1. Introduction

PbS and PbTe belong to the class of A4 B6 narrow-gap semiconducting materials with covalent bonds. Electrons can gain optical, thermal or electrical energy to transfer and share in the conduction process. PbS and PbTe semiconductors are promising in many applications such as IR detectors, thermoelectric devices, gas sensors, environmental recycling and solar cells^[1–4]. Different methods have been used for the preparation of nanosized PbTe and PbS^[5–8]. Susan et al.^[9] have prepared dendrite-like PbTe powdered nanostructures. Poudel et al.^[10] have prepared PbSe and PbTe nanomaterials by a hydrothermal method. They observed nano-boxes and microcrystals of different morphologies such as micro-flowers, semi-micro-flowers, and cubic nanoparticles. Chongjian et al.^[11] have prepared PbTe_1-xSe_x nanowires for thermoelectric applications. Star-shaped nanocrystals, prepared by a hydrothermal method, were obtained by Masoud Salavaty et al.^[5]. Kingumadevi et al.^[12] have prepared PbTe nanocubes, worm-like structures and nano-particles by a common evaporation technique. Clemens Burda et al.^[13] have studied the chemical and physical properties of nanocrystals for various materials with different shapes. The dependences of conductivity and the dielectric behavior on the frequency give us useful information about the conduction mechanism of the materials under investigation. The AC measurements can offer useful information from both the fundamental and technological points of view^[14–16]. The most common results on semiconducting nanocrystallites are a significant increase in the energy band gap (E_g), which is an important parameter for semiconductors, and a drastic reduction in the dielectric constant ε when the crystallite size approaches a few nanometers^{[15, 16]}. Decreasing ε causes an increase in the Coulomb interaction energy between electrons or holes and ionized shallow impurities. Consequently, it can significantly modify the optical absorption and the electrical properties of a nano scale device^{[17, 18]}. The goal of this work is to study the structural and electrical properties of the prepared PbS and PbTe nanomaterials since their large surface-to-volume ratios have deep effects on most chemical and physical properties.

2. Experimental

Both PbS and PbTe were prepared by a simple hydrothermal method. The starting materials for PbTe, polyethylene glycol (PEG, molecular weight: 20 000), sodium hydroxide (NaOH), sodium tellurite (NaTeO₃), sodium sulfite (NaSO₃), lead acetate trihydrate (PbAc) and hydrazine hydrate, were purchased from Aldrich and/or Alfa-Aesar. In a typical synthesis of PbTe face-open nano-boxes, 50 mg of PEG and 2.4 g of NaOH were added to 50 ml of de-ionized water. After a few minutes of stirring, 1 mmol each of NaTeO₃ and PbAc were added to the solution and stirred until the reactants were dissolved completely. 10 ml of hydrazine hydrate was finally added to the solution and transferred into a Teflon-lined autoclave. The sealed vessel was kept in a furnace at a temperature of 170 °C for 10 h and then cooled down to room temperature. The product was washed several times with water. PbS, synthesized by 3.2 mmol of Pb(Ac)₂·3H₂O, and 80 mmol of thiourea were added to 80 mL of deionized water under stirring. The mixture was transferred to a Teflon-lined stainless steel autoclave of 100 mL capacity and heated at 170 °C for 12 h, and then cooled to room temperature. The resulting precipitates were collected by centrifugation and washed many times by deionized water and ethanol to remove possible impurities, then dried at 70 °C for 10 h.

The morphology and compositional analysis of PbS and PbTe were examined using a scanning electron microscope (QuntaFEG 250). