Effect of rare-earth (Re3+) ions on the optoelectrical parameters of lithium sulfate monohydrate crystals

doi:10.21203/rs.3.rs-1653474/v1

The effect of rare-earth Eu³⁺ and Tb³⁺ ions on the optoelectrical parameters of lithium sulfate monohydrate (LSMH) crystals grown by slow evaporation solution growth technique was observed. The information extracted from transmittance and reflectance spectra proved helpful in determining refractive index, extinction coefficient and various other parameters for these crystals. The effect on the volume/surface energy loss functions, optical energy gap, band tail width and tangent loss factor due to doping in these crystals were discussed. The impact of rare-earth ions on dispersion parameters derived from Wemple-DiDomeenico single oscillator model, Cauchy dispersion relation and on optoelectrical parameters derived from Drude theory was also studied. The other different parameters like optical conductivity, electronic polarizability, nonlinear refractive index, optical electronegativity, nonlinear optical susceptibility, optical and chemical hardness also get affected by doping of rare-earth ions. The relation between group and phase velocity in terms of group index in these crystals was also discussed. However, some empirical and semi-empirical rules have been used to calculate the value of the refractive index.

Slow evaporation

optoelectrical

transmittance and electronic polarizability.

In a broader perspective the electronic, optical, mechanical and surface morphological properties get affected by doping which is associated with the character of the dopant atom and host material. Recently, it has been observed the effect of rare earth ions on the non-linear optical (NLO) property and crystallinity of various inorganic and semi inorganic optical materials [1–3]. Due to remarkable progress in various areas like displays, lasers, electronics, fluorescent sensing, photonics and optical amplification substantial interest has been addicted to the luminescent materials used for optical application [4–7]. Optical and electro-optic processes are mainly associated with the changes that light undergoes upon interacting with a particular substance. The macroscopic and microscopic properties of the material like the nature of its surface morphology and its electronic structure influences its optoelectrical properties [8]. The optical properties of a substance like the reflectance, transmittance and absorbance are often used to probe other important optical constants, such as refractive index and extinction coefficient. The optical properties of solids provide a vital tool for studying energy band structure, impurity levels and localized defects [9]. The large band gap materials apart from finding their utilization in many electronic and optoelectronic devices but the size of the band gap is also vital in determining the many optoelectrical properties in materials. The reason behind the application of materials in higher power electronic appliances which are competent enough to bear high electric fields prior to breakdown is also the wide band gap [10]. It has been observed that in these materials the band gaps rely on the width of the gap as well as on the electronegativity of the constituent atoms in the crystal lattice. The wide band gaps have been correlated with the small inter-atomic spacing’s and the large differences of the electronegativity in the constituent atoms of these materials [11]. While exploring the optoelectrical property of materials, the frequency-dependent complex dielectric constant which is directly associated with the energy band structure of solids can be determined from the extinction coefficient (k) and refractive index (n). In the selection of materials for the optical application and in determining the various other optoelectrical parameters which help design new optical devices the refractive index play a vital role [12, 13].

Lithium sulfate monohydrate (LSMH) belongs to the family of alkali sulfates and is confirmed to be a suitable host material because of its adequate mechanical, thermal and chemical stabilities [14, 15]. Due to the wide band gap, high transmittance and desired refractive index of LSMH in the ultra-violet visible region puts forward them as a suitable material for utilization in optoelectronic devices. From the literature survey, it has been found that various alkali sulfates were used as phosphors in the powder form after rare earth ion doping [16, 17]. The luminescence of Eu³⁺ ion in Li/Na sulfates was also studied where they act as activators [18, 19]. Recently LSMH crystals were doped with cadmium and manganese ions to study its laser damaged threshold, NLO, photoluminescence and dielectric properties [20, 21]. The effect of Vo²⁺ ion on the structural, optical, dielectric and EPR properties of LSMH crystals for optoelectronic applications was also studied [22]. The effects of vitamin-C on the structural, thermal and optical properties of LSMH were studied to explore the SHG and its laser application [23]. Apart from these findings, LSMH crystals were found to exhibit some special features and can be used as molecular switches, dielectric switches, sensors and diodes for various applications [24–26].

It was observed that less heed has been given to investigate the various optoelectrical parameters like linear/non-linear optical refractive index (n/n₂), optical extinction coefficient (k), dispersion parameters, Lorentz-Drude parameters, chemical/optical hardness, optical conductivity and electronic polarizability after literature survey on LSMH crystals. As LSMH crystal has a wide optical band gap of 4.49 eV [27], hence such parameters should be investigated to find the application of the LSMH crystals in various optoelectronic devices. In our previous work, we study the effect of Eu³⁺ and Tb³⁺ ions on the structural, dielectric and optical properties of LSMH crystal and attempted to explore the possibility of using them as red and green light-emitting photoluminescent materials (Phosphors) [27]. In the present work, we preceded further the previous work to study the effect of the rare-earth ions (Eu³⁺ and Tb³⁺) on the various optoelectrical parameters of LSMH crystals. This in depth study will help to explore more vividly their possible application in various optoelectronic devices.

