Empirical Modeling of Electron Transport in Fe/Ti Layered Double Hydroxide Using Exponential, Gaussian and Mixed Gauss–Exponential Distribution

Fe/Ti-layered double hydroxide (LDH) has been hydrothermally prepared and characterized using X-ray diffraction, scanning electron microscopy, atomic force microscopy, Fourier transform infrared spectroscopy, and UV–visible diffuse reflectance spectroscopy for evaluation of its structure, morphology, and optical properties. The purpose of doping Ti4+ with Fe3+ toward the synthesis of Fe/Ti LDH is to extend the absorption of the nanomaterial to longer wavelength, which is known to exhibit higher electron transport performance. To provide a practical realization, electron transport modeling across the band gap has been interpreted using exponential, Gaussian, and mixed Gauss–exponential distribution. The conduction band energy (EC) has been calculated by using the observed values of band gap (Eg) and ξ-potential of the LDH. A detailed study has been undertaken to investigate the pattern of theoretical density of the LDH on the basis of unknown (EC = 0) and known (calculated) values of EC. Fermi–Dirac statistics has been used extensively for estimating the occupancy probability of electron (e–)–hole (h+) pair formation within the valence and conduction bands, respectively, with different temperatures, as well as for given energy levels. Monte Carlo simulations have also been performed to evaluate the suitability of the choice of the model, on the basis of the probability of availability of e–s within the conduction band. To provide a practical realization of the suggested models, electronic transition across the band gap of Fe/Ti LDH has been extensively investigated.


INTRODUCTION
The electron transport phenomenon occurs with the excitation of semiconductors by photons. 1,2 The semiconducting properties of layered nanomaterials have been widely explored due to their fast mobility of charge carriers. The irradiation of semiconducting nanomaterials by light having energy sizeable to the band gap energy of the nanomaterial promotes e − s to the conduction band (CB), thereby generating h + s within the VB These generated e − −h + pairs may recombine, releasing energy in the form of heat resulting no chemical changes within the nanomaterial. Without recombination, if the e − s and h + s move to the surface, then the nanomaterial is expected to attain semiconducting property and can participate in a variety of redox reactions with adsorbed organic/inorganic molecular or ionic species. The redox reactions taking place over the surface of the semiconducting nanomaterials form the basis of photochemical reactions. The mechanism of e − −h + hopping conduction model for basic semiconduction process is presented in Figure 1A. The usefulness of the semiconduction mechanism could be demonstrated with the help of h VB + within the catalyst surface that induces oxidative decomposition of the material. The semiconductor materials also exhibit catalytic activities due to effective e − −h + pair separation. The phenomenon of photosensitization in a basic semiconductor nanomaterial is shown in Figure 1B.
Layered double hydroxides (LDHs) possess inorganic l a y e r e d m a t r i c e s , g e n e r a l l y r e p r e s e n t e d a s [M 1−x II M x III (OH) 2 ] x+ (A n ) x/n ·yH 2 O, where M II and M III represent di-and trivalent metal cations, respectively, and A n− is a charge-balancing anion. LDHs comprise brucite-like layers, having M 2+ ions (e.g., Mg 2+ , Fe 2+ , Co 2+ , Cu 2+ , Ni 2+ , or Zn 2+ ) octahedrally coordinated by −OH groups and a fraction (x) of M 3+ ions (e.g., Al 3+ , Cr 3+ , Ga 3+ , In 3+ , Mn 3+ , Ti 3+ , or Fe 3+ ) that isomorphously substitute M 2+ ions in brucite-like layers and, hence, result in positively charged layers. The positively charged brucite layers are counterbalanced by the interlayer anions (A n− ). H 2 O molecules exist within the interlayer galleries. The −OH groups are directed toward the interlayer spacing and may have been engaged in H-bonding with H 2 O molecules as well as with the anions. The relatively weak H-bonding between the brucite layers enable LDHs to exhibit an expanding nature. LDHs are also considered as doped semiconductors having compositional variability, and hence they find extensive applicability as novel materials with efficient photochemical functions. 3−5 The semiconduction mechanism exhibited by LDHs occurs as a result of the transfer of e − s from the valence band (VB) to the conduction band (CB). The purpose of doping Ti 4+ with Fe 3+ toward the synthesis of Fe/Ti LDH is to create band states, thereby favoring electronic transition and resulting in narrowing the band gap, which is known to exhibit higher photochemical performance. Moreover, Fe 3+ also acts as a photogenerated electron trapper, since the energy level for Fe 3+ ions is above the valence band edge of Ti 4+ ions, therefore inducing higher electron transport efficiency. Fe 3+ also possesses a stable halffilled (3d 5 ) electronic configuration, and the trapped charge could easily transform Fe 3+ to Fe 2+ and could also migrate to the surface for participation in various photochemical reactions. The present work portrays the synthesis of Fe/Ti LDH by a single-step hydrothermal route and its characterization for evaluation of structure, morphology, and optical properties. To provide a practical realization of electron transport across Fe/Ti LDH, depending on the probability of availability of e − s in the CB, theoretical modeling of e − transport across the band gap of the LDH has been established using Gaussian, exponential, and mixed Gauss−exponential distributions by adopting the Monte Carlo simulation procedure. The electronic transition across the band gap of the LDH has been demonstrated for realization of the suggested models.

