh shift below the optical bandgap absorption region, as well as the quasi-periodic oscillations above the fundamental bandgap of TlGaSe2, were observed. The measured changes in optical light absorption of the TlGaSe2 sample were revealed after poling processing. The poling technique is used to produce the built-in internal electric field within the TlGaSe2 semiconductor. The frozen-in internal electric field in TlGaSe2 was experimentally monitored through changes in the lineshape of the absorption spectra at the fundamental band edge. The observed results are accurately fitted with the theoretical lineshape function of the Franz–Keldysh absorption tail below the bandgap of TlGaSe2 and quasi-periodic oscillations above the bandgap. A good agreement between the theoretical and experimental results was observed. The present study demonstrated that the Franz–Keldysh effect can be used to identify and characterize the localized internal electric fields originating from electrically active native imperfections in the TlGaSe2 crystals.


Introduction
Layered semiconducting materials of thallium-gallium diselenide (TlGaSe 2 ) belong to a group of atomically twodimensional materials with the common chemical formula T1 + ( M 3+ X 2− 2 ) − , where M is a metal atom from column III of the periodic table (Ga, In) and X is the chalcogenide atom * Author to whom any correspondence should be addressed.
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The overall lattice structure of T1 + ( M 3+ X 2− 2 ) − layered semiconductors is actually also identical [1,[8][9][10][11].The twodimensional anionic framework of all compounds can be builtup from corner-linked M 4 X 10 molecular units (polyhedron) that in turn are comprised of four elementary MX 4 tetrahedra linked together through edge-sharing chalcogen atoms.For example, the fundamental molecular unit of TlGaSe 2 is a Ga 4 Se 10 polyhedron.The layers in the ab-atomic plane are formed by the bonding between edge-sharing polyhedrons of Ga 4 Se 10 .The crystallographic atomic arrangement of TlGaSe 2 consists of tetragonal anionic layers stacked alternately along the c-axis.The chemical bonding character between atoms within the layer is not purely covalent but also includes a significant ionic type part.Two adjacent layers are bonded together by weak van der Waals forces.The layer structure of the TlGaSe 2 single crystal is constructed from two types of alternating layers, each of which is rotated by a right angle with respect to the previous one.As a result, trigonal prismatic voids occur and Tl + ions fill these voids.Two types of nearly planar atomic chains along the [110] and [1 10] crystallographic lines are built from monovalent Tl + ions.Se − 2 ions surround each monovalent Tl + ion.The TlGaSe 2 monolayer exhibits the tetragonal point group symmetry of D 2d .Thus, the room-temperature crystalline structure of bulk TlGaSe 2 belongs to the centrosymmetric monoclinic C 6 2h space group with the lattice parameters: a = 10.772Å, b = 10.771Å, c = 15.63Å, β = 100 • , where β is a monoclinic angle.Each monoclinic unit cell includes Z = 16 formula units and 64 atoms [1,[8][9][10][11].
Intrinsic structural imperfections are another common feature of all T1 + ( M 3+ X 2− 2 ) − layered dichalcogenides [12][13][14][15][16][17][18][19][20][21][22][23][24][25].The common intrinsic defect in the Tl-based layered material family is probably chalcogen vacancy-related defects, which may be formed during the sample growth due to their relatively low formation energy (very fast evaporation) [26].The identification of and an understanding of the type and role of intrinsic defects on the electronic properties of T1 + ( M 3+ X 2− 2 ) − , as well as their control, are technologically important in a wide range of applications.Six dominant intrinsic deep carrier traps are commonly observed in all undoped TlGaSe 2 samples grown using the Bridgman-Stockbarger method [13,14,16,[18][19][20][21].Previously, the photo-induced current transient spectroscopy (PICTS) technique coupled with a thermally stimulated current (TSC) method was used to identify deep-level traps in the high-resistivity (semi-insulating) TlGaSe 2 [16,[19][20][21].The aforementioned intrinsic deep-level traps were labeled as A1, A2, A3, A4, A5, and A6 [13,14,23].It is revealed that these native deep impurities inside the TlGaSe 2 sample are crucial for understanding the different ambiguous transport properties of the material [15-17, 22, 24-27] Chalcogenide vacancies are the most preferable intrinsic defects in Tl-based van der Waals dichalcogenides.In the undoped (pristine) TlGaSe 2 layered compound, Se vacancies are energetically favorable to the other kinds of native defects due to their relatively low formation energy (very fast evaporation) [28,29].These Se vacancies are unavoidably formed during the single-crystal preparation using a vertical Bridgman-Stockbarger method.The development of a technology to control Se chalcogen vacancies during the growth of TlGaSe 2 crystals is a crucial task, because Se vacancies essentially affect the electronic and optical properties of this semiconductor.Unfortunately, a detailed study of the growth conditions of TlGaSe 2 in relation to the control of Se-vacancy density during the growth process is lacking in the literature.Accordingly, details of the synthesis process (i.e.temperature, precursors, etc), which defines the type and concentration of native defects in TlGaSe 2 layered crystals, have been unknown until now.
The studies of native defects in pristine Tl-based ternary dichalcogenides, such as TlGaSe 2 layered semiconductors, have been started only recently [12,23,25,30].According to the literature, the TlGaSe 2 layered crystal (as well as other low-dimensional transition-metal and Tl-based metal dichalcogenides) is always a p-type semiconductor with hole conduction behavior.A great deal of effort has been made to achieve reliable n-type conductivity in this material.Attempts to perform the conversion from p-type to n-type conduction via doping techniques do not alter its conduction type.Even the addition of metal impurities at large concentrations cannot switch the conduction type of TlGaSe 2 chalcogenides from p-type to n-type.It is believed that native Se vacancies are responsible for the p-type conductivity of pristine TlGaSe 2 samples.Indeed, the Se (V 2+ Se ) vacancies within the TlGaSe 2 compound always act as efficient traps (but not as dopants or recombination centers) for electrons, reducing their concentration and lifetime.As a result, Se-vacancy defective TlGaSe 2 bulk crystals always show p-type behavior.Thus, the unintentional p-type conductivity of TlGaSe 2 is presumably associated with the presence of intrinsic deep-level traps localized in the bandgap above the valence band maximum that have originated mainly from Se vacancies.
It is well known that undoped TlGaSe 2 is a partially self-charged compensated semiconductor with high resistivity and has low carrier concentration at room temperature [31].Therefore, the electrons on the donor levels that are partially compensating a number of holes on the acceptor levels must also be present in the TlGaSe 2 .The existence of such native donor levels directly below the conduction band minimum has been experimentally confirmed by our previous work in the pristine TlGaSe 2 sample [31].The energy levels of these donors are found to be quite shallow; so they cannot lead to appreciable scattering or trapping of the electrons [31].Presumably, Se antisite or interstitial defects could create the lower-energy shallow donor-like defects in the electronic bandgap of TlGaSe 2 .Vacancies at (Ga or Tl) cation sites could promote the formation of acceptor-like energy levels with deep defect activation energies.It is likely that the dominant acceptor levels in TlGaSe 2 have nearly the same energy levels inside the electronic bandgap as the deep-level trap centers.
Memristive (resistive-switching) memory effects have been recently observed by authors in nominally undoped bulk TlGaSe 2 samples [32,33].Diode-type nonlinear currentvoltage characteristics in metal-TlGaSe 2 -metal structures have been revealed.In these structures, the rectification direction depends only on the polarity of previously applied bias to the electrodes and is independent of the electrode materials.This phenomenon has been understood within the framework of electromigration of charged entities (native Se vacancies or Se ions) leading to the formation of a thin insulator layer near the biased TlGaSe 2 sample surface [32,33].
In TlGaSe 2 , under the external perturbations, the transformation of the Urbach tail from the thermal disorder (or temperature-independent contribution to the Urbach tail, which is typical of crystalline semiconductors) type to that related to the structural disorder (or temperature-dependent contribution to the Urbach tail, which is typical of amorphous semiconductors and non-crystalline solids) has been found [34].
Finally, the Staebler-Wronski effect in the nominally undoped TlGaSe 2 layered semiconductor, related to a significant drop in conductivity of the sample after prolonged illumination by intense white light, has been revealed in [17].Thus, it may be assumed that the presence of a rather strong built-in internal bias field in this crystal, generated by the native defects (vacancies) ionized under external perturbations, is most probably the origin of structural disorder responsible for the specific Urbach tail and Staebler-Wronski effect.
In this paper, modifications in the optical absorption band edge of an ∼200 µm thick TlGaSe 2 sample induced by the Franz-Keldysh effect were investigated at different temperatures from ∼80 K to ∼300 K.A typical manifestation of the Franz-Keldysh effect near the band-edge optical absorption region of semiconductor materials due to the tilt of the energy band edges, which is usually observed under the influence of an applied static electric field, was experimentally monitored in TlGaSe 2 layered crystal after its electrical poling.The existence of an internal built-in electric field in poled TlGaSe 2 generated by ionized deep impurity centers was confirmed from the fundamental optical absorption behavior.The absorption tail below the fundamental edge and quasi-periodic oscillations above the bandgap of TlGaSe 2 as a function of the incident light wavelength were computed in terms of the Franz-Keldysh mechanism.The Franz-Keldysh theoretical model applied to the absorption edge of TlGaSe 2 gave results that were in very good agreement with the experimental values.The quantitative comparison of measured and numerically simulated data allowed the estimation of an average internal field within TlGaSe 2 due to the carrier trapping phenomena.
Our findings open the way to the use of TlGaSe 2 layered crystal as a promising material for electro-absorption devices.The energy band edge shifts observed in pristine TlGaSe 2 crystal under an electric field, either internally built-in or externally applied, may find applications as electro-optical switches, as required for practical realization of optical transistors [35,36], as well as in a wide array of applications, including light-harvesting and light-emitting optoelectronic devices.