In this work, the chemicals used were of analytical grade and were purchased from Sigma-Aldrich Chemicals Company. By using the method of recrystallization process the lithium sulfate monohydrate was purified. By adding a 0.1 mole% of Eu³⁺ and Tb³⁺ ions to the saturated solution of LSMH and then allowed to crystallize by a slow evaporation method. For doping purposes, we have used europium and terbium sulfate and the doping concentrations were kept low. Above saturation temperature of nearly 5⁰C the solution was heated and then filtered, and stirred continuously to obtain homogeneity. Pristine and rare-earth doped crystals of LSMH of a given size (Fig. 1) were gathered after 4–5 weeks kept in a dust-free atmosphere in a constant temperature bath. The crystal growth steps for the current study are depicted in Fig. 2. In our previous work these crystals were already studied by using single crystal/powder X-ray diffraction (XRD), Energy dispersive X-ray analysis (EDAX), Infrared (IR) spectroscopy, optical (UV-Visible/Photoluminescence) and dielectric spectroscopic analysis measurements [15, 27]. Dual-beam Varian Carry 500 spectrophotometer was used to study the optical parameters from absorbance, transmittance and reflectance spectra of pure and doped LSMH crystals.

3.1 Transmittance, reflectance and absorbance spectra

For the optical study, the transmittance T(λ), reflectance R(λ) and absorbance A(λ) spectra of these pure and doped LSMH crystals were measured along a-axis (100 plane). From T(λ) and R(λ) (inset) spectra of the pure and doped LSMH crystals as shown in Fig. 3, it has been seen that in the entire visible region (400–800 nm) the crystals show high transmittance (nearly 73%). However, it has been observed that the transmittance decreases with the doping of rare earth ions but the optical nature remains intact. Similar nature of reflectance spectra was observed as the reflectance increases with the doping of rare-earth ions in these crystals. The viable cause for the reduction in transmittance of these grown crystals is attributed to the creation of structural disorder and results in the generation of localized states when doped with Eu³⁺ and Tb³⁺ ions [28]. It has been revealed that the different types of disorders like structural and compositional defects can be associated with the charge localization presence in materials [29]. The micro-structural character got influenced because of doping once the structural defects are developed in these crystals.

The low absorbance (Fig. 4) and enhanced transmittance (Fig. 3) as shown by the crystals offer their viable usefulness for various NLO applications. The slight shift in the absorption edge in the direction of higher wavelength and increase in absorbance was due to the doping of these crystals. For the calculation of absorption coefficient (α) the relation which relates T with the α is given by;

$$\alpha =\frac{2.303\text{l}\text{o}\text{g}\left(1/T\right)}{d}$$

1

where T and represents the transmittance and thickness of the crystal respectively. In the case of direct band gap materials for the calculation of band gap we use the following Tauc relation between the α and photon energy (hν);

$$\alpha =\frac{A\sqrt{h\upsilon -{E}_{g}}}{h\upsilon }$$

2

Where A is a constant and E_g is the optical band gap of the crystal. The inset Fig. 4 shows the variation of (αhν) ² versus hν and E_g was determined from the linear portion after extrapolation. The intercept on the energy axis provides the value of E_g (direct) for pure and doped LSMH crystals. From the plot, it was revealed that E_g of these crystals shows a decreasing trend with doping and is in agreement with the reported values [14]. Table 1 shows the determined values of E_g of these crystals. The creation of defects in such types of crystals produces structural disorder as a consequence of [31]. The defect states induced in the band gap, which is appearing as band edge and hence reducing the optical band gap [27, 32]. The wide band gap and elevated transmittance depicted by LSMH crystals projected them as desired candidates for optoelectronic application [33–35].

The doping of materials generally results in a phenomenon called band gap narrowing and band tailing. In the case of band tailing the band edges at valance band (VB) and conduction band (CB) are not well-defined cutoff energies, and it has been observed that above and below CB there are electronic states. From the band edges, the density of these states drops steeply with energy. In the present direct band gap crystals once these electrons get excited from VB below the Fermi energy (E_f) having a nearly parabolic band to the tail states below CB having a density of states showing exponential decrease with energy away from CB. Hence such excitations show that how α depends exponentially on hν and this dependence is known as the Urbach rule given by [36, 37];

$$\alpha ={\alpha }_{0}e\text{x}\text{p}\left(\frac{\text{h}{\upsilon }}{{\text{E}}_{\text{u}}}\right)$$

3

$$ln\alpha =ln{\alpha }_{0}+\frac{\text{h}{\upsilon }}{{\text{E}}_{\text{u}}}$$

4

where α₀ is a material-dependent constant, E_u is called Urbach band tail width (Urbach energy/optical activation energy) and is also a material-dependent constant. From the plot of ln(α) versus hν (Fig. 5) near the absorption edge, the band tail width can be determined from the slope of the linear region of the plot for pure and doped LSMH crystals. The values of E_u calculated from this plot are given in Table 1. With doping the values of E_u increases from 1.37 to 1.43 and also E_u shows an increasing trend with a decrease in band gap in grown crystals. Due to doping the defects/disorders introduced in such systems enhance the localized states near or at the CB which results in increase in E_u [31, 38]. However, exponential band tailing can reveal the observed Urbach tail of ln(α) versus hν, it is also viable to describe the absorption tail response to high internal fields emerging from ionized dopants or defects.