CHARACTERIZATION RESULTS
2.1. X-ray Diffraction (XRD) Analysis. XRD patterns confirmed the material to be Fe/Ti LDH ( Figure 2). The strong reflection peaks at (003), (006), (009), (110), (113), (100), (0111), (018), and (1013) represent the characteristic hexagonal phase of LDHs. 2−5 The d-spacing (d 003 ) was 0.70 nm (2θ = 12.69°), consistent with the previous reports of hexagonal phase LDHs with interlayer CO 3 2− and H 2 O molecules. Since Fe/Ti LDH exhibited similar layer thickness to normal LDHs, it is most likely that planar orientation of anions has been exhibited mainly by CO 3 2− and H 2 O molecules, in the interlayer gallery of Fe/Ti LDH (Figure S1.1; Supporting Information). 3,6 The incorporation of TiCl 4 during the aqueous phase LDH synthesis has generated TiO 2 in its anatase phase, which is evident from the (110) and (101) diffraction peaks present at 2θ values of 27.99 and 33.20°, respectively, in the X-ray diffractogram of Fe/Ti LDH. The existence of the characteristic peaks of Ti (anatase) phase within the X-ray diffractogram further indicates the successful doping of Ti within the synthesized LDH material. Moreover, TiO 2 , in anatase form, exhibits high charge transfer efficiency and introduces defect states within the LDH. 7,8 It also helps in narrowing the band gap. The narrow and sharp diffraction peaks indicate the existence of well-defined crystalline phases in Fe/Ti LDH. The X-ray crystallographic parameters are discussed in S1 (Supporting Information).

Scanning
Electron Microscopy (SEM)−Energy-Dispersive X-ray (EDX) Analysis. SEM and EDX mapping of the LDH sample have been performed to have an insight into the morphology of the LDH nanomaterial. The SEM image ( Figure 3) confirmed the presence of ultrafine regular hexagonal-shaped nanolayers stacked one over the other representing the lamellar phase of LDH materials. 3,10 The regular hexagonal particles existed due to the subsequent crystallization of LDH incorporated with simple interlayer anions and H 2 O molecules owing to hydrothermal treatment. The existence of distorted hexagonal particles is most likely due to the ultrasonic treatment of the sample prior to SEM imaging. The X-ray elemental maps of the Fe/Ti LDH sample are presented in Figure 4. The Fe/Ti weight ratio by elemental mapping was found to be 2.125:1, nearly equivalent to the stoichiometric ratio of 2:1 Fe/Ti LDH (corresponding to the weight ratio Fe/Ti = 2.33:1) of Fe and Ti taken during the hydrothermal synthesis.