Experimental section
The starting Tl, Ga, and Se chemical elements were purchased from Alfa Aesar China (Tianjin) Co. Ltd, with a purity grade not less than 4 N (∼99.99%).All chemicals were used without further purification.The raw materials were taken for the polycrystalline TlGaSe 2 material.The synthesis process was carried out in a closed conical quartz ampule.Initially, the quartz ampule (tube), with an inner diameter of ∼10 mm and size of ∼100 mm, was sequentially cleaned chemically with a mixture of concentrated nitric and hydrochloric acids with a 1 × 1 molar ratio for 24 h.Then, the ampule was washed with acetone and ethanol for a similar period of time.Finally, the ampule was rinsed thoroughly with double-distilled water and gently dried.
The quartz tube was loaded with Tl and Ga shots, together with the appropriate amount of Se powder to match the stoichiometry.The polycrystalline ingot of TlGaSe 2 was synthesized by direct reaction of the constituent elements in the ampule evacuated to a pressure of ∼10 −5 Torr.The sealed ampule was smoothly heated up to ∼1100 K (this temperature is slightly above the melting point of the TlGaSe 2 compound) in a horizontal furnace and dwelt at this temperature for about 2 days for full mixing of the melting elements.Then, the molten load was slowly furnace-cooled to ∼750 K and finally removed from the furnace and then cooled in air.
A single crystal of TlGaSe 2 was grown using the modified Bridgman-Stockbarger method.The polycrystalline ingot of TlGaSe 2 was firstly grounded in an agate mortar into fine powders.Then, powders were introduced directly into the quartz container using a spatula.The ampule for TlGaSe 2 crystal growth was specially designed from an ∼10 mm in diameter and ∼100 mm in length quartz tube featuring a small coneshaped tip at its bottom.The ampule was coated on the inside with a pyrolytic graphite layer to prevent ingot adherence to the inner wall of the quartz container after growth.After this, the quartz ampule was evacuated and sealed at ∼2 × 10 −6 Torr pressure.
TlGaSe 2 single crystals were grown using a Bridgman-Stockbarger technique from the melt in a vertical resistance furnace with an axial temperature gradient.A Bridgman twozone furnace consisted of regions both above and below the melting point of TlGaSe 2 (∼1053 K).The temperature gradient at the growth interface between the upper and lower zones was automatically controlled and kept at ∼15 K cm −1 .The growth ampule was tightly fitted in the ampule holder and fixed to the upper part of a Bridgman two-zone furnace by a puller.To minimize the risk of the ampule explosion due to a high Se vapor pressure at high temperature, the furnace was heated from room temperature to ∼500 K in 12 h and from ∼500 K to the melting point of the TlGaSe 2 compound in ∼30 h and held at ∼1100 K for ∼20 h until the TlGaSe 2 polycrystalline compound was fully melted.The growth process was carried out by lowering the ampule through the temperature gradient zone at a rate of ∼5 mm d −1 with stepper translation motion.The TlGaSe 2 single crystal was grown by self-nucleation.The initial nucleation was started inside the tip at the bottom when it passed through the temperature-gradient zone, and the TlGaSe 2 crystal continued to grow until the whole ampule was moved into the low-temperature zone.After completing the crystal growth process, the ampule was allowed to slowly cool to room temperature at a rate of ∼5 K h −1 .The as-grown TlGaSe 2 single crystal was carefully removed from the growth ampule by breaking it.Two TlGaSe 2 ingots were produced, each having a weight of ∼10-12 g, a diameter of ∼6-8 mm, and a height of ∼8-10 mm.
TlGaSe 2 specimens for the measurements were prepared from the middle of ingot heights.The center of each grown ingot was gently cleaved along the cleavage plane (001) into ∼200 µm thick wafers using a razor blade.Several TlGaSe 2 samples had natural mirror-like cleavage faces.No further polishing and cleaning treatments were required for freshly cleaved samples because the mirror-like cleavage surface planes exhibited excellent optical quality and had no cracks or voids.
Two freshly cleaved platelets parallel to the layers with ∼5 × 5 mm 2 surface area and ∼200 mm thickness were prepared from different TlGaSe 2 ingots for optical absorption measurements.The TlGaSe 2 sample was mounted on a vertical Au-plated copper holder attached to a cold finger of a Janis closed-cycle helium optical cryostat with quartz windows.The sample holder was specially designed for optical transmittance measurements.Thermal contact between the sample and the cryostat holder was made by vacuum grease.The temperature was operated with a Lake Shore-340 temperature controller.A heater was installed near the sample.The temperature was precisely measured with a DT-470 silicon diode thermosensor mounted on the cold finger.The cryostat makes it possible to stabilize the temperature within ±10 mK temperature fluctuations.
A UV-visible Triax-550 type single-grating monochromator with ∼0.2 Å spectral resolution was employed to record the optical transmission spectra at different temperatures ranging from ∼20 K to ∼300 K with ∼10 K steps.Each spectrum was measured over the wavelength range between ∼400 and ∼900 nm.Optical transmittance spectra were measured in the direction perpendicular to the layer planes during the heating of the sample.A commercial 300 W Thermo-Oriel highpressure xenon arc lamp operated continuously was used as the radiation source.The white beam from the spectral lamp was focused onto the monochromator entrance port.The slits were fixed in position ∼0.5 mm.The dispersed monochromatic light with a small spot size (smaller than the size of the pinhole ∼1 mm in diameter) was directly focused onto the sample surface using optical elements.We checked that the light beam did not practically heat the sample by comparing the spectra measured at a given temperature with variable light intensity.Here, we note only that in our experimental setup, the incident light passed through the sample and traveled through the two opposite windows of the cryostat chamber enclosed by quartz glasses between which the sample was held.The intensity of the monochromatic incident light and the light transmitted through the specimen were measured by a Hamamatsu R-1527 photomultiplier tube.The photomultiplier tube was positioned to detect more scattered transmission rays.The experimental setup was fully computer controlled.
The optical absorption coefficient of TlGaSe 2 was evaluated from the measured transmission spectra.To evaluate the effects from the cryostat windows and other optical components, the transmittance spectra of the quartz windows were also recorded at corresponding temperatures.The spectral dependence of the optical absorption coefficient (α (hν)) of TlGaSe 2 at each wavelength can be calculated from experimental transmittance according to the following relationship [37,38]: where T is the measured transmittance, d ∼ 200 µm is the sample thickness, and R is the reflectivity.For TlGaSe 2 at nearly normal light incidence the reflection coefficient can be considered as constant (R does not change more than ∼2%) over the spectral range of our interest.Moreover, the value of the thermal expansion coefficient of TlGaSe 2 is quite small (∼10 −6 K −1 ) and, therefore, the temperature dependence of the sample thickness can be neglected in the analysis.At the same time, our ellipsometry measurement of the reflection spectrum of TlGaSe 2 showed that the absolute magnitude of the reflectivity is negligibly small throughout the whole measured spectral region.Thus, the absorption coefficient of the TlGaSe 2 specimen at each fixed temperature can be obtained from the measured transmittance using the Beer-Lambert formula [37,38]: where I 0 (hν) and I (hν) are the intensities of the incident radiation and transmitted light passing through the specimen at each wavelength, respectively.
To investigate the optical absorption properties of the previously poled TlGaSe 2 , electrical contacts to the sample were established in a sandwich configuration perpendicular to the crystal planes.Due to the difficulty of attaining mirror-like surfaces through mechanical cutting and optical quality polishing processes, establishing metallic contacts on the lateral surfaces of the TlGaSe 2 sample was not preferred in this investigation.For the poling purposes, a large-area rectangular-shaped Au layer with ∼200 nm thickness was first deposited as the bottom electrode in vacuum at ∼4 × 10 −6 Torr on the freshly cleaved surface of the sample through a stainless steel mask.Then, a semi-transparent circular Au electrode of diameter about ∼3 mm and a transmissivity of ∼80% in an ∼400-900 nm spectral window was formed by evaporation onto the front mirror surface of the sample through another shadow mask.As the metal electrode, Au was preferred since the work function of Au is greater than the electron affinity of TlGaSe 2 ; hence, an ohmic contact arises.Additionally, Au does not form an oxidizing layer.Thin copper wires were soldered to the electrodes by a high-purity silver paste drop for circuit connection.The sample was then mounted on the sample holder, which consisted of two electrically isolated pins.
Electric poling was performed during sample cooling in darkness using a Keithley 6517A electrometer and ∼100 KΩ series resistance placed in series with the thin film capacitor, which limited the maximum dc current in the poling process.A fixed bias electric voltage of ∼15 V was applied at room temperature and switched off at predefined low temperatures.Poling aims to obtain an internal electric field inside the semiinsulating TlGaSe 2 material, which may arise from different ionized deep-level defect levels.
For TSC and dark conductivity measurements, another TlGaSe 2 sample cleaved from the same ingot was used.For electrical contacts, the gap geometry configuration was applied.In this in-plane configuration, the electrical resistivity of the TlGaSe 2 layered semiconductor is at least two orders of magnitude smaller than the corresponding value measured perpendicular to the layer direction.This is why we preferred this electric field direction during TSC measurements.
Two electrodes were thermally deposited onto the top mirror surface of the sample, leaving a sensitive distance of about ∼3 mm between them.The electrodes were formed at two opposite edges of the front surface of the sample by evaporating Au in a ∼10 −6 Torr vacuum chamber.Thin copper wires of diameter ∼10 µm were attached to the electrodes for electrical circuit connection.These contacts were also found to be ohmic.
The dark conductivity of the TlGaSe 2 sample was determined by applying a constant voltage of ∼30 V to the electrodes from a computer-controlled Keithley 6517A electrometer and measuring the corresponding current with a Keithley 6485 digital picoammeter.The electrical conductivity data were collected during the heating process by a proportional -integralderivative (PID) temperature controller, providing a constant heating rate of ∼5 K min −1 .
The experimental procedure to perform TSC measurements consists of two states.Firstly, the sample was cooled in the dark from room temperature to T 0 ∼ 80 K and kept at this temperature for about 30 min.Next, the sample was illuminated for ∼10 min by a 100 W halogen lamp.Then, the light excitation was switched off, and the sample was left in the dark for approximately 5 min before heating.A dc voltage of ∼7 V was applied to the sample and the latter was heated in the dark until ∼300 K with a linear rate of β = dT/dt ∼ 7 m s −1 and monitored by a PID temperature controller.Thus, the temperature as a function of time was increased as T = T 0 + βt, in which t is given in seconds.TSC spectra were recorded with a Keithley 6517 programmable electrometer.The graphical language LabVIEW was installed on a computer, which was employed to record and control the output signals from the temperature controller and the electrometer continuously via General Purpose Interface Bus (GPIB) connection.The measured TSC data, i.e. the difference between dark current and TSC scans, was afterward analyzed using appropriate software.
Scanning electron microscopy (SEM, Philips XL 30 SFEG) was used to produce images that contain information about the sample's surface topography.The SEM images were acquired via a JEOL camera.SEM micrographs were obtained using a vacuum SE detector, where the electron acceleration voltage of the incident beam was ∼20 KV and the sample was kept typically at ∼10 −5 Torr inside the SEM chamber.The chemical compositional characterizations of the TlGaSe 2 samples were measured via SEM fitted with an energy dispersive x-ray (EDX) detector.The SEM-EDX measurements were carried out at room temperature.EDX detection was performed using an electron probe of ∼0.8 nm diameter and ∼0.5 nA current.The extracted EDX spectra were quantified using the INCA software package (Oxford Instruments).