3.2 Refractive index and extinction coefficient

By using the theory of reflectivity of light refractive index (n) and extinction coefficient (k) can be determined from the R(λ) and T(λ) spectra by using the given relations [39];

$$\text{R}= \frac{{(\text{n}-1)}^{2}+{\text{k}}^{2}}{{(\text{n}+1)}^{2}+{\text{k}}^{2}}$$

5

For insulating materials k = 0, hence above equation can be written as;

$$\text{R}= \frac{{(\text{n}-1)}^{2}}{{(\text{n}+1)}^{2}}$$

6

$$k= \frac{\alpha \lambda }{4\pi }$$

7

Here α is the absorption coefficient and λ is the wavelength. In the case of non-magnetic materials, the relation between refractive index and relative permittivity proved to be useful for correlating the electrical and optical properties of materials at any desired wavelength. Apart from dispersion an electromagnetic wave while passing through a material medium suffers attenuation that is it loses its energy due to the creation of phonons (lattice vibrations), photo-generation, free carrier absorption and scattering. In these, materials, the refractive index is a complex function of frequency. The variation of n and k with wavelength (300-800nm) for pure and doped LSMH crystals is shown in Fig. 6. At a low wavelength, n shows high value which is possibly due to the absorption of radiation and resonance effect in this range by the crystals. At higher wavelength λ > 400 nm, n shows normal dispersion as its value decreases and is found to be around 1.79 (pristine) at 800 nm. Also, the value of n decreases with rare-earth ion doping in these crystals. The lower value of k as shown in inset Fig. 6 is attributed to the low attenuation due to absorption and scattering though the wavelength range. A similar type of response is exhibited by other crystals of this family in our earlier work [40]. It has been observed that there is a correlation between the n and E_g, the value of n increases with decreases in band gap that is the crystals with higher refractive index have narrow band gaps. In other terms, enhancement in the absorption coefficient for photon energy near and above the E_g also results in the increase in n in this region [8].

3.3 Cauchy and Wemple-DiDomeenico (WDD) dispersion relation

In the case of transparent materials like insulators and glasses which exhibits very small optical absorption we can use Cauchy’s dispersion equation in the following form as;

$$n\left(\lambda \right)=A+\frac{B}{{\lambda }^{2}}+\frac{C}{{\lambda }^{4}}+\cdots \cdots \cdots$$

8

where A, B and C are material dependent fitting parameters. A is a dimensionless parameter when $\lambda \to \infty$ then$n\left(\lambda \right)\to \infty$ and B (nm²) is associated with the amplitude and curvature of n for the medium wavelength in the visible region. Parameter C (nm⁴) also affects the curvature and amplitude of n in the lower wavelength UV range. However, in general, it has been observed that$0<\left|C\right|<\left|B\right|<1<A$.

For normal dispersion of refractive index in case of different optical glasses in the visible region of the wavelength spectrum we mostly use this Cauchy’s dispersion equation. For simplification purpose, only the first two terms of this Eq. (8) have been taken into account to fit the refractive index to a two-term Cauchy’s relation in the wavelength region (300-800nm) [41].

$$n\left(\lambda \right)=A+\frac{B}{{\lambda }^{2}}$$

9

Figure 7 shows Cauchy’s fitting of the dispersion relation for pure and doped LSMH crystals. The values of A and B determined from Cauchy’s fitting are tabulated in Table 1. The dispersion of n in terms of λ and frequency (ω) of the incident light is expressed as;

$$n={a}_{0}+{a}_{2}{\lambda }^{-2}+{a}_{4}{\lambda }^{-4}+{a}_{6}{\lambda }^{-6}+ \dots \dots \dots \lambda >{\lambda }_{h}$$

10

$$n={n}_{0}+{n}_{2}{\lambda \omega }^{2}+{n}_{4}{\omega }^{4}+{n}_{6}{\omega }^{6}+ \dots \dots \dots \omega >{\omega }_{h}$$

11

where $\hslash \omega$ = incident photon energy, $\hslash {\omega }_{h}=hc/{\lambda }_{h}$ is the optical excitation thrush hold (energy band gap). Here a₀, a₂. .. . and n₀, n₂. .. . are constants. Over a wide range of energies Cauchy’s relation in terms of photon energy is written as;

$$n={n}_{-2}{\left(\hslash \omega \right)}^{2}+{n}_{0}+{n}_{2}{\left(\hslash \omega \right)}^{2}+{n}_{4}{\left(\hslash \omega \right)}^{4}$$

12

In the case of insulating and semiconductor materials refractive index shows a decreasing trend with an increase in the band gap. Different empirical and semi-empirical rules and expressions have been used to relate n and E_g.

