Structural properties and variable-range hopping conductivity of Cu₂SnS₃

The synthesis of CTS samples was carried out by pyrolytic decomposition method. The essence of this method is the formation of new compounds by the decomposition of corresponding salts at elevated temperatures. The aqueous solution (a mixture of water and methanol) contains the precursors of CTS samples such as copper chloride (CuCl₂·2H₂O) for the copper, tin chloride (SnCl₂·2H₂O) for the tin and thiourea (CS(NH₂)₂) for the sulfur. It can be seen that the molar ratio of sulfur to metals S/(Cu + Sn) in CTS formula is 1, but due to the fact that elemental sulfur can be easily evaporated at high temperatures, the amount of sulfur in all samples was chosen twice as much as the stoichiometric requirement. After a slow drying process at 75 °C, the obtained powder was pulverized in a mortar and annealed in a vacuum tube furnace at 400 °C for half an hour. To study the effect of the varying molar ratio of Sn/Cu on the formation and characteristic properties of CTS samples, the molar ratio of Sn/Cu was chosen to be 0.2, 0.4, 0.5 and 0.6 for samples (CTS1), (CTS2), (CTS3) and (CTS4), respectively.

The chemical composition of the obtained materials was studied using energy dispersive x-ray spectroscopy. The structure and phase composition of the obtained material were determined by x-ray diffraction using Rigaku Ultima IV diffractometer (Japan) with a Cu-K_α source (wavelength λ = 1.54056 Å), without a monochromator, with detector in the range of 2θ = 10–95 deg. in increments of 2θ = 0.02 deg. and scanning speed 2 deg. per minute. Although x-ray diffraction has been widely used as a powerful tool to determine the structure of CTS sample, it could not resolve completely the different phases with similar structures that may be present. Also, the presence of defects may affect the position of the peaks in the diffraction pattern. Furthermore, an additional structure characterization was done by the Raman microscope XploRA ONE with an excitation wavelength of 532 nm (2.34 eV).

In order to study the conductivity of all CTS samples, the obtained powders were pressed into pellets with a parallelepiped shape under a pressure of 0.4 MPa using a laboratory hydraulic press PLG-20. Then the pellets were additionally annealed at a temperature of 200 °C for half an hour. The temperature dependence of electrical resistivity was measured by the standard six probe technique using the Janis CCS-350S cryostat, the 8200 helium Compressor, LakeShore Model 331 Temperature Controller and the Agilent Digital Multimeter. The contacts were made with a silver paste. The resistivity measurements were performed in the temperature range of 10-300 K with a step of 2 K and recorded by Labview program.

The elemental composition of all the CTS samples obtained using energy dispersive x-ray spectroscopy (EDS) is shown in table 1. From table 1 it is seen that even though the amount of sulfur (S) in the precursors was twice as much as stoichiometric requirement, sulfur deficiency was observed in all synthesized samples. In addition, it is worth mentioning that the loss of tin (Sn) during the high-temperature annealing process was also observed. This fact was found to be in accordance with references [2, 9]. The elemental composition analysis showed that CTS4 is probably the best sample, since it was found to have near-stoichiometric composition. Considerable deviation from the stoichiometry of other samples suggests the presence of unwanted phases.

Figure 1 shows the powder x-ray diffraction patterns of the studied samples with different molar ratios of Sn/Cu. The structure and the cell parameters of the synthesized samples were determined by the Rietveld method using the FULLPROF program [12]. The diffraction data refinement showed that the compounds crystallized in a tetragonal structure with symmetry $I\overline{4}2m.$ Results from x-ray diffraction analysis revealed that the CTS2, CTS3, CTS4 samples were single-phase. At the same time, the presence of a secondary phase Cu₂S with space group $Fm\overline{3}m$ had found in the CTS1 sample. Although the presence of a secondary phase in the sample (CTS1) was undesirable, but not surprising, since copper-rich growth conditions may promote the formation of different copper sulfide [13]. The lattice parameters and average crystallite size of the CTS compounds determined by x-ray diffraction analysis are shown in table 2.

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The average crystallite size in table 2 was calculated using the Debye–Scherrer equation:

$$\begin{eqnarray}&&D=n\lambda /\beta \,\cos \,\theta \end{eqnarray} \tag{ 1 }$$

where D is the crystallite size in angstroms; n is a dimensionless shape factor, with a value close to unity; λ is the wavelength of x-ray radiation in angstroms; β is the line broadening at half the maximum intensity; θ is the Bragg angle in radians [14]. It can be seen from table 2 that with variation of the molar ratio Cu/Sn from 0.2 to 0.6, the average crystallite size ranges from 75.60 Å to 60.46 Å.