Franz-Keldysh effect. Background theory
The Franz-Keldysh optical absorption spectrum in the presence of the applied dc electric field potential (V) is found to be [39][40][41]: where ω is the angular frequency of light, c is the speed of light in vacuum, h is the Plank constant, n is the refractive index, d cv is the interband dipole matrix element, µ is the reduced effective mass for the optical transitions, E g is the semiconductor bandgap, and Ai(ξ) is the Airy function of the ξ = (E g − hω) argument.Prime in equation ( 3) denotes differentiation with respect to the argument.
Using the asymptotic expansion of the Airy function [39,42], the below-gap Franz-Keldysh absorption coefficient (hω ≲ E g ) can be approximated by an exponential function [39,42]: where Usually, the wavelength dependence of n near the absorption edge of conventional semiconductors is negligibly small; therefore, K may be taken as a constant.Thus, the Franz-Keldysh effect must be exhibited as an almost exponential decay tail below the bandgap, where the absorption normally does not occur in the absence of an external electric field.The Franz-Keldysh absorption coefficient as a function of photon energies at the higher energy side near the fundamental bandgap (hω > E g ) is found to be [39,42]: The Franz-Keldysh absorption spectrum above the band edge must be characterized by the periodical oscillations with decreasing period and amplitude when the photon energy increases for the constant applied electric field.Additionally, the period of the Franz-Keldysh oscillations must depend on the electric-field magnitude.We also notice that the true profile of the Franz-Keldysh oscillations should be quasi-periodical because the Airy functions are not strictly periodic [42,43]  The physical mechanism of the Franz-Keldysh effect can be easily understood with the aid of figure 1.Under zerofield conditions, the valence and conduction bands of a perfect semiconducting compound are flat and separated by a constant energy bandgap, E g .However, under a constant electric field, both electronic band edges are tilted due to a spatial gradient electrostatic potential.The vertical distance from the valence band to the conduction band is still E g , but since the band edges are no longer horizontal, the band states can tunnel out into the gap.Moreover, the wave function of Bloch electrons in semiconducting crystals modifies into an Airy function under the influence of the applied electric field.Consequently, optical transitions between valence and conduction bands are now possible for energies smaller than E g , which are interpreted as photon-assisted tunneling through the forbidden band region.This tunneling effect is described by the exponential-like asymptotic function, equation ( 4), and represents a red-shift of the absorption edge, which is a manifestation of the Franz-Keldysh effect below the bandgap (hω ⩽ E g ).Quasi-periodic optical absorption oscillations above the energy bandgap (hω > E g ) are the resonance effects, whenever the distance the electron travels equals an integer number of de Broglie wavelength [44].Theoretically, this manifestation of the Franz-Keldysh effect can be described by taking into account equation (5).