In Moss rule, n and E_g are correlated by;

$${n}^{4}{E}_{g}=K=\text{c}\text{o}\text{n}\text{s}\text{t}\text{a}\text{n}\text{t} (\approx 100 \text{e}\text{V})$$

13

In Herve-Vandanme relationship n and E_g are related as;

$${n}^{2}=1+{\left(\frac{A}{{E}_{g}+B}\right)}^{2}$$

14

Here A and B are constants (A ≈ 13.6 eV and B ≈ 3.48 eV). The value of n calculated from these empirical/semi-empirical equations (13) and (14) are in agreement with the values calculated from the experimental data for pure and doped LSMH crystals. The calculated values are given in Table 1.

It is possible to determine the group index (N_g) in these crystals which are given by the ratio of the velocity of a group of waves in free space to the dielectric medium (N_g = C/V_g). N_g simply represents the factor by which the group velocity (V_g) of a group of waves in a dielectric material medium is decreased with respect to the transmission in the free space. The N_g can be evaluated from n by using the given equation;

$${N}_{g}=n-\lambda \frac{dn}{d\lambda }$$

15

The inset Fig. 7 shows the variation of the N_g with λ for pure and doped LSMH crystals. From the plot, it was observed that N_g > n for these crystals. Since n is defined in terms of phase velocity V_P asit means that group velocity of waves is less than phase velocity (V_g < V_p) in these crystals and shows normal dispersion [8].

The dispersion parameters of optical materials provide useful information for their application in different optical dispersion and optical communication devices. The Wemple-DiDomeenico (WDD) dispersion relation is a semi-empirical relation based on a single effective oscillator model to determine the refractive index, dispersion energy and single oscillator energy of different materials for photon energies below the inter-band absorption edge given as [42];

$$\left({n}^{2}-1\right)=\frac{{E}_{d}{E}_{o}}{\left[{{E}_{O}}^{2}-{\left(hv\right)}^{2}\right]}$$

16

$${\left({n}^{2}-1\right)}^{-1}=\frac{{E}_{o}}{{E}_{d}}-\frac{{\left(hv\right)}^{2}}{{E}_{d}{E}_{o}}$$

17

Here E_o (single oscillator energy) and E_d (dispersion energy) are vital parameters for controlling this WDD model. It has been observed that E_d is the measure of the mean strength of inter-band optical transitions such that${ E}_{d}=\beta {N}_{c}{Z}_{a}{N}_{e}\left(eV\right)$. Here N_c is the effective coordination number (N_c = 6 in LSMH), Z_a represents anion valency (Z_a = 1 in LSMH) and N_e is the iconicity of the material that is the effective number of valence electrons per anion excluding the cores (N_e = 8 in LSMH). β is a constant (β_i = 0.26 ± 0.04 eV if inter-atomic bonding is ionic) and (β_c = 0.37 ± 0.05 eV for covalent bonding). In the present case, the LSMH crystals are ionic crystals and after substitution of these values, E_d was calculated to be around 11.52 eV (pristine LSMH). Empirically the value of E_o is calculated by using the relation${ E}_{0}=D{E}_{g}$, where D (constant) ≈ 1.5 and E_g is the direct band gap. The information acquired from the E_o is different from the E_g of the materials. E_o gives quantitative information on the overall band structure of the material or the average electronic energy gap of transition [43, 44].

From the plot of E² versus (n² -1)⁻¹ as shown in Fig. 8 the dispersion parameter E_o and E_d can be calculated and the values are given in Table 1. The value of E_d = 11.53 eV calculated from this plot agrees with the value determined from ${ E}_{d}=\beta {N}_{c}{Z}_{a}{N}_{e}(=11.52 eV)$equation for pure LSMH crystal. The condition for the calculation of these parameters from this model is that n should show inverse relation with E² (normal dispersion).

By using the concept of Sellmeier oscillators it is possible to determine the value of single oscillator strength (S_o) and oscillator wavelength (λ_o) by using the the relation for refractive index at low energies given as [45];

$${\left({n}^{2}-1\right)}^{-1}=\frac{1}{{S}_{o}{{\lambda }_{o}}^{2}}-\frac{1}{{S}_{o}{\lambda }^{2}}$$

18

From the inset plot between (n²-1)⁻¹ and (λ)⁻² as shown in Fig .8 we can get the value of S_o and λ_o from the intercept (1/λ_oS_o) and slope (1/S_o) of the straight portion of the graph. These values for pure and LSMH crystals are given in Table 1. The value of the static refractive index (n₀) can be determined from Eq. (17) as $E=h\upsilon \to 0$ by substituting the values of E_o and E_d.