Raman spectroscopy is a useful complementary technique to analyze the phase composition of the obtained samples. Raman scattering spectra of all the CST 1, 2, 3 and 4 samples are shown in figures 2(a)–(d), respectively. Three peaks at 296 cm⁻¹, 336 cm⁻¹ and 351 cm⁻¹ presented in the Raman spectrum of all four samples are attributed to tetragonal CTS phase [15–17]. On the other hand, the peak at 231 cm⁻¹ indicates the presence of Sn₂S₃ phase in CTS1, CTS2 and CTS4 samples [18]. The peak at 160 cm⁻¹ observed in the Raman spectrum of the CTS3 sample is attributed to SnS phase [17]. In agreement with x-ray diffraction analysis and the reported Raman data in literature [16, 17], the peak at 468 cm⁻¹ in CTS1 sample is assigned to Cu₂S phase. Thus, the peaks corresponding to the tetragonal CTS phase were observed in all four samples in the Raman scattering spectra. However, the samples were found to contain SnS, Sn₂S₃ and Cu₂S as secondary phases along with tetragonal CTS. The absence of secondary phases (excluding Cu₂S) in the x-ray diffraction pattern could be due to the following two reasons: first, the amount of the secondary phase in the samples was too small to be determined by x-ray diffraction. Second, the secondary phase might be present in amorphous form. In conclusion, by using x-ray diffraction and Raman spectroscopy, it could be confirmed that the synthesized samples mainly consist of tetragonal CTS phase.

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The resistivity measurements of CTS samples were performed in the temperature range of 10 to 300 K. In general, the conductivity increases with rising temperature indicating the semi-conductive behavior of all obtained samples. Figure 3(a) exhibits the variation of resistivity of CTS samples with the inverse of absolute temperature. The observed increase in electrical resistivity with an increment in the Sn/Cu molar ratio from 0.2 to 0.6 (except the CTS3 sample) could be associated with a decrease in the crystallite size (table 2) and with copper-poor conditions in the samples. Similar observations have been reported in literature [19–21]. According to [19–21] in polycrystalline semiconductors, the electrical resistivity of samples increases with a decrease in the crystallite size due to a decrease in charge carrier mobility. Furthermore, from figure 3 it was observed that two regions with different slopes in the plot suggest the presence of two different conduction mechanisms in the investigated temperature range.

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The conduction mechanism of CTS samples at high temperatures was obtained by fitting the experimental data to the following universal equation [22]:

$$\begin{eqnarray}&&\rho \left(T\right)={\rho }_{0}\exp \left(\displaystyle \frac{{E}_{act}}{{k}_{B}T}\right)\end{eqnarray} \tag{ 2 }$$

where ρ₀ is a pre-exponential factor, E_act is the conductivity activation energy and k_B is the Boltzmann constant. The variation of resistivity ρ in the high-temperature region can be explained by considering three possible types of conduction mechanism: activation of the main carriers from the level formed by defects to the valence band; excitation through the 'mobility edge' in non-delocalized (extended) states and nearest-neighbor hopping (NNH) [22]. The basic difference between the band and NNH conductions is essentially that the pre-exponential factor ρ₀ does not depend on temperature in the first case, while in the second case ${\rho }_{0}=AT,$ where A is a proportionality constant independent of temperature. Using equation (2), activation energy values of the band (NNH) conduction can be estimated from the slopes of $\mathrm{ln}\left(\rho \right)$ versus (1/T) and $\mathrm{ln}\left(\rho /T\right)$ versus (1/T). If the thermodynamic energy of the system is estimated as ${k}_{B}T$ in the temperature range of 150-300 K, it will take values from 13 meV to 25 meV. The activation energies estimated from the linearization of $\mathrm{ln}\left(\rho \right)$ versus (1/T) in the case of band conduction were found not to exceed 9 meV, which were too small compared to the thermodynamic energy of the system. Thus, the electrical conduction mechanism of CTS samples at high temperatures more likely obeys the NNH model.

In the low-temperature region, the carrier conduction of studied samples is due to a hopping mechanism. In this case, the temperature dependence of semiconductor resistivity is described by the following relation [23]:

$$\begin{eqnarray}&&\rho \left(T\right)={A}_{p}{T}^{m}\exp {\left(\displaystyle \frac{{T}_{0p}}{T}\right)}^{p}\end{eqnarray} \tag{ 3 }$$

where A_p is a constant pre-exponential factor, T_0p is the characteristic temperature depending on the hopping mechanism; a pair of constants p and m determines the type of hopping conductivity. Hopping over nearest-neighbors conductivity corresponds to the value of p = 1, while the variable range hopping (VRH) conductivity governed by the Mott and Shklovskii–Efros mechanisms are characterized by p = 1/4 and p = 1/2, respectively. The values of p and m corresponding to the various mechanisms of hopping conductivity are given in table 2 in [23].

