X-ray peak profile analysis of Sb₂O₃-doped ZnO nanocomposite semiconductor

XRD patterns of ZnO doped with different concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ are shown in figure 1. The diffraction peaks at $2\theta ={\rm 31}{\rm .65}{}^\circ {\rm ,34}{\rm .36}{}^\circ {\rm ,36}{\rm .21}{}^\circ {\rm ,47}{\rm .44}{}^\circ {\rm ,56}{\rm .46}{}^\circ$ and ${\rm 62}{\rm .80}{}^\circ$ are marked by their miller indices (100), (002), (101), (102), (110) and (103), corresponding to ZnO (space group p63mc, JCPDS no. 36-1451) indicating that the phase of the sample was wurtzite structure. XRD peaks corresponding to the ${\rm Z}{{{\rm n}}_{7}}{\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{12}}$ (JCPDS no. 74-1858) phase were observed, which indicates that the structure shift from mono phase to heterophase. The intensity of these peaks increased with the increase of doping concentration of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ in ZnO. The second phase $\alpha {{\rm \text -}Z}{{{\rm n}}_{{\rm 7}}}{\rm S}{{{\rm b}}_{{\rm 2}}}{{{\rm O}}_{{\rm 12}}}$ plays an important role in the microstructure development [15]. The intensity of the diffraction peaks (100) and (101) decreased and their full width at half-maximum (FWHM) increased with the increase of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doping concentration in ZnO. Such changes in crystallinity might be the result of changes in the atomic environment due to impurity doping on ZnO samples. No change in the crystalline structure was noticed, which suggests that the most Sb atoms were incorporated in ZnO wurtzite lattice. Sb dopant in ZnO lattice was expected to substitute Zn atom and connect two vacancies of Zn to form a ${\rm S}{{{\rm b}}_{{\rm Zn}}}{{\rm \text -}2}{{{\rm V}}_{{\rm Zn}}}$ complex [16]. According to Xiu et al [17] the complex ${\rm S}{{{\rm b}}_{{\rm Zn}}}{{\rm \text -}2}{{{\rm V}}_{{\rm Zn}}}$ could be the explanation of strong $p{{\rm \text -}}$type conductivity in Sb doped ZnO films. This result is attributed to a small lattice mismatch between radii of ${\rm Z}{{{\rm n}}^{{\rm 2+}}}$ (0.074 nm) and ${\rm S}{{{\rm b}}^{{\rm 3+}}}$ (0.076 nm), and it indicates that Sb ions systematically substituted Zn ions without deteriorating its crystal structure.

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The average crystallite size $\left(D \right)$ was calculated using Scherrer's formula [18]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle D=\frac{K\lambda }{{{\beta }_{hkl}}\cos \theta }\nonumber \end{align} \tag{ 1 }$$

where $D$ is crystallite size, $K\left( =0.94 \right)$ is shape factor and $\lambda \left( =0.154\ {\rm nm} \right)$ is the wavelength of ${\rm Cu{\text -}}{{K}_{\alpha }}$ radiation. The crystallite size estimated from Scherrer formula is found to be decreased from 56 to 32 nm with the increase of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ content in ZnO. The decrease of crystallite size correlates with a large developed surface of grain boundaries, thus leading to a larger scattering effect. Another reason for the decreased crystallite size values may be due to the drag force exerted by the dopant ions on boundary motion and grain growth. The increase of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doping progressively reduces the concentration of zinc in the system. Thus the diffusivity is decreased in ZnO, which results in a suppressed grain growth of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO samples. At the same time, the substituted Sb ions provide a retarding force on the grain boundaries. If the retarding force generated is more than the driving force for grain growth due to Zn, the movement of the grain boundary is impeded [19, 20]. This in turn gradually decreases crystallize size with increasing ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ concentration. There exist reports of similar trend in some Mn doped ZnO systems [21, 22].

The lattice parameters $a$ and $c$ of ZnO doped with different concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ were calculated from the positions of the (100) and (002) peaks, respectively using the formulas as reported in our previous work [23]. The lattice constants calculated from XRD data for ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO composite samples are close to the lattice constants given in the standard data (JCPDS no. 79-2205, 80-0075). The change in the lattice parameters of ZnO host material depends on the ionic radii of the impurity that substitute the Zn ions at the lattice site [24]. In case of Sb doping the ionic radii of the ${\rm S}{{{\rm b}}^{{\rm 3+}}}$ (0.076 nm) is larger than ${\rm Z}{{{\rm n}}^{{\rm 2+}}}$ (0.074 nm). If ${\rm S}{{{\rm b}}^{{\rm 3+}}}$ ion substitutes ${\rm Z}{{{\rm n}}^{{\rm 2+}}}$ ion in ZnO host lattice, then the variation in the lattice constants is expected due to ionic radii difference which has also been reported in the previous literature [25]. The increase in lattice constant values with increasing ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ concentration in ZnO is due to interstitial position of Sb ions in ZnO lattice.