Table 1

The band gap, Urbach energy and dispersion parameters for pure and doped LSMH crystals.
The estimated parameters	Crystal
The estimated parameters	LSMH	LSMH: Eu³⁺	LSMH: Tb³⁺
Optical band gap E_g (eV) Urbach energy E_u (eV) Oscillator energy E₀ (eV) Dispersion energy E_d (eV) Static dielectric constant ε₀ Static refractive index n₀ Oscillator wavelength λ₀ (nm) Oscillator Strength S₀ (m^− 2) Cauchy constants A B Refractive index (n) Moss rule Herve-Vandamme relation	4.49 1.377 5.41 11.53 3.13 1.76 230 4.03× 10^− 5 1.78 0.330 2.144 1.992	4.46 1.417 5.20 11.30 3.17 1.78 242 3.70× 10^− 5 1.80 0.297 2.150 1.998	4.38 1.438 5.01 11.11 3.22 1.79 258 3.33× 10^− 5 1.83 0.133 2.155 2.008

3.4 Optoelectrical constants

For the determination of optoelectrical constants the complex dielectric constant (ϵ^*) performs vital job. The ϵ^* is expressed as;

$${ϵ}^{*}={ϵ}_{1}+{iϵ}_{2}$$

19

where ϵ₁ (real part) and ϵ₂ (imaginary part) of dielectric constsnt are associated with the dispersion and absorption/scattering of electromagnetic waves while passing through the dielectric medium ( crystals) respectively. The value of ϵ₁ and ϵ₂ can also be determined from n and k by using the given relations [46];

$${ϵ}_{1}={n}^{2}-{k}^{2}$$

20

$${ϵ}_{2}=2nk$$

21

Figure 9 shows the variation of ϵ₁ and ϵ₂ (inset) with the incident photon wavelength for pure and doped LSMH crystals. From these plots, it has been observed that ϵ₁ > ϵ₂ as n > k and the small value of k results in the decrease of nk product which in turn reduces the value of ϵ₂ in comparison to ϵ₁. Up to λ ≈ 450 nm both the real (ϵ₁) and imaginary (ϵ₂) parts of the dielectric constant show a decreasing trend while above that wavelength ϵ₁ seems to remain constant while as ϵ₂ increases slightly at higher wavelengths which gives us a clear idea about the interaction of incident photons with the charge carriers (free electrons) in these crystals. In Fig. 9 ϵ₁ and ϵ₂ (inset) shows an evident enhancement with the doping which becomes the possible reason for the decrease in the transmission/propagation of em waves in these crystals due to dissipation and absorption effects.

A low-frequency em wave can interact with free charge carriers in material and can drift these carriers. It has been seen that the loss of energy takes place from the em wave to the lattice vibrations (phonons) by the carrier scattering process when the interaction occurs. In the non-absorbing (transparent) region, the variation of ϵ₁ = n² with λ² and ϵ₂ with λ³ can be observed by using the Drude model and are given by the following equations [47, 48];

$${{\text{ϵ}}_{1}=\text{n}}^{2}= {\text{ϵ}}_{\text{L}}-\frac{1}{4{{\pi }}^{2}{{\epsilon }}_{0}}\left(\frac{{\text{e}}^{2}}{{\text{c}}^{2}}\right)\left(\frac{{\text{N}}_{\text{o}\text{p}\text{t}}}{{\text{m}}^{\text{*}}}\right){{\lambda }}^{2}$$

22

$${\text{ϵ}}_{2}=\frac{1}{4{{\pi }}^{3}{{\epsilon }}_{0}}\left(\frac{{\text{e}}^{2}}{{\text{c}}^{3}}\right)\left(\frac{{\text{N}}_{\text{o}\text{p}\text{t}}}{{\text{m}}^{\text{*}}}\right)\left(\frac{1}{{\tau }}\right){{\lambda }}^{3}$$

23

where ϵ_L = lattice dielectric constant, e = electronic charge, ε₀ = permittivity of free space, N_opt/m^* = ratio of free carrier concentration to the effective mass, c = velocity of light and τ = relaxation time.

Figure 10 shows the variation of n² with λ² for pure and doped LSMH crystals. From the slope and intercept after extrapolation, the value ϵ_L and N_opt/m^* can be determined. Also from the inset of Fig. 10 the plot of ϵ₂ versus λ³ helps in evaluating the value of τ from the slope after using the value of N_opt/m^* in it. The other associated optical constants like optical mobility (µ_opt) and optical resistivity (ρ_opt) can be determined by using the relations as [49];

$${\mu }_{opt}=\frac{e\tau }{{m}^{*}}$$

24

$${\rho }_{opt}=\frac{1}{e{\mu }_{opt}{N}_{opt}}$$

25

The value of these determined parameters ϵ_L, N_opt/m^*, τ, µ_opt and ρ_opt are listed in Table 2. It has been observed that the value of lattice dielectric constant (ϵ_L) varies from the value of static dielectric (ϵ₀) which may be attributed to the free carrier absorption [50]. The value of ϵ_L increases with doping while other parameters also show an evident change in their values as a result of doping with rare-earth ions.