SEM images and composition characterization
A room-temperature SEM micrograph of the freshly cleaved basal surface and a sideward view of the pristine TlGaSe 2 sample with thickness of ∼180 µm prepared for optical measurements are shown in figures 2(a) and (b), respectively.The SEM imaging presented in figure 2(a) emphasizes that the scanned area is a highly smooth and homogeneous surface without topographic contrast, islands, grains, surface textures, and sharp edges.The single-crystalline morphology of the TlGaSe 2 sample can be clearly observed in figure 2(a).The SEM image displayed in figure 2(b) demonstrates that the sideward profile of the described specimen is characterized by a typical layered morphology, which clearly evidences the layered crystalline nature of the TlGaSe 2 compound.
To determine the chemical composition of the studied sample, EDX microanalysis was performed at room temperature.The EDX signal distributions corresponding to various elements in a sample are displayed in figure 2(d).The EDX data were collected from at least four independent areas of identical dimensions on the sample's surface, imaged in figure 2(a), to ensure the reproducibility of the recorded results.The results of EDX analysis revealed an acceptable correlation between major Tl, Ga, and Se elements with average atomic ratios 24.76:24.63:48.93 at % that closely match the stoichiometric Tl/Ga/2Se proportion.As displayed in figure 2(d), the presence of a small number of oxygen atoms on the surface of the sample was unambiguously found from the EDX elemental mapping studies with a total contaminant of less than ∼1.7 at %. Thus, the EDX and SEM results prove that the TlGaSe 2 sample is chemically and morphologically homogeneous.

DC dark-current measurement
The temperature dependence of the dc dark current (i dark ), measured for the TlGaSe 2 sample in the range of ∼10 K-300 K in the heating process, is plotted in figure 3. The thickness of the sample is of the order of 400 µm.The applied voltage is ∼30 V.The temperature dependence of the dc dark current was investigated to identify the average activation energy of the dominant defect levels within the bandgap of the TlGaSe 2 sample to control the electrical transport over the aforementioned range of temperatures.Figure 3 shows an exponential increase in dark current with increasing temperature in the range ∼150 K-300 K, which is attributed to the increase in the total carrier density.According to [45], the dark current depends on the temperature according to the relation: where i 0 is the pre-exponential factor, E a is the thermal activation energy in a given temperature range (denoted by the letter a), k B is the Boltzmann constant, and T is the absolute temperature.The value of the thermal activation energy was extracted from the slopes of the Arrhenius plots of ln(i dark ) versus 1/k B T, as presented in the inset of figure 3. The slope of the fitted straight line in the temperature region of ∼178 K-300 K is found to be approximately ∼0. 25 eV.The calculated thermal activation energy for the measured sample should be attributed to a deep-level acceptor defect (room-temperature bandgap of TlGaSe 2 is ∼2.1 eV).It should be noted here that TlGaSe 2 is a p-type material.This means that the revealed deep defect level with ionization energy of ∼0.25 eV could conceivably be located somewhere within the forbidden gap of the sample above the valence band maximum, and should be attributed to native defects and/or background impurities in the pristine TlGaSe 2 single crystal.