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The difference between the stoichiometric weight and that detected by X-ray elemental mapping could be due to the washing of the LDH after hydrothermal synthesis. Further, the EDX analysis showed the existence of a very little amount of S, which might be due to the presence of impurities in the precursors used in the preparation of LDH. The detection of C corresponds to the interlayer CO 3 2− species (as observed in the Fourier transform infrared (FT-IR) analysis) of the LDH, which appears to be due to the use of urea as a basic precipitant in the LDH synthesis. The detection of O corresponds to the −OH groups that stabilize the brucite sheets and the CO 3 2− ions and H 2 O present within the brucite sheets of Fe/Ti LDH. However, the mapping does not indicate the presence of N and Cl as these elements transform to form NH 3 and NH 4 Cl during the hydrothermal synthesis (in which urea and TiCl 4 are used as precursors) and were removed by washing the LDH sample.
2.3. AFM Analysis. Tapping noncontact AFM has been used to investigate the morphology and the total thickness along c-axis represented by the height profile. The AFM image ( Figure 5A) also confirms the existence of hexagonal platelets, consistent with the SEM observations. 9−11 For AFM analysis, the LDH sample has been dispersed in EtOH by ultrasonication for ∼5 min so that the stacked hexagonal layers, observed under scanning electron microscopy, get opened to single hexagonal nanosheets or at least to two-to three-stacked hexagonal nanosheet crystallites. Two slightly distorted hexagonal nanosheets are clearly observed in the 3D and 2D height profile images ( Figure 5A,B). The distortion of hexagonal surfaces may occur due to the prolonged ultrasonication process. The two height tracers reveal the average height of the nanosheets to be ∼2 nm ( Figure 5C), which corresponds to at least four stacked nanosheets (Figure S1.1; Supporting Information).
The surface topography ( Figure 5A) also reveals two crests and troughs, among which two crests seemed to be very much prominent over the hexagonal sheets, which is most likely due to the staking of the nanoparticles one over the other, i.e., the growth of nanocrystal lattice in Z (or c) direction. The height profile reveals the total height of the LDH sample to be ∼24 nm.
2.4. FT-IR Analysis. The FT-IR results are found to be comparable to the LDHs containing interlayer anions and H 2 O molecules ( Figure 6). The broad IR band at 3467 cm −1 could be assigned to −OH stretching vibrations corresponding to the surface −OH groups and interlayer H 2 O molecules being observed at lower wavenumber in the LDH rather than assigned for free H 2 O at 3600 cm −1 . The shoulders at 2973 and 2848 cm −1 might be due to the H-bonding among the surface −OH groups, interlayer H 2 O, and CO 3 2− anions present in the LDH system. The sharp peak at 1800 cm −1 appearing in the spectra could be due to the CO str vibrations, which correspond to the existence of interlayer CO 3 2− species. The absorption band at 1635 cm −1 could be attributed to the H−O−H bending vibrations. The bands at 1549 and 1125 cm −1 might be due to the asymmetric and symmetric vibrations, respectively, i.e., C−O str vibrations of CO 3 2− species present in the system. Moreover, the bands at 1418 cm −1 (ν 3 ), 1187 cm −1 (ν 1 ), and 1122 cm −1 (ν 2 ) might also be due to the bending vibrational modes of interlayer CO 3 2− species and the band at 865 cm −1 might be due to the in-plane quadrant bending vibration mode of CO 3 2− . The bands at 720 and 614 cm −1 confirm the presence of metal− hydroxyl (M−OH) and metal−oxygen (M−O) groups associated with the Fe/Ti LDH. 3,6−9 The existence of these IR bands indicates the successful preparation of Fe/Ti LDH and the presence of CO 3 2− and H 2 O molecules in its interlayer galleries.
2.5. UV−Visible Diffuse Reflectance Spectroscopy (UV−vis DRS) Analysis. UV−vis DRS was employed for investigating the coordination state and nature of bonding of Fe and Ti within the LDH (Figure 7). The strong absorption peak at ∼320 nm extending up to 450 nm could be indexed to Fe(II) coordinated to CO 3 2− within the interlayer galleries. A shoulder starting at ∼455 nm and extending up to 650 nm is most likely due to the presence of Ti n+ within the brucite sheets or may be a result of supramolecular guest−guest or guest−host interactions. 12−15 The band gap of the LDH has been calculated using the Tauc equation 3,6 from the UV− visible diffuse reflectance spectra where E g represents the optical band gap, hν represents photon energy, K is a constant, and n depends on the nature of the transition. In fact, n assumes values of , , 2 , and 3 for direct allowed, direct forbidden, indirect allowed, and indirect forbidden transitions respectively. In this case, the best fit of (αhν) 1/n versus E (eV) was obtained for = n 1 2 , suggesting allowed direct transitions across the energy band gap. The extrapolated value (the straight line to the X-axis) of hν at α = 0 gives a band gap value of 2.65 eV for the LDH (Figure 7 (inset plot)). The narrow band gap exhibits the LDH to possess remarkable semiconducting property. Moreover, the band gap is most likely due to the direct electronic transition from oxygen 2p to the metal ns or np levels. This also suggests that e − −h + pairs could be photogenerated within the LDH by irradiation with light of longer wavelength. The narrow band gap of the LDH might also be due to the gap states induced by Ti n+ species, associated with oxygen vacancies within the layered structure.