The other Drude parameters like plasma frequency (ω_p) of charge carriers and damping frequency (ω_d) can also be determined for such materials by using the equation which relates dielectric constant (ϵ₁) with ω_p and ω_d [51–53] as;

$${\text{ϵ}}_{1}={\text{ϵ}}_{\text{L}}-\frac{{{\omega }}_{\text{p}}^{2}}{{{\omega }}^{2}}$$

26

$$\frac{1}{{{\omega }}^{2}(1-{\text{ϵ}}_{1})}=\frac{{{\omega }}_{\text{d}}}{{{\omega }}^{2}{{\omega }}_{\text{p}}^{2}}+\frac{1}{{{\omega }}_{\text{p}}^{2}}$$

27

The variation of ϵ₁ with 1/ω² is shown in Fig. 11 for pure and doped LSMH crystals. From the intercept and slope of the linear portion of the graph, we determined the value of ϵ_L (high-frequency dielectric constant) and ω_p respectively. The values of ϵ_L determined from equations (26) and (22) are in agreement. Inset Fig. 11 shows the plot between $\left[\frac{1}{{{\omega }}^{2}(1-{\text{ϵ}}_{1})}\right]$ and $\left[\frac{1}{{{\omega }}^{2}}\right]$ for these crystals. Again from the slope $\left[\frac{{{\omega }}_{\text{d}}}{{{\omega }}_{\text{p}}^{2}}\right]$ and intercept $\left[\frac{1}{{{\omega }}_{\text{p}}^{2}}\right]$ we have determined the values of ω_p and ω_d given in Table 2. The effect of doping on ω_p and ω_d is evident from the plot and their determined values. Also, the value of ω_p calculated from equations (26) and (27) shows slight variation but is of the same order.

3.5 Volume, surface and dielectric energy losses

The characteristic energy loss is observed in materials when the charge carriers (electrons excited due to plasma oscillations) propagate through them. The optical properties of these materials have a direct association with these energy losses via the dielectric constant. Most probably the electrons will lose energy in two possible ways one while passing within the material called volume energy loss function (VELF) and the other on the surface of a material termed as a surface energy loss function (SELF). The relation of ϵ₁ and ϵ₂ with VELF and SELF is given by equation [54, 55];

$$\text{V}\text{E}\text{L}\text{F}=\frac{{\text{ϵ}}_{2}}{\left({\text{ϵ}}_{1}^{2}+{\text{ϵ}}_{2}^{2}\right)}$$

28

$$\text{S}\text{E}\text{L}\text{F}=\frac{{\text{ϵ}}_{2}}{\left[{\left({\text{ϵ}}_{1}+1\right)}^{2}+{\text{ϵ}}_{2}^{2}\right]}$$

29

Figure 12 shows the variation of VELF and SELF (inset) with the photon energy for pure and doped LSMH crystals. From these plots, it was observed that both VELF and SELF shows an inverse relation with photon energy and also both of them decrease with doping. However, it was also evident that VELF > SELF for pure and doped LSMH crystals indicates that most of the energy loss takes place within the material.

One more loss that arises in dielectric materials due to phase difference when an em wave propagates in a medium at a particular frequency is the dielectric loss factor (tanδ). Actually, it is relative loss which is the ratio of energy lost to the energy stored in a material and is given by the ratio of imaginary (ϵ₂) to the real (ϵ₁) part of dielectric constant [56].

$$\text{t}\text{a}\text{n}{\delta }=\frac{{\text{ϵ}}_{2}}{{\text{ϵ}}_{1}}$$

30

The variation of tanδ with photon energy and the effect of doping on tanδ for pure and doped LSMH crystals are shown in Fig. 13. It has been revealed that tanδ decreases with photon energy. However, it was seen that tanδ increases with the doping and indicated that there is an increase of loss with doping which may be due to the incorporation of Eu³⁺ and Tb³⁺ ions into the LSMH crystal lattice. As we have observed a dielectric loss peak in the tanδ versus energy graph, hence the systems exhibit the properties of dipolar systems and it is suggested that LSMH systems do not show low-frequency dispersion (LFD) behaviour because loss peak was observed in the dielectric dispersion [57].

3.6 Optical conductivity and electronic polarizability

The optical conductivity is associated with the dielectric function of materials that is with the dielectric loss and refractive index which is responsible for conductivity and propagation of light in materials. It has been observed that absorption coefficient (α), refractive index (n), incident photon frequency (ω) and extinction coefficient (k) are some of the important parameters accountable for the optical conductivity given by;

$${\sigma }= \frac{{\alpha }\text{n}\text{c}}{4{\pi }}$$

31

Here c is the velocity of light. Figure 14 shows the graph of optical conductivity versus wavelength for pure and doped LSMH crystals. At low wavelength (high energy) the high absorbing behaviour shown by these crystals and the possibility of excitation of electrons by the incoming photons is attributed to the enhancement in optical conductivity. Also, the optical conductivity increases with doping in these crystals and such kind of behaviour is shown by various crystals mentioned in literature [38, 58].