TSC measurements
To identify and characterize the deep-level trapping centers in the TlGaSe 2 sample, TSC measurement was carried out over the temperature range of ∼20 K-300 K.The experimental TSC spectrum of TlGaSe 2 is shown in figure 4. For the TSC experiments, the sample was first cooled to ∼20 K and illuminated through the area between the electrodes by a 300 W white light from a xenon lamp and focusing optics for bias light illumination.The illumination time of illumination of ∼10 min was found to be sufficient to fill the native deep-level traps in TlGaSe 2 .The Au electrodes were made on the front side of the sample (in the same manner as for a dark-current measurement) to measure the TSC.Two ∼0.5 mm diameter Cu wires were attached to the Au contacts by silver paste (used as a glue metal) for the electrical connection of the electrodes to the picoammeter.After keeping the sample in the dark at the photoexcitation temperature for about 20 min, it was then heated at a constant rate of β = 7 K min −1 while the current through the TlGaSe 2 was recorded under a driving voltage of ∼7 V.It should be noted that well-separated TSC peaks were revealed only in the temperature range of ∼90 K-150 K in the TlGaSe 2 sample irradiated by light.Therefore, it can be concluded that the TSC peaks presented in figure 4 are due to photoexcited free carriers captured by the intrinsic defects in the pristine TlGaSe 2 single crystal.
As seen in figure 4, the TSC (I TSC ) curve of TlGaSe 2 exhibited several peaks.Three maxima at ∼98.5 K, ∼119.5 K, and ∼129 K, and a shoulder at ∼140 K emerged.To evaluate the activation energy of traps from the obtained experimental TSC spectra, a curve fitting method based on the slow retrapping model was introduced [20,21,[46][47][48][49][50][51][52][53][54][55].The following equation is used to describe the complete lineshape of i TSC as a function of temperature [46][47][48][49][50][51][52][53][54][55]: where A ∼ 2 mm 2 is the electrode's effective area, V = 7 V is the applied dc bias voltage, and d ∼ 3 mm is the distance between the electrodes.In equation ( 7), i γ TSC is the current contribution from the γth TSC peak, which can be calculated from the equation given as follows [20,21,[46][47][48][49][50][51][52][53][54][55]: Here, ν γ is the attempt-to-escape frequency of a trapped electron, n t0 is the initial density of filled traps, τ is the lifetime of free carriers, e is the elementary charge, µ is the carrier's mobility, E mγ is the energy depth (or energy difference between the transport energy level and the trap energy) of the γth trap, β is the constant heating rate (T = T 0 + βt), t is the heating time, and T 0 = 20 K is the temperature at which the trap-filling procedure was carried out.A relationship connects the temperature T and time t: dT = βdt.The attempt-to-escape frequency is related to the capture cross section of the γth trap (S tγ ) via the expression [20,21,[46][47][48][49][50][51][52][53][54][55]: where is the effective density of transporting states, m * is the charge carrier (hole) effective mass value estimated as ∼0.14 m e (m e is the electron rest mass) [47,53], and υ th = √ 3k B T/m * is the thermal velocity of the charge carrier (hole).The TSC spectrum is also helpful in clarifying the total charge (Q γ ) released (or accumulated under photoexcitation) by an individual trap.The number of released carriers can be revealed as the area under the γth TSC peak.Meanwhile, Q γ is related to the concentration (density) of the corresponding trap center (N tγ ) by the equation [46,[49][50][51][52]: where e is the elementary charge, Φ is the surface area where the incident light falls on the sample surface, L is the specimen thickness, and G is the photoconductivity gain.Figure 4 displays the experimental TSC curve (open circles) and numerical fits obtained in the present work (solid lines).The TSC spectrum was deconvoluted into seven discrete component peaks labeled from 1 to 7. The black solid line in figure 4 denotes the shape of the best fitted TSC curve obtained due to superposition of various fractions of seven peaks.
Each peak in the TSC spectrum corresponds to a certain defect level in the bandgap of TlGaSe 2 .The temperature T mγ at which the current maximum occurs, the thermal ionization energy (depth) of the corresponding trap, their capture cross sections, and the concentration of each individual trapping center are summarized in table 1.Note that although we were able to accurately define the characteristic defect-related parameters (depth, capture cross section, density, etc) for each intrinsic trapping center in TlGaSe 2 by analyzing the temperature dependence of the TSC spectrum, their type (donor or acceptor) cannot be determined from TSC measurement.