The volume of the ZnO hexagonal cell was calculated using the formula

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle V=0.866{{a}^{2}}c.\nonumber \end{align} \tag{ 2 }$$

The Zn–O bond length was calculated from the formula [26, 27]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle V=\sqrt{\frac{{{a}^{2}}}{3}+{{(0.5-u)}^{2}}{{c}^{2}}}.\nonumber \end{align} \tag{ 3 }$$

In wurtzite structure, the parameter $u$ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle u=\frac{{{a}^{2}}}{3{{c}^{2}}}+0.25.\nonumber \end{align} \tag{ 4 }$$

The degree of crystallinity ${{X}_{c}}$ was calculated using the following equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{X}_{c}}={{\left( \frac{0.24}{{{\beta }_{002}}} \right)}^{2}},\nonumber \end{align} \tag{ 5 }$$

where ${{\beta }_{002}}$ is the full width at half maximum (in degrees) of (002) Miller's plane. The unit cell volume is calculated for all samples using lattice parameters. The values of the unit cell volume are increasing with increasing ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ concentration in ZnO which led to increase the bond length. The parameter $u$ represents the relative position of two hexagonal close-packed sublattices i.e. the position of the anion sublattice with respect to the cation sublattice. The enhancement of parameter $u$ represents the softer Zn-O bond along the $c{{\rm \text -}}$axis direction.

The specific surface area of the crystallites of the samples was determined from XRD data. The specific surface area is a material property of solids which measures the total surface area of the crystallites present in per unit of mass. It is particularly significant for adsorption, heterogeneous catalysis, and reactions on surfaces. The specific surface area can be calculated by Sauter formula [28]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S=\frac{6\times {{10}^{3}}}{{{D}_{p}}\rho },\nonumber \end{align} \tag{ 6 }$$

where $S$ is the specific surface area, ${{D}_{p}}$ is the size of the particle and $\rho$ is the density of bulk ZnO which equals to $5.606\ {\rm g}\ {\rm c}{{{\rm m}}^{-3}}$. The specific surface area of the samples increases with the increase of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ concentration in ZnO. The increase in specific surface area is due to presence of pores (as we noticed in SEM images) which leads to decrease in particle size. The structural parameters of different concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO estimated from x-ray diffraction data are given in table 1.

Table 1. Microstructural parameters of ZnO doped with different concentrations of ${\rm S}{{{\rm b}}_{{\rm 2}}}{{{\rm O}}_{{\rm 3}}}$.

% of ${\rm S}{{{\rm b}}_{{\rm 2}}}{{{\rm O}}_{{\rm 3}}}$ in ZnO	$D$ (nm)	$a$ ${\rm ({}\mathring{\rm A})}$	$c$ ${\rm ({}\mathring{\rm A})}$	$c/a$	$V$ ${\rm (}{{{{\rm }\mathring{\rm A} }}^{3}}{\rm)}$	$L$ ${\rm ({}\mathring{\rm A})}$	${{X}_{c}}$	$S$ $\left( {{{\rm m}}^{{\rm 2}}}{\rm /g }\times{\rm 1}{{{\rm 0}}^{{\rm 3}}} \right)$
2	62.3	3.261	5.226	1.602	48.12	1.98	0.6663	17.1
4	57.2	3.361	5.272	1.568	51.57	2.03	0.6486	18.7
6	49.4	3.414	5.312	1.555	53.61	2.06	0.5521	21.6
8	46.2	3.452	5.386	1.560	55.58	2.08	0.4162	23.1
10	44.1	3.486	5.412	1.552	56.95	2.10	0.3376	24.2

The broadening of XRD pattern is attributed to the crystallite size-induced or strain induced broadening. The significance of peak broadening of the sample evidence the large strain associated with the powder and grain refinement. The instrumental broadening $\left( {{\beta }_{hkl}} \right)$ of the diffraction peak was corrected using the equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\beta }_{hkl}}={{\left( \beta _{{\rm obs}}^{2}-\beta _{{\rm inst}}^{2} \right)}^{1/2}}.\nonumber \end{align} \tag{ 7 }$$

The strain induced in powders due to crystal imperfection and distortion was calculated using the formula [29]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varepsilon =\frac{{{\beta }_{hkl}}}{4{\rm tan}\theta }.\nonumber \end{align} \tag{ 8 }$$

The strain and particle size contributions to x-ray peak broadening are independent to each other and both have a Cauchy-like profile, the observed line breadth is simply the sum of equations (1) and (8)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\beta }_{hkl}}=\frac{K\lambda }{D{\rm cos}\theta }+4\varepsilon {\rm tan}\theta .\nonumber \end{align} \tag{ 9 }$$

By rearranging the above equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\beta }_{hkl}}=\frac{K\lambda }{D}+4\varepsilon {\rm sin}\theta .\nonumber \end{align} \tag{ 10 }$$

The above equations (9) and (10) are Williamson-Hall equations.