The non-linear optical NLO behaviour shown by materials is mainly controlled by average electronic polarizability which is a vital characteristic of any material used for NLO applications. The information about the nature of materials (covalent/ionic) and other related optical parameters can also be obtained from polarizability [59, 60]. The behaviour shown by electron cloud while interacting with the applied field in the materials is expressed by electronic polarizability (α_e). From dielectric constant α_e can be calculated by using Clausius-Mossoti equation [61] as;

$${{\alpha }}_{\text{e}}=\left[\frac{{{\epsilon }}_{1}-1}{{{\epsilon }}_{1}+2}\right]\left[\frac{\text{M}}{{\rho }}\right]\times 0.395\times {10}^{-24}{\text{c}\text{m}}^{3}$$

32

Here M = molecular weight and ρ = density of crystals. In the wavelength range of 300–800 nm α_e changes from 20.38×10^− 24 to 10.88×10^− 24 cm³ as shown in Fig. 15 for pure and doped LSMH crystals. α_e can also be determined by using empirical relation given by Reddy et al [62] in which they use band gap parameter as;

$${{\alpha }}_{\text{e}}= \left[1-\frac{\sqrt{{\text{E}}_{\text{g}}}}{4.06}\right]\frac{\text{M}}{{\rho }} \times 0.396\times {10}^{-24} {\text{c}\text{m}}^{3}$$

33

From this Eq. (33) the value of α_e = 12.34×10^− 24 cm³ was calculated for pure LSMH crystal after substituting the value of E_g =4.49 eV and this value of α_e is in agreement with the Clausius-Mossoti Eq. (32). The same agreement was observed in the case of doped crystals and also α_e increases with doping in these crystals.

3.7 Optical electronegativity and Optical susceptibility

The optical electronegativity (η_opt) is also one of the important parameters while discussing other different physio-chemical parameters of the materials put forward by Duffy [63]. Electronegativity is used to describe the nature of bonding in materials and is used for their application in optoelectronics. The relationship established between refractive index (n) and η_opt is stated as [64];

$${\eta }_{\text{o}\text{p}\text{t}}={\left[ \frac{A}{n} \right]}^{1/4}$$

34

where A is a dimensionless constant ≈ 25.54 for all materials [65]. In semiconductors mostly have covalent nature which is attributed to the smaller value of η_opt, while as in the case of ionic crystals the ionic nature is attributed to the larger value of η_opt [66]. In the present case, the values of η_opt calculated from the above equation seem to decrease with doping from 1.95–1.93 and the values are given in Table 2. The decreasing trend observed in η_opt is due to the increase in the value of n as η_opt shows inverse relation with n. thus ionic nature attributed by these pure and doped LSMH crystals is related to the larger values of η_opt and comparatively smaller values of n shown by these crystals [65].

The non-linear optical (NLO) nature exhibited by the materials is directly associated with its polarization. The hyperpolarizability values can be used to discuss the slight distortion of crystal structure due to the doping of crystals. As hyperpolarizability shows a direct propositional relation with the imaginary part of third-order susceptibility Im(χ³) and is used to observe the NLO nature of materials [67]. The value of Im(χ³) can be calculated by using the value of dispersion parameters (E₀, E_d) and refractive index (n) in association with Miller’s generalized rule as [67];

$$Im\left({\chi }^{3}\right)= {A}_{0}\left(\frac{{E}_{d}}{{E}_{0}}\right)/{\left(4\pi \right)}^{4}={A}_{0}{\left(\frac{{{n}_{0}}^{2}-1}{4\pi }\right)}^{4}$$

35

Where A₀ = 1.7×10^− 10 esu and n₀ is the static refractive index whose value can be evaluated by setting $E\to 0$ in the Eq. (17) as;

$${n}_{0}={\left(1+\frac{{E}_{d}}{{E}_{0}}\right)}^{1/2}$$

36

The value of non-linear refractive index n₂ can be evaluated by using the relation [67];

$${n}_{2}^{2}=\frac{12\pi {\chi }^{3}}{{n}_{0}}$$

37

It has been noticed that the values of optical non-linear susceptibility χ³ and non-linear refractive index n₂ increase with doping and shows that the NLO behaviour of these crystals also gets affected by doping.

Table 2

The values of various optoelectrical parameters calculated for pure and doped LSMH crystals.
The estimated parameters	Crystal
The estimated parameters	LSMH	LSMH: Eu³⁺	LSMH: Tb³⁺
Lattice dielectric constant ϵ_∞ (ϵ_L) The ratio (N_opt/m*) ×10⁵⁷ m^− 3Kg^− 1 Optical charge carriers, N_opt ×10²⁶ m^− 3 The relaxation time, τ × 10^− 12 Sec Optical mobility, µ_opt m²/V.s Optical resistivity, ρ_opt × 10^− 9 Ω m The plasma frequency ω_p × 10¹⁵ Hz (from Eq. 26) The plasma frequency ω_p × 10¹⁵ Hz (from Eq. 27) The damping frequency ω_d × 10³¹ Hz Electronic polarizability α_e × 10²⁴ cm³ Optical electronegativity η_opt Non-linear optical susceptibility χ³ × 10^− 13 Non-linear refractive index n₂×10^− 12	3.84 1.29 5.16 6.31 2.52 4.79 1.94 10.25 5.29 20.22 1.950 1.32 2.83	4.19 1.80 7.21 11.79 4.71 1.83 2.29 9.07 4.11 20.28 1.940 1.51 3.19	4.65 2.45 9.85 29.79 11.91 0.54 2.69 8.43 3.44 20.40 1.930 2.61 3.39