Optical absorption measurements
Optical absorption spectra of the TlGaSe 2 sample as a function of the incident photon energy (hν) measured at ∼80 K are plotted in figure 5. Since the electric current peaks in TSC measurements were only observed at temperatures between ∼80 K and 150 K, one would assume that changes in the optical absorption spectrum of TlGaSe 2 , induced by the intrinsic defects charged during the electric field poling, may be revealed in the aforementioned temperature range.We can see that the optical absorption coefficient (α) of the same sample measured immediately after the electric field poling shows a significant change, which could have originated from the formation of the built-in electric field inside the TlGaSe 2 .
The intrinsic deep-level traps in TlGaSe 2 that become charged during the electric field poling process may contribute to the retention of a frozen internal electric field exclusively within the sample [15,17,22].We note that the findings presented in the current study are fully identical to some of our previous transport results on the TlGaSe 2 layered semiconductor that were observed after an applied electric field poling, even at a low dc biases [15,17,22].The latter results were explained by a dc electric field imprinted within the mentioned material [15,17,22].Furthermore, the charge carriers trapped in the native defects or contaminations in the pristine TlGaSe 2 crystal when it was previously cooled under the externally applied poling voltage from high temperatures are presumed to give rise to an electric field frozen within the compound [15,17,22].Here, we demonstrate, for the first time to our knowledge, that the formation of the permanent electric field residing exclusively within the TlGaSe 2 crystal after electric-field poling can be confirmed by the Franz-Keldysh effect.
By analyzing the above optical data, we can obtain information about modifying the fundamental absorption edge of the TlGaSe 2 crystal induced by the electric field poling.Although the undoped TlGaSe 2 semiconductor is a direct bandgap material with a fundamental absorption edge ranging between about E d g ∼ 2.0-2.2eV at ambient temperature and pressure, some researchers have assumed that TlGaSe 2 is an indirect bandgap semiconductor with a bandgap width varying from ∼1.8 to ∼2.0 eV [25,[56][57][58][59].As is well known, phononassisted optical transitions occur in indirect bandgap semiconductor materials [56][57][58][59].In contrast to the indirect energy bandgap model, it has been recently suggested that intrinsic imperfections of the undoped TlGaSe 2 crystal, such as Se vacancies, may introduce the defect-induced electronic levels inside the bandgap of TlGaSe 2 with the absolute energy minimum located away from the center of the Brillouin zone [25].One can conclude that native defects in pristine bulk TlGaSe 2 cause this ternary compound to exhibit indirect-like bandgap properties, whereas a few layers of TlGaSe 2 display direct bandgap characteristics [25].
In the present study, we explore and discuss the contribution of the internal electric field created by the trapped charges The experimental optical absorption (α) spectra of TlGaSe 2 as a function of the incident photon energy (hν) recorded in the transparent regime after cooling of the sample from room temperature to ∼80 K.The black curve was obtained when the sample was cooled from room temperature to ∼80 K at zero applied bias.The red curve was recorded when the sample was previously cooled from room temperature to ∼80 K under the influence of an external electric field.Both spectra were measured at ∼80 K.The bias was kept at the constant poling voltage of 15 V, where the positive poling potential during the cooling process was applied to the top sample surface.The red and black curves were obtained from the same sample in two separate thermal cycles from room temperature down to low temperature.Optical absorption spectra are shifted for the sake of clarity of the poling effect.
on the absorption coefficient of TlGaSe 2 by considering the optically allowed transitions at the direct energy bandgap only.As a consequence, we have analyzed the (αhν) 2 versus photon energy (hν) for the sample with the help of the equation [37][38][39]: where h is Planck's constant, ν is the frequency of the photon, E d g is the bandgap energy for direct allowed transitions, and B is the proportionality constant.The plot (hν × α (hν)) 2 vs hν for the unpoled TlGaSe 2 sample is depicted in figure 6.The best fit of the experimental data is also shown in figure 6 by a blue line.The x-intercept of the linear region of the plot indicates that an optically obtained direct bandgap value of the unpoled TlGaSe 2 sample is found to be E d g ∼ 2.05 eV.The corresponding optical absorption spectrum fitted for the initially poled TlGaSe 2 sample in the energy range of hν ≲ E d g (below the direct bandgap) is presented in figure 7(a).Following equation ( 4), the measured optical absorption coefficient α (hν) for photon energies that is somewhat less than E d g is theoretically modeled by a simplified formula: where Ξ and β are fitting parameters.The analysis of the experimental optical absorption curve measured on the initially poled TlGaSe 2 sample is presented in figure 7.For illustrative purposes, we separately plot the theoretical absorption curves related to the sub-direct bandgap photon absorption spectrum (hν ≲ E d g ) and the above one (hν ≳ E d g ) in figure 7 in the insets (a) and (b), respectively.The optical absorption spectrum of the poled TlGaSe 2 sample for photon energies that is somewhat less than E d g bandgap fitted using equation ( 12) is depicted in the inset (a) by the solid black line.Mathematically, the above expression describes the exponential tail for sub-bandgap light absorption, which is in good agreement with the measured optical data.We believe that the built-in internal electric field created inside the TlGaSe 2 crystal due to the poling procedure gives rise to an exponential absorption tail stated by the Franz-Keldysh effect in the sub-bandgap light absorption region.An imprint electric field could induce tunneling of electrons from the valence band into the conduction one in the poled TlGaSe 2 crystal for photon energies of hν ≲ E d g .By adjusting expression (11) to the experimental optical data, numerical values of β and Ξ fitting parameters were estimated (given in figure 7).
>The inset (b) in figure 7 demonstrates the theoretical absorption curve (blue line) calculated for the poled TlGaSe 2 sample in the photon energy range above the direct bandgap energy, that is hν ≳ E d g .The absorption coefficient lineshape for this spectral range induced by the imprint potential within the TlGaSe 2 sample is calculated using the asymptotic form of the Airy function given by equation (5).The appearance of characteristic quasi-periodic oscillations (the so-called Franz-Keldysh oscillations) in the optical absorption coefficient above the fundamental bandgap associated with an external electric field applied to a semiconductor is analytically predicted by equation ( 5) [39,42].
The open red circles in the inset (b) in figure 7 show the experimentally measured changes in the optical absorption coefficient of the poled TlGaSe 2 sample in the spectral region of hν ≳ E d g , which unambiguously demonstrate oscillating behavior predicted by the Franz-Keldysh theory [39][40][41][42].The fit of the experimental absorption spectrum according to equation (5), shown in the inset (b) by the blue solid line, clearly indicates that the calculated curve is in good agreement with the measured optical data.Importantly, a curve fit of the functions, equations ( 5) and (12), to the measured optical absorption spectrum at the lower and higher energy sides of E d g provided the possibility to estimate the direct bandgap energy value of the initially poled TlGaSe 2 sample.We have found that the initial direct optical bandgap of the TlGaSe 2 crystal is unequivocally decreased from ∼2.05 to ∼2.0 eV due to the built-in potential.Thus, the direct bandgap narrowing on approximately ∼50 meV and resulting in the red-shift of the optical absorption edge of the TlGaSe 2 sample due to electrical poling is a distinctive manifestation of the Franz-Keldysh effect [39][40][41][42].