ELECTRON TRANSPORT MODELING ACROSS THE
BAND GAP OF FE/TI LDH An investigation into the electron transport across the band gap of the Fe/Ti LDH was done using the multielectron Monte Carlo simulation procedure by incorporating the value of conduction band energy (E C ) of Fe/Ti LDH (which, in

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Article turn, is calculated using the value of direct band gap (E g ) (calculated using Tauc plot; obtained through UV−visible DRS analysis)) to the energy density equation for determination of theoretical energy density.
To understand the theoretical concept of electron transport across the semiconductor LDH, it is assumed that the defects/ traps are uniformly distributed in a cubic (100 × 100 × 100) box with periodic boundary condition. 16−22 The exponential distribution has been employed for interpretation of electron transport across Fe/Ti LDH. Moreover, the electronic movement across the LDH has also been interpreted by applying Gaussian and mixed Gauss−exponential distribution for the first time in this work. A comparative study of the theoretical models has also been presented herein. The Monte Carlo procedure has been adopted for providing the simulation-based realization of electron transport. Therefore, a system is considered to possess N number of electrons, where N c and N v denote the number of electrons in the CB and VB, respectively, so that the total number of electrons is represented by

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Article At a given temperature T (K), the shifting of electrons from VB to CB depends upon the incident energy on the LDH. The energy (eV) incorporated to the system is denoted by E = ε at the prefixed temperature T. Let N c (ε, t) (=N(ε)) represent the number of electrons shifted from VB to CB. Consequently, the number of electrons remaining in the VB at the same temperature would be N v (ε) = N − N c (ε) (from eq 3). The availability of N c (ε) electrons in the CB at a given temperature and energy ε is assumed to follow a binomial distribution (N, p ε ), which is represented as where p ε denotes the probability that at energy level ε in the CB would be populated by at least one electron. Hence, the component (1 − p ε ) = q ε denotes the probability of nonmovement electrons to CB from VB (which is equivalent to the condition when no h + ions are formed in the VB). Hence, at the given energy level, the mean number of e − s, [E(N c (ε))], transferred to CB at constant temperature and its variance, V(N c (ε)), which was obtained using eq 4, have been presented herein as Moreover, the occupancy of e − at CB has been determined by the probability of transfer of at least one e − from the VB to CB, at the energy level E = ε for the given fixed temperature T. The energy E = ε delivered to the system is supposed to be divided