3.8 Optical and chemical hardness

The role of material hardness is important in describing the strength of materials for different applications in terms of atomic/molecular and mechanical stability. The parameter which develops a relation between optical, chemical, and mechanical hardness of materials is the band gap and the association of these parameters with polarizability is important for discussing their optical applications. According to Pearson, the polarizabilities of acids and bases are directly related to chemical hardness [68]. The hard acids and bases differ from soft acids and bases in terms of low and high polarizibility shown by them respectively. In the majority of materials, it has been observed that chemical hardness is equal to half of their band gap and also show a correlation with electronic polarizability (α_e) [69]. The optical hardness can be determined from the LUMO-HOMO band gap of materials and is capable of estimating the n besides having affiliation with α_e [38, 70, 71]. The optically soft materials show a high value of n and a low value of the velocity of light in them. The relationship developed between n and α_e of material from Lorenz-Lorentz equation is stated as;

$$\frac{{n}^{2}-1}{{n}^{2}+2}= \frac{4\pi }{3}\left(\frac{{\alpha }_{e}}{V}\right)$$

38

where V is the volume. The link of optical hardness with polarizability is exhibited in terms of n and this optical hardness shows an inverse relation with n or α_e. The band gap parameter is the bridge to develop a relationship between chemical hardness and polarizability as [72];

$${\alpha }_{e}= \frac{N}{m}{\left(\frac{he}{2\pi {\Delta }}\right)}^{2}$$

51

where Δ = 2×chemical hardness = band gap, h = Planks constant, e = electron charge. Thus, the inverse link shown by polarizability with the chemical and optical hardness specifies that both of them enhance or diminish together with α_e. The α_e shows an increasing inclination with doping in these crystals, hence the optical and chemical decreases in these crystals with doping.

The optoelectrical parameters of LSMH crystals were studied under the influence of Eu³⁺ and Tb³⁺ ions for optoelectronic application. To determine the optical properties in these crystals reflectance R(λ), transmittance T(λ) and absorbance A(λ) spectra were analyzed. The impact of doping is observable from the enhancement in the perturbed atoms and imperfections in the structural bonding which results in the decrease in band gap (E_g) and increase in the Urbach energy (E_u) in such crystals. Wemple-DiDomeenico (WDD) single effective oscillator model and Cauchy dispersion relation were used to determine various dispersion parameters. The value refractive index (n) calculated from different empirical and semi-empirical relations was found to be in agreement with the experimental values. From the group index (N_g) of these crystals, it was found that the group velocity of waves propagating in these crystals is lesser than phase velocity and hence shows normal dispersion of waves in them. From n and k the ϵ₁ and ϵ₂ of the complex dielectric constant were determined to discuss the mechanism of interaction of free electrons with the incident photons in these crystals. Drude free electron model was used to reveal the effect of doping on the other optoelectronic parameters such as ϵ_L, N_opt, µ_opt,ρ_opt and τ. It has been observed that the values of VELF and SELF decreases with the photon energy but increase with doping such that VELF > SELF in these crystals. The dielectric loss factor (tanδ) in LSMH crystals apart from showing a variation with the incident photon energy and doping but also shows a loss peak, and hence exhibits the properties of dipolar systems. The plasma frequency and damping frequency also shows a strong dependence on the doping and increase and decreases with doping respectively. The optical conductivity and electronic polarizability (α_e) show an increasing trend both with photon energy and doping in these crystals. The values of electronic polarizability calculated by using the Clausius-Mossoti equation and an empirical relation by using values of band gap were found to be in agreement. The optical electronegativity (η_opt) decreases from (1.95 to 1.93) while as NLO susceptibility χ⁽³⁾ increases from (1.32×10^− 13 to 1.61×10^− 13) in these crystals with doping. The ionic nature of these crystals is confirmed from the evaluated values of η_opt (relatively high) and n₂ (relatively small). The non-linear refractive index n₂ also increases with doping and hence NLO behaviour exhibited by these crystals also gets affected. The inter-linkage of various parameters like α_e, E_g and n in these crystals reveals that both chemical and optical hardness shows decreasing trend with doping.

Acknowledgement

One of the authors FAN is highly thankful to Department of Physics, S.P. College Srinagar for their support. FAM would like to thank UGC for financial support.

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Effect of rare-earth (Re3+) ions on the optoelectrical parameters of lithium sulfate monohydrate crystals

Status:

Journal Publication

Version 1

Abstract

Figures

1. Introduction

2. Materials And Methods

3. Result And Discussion