Discussion
A real crystal structure of the undoped TlGaSe 2 layered semiconductor always contains different types of native or intrinsic defects (contamination), which are inevitably introduced into the TlGaSe 2 host due to an uncontrolled deviation of important synthesis parameters (such as the melt stoichiometry, amount of starting elements, ampule volume/mass of charge ratio, synthesis temperature, vapor pressure, thermodynamics and kinetics of chemical reactions, ampule shape, and heating and cooling rates) from optimized parameters during crystallization processing steps.Although the concentrations of the latter are low enough, they can greatly affect the electronic properties of the studied material.For example, the electrical conductivity of the unintentionally doped bulk TlGaSe 2 crystal is always p-type.The latter has generally been associated with the presence of native imperfections in the lattice structure of the TlGaSe 2 semiconductor.From a review of the literature, there has been no systematic study on appropriate intrinsic defect formation mechanisms within TlGaSe 2 , especially concerning charged native defects.It is demonstrated that the Se vacancies are the most dominant defect type in the pristine TlGaSe 2 crystal [12,25].
Electronically, undoped TlGaSe 2 samples have extremely high resistivity (low carrier density) in the dark, even at room temperature [31,33,[60][61][62].It is presently accepted that the self-compensation of native deep-level acceptor states in TlGaSe 2 with intrinsic shallow donors is responsible for the low carrier density and poor electrical conductivity of this material [31].Based on the experimental data obtained from TSC measurements, we were able to identify the intrinsic defects (marked as 4 and 7 in figure 4 and table 1) with almost coinciding thermal activation energies of ∼0.29 and ∼0.27 eV as deep-level acceptor states controlling the temperature dependence of the dc conductivity (dark current) of the undoped TlGaSe 2 layered semiconductor from ∼10 K to ∼300 K.It is noteworthy that the native deep-level defect, whose thermal activation energy is located in the range of ∼0.19-0.23 eV within the bandgap, has always been detected through PICTS measurements in a series of undoped TlGaSe 2 samples selected from different batches [14,[18][19][20][21][22][23].This native deep-level defect, labeled as A2 in the PICTS experiments [14,[18][19][20][21][22][23], is believed to be the dominant acceptor state of the TlGaSe 2 layered semiconductor, which is very similar to the 4 and 7 intrinsic defects observed in this study.
Intuitively, we suspect that that other groups of intrinsic deep-level defects detected in the present TSC study and labeled as 2, 3, 5, and 6 are the electrically active defect centers of the undoped TlGaSe 2 sample.We would like to highlight that such a conclusion is suggested by comparing experimental optical absorption spectra of the same undoped TlGaSe 2 sample in its unpoled and poled states.The presence of charged defects and space charges is needed to form a built-in internal electric field inside the previously poled TlGaSe 2 .It is very important that the internal bias field in the TlGaSe 2 single crystal sample can result from external electric-field poling only.Furthermore, our results indicated that an electrical poling process must involve sample cooling through the Curie temperature.This means that a process of alignment or reorientation of dipoles related to charged native structural defects in TlGaSe 2 must be realized when the electrostatic field is applied to the sample during external poling.This experimental fact is very critical for understanding the scenario of a built-in electric-field formation in TlGaSe 2 .Keeping this in mind, the nature of the observed intrinsic 2, 3, 5, and 6 deep-level defects within the bandgap of TlGaSe 2 may be examined.
It is immediately apparent from the above results that the charged 2, 3, 5, and 6 deep-level defects cannot contribute to the temperature dependence of charge carrier transport (conductivity) and concentration distributions in the undoped TlGaSe 2 sample.Nevertheless, they may affect the carrier lifetime.We predict that these defect states are deep-level traps in the bandgap of TlGaSe 2 .
Three plausible physical scenarios can be put forward to interpret the occurrence of a built-in internal electric field due to charged defects in TlGaSe 2 single crystal, induced by electric-field poling.The first mechanism is based on the drift (electromigration) of constituent ions (or vacancies) in TlGaSe 2 induced by the prior electric field poling.The ion (vacancy) drift (electromigration) may generate a nearly stable internal electrical field inside the TlGaSe 2 layered crystal, which must screen an applied external poling field.As a result, a quasi-equilibrium electret state with imprinted electrical potential inside the undoped TlGaSe 2 crystal could be achieved after removing an applied poling voltage, especially at low temperatures.
According to the second model, the ions (or vacancies) that have drifted (migrated) under the influence of an external poling field may strongly affect the density of native deeplevel traps and modify the distribution of their energy levels in the TlGaSe 2 bandgap.Therefore, the appearance of a quasi-stable internal bias field that occurs inside the initially poled TlGaSe 2 sample and freezes at low temperatures can be expected.
In contrast to the above two models related to the ion (vacancy) migration during the poling process, the model of defect dipoles alignment under application of the poling electric field giving rise to an experimentally observed internal bias field within the TlGaSe 2 single crystal seems to be more reasonable.We can propose that these native dipole defect complexes are formed from doubly positively charged Se vacancies (marked as V 2+ Se ) and two negatively charged metal vacancy centers labeled as V − Tl and V 3− Ga , respectively.These vacancies are believed to always be present in the as-grown TlGaSe 2 crystals due to stoichiometric deviations of constituent elements from the ideal 1:1:2 atomic ratio of Tl/Ga/Se.The stoichiometric deviation of three majority elements from the standard proportion in the above-discussed TlGaSe 2 samples was supported by EDX microanalysis.As expected, the TlGaSe 2 compound tolerates off-stoichiometric composition due to the high vapor pressure of the Se component and the lower melting temperature of the pure Ga precursor.Additionally, the air-sensitive Tl precursor may lead to problems during syntheses and crystal growth (for example, Tl oxidation, and the chemical reactions between Tl-oxide and other components).By heating an evacuated growth quartz ampule during the modified vertical Bridgman process, the Se and Ga evaporate to the colder parts of the ampule, and a nonreacting portion of the Se and Ga atoms in the colder part of the ampule always leads to a small deficit of Se and Ga ions in the structure of the TlGaSe 2 compound.Thus, the layered structure of this compound is very prone to the formation of cationic and anionic vacancies, which are unavoidable key defects in TlGaSe 2 .
The monoclinic symmetry of pristine TlGaSe 2 is characterized by two crystallographically independent sites for the Se atoms in the unit cell of the compound.The Se vacancies in these symmetrically nonequivalent anionic positions are likely to be native charged deep-level centers characterized by a similar activation energy and capture cross section.As a result, the two peaks revealed by the TSC method in the temperature range ∼120 K-128 K (labeled as 5 and 6 in figure 7) are attributed to different Se vacancies in the undoped TlGaSe 2 sample.The corresponding analysis indicates that these two defect levels have close activation energies of 0.30 and 0.32 eV within the bandgap of TlGaSe 2 and the same order of capture cross section ∼10 −15 cm 2 , respectively.
Among the defects shown in figure 7, peak 2 is probably associated with a Tl vacancy (V − Tl ).The best TSC fit gives a thermal activation energy of ∼0.37 eV with the larger capture cross section value of the order of ∼10 −9 cm 2 .We intuitively suggest that this structural defect in the undoped TlGaSe 2 sample can be attributed to Tl vacancy because the ionic radius of T1 + in the TlGaSe 2 lattice is too large (∼1.15 nm) in comparison with the Ga 3+ ion (∼0.062 nm) [63].Finally, the origin of peak 3 in figure 7 is likely attributed to a Ga vacancy (V 3− Ga ) acting as a triple acceptor type native deep-level defect activated in the temperature ranges of ∼105 K-115 K.More specifically, the V 3− Ga is possibly a charged vacancy-type defect with an energy level of ∼0.32 eV inside the energy bandgap of nominally undoped TlGaSe 2 close to the valence band with a capture cross-section of ∼10 −13 cm 2 .Thus, it can be proposed that the built-in electric field induced in the undoped TlGaSe 2 sample due to external poling is formed by cooperative alignment of native dipolar defect complexes composed from 2V 2+  Se ↔ . The question is: what are the native deep-level defects in the undoped TlGaSe 2 sample that are only activated in the temperature range ∼100 K-140 K? As mentioned previously, the undoped TlGaSe 2 single crystal undergoes two phase transformations at lower temperatures [11].The first one near T i ∼ 120-130 K corresponds to the incommensurate (INC) phase transition from the high-temperature paraelectric phase.The second one at T c ∼ 100-110 K is the phase transition to the ferroelectric phase.The INC-phase transition is accompanied by modulation of the crystal structure of the paraelectric phase with a modulation vector k = (δ; δ; 0.25), where δ ∼ 0.02 is the incommensurability parameter.Upon further cooling, δ linearly decreases from 0.012 to zero at T c , and a commensurate ferroelectric with quadrupling of the unit cell with the modulation period at k = (0; 0; 0.25) takes place.The extremely large dielectric constant value of crystals due to the lack of translational lattice periodicity must be placed at the temperature region inside which the INC phase exists.It should be pointed out that TlGaSe 2 layered crystal exhibits a large dielectric constant value of approximately ε ∼ 1500 inside the temperature interval of ∆T = T i − T c ∼ 20-30 K.Such a medium with a larger dielectric constant (the dielectric environment) can provide an intrinsic driving force for native deep-level defects activation because the activation energies of these defects are extremely sensitive to dielectric screening by the surrounding environment.Our results indicate that the presence of a built-in electric field inside an undoped TlGaSe 2 layered semiconductor induced by an external poling process can be simply detected through the Franz-Keldysh effect.
A direct and unambiguous identification of native vacancy type defects and/or dipolar vacancy complexes in TlGaSe 2 layered crystal is still an open question.It has been demonstrated that the native isolated vacancies as well as vacancy complexes in some wide-bandgap semiconductor materials can be directly identified using a positron annihilation lifetime spectroscopy method [64,65].To date, identification of isolated chalcogen vacancies (monovacancies), which could be an abundant defect in all two-dimensional TlMX 2 wide-bandgap dichalcogenides, or metal-chalcogen vacancy complexes at the atomic level is also possible by performing a combination of experiments, such as x-ray photoelectron spectroscopy, Auger electron spectroscopy, and electron paramagnetic resonance.We will present the results of these experiments in our next publications.