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Article into "m" nonoverlapping equal subintervals, where m is considered to be a positive integer, then each corresponding subinterval ΔE (=ε/m) could be assumed to be an infinitesimally small energy level. The probability of transfer of an e − to CB from VB in the energy interval ΔE would therefore be where P C represents the nonoverlapping energy gap and also the sum of the probabilities of the presence of e − s in the CB within the g energy gaps. In eq 6, "g" takes all integer values in the interval [k, m], where k ≤ m. Therefore, considering the different conditions, three formalized models are presented for interpretation of e − transport 27 across Fe/Ti LDH. Model 1: When m → ∞ (i.e., the energy gap is sufficiently large) and p ε → 0 (i.e., the probability of availability of electron in CB is very small and hence Np ε would also be small), for minimum number of e − s at CB, k = 1. Hence, eq 7 has been approximated using Poisson distribution with parameter (Np ε ε) as 27 where F expo (ε) represents the cumulative density function of exponential energy gap with parameter Np ε = 1/ε. Model 2: When m is very large, i.e., m → ∞, but neither of , and (1 − p ε *) are close to zero, the binomial distribution could be approximated by Gaussian distribution with mean (μ) = mp ε * and variance σ 2 = mp ε *(1 − p ε *). When m is very large and large energy gaps are expected to exist within the LDH between the VB and the CB, eq 7 can be approximated using Stirling's approximation as 27 where ρ represents the normality constant and ε ̅ = 1/Np ε is the mean value of the statistical energy gap. Since the energy gap g is sufficiently large, the summation (7) could be changed to the integral form as 27 The mean numbers of e − s in the CB for the prefixed energy ε is represented by Np ε (a constant) and g = [Np ε ε] (the greatest integer part of Np ε ε), and eq 10 for the given Np ε and the statistical energy gap g could be approximated as Gaussian distribution with μ = ε ̅ and σ 2 = ε ̅ /Np ε , and the density function for energy (ε) would be represented as 28,29 Model 3: If medium number of e − s are transferred (Np ε ) to CB, then e − distribution would neither be determined by Gaussian method nor by exponential model. In this situation, the mixture of these two distributions could be considered as an alternative model. The analytical expression for mixed Gauss−exponential density function is considered to be a convex combination of Gaussian and exponential distributions, which is of the form 28,29 where α is the incidence probability. The threshold energy required for e − transport across the band gap of the LDH (from VB to CB) at a given temperature T is represented by E C . Hence, if the supplied energy E > E C , then the e − movement from VB to CB is possible. Moreover, E C is considered to be fixed for an LDH or any semiconductor system for all temperatures. Hence, probability π ε (T,ε c ) of e − transfer from VB to CB at temperature T and energy E, such that E > E C = ε c , has been obtained using Fermi−Dirac statistics using the equation 28,29 where k B is the Boltzmann constant, E C = ε C represents the conduction band energy, and T represents temperature in Kelvin scale. The approximation for p ε , given by Fermi−Dirac statics π ε (T, ε c ) (eq 14), is presented in Figure 8.
The analytical expressions of the models for the statistical energy gap based on exponential, Gaussian, and mixed Gauss− exponential density functions, based on given E C and T, are of the form

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Article ε ε α ε ε α ε ε (1 ) ( , ) The pattern of energy density is approximated under different models through the Monte Carlo technique. The Monte Carlo process provides an empirical summary of the density using the simulated values, which shares the same distributional properties. To understand the concept of e − transport across the semiconductor LDH under different suggested models, it is assumed that the defects/traps are distributed according to the choice of the model. 12−15,30−32 The pattern of distribution of the number of traps on different energy and temperature scales within the LDH are presented herein. Let N t (=1.0 × 10 6 ) denote the number of traps, assumed to be distributed uniformly in a 100 × 100 × 100 cubic box, with periodic boundary condition being considered; k B (=1.38 × 10 −23 J K −1 ) represents Boltzmann constant; T o is the standard temperature (in Kelvin); and E C denotes the conduction band energy of the LDH. E C defines the minimum amount of energy required to transfer an electron from VB to CB. To incorporate the situation considered for model 1, the energy gap has been assumed to be sufficiently large and therefore the probability of availability of electron in CB is considered to be very small. This is possible only if E C is comparatively very low, and for consideration of this extreme situation, it is assumed that E C = 0 (unknown). On the other hand, to incorporate a more realistic condition, where E C has been calculated on the basis of the parameters under the experimental setup, it is denoted as E C (calculated). In the present study, the density of the LDH has been approximated 32 for both the conditions and the pattern of density is approximated for conduction band energy (E C ) and valence band energy (E V ), which has been calculated using the following equations reported by Zhang et al. 36 where χ and E g represent the absolute electronegativity and band gap energy of the LDH, respectively; PZZP is the pH (7.2) at which Fe/Ti LDH exhibits zero ζ-potential (S2; Supporting Information); and pH* is the experimentally observed pH 7.2, in which the surface of the material remains neutral. The χ oxide (eV) value was calculated by using the following set of equations: where χ (eV; elements) is the electronegetivity of elements in eV scale; E IE and E IE are, respectively, the first ionization energy and first electron affinity of the elements in eV scale; and χ oxide (eV) represents the absolute electronegetivity of the oxide in eV scale. The approximated formula of the mixed metal oxide is calculated using SEM/EDX data and is found to be Fe 0.67 Ti 0.33 O 1.55 (S3; Supporting Information), and for this mixed oxide, χ oxide (eV) is calculated as 5.77 eV (S4;  Figure 9A), the distribution of trap numbers, based on model 1, showed that with increase in temperature and energy below ∼0.45 eV, the trap numbers show an increasing pattern, and above that energy, it starts decreasing and is observed to be stabilized around 4 × 10 5 . Under model 2, for E C = 0, the trap numbers ( Figure 9B) show an increasing trend with increase in temperature and energy. When E C = −4.45 eV ( Figure 9C), the trap number based on model 1 shows a decreasing trend; on the other hand, Figure 9D, based on model 2, shows an increasing trend in trap numbers with increase in temperature and energy. Figures 10 and 11 depicts the changing pattern in trap numbers with changes in temperature and energy in model 3 at E C = 0 and −4.45 eV, respectively, for different values of incidence probability, say α. Here, as α, temperature, and energy below ∼0.45 eV increase, the trap numbers show an increasing trend (Figure 10), and above 0.45 eV energy level, the trap numbers show a decreasing pattern. In Figure 11, where E C = −4.45 eV, it is observed that with increase in α and temperature, the trap numbers increase toward lower energy levels.