In our work, the hopping conductivity mechanism of investigated samples in the low-temperature region was determined by analysis of the local activation energy E_loc given by the following equation [22, 23]:

$$\begin{eqnarray}&&{E}_{loc}\left(T\right)=\displaystyle \frac{d\left[\mathrm{ln}\,\rho \left(T\right)\right]}{d{\left[{k}_{B}T\right]}^{-1}}\end{eqnarray} \tag{ 4 }$$

Then equation (4) can be presented in the form:

$$\begin{eqnarray}&&\mathrm{ln}\left(\displaystyle \frac{{E}_{loc}}{{k}_{B}T}+m\right)=\,\mathrm{ln}\,p+p\,\mathrm{ln}\,{T}_{0p}+p\,\mathrm{ln}\left(\displaystyle \frac{1}{T}\right)\end{eqnarray} \tag{ 5 }$$

The left-hand side of equation (5) represents a linear function of $\mathrm{ln}\left(1/T\right)$ for a single value of m. The value of p can be found from the slope of this plot. Therefore, the determination of hopping conductivity mechanism is carried out as follows: the dependence of $\mathrm{ln}\left[{E}_{loc}/\left({k}_{B}T\right)+m\right]$ versus $\mathrm{ln}\left(1/T\right)$ was plotted for fixed values of m. Then the value of p was determined for each value of m, that gives the function of p(m). Using this function, one can choose a proper pair of m and p in agreement with those in [23]. The plot of p(m) for each studied sample was shown in figure 4.

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From figure 4, in order to determine the mechanism of conduction, the best fit is obtained using m = 1/4 and p = 1/4 for all samples. As a result, the behavior of the electrical resistivity of the samples CTS 1, 2, 3 and 4 in the low-temperature range corresponds to the Mott mechanism of variable range hopping conductivity. Figure 3(b) shows the plot of $\mathrm{ln}\left(\rho /{T}^{1/4}\right)$ versus T^−1/4 of the investigated samples CTS1-CTS4. The linear part of the plots in figure 3(b) yields the value of the characteristic temperature T₀₄ as well as the upper limit of the Mott VRH conductivity regime T_v and the lowest temperature T_m, which are listed tin table 3. The dependence of $\mathrm{ln}\left(\displaystyle \frac{{E}_{loc}}{{k}_{B}T}+\displaystyle \frac{1}{4}\right)$ versus ln(1/T) for all samples was shown in figure 5. It can be seen from figure 5 that the data plots can be fitted to a linear relationship, giving p = 1/4. In addition, such plots yield the temperature intervals (T_v, T_m), which are close to the corresponding characteristic parameters in table 3.

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The hopping conduction mechanism of the investigated compounds in the temperature intervals (T_v, T_m) was also confirmed by minimizing the percentage of deviation (Dev) given by the equation (6).

$$\begin{eqnarray}&&Dev\left( \% \right)={\left[\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}\displaystyle \frac{100}{{\rho }_{i}}{\left({\rho }_{0}\exp \left[{\left({T}_{0}/T\right)}^{p}\right]-{\rho }_{i}\right)}^{2}\right]}^{1/2}\end{eqnarray} \tag{ 6 }$$

where n is the number of experimental data points [24]. Construction of the Dev function at different values of p allows to determine the type of hopping conductivity. With this method, the variation of Dev (%) as a function of the parameter p for all the investigated samples is shown in figure 6. The best value of p is the one which gives a minimum of the percentage deviation of the fitted curve to experimental points ρ_i. It is clear that the best fits are obtained for the value of p around 1/4. The percentage deviation of the calculated values from the experimental points of CTS1-CTS4 sample was only 0.6%, 1.3%, 2.7% and 3.6%, respectively. The slight percentage deviations indicate that in this temperature range, the electrical conduction mechanism obeys the Mott VRH model. Thus, observed data for the variation of conductivity in the temperature interval of 10–300 K can be explained by amalgamation of two different conduction mechanisms. In the high-temperature region, the conductivity is governed by the NNH model, whereas in the low-temperature range, the conductivity of the samples is described by the Mott VRH conduction mechanism. This crossover from the NNH to the Mott VRH conduction mechanism was also observed in Cu₂ZnSnS₄ material with tetragonal structure [25, 26].