Conclusions
In summary, we successfully synthesized high-quality TlGaSe 2 polycrystals and grew optically pure single-phase crystalline ingots using a modified Bridgman-Stockbarger method.The samples exhibited excellent homogeneity, as confirmed by EDX measurements.Our optical absorption experiments revealed the Franz-Keldysh effect in the poled sample under an external electric field, which was absent without the poling process.This suggests that an externally induced built-in electric field is a prerequisite for the Franz-Keldysh effect in initially poled TlGaSe 2 .
To understand the origin of the internal bias field in TlGaSe 2 , we investigated deep-level defects within the crystal.Dark-current investigations with Au electrodes showed that native acceptor defect centers, approximately 0.25 eV above the valence band, control the dark current between ∼178 and 300 K. Using TSC spectroscopy, we identified seven defect energy levels, where defects 4 and 7, with activation energies of ∼0.27 and ∼0.29 eV, likely influence the dc electrical properties.
Four native defects (2, 3, 5, and 6) are likely charged deeplevel traps, with defects 5 and 6 possibly attributed to Se vacancies at nonequivalent sites.Defects 2 and 3 appear to be associated with Tl and Ga vacancies, respectively.Complexes formed by these native defects may contribute to the internal electric field in the undoped TlGaSe 2 crystal after external poling, particularly at lower temperatures, due to the alignment of dipolar defects induced by poling.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently

Figure 1 .
Figure 1.A schematic picture describing the Franz-Keldysh effect due to the sailing of wave functions into the forbidden gap.

Figure 2 .
Figure 2. (a) An SEM image of the cleaved ab-plane of the TlGaSe 2 specimen emphasizing the lack of surface imperfections and inhomogeneities.The SEM image was recorded at room temperature prior to deposition of the conducted electrodes.The scale bar is 5 µm with a magnification of 5000×; (b) a representative cross-sectional SEM micrograph of the same sample unambiguously showing the layered structure.The scale bar is 2 µm with a magnification of 10000×.(c) An optical photograph of a thick (∼2.5 mm), plate-like TlGaSe 2 tablet obtained after standard mechanical cleavage from the grown crystal ingot.Different specimens for optical and transport measurements were prepared from this wafer.A scale interval of 1 mm is presented for wafer size comparison; (d) the corresponding EDX elemental composition information acquired on the surface area is displayed in part (a).

Figure 3 .
Figure 3. Temperature dependence of the pure dark current for the TlGaSe 2 sample measured upon a bias voltage of 30 V. The heating rate is 5 K min −1 .The sample thickness is ∼400 µm.Inset: An Arrhenius plot of the pure dark current versus reciprocal temperature 10 3 T showing the thermal activation energy Ea = 0.25 eV.

Figure 4 .
Figure 4.The TSC spectrum of the TlGaSe 2 sample after subtracting the dark current in the temperature range of ∼80 K-150 K.The circles are experimental data and the black solid line is the envelope of the deconvolution curves 1-7.

Figure 5 .
Figure 5.The experimental optical absorption (α) spectra of TlGaSe 2 as a function of the incident photon energy (hν) recorded in the transparent regime after cooling of the sample from room temperature to ∼80 K.The black curve was obtained when the sample was cooled from room temperature to ∼80 K at zero applied bias.The red curve was recorded when the sample was previously cooled from room temperature to ∼80 K under the influence of an external electric field.Both spectra were measured at ∼80 K.The bias was kept at the constant poling voltage of 15 V, where the positive poling potential during the cooling process was applied to the top sample surface.The red and black curves were obtained from the same sample in two separate thermal cycles from room temperature down to low temperature.Optical absorption spectra are shifted for the sake of clarity of the poling effect.

Figure 6 .
Figure 6.A plot of (αhν) 2 versus photon energy (hν) calculated for the investigated sample in its initial state that is without any inscribed electric field present.The blue line represents the linear fit of the optical data.The intersection of the linear fit with the energy axis determines the direct bandgap energy (E d g ) value shown inside the figure.The optical absorption spectrum was measured at low temperature (∼80 K).

Figure 7 .
Figure 7. Measured (red open circles) and calculated (dark and blue lines) optical absorption spectra of the initially poled TlGaSe 2 sample.The electrical poling was carried out in the dark by applying an external bias voltage of ∼15 V (electric field ∼750 cm −1 ) during cooling from room temperature to ∼80 K.When the temperature reached around 80 K the external dc voltage was removed.The corresponding optical absorption curve was recorded by the heating process.The inset (a) shows comparisons between the experimental data and simulation results for the sub-direct bandgap photon absorption region.The exponential low energy absorption profile (dark line) is calculated according to equation (12).The theoretical curve captures an experimentally observed absorption tail in the region of hν ≲ E d g predicted by the Franz-Keldysh effect reasonably well.Numerical values of β and Ξ fitting parameters are given in the figure.Plots of the calculated (blue line) and the experimentally observed changes in the optical absorption as a function of photon energy in the spectral region of the Franz-Keldysh oscillations are shown in inset (b).The oscillatory behavior above the bandgap is calculated according to equation(5).Good agreement between the theoretical prediction and our experimental data for hν ≳ E d g is seen.

Table 1 .
The characteristic trap-related parameters, such as peak temperatures, their activation energies (depth), capture cross sections, and the concentration of each trapping center, evaluated from the TSC experiment for TlGaSe 2 .