CONCLUSIONS
Thus, Fe/Ti LDH exhibited hexagonal morphology with narrow band gap within the layered framework. The occurrence of e − transport across the semiconducting LDH has been confirmed using UV−visible diffuse reflectance spectroscopy analysis. Accordingly, empirical models based on exponential, Gaussian, and the mixed Gauss−exponential distributions successfully justified the concept of e − transport across the band gap of the LDH nanomaterial. Fermi−Dirac statistics successfully estimated the probability of transfer of e − from VB to CB at a given temperature and for given energy. The simulated results based on model 1 reveals that exponential distribution to be appropriate under the conditions where the energy gap is sufficiently large and the probability of availability of electron in CB is very small. Moreover, the present work also reveals the importance of Gaussian (when the energy gap is sufficiently large and probability of availability of electron is not close to zero) and mixed Gauss−exponential models (when the medium number of electrons are transferred to the CB) for interpretation of the e − transport phenomenon across the LDHs.   5.3. Characterizations. Powder X-ray diffraction (XRD) pattern of Fe/Ti LDH was carried out using a PANalytical X'Pert PRO diffractometer equipped with monochromatic Cu Kα X-ray radiation (λ = 0.15418 nm) at 25 mA and 35 kV with a scanning rate of 4°min −1 along the 2θ range of 5−80°.

EXPERIMENTAL
The morphological study and the elemental composition of Fe/Ti LDH were evaluated with Zeiss Gemini SEM (5 kV)/ EDX (15 kV). Gold coating of the specimen was done at ∼6 nm min −1 (at a pressure of 7 × 10 −2 Pa and a current 20 mA) with Quorum (Q 150R ES). Atomic force microscopy (Oxford Instruments; Cypher Asylum Research, equipped with ARC2 controller) was employed in noncontact tapping mode using a Si tip of 10 nm radius to obtain information about the surface topography of Fe/Ti LDH. The Fe/Ti LDH sample is dispersed in ethanol (EtOH) by ultrasonication for ∼5 min, and the dispersion was spread over a glass slide (10 × 10 mm 2 ) with a micropipette and dried such that the stacked hexagonal layers, observed in the SEM images, open up to one single nanolayer or at least two to three-stacked nanolayers. The dispersed solution was then spread over a square glass slide with a micropipette and then allowed to dry in air. Fourier transform infrared (FT-IR) spectroscopy was conducted with a Shimadzu FT-IR 3000 spectrometer. The LDH sample was mixed with KBr in weight ratio 1:100 and compressed to form pellets for analysis. UV−visible diffuse reflectance spectroscopy of the LDH sample was conducted with a Hitachi U4100 spectrometer, equipped with integrating sphere attachment having BaSO 4 background.

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