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The transition from the NNH to the Mott VRH conduction is often stimulated by increasing the disorder in crystalline structures or lowering the temperature, since this condition is energetically advantageous for the making jumps with end points outside the closest center [22]. On the other hand, the VRH transfer takes place when the Fermi energy, μ, lies in the range of energies where states are localized, and the material is called as 'Fermi glass' [22, 27]. In other words, this material has a finite density of states (DOS), g(μ), around the Fermi level, but in which there is enough disorder for states there to be localized [22, 28]. It means that the Fermi energy lies close to one of the edges of the impurity band (the acceptor band, in our case of the p-type semiconductor). First, this smallness of DOS near the edges of the acceptor band leads to difficulty in finding a closest neighboring site with low activation energy [28]. Second, if the critical energy E_C, called the 'mobility edge' separate localized from non-localized (extended) states, the impurity states will be extended in the interval (−E_C, E_C) around the center of the impurity band and localized outside this interval (see figure 7). If μ lies within intervals of the non-localized states (at −E_C < μ < E_C) the metallic conduction is expected. Hence, the VRH conduction sets in only for μ lying close to the one of the acceptor band edges. When the Fermi energy crosses the mobility edge (if the density of charge carries or degree of disorder in the system can be varied), that $\left|\mu -{E}_{C}\right|$ changes sign, an Anderson 'metal-insulator transition' (MIT) occurs [22, 27].

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Microscopic properties of the localized carries, as well as some important macroscopic parameters of CTS samples were further studied by investigation Mott VRH conduction. The Mott VRH characteristic temperature T₀₄ in equation (3) is determined as ${T}_{04}=\beta /\left[{k}_{B}g\left(\mu \right){a}^{3}\right],$ where $\beta \approx 21$ is number constant for Mott VRH conduction and a is the localization radius of charge carriers $a={a}_{0}{\left(1-N/{N}_{C}\right)}^{-\upsilon }$ [29]. Here, N is the concentration of states involved in the hopping conduction, N_C is the critical acceptor concentration of the MIT, a₀ is the localization radius far from the MIT [22] (i.e., at $N\ll {N}_{C}$) and $\upsilon \approx 1$ is a critical exponent [29].

The linear part of the plot of $\mathrm{ln}\left(\rho /{T}^{1/4}\right)$ versus T^1/4 in figure 3(b) yields the value of the characteristic temperature T₀₄, as well as the onset (T_V) and the lowest temperature (T_m) of the Mott VRH conduction regime, which are represented in table 3 for all investigated samples CTS1-CTS4. The semi-width of the impurity band, determined from the equation [22] $W\approx 0.5{k}_{B}{\left({T}_{\upsilon }^{3}{T}_{0}\right)}^{1/4}$ is also exhibited in table 3. From table 3, the highest values of T₀₄ and W reflect a strong lattice disorder in CTS1 sample. The near-stoichiometric composition of CTS4 sample might be connected to the reduced value of W and to smaller lattice disorder in this sample. As can be seen in table 3, the CTS3 sample has higher values of T₀₄ and W in comparison with the CST2 and CTS4 samples, it is reasonable to suppose that the secondary phase SnS made the lattice disorder increasing in the CST3 sample. However, the microscopic parameters of sample were not be influenced significantly as the percentage of this secondary phases was very small. The effect of lattice disorder can also be seen from the change in the slope of the $\mathrm{ln}\left(\rho /{T}^{1/4}\right)$ versus T^1/4 plot, as shown in figure 3(b), since the slope is sensitive to the defect concentration in the sample. Note that that the maximum value of slope was obtained for CTS1 sample indicates the largest concentrations of defects. This result is in agreement with the collected data in table 3 and the discussion above.

If the DOS is approximated with the rectangular shape [28, 30, 31], $g\left(\mu \right)\approx N/\left(2W\right),$ the values of relative acceptor concentration N/N_C then the relative value of localization radius a/a₀ will be obtained by the following equation at $\upsilon =1:$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{T}_{04}}{{T}_{\upsilon }}\right)}^{1/4}\approx 4{\beta }^{1/3}{\left(\displaystyle \frac{{N}_{C}}{N}\right)}^{1/3}{\left(1-\displaystyle \frac{N}{{N}_{C}}\right)}^{\upsilon }\end{eqnarray} \tag{ 7 }$$