To make Williamson-Hall analysis, a plot is drawn with $4\sin \theta$ along the $x{{\rm \text -}}$axis and ${{\beta }_{hkl}}\cos \theta$ along the $y{{\rm \text -}}$axis for all orientation peaks of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO nanoparticles as shown in figure 2. The crystalline size $(D)$ was estimated from the $y{{\rm \text -}}$intercept and the slope of the linear fit to the data gives the value of strain $(\varepsilon)$. Equation (10) represents the uniform deformation model, where the strain $(\varepsilon)$ is considered to be uniform in all crystallographic directions. It is clear that there is an increase in the lattice strain with increasing doping concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ in ZnO. The uniform deformation model for different concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO is shown in figures 2(a)–(e).

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In the uniform stress deformation model (USDM) there is a linear proportionality between stress and strain given by $\sigma =Y\varepsilon$ which is known as the Hook's law within the elastic limit. In this relation, $\sigma$ is the stress of the crystal and $Y$ is the Young's modulus. Hook's law is a reasonable approximation to estimate the lattice stress. In USDM, the Williamson-Hall equation is modified by substituting the value of strain $\left( \varepsilon \right)$ in the second term of equation (5) which yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\beta }_{hkl}}\cos \theta =\frac{K\lambda }{D}+4\frac{\sigma }{{{Y}_{hkl}}}\sin \theta .\nonumber \end{align} \tag{ 11 }$$

${{Y}_{hkl}}$ is the Young's modulus in the direction perpendicular to the set of crystal lattice plane $\left(hkl \right)$. The uniform deformation stress is estimated from the slope of the line plotted between $4{\rm sin}\theta /{{Y}_{hkl}}$ and ${{\beta }_{hkl}}{\rm cos}\theta$, and crystallite size $\left(D \right)$ from the $y{{\rm \text -}}$intercept as shown in figure 3.

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The strain can be measured if ${{Y}_{hkl}}$ of hexagonal ZnO nanoparticles is known. The Young's modulus ${{Y}_{hkl}}$ for hexagonal crystal phase is related to their elastic compliances ${{S}_{ij}}$ as [30, 31]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{Y}_{hkl}}=\frac{{{\left[ {{h}^{2}}+\frac{{{\left(h+2k \right)}^{2}}}{3}+{{\left( \frac{al}{c} \right)}^{2}} \right]}^{2}}}{{{S}_{11}}{{\left( {{h}^{2}}+\frac{{{\left(h+2k \right)}^{2}}}{3} \right)}^{2}}+{{S}_{33}}{{\left(\frac{al}{c} \right)}^{4}}+\left(2{{S}_{13}}+{{S}_{14}} \right)\left({{h}^{2}}+\frac{{{\left(h+2k \right)}^{2}}}{3} \right){{\left(\frac{al}{c} \right)}^{2}}},\nonumber \end{align} \tag{ 12 }$$

where ${{S}_{11}}$, ${{S}_{13}}$, ${{S}_{33}}$ and ${{S}_{44}}$ are the elastic compliances of ZnO and their values are $7.858\times {{10}^{-12}}$, $2.206\times {{10}^{-12}}$, $6.940\times {{10}^{-12}}$ and $23.57\times {{10}^{-12}}\ {{{\rm m}}^{2}}\ {{{\rm N}}^{-1}}$, respectively [32, 33]. The elastic compliances ${{S}_{11}}$ and ${{S}_{33}}$ represent the longitudinal coefficients along the [1000] and [0001] directions, while the ${{S}_{44}}$ dominate the transverse modes along the [0001] and [1000] directions. Additionally, ${{S}_{13}}$ describes the velocity of modes in low symmetrical directions like [0011]. The Young's modulus, ${{Y}_{hkl}}$ value for hexagonal ZnO is $\sim 127\ {\rm GPa}$.

When the strain energy density $\left(u \right)$ is considered, all the constants of proportionality associated with the stress-strain relation are independent. According to Hooke's law, the energy density $u$ (energy per unit volume) as a function of strain is $u={{\varepsilon }^{2}}{{Y}_{hkl}}/2$. Thus the equation (11) can be modified to the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\beta }_{hkl}}{\rm cos}\theta =\frac{K\lambda }{D}+\left(4{\rm sin}\theta {{\left(2u/{{Y}_{hkl}} \right)}^{1/2}} \right).\nonumber \end{align} \tag{ 13 }$$

The slope of the line plotted between ${{\beta }_{hkl}}\cos \theta$ and $4\sin \theta {{\left(2/{{Y}_{hkl}} \right)}^{1/2}}$ gives the value of uniform deformation energy density. The lattice strain can be evaluated by knowing the ${{Y}_{hkl}}$ values of the sample. The value of $u$ was calculated from the slope and the crystallite size $\left(D \right)$ is estimated from the $y{{\rm \text -}}$intercept of linear fit W-H equations modified assuming UDEDM and the corresponding plots are shown in figure 4.

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From equations (11) and (13), the deformation stress and deformation energy density are related as $u={{\sigma }^{2}}/{{Y}_{hkl}}$. It is to be note that though both equations (11) and (13) are taken into account in the anisotropic nature of the elastic constant, they are essentially different. This is because in equation (7), it is assumed that the deformation stress has the same value in all crystallographic directions allowing $u$ to be anisotropic, while equation (12) is developed by assuming the deformation energy to be uniform treating the deformation stress $\left(\sigma \right)$ to be anisotropic.

The scattering of the points away from the linear expression is lesser for figure 2 as compared with figures 3 and 4. The results obtained from W-H models (UDM, USDM and UDEDM) are summarized in table 2. It can be noted that the values of the average crystallite size obtained from the UDM, USDM and UDEDM are in good agreement with the values obtained from Scherrer's formula. It was observed that the strain and stress values increased with decreasing average crystallite size. This study reveals the importance of models in the determination of crystallite size of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO nanoparticles. Thus all the three models are found to be suitable for the determination of crystallite size.

Table 2. Estimated microstructural parameters of ZnO doped with ${\rm S}{{{\rm b}}_{{\rm 2}}}{{{\rm O}}_{{\rm 3}}}$.

% of ${\rm S}{{{\rm b}}_{{\rm 2}}}{{{\rm O}}_{{\rm 3}}}$ in ZnO	Williamson-Hall method
	UDM		USDM			UDEDM
	$D$ (nm)	$\varepsilon$ $(\times {{10}^{-3}})$	$D$ (nm)	$\varepsilon$ $(\times {{10}^{-3}})$	$\sigma$ (MPa)	$D$ (nm)	$\varepsilon$ $(\times {{10}^{-3}})$	$\sigma$ (MPa)	$U$ $\left({\rm kJ}\times {{{\rm m}}^{-3}} \right)$
2	62.3	0.330	61.8	0.349	42.4	61.8	0.405	64.1	32.4
4	57.2	0.410	57	0.419	53.3	57.3	0.510	64.7	33
6	49.4	0.487	49.2	0.502	63.3	49.3	0.556	70.6	39.3
8	46.2	0.538	45.8	0.558	70	45.8	0.585	74.2	43.4
10	44.1	0.680	44.4	0.703	89.3	45.3	0.680	83.8	55.4

Figures 5(a)–(e) shows the SEM images of different concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO. The surface morphologies of the samples appeared to be smooth and dense with a lot of pores. It is also observed that the surface of the sample is dense up to 6% doped ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ in ZnO, as ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ content is increased up to 10 ${\rm at}\%$, the surface is bumpy and rough. Figures 6(a)–(e) shows the EDX spectra of different concentrations of ${\rm S}{{{\rm b}}_{2}}{{{\rm O}}_{3}}$ doped ZnO. The spectra revealed the presence of Zn, Sb and O elements along with their atomic percentage composition. The presence of O ${{K}_{\alpha }}$ peak at 0.56 keV, Zn ${{L}_{\alpha }}$ peak at 1.01 keV, Zn ${{K}_{\alpha }}$ peak 8.68 keV, Sb ${{L}_{\alpha }}$ peak at 3.64 keV was observed. To study the distribution of Sb in the nanostructures, we performed elemental mapping for the samples. It is noticed that the distribution of zinc in the sample was homogeneous, oxygen and antimony remained inhomogeneously distributed. With the increase of Sb concentration, the concentration of zinc decreased significantly. We believe the incorporated Sb atoms substitute Zn atoms from their lattice sites in ZnO nanostructures.

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