The obtained values of $N/{N}_{C}\approx 1$ and $a\gg {a}_{0}$ (table 3) are consistent with the proximity of the investigated samples to MIT. In doped semiconductors ${a}_{0}\approx {a}_{B},$ where ${a}_{B}$ is the Bohr radius. They are given with the relations [22]:

$$\begin{eqnarray}&&{a}_{0}=\hslash {(2m{E}_{0})}^{-1/2}\,{\rm{and}}\,{a}_{B}={\hslash }^{2}{\kappa }_{0}{\left(m{e}^{2}\right)}^{-1}\end{eqnarray} \tag{ 8 }$$

where E₀ is the energy of the DOS peak in the impurity band, κ₀ is the dielectric permittivity far from the MIT, and $m\approx 0.2{m}_{0}$ is the effective mass of the charge carries [32, 33]. Therefore, the ratio of a/a_B can be also calculated as $a/{a}_{B}={\left(1-{E}_{C}/\mu \right)}^{-\upsilon }$ [25] by putting $\upsilon =1,$ $\mu \approx W$ and using the expression ${E}_{C}\approx W-{V}_{0}^{2}/\left[4\left(z-1\right)J\right],$ here ${V}_{0}\approx 2W,$ $z=6$ is the number of nearest neighbors and $J={J}_{0}\exp \left(-R/{a}_{B}\right)$ is the overlap integral [22]. $R\approx {\left(4\pi N/3\right)}^{-1/3}$ is the half of the average distance between acceptors, where $N=\left(N/{N}_{C}\right)\times {N}_{C},$ the values of N/N_C are taken from table IV and N_C from the Mott criterion ${{N}_{c}}^{1/3}{a}_{0}\approx 0.25.$ J₀ is the prefactor ${J}_{0}\approx \displaystyle \frac{{e}^{2}}{{\kappa }_{0}{a}_{B}}\left[\displaystyle \frac{3}{2}\left(1+\displaystyle \frac{R}{{a}_{B}}\right)+\displaystyle \frac{1}{6}{\left(\displaystyle \frac{R}{{a}_{B}}\right)}^{2}\right]$ [22]. Therefore, the only unknown parameter to calculate the ratios of a/a_B is κ₀. But the value of κ₀ can be found by fitting the value of a/a_B to the data of a/a₀ in table 3 with minimizing the standard deviation SD:

$$\begin{eqnarray}&&SD={\left\{\displaystyle \frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left[{\left(\displaystyle \frac{a}{{a}_{0}}\right)}_{i}-{\left(\displaystyle \frac{a}{{a}_{B}}\right)}_{i}\right]}^{2}\right\}}^{1/2}\end{eqnarray} \tag{ 9 }$$

where n is the number of samples [28, 31].

At first the best fit for all samples (n = 4) yields high values of the standard deviation SD = 0.83. It is noticed that the difference in structure between the CTS1 and the others sample (observed from x-ray diffraction analysis) may lead to the difference in the dielectric permittivity of this material. Hence, in order to increase the fitting quality, we fit a/a_B to a/a₀ using only data for CTS2, CTS3 and CTS4 samples (n = 3). In this case, the best fit yields κ₀ = 10.7 at a much smaller value of SD = 0.14. The obtained values of a/a_B are presented in table 3. The values of ${a}_{0}\approx {a}_{B}\approx 21.1\,{\rm{\mathring{\rm{A}} }}$ and ${E}_{0}\approx 42$ meV are calculated from equation (8). The estimated value of E₀ was two times higher than the acceptor energies of 22.9 and 24.8 meV for Cu-poor/S-rich CTS [34], but quite close to the acceptor energies of 43 meV for CuInS₂ [35] and was comparable to energies of 50-60 meV for Cu₂ZnSnS₄ [25, 36], the critical acceptor concentration of the MIT was found to be ${N}_{C}\approx 6.9\times {10}^{17}$ cm⁻³. A number of studies have shown that CTS material often has hole concentration in the range of 10¹⁷−10¹⁹ cm⁻³ [2, 5, 37]. This high level of carrier concentration might lead to an increase in tunneling enhanced recombination within the absorber layer and at the heterojunction interface [38]. Thus, the calculated values of hole concentration for our CTS samples seem to be more suitable for a photovoltaic absorber material. The values of DOS evaluated with the expression $g\left(\mu \right)\approx N/\left(2W\right)$ and E_C for all samples are collected in table 3. The obtained values of κ₀ = 10.7 is close to the reported value of κ₀ = 9.6 in literature [33, 39]. The deviation from these values may be related to an indirect way to determine the value of κ₀ and the rectangular approximation of DOS. However, this small deviation still supports the validity of obtained microscopic parameters shown in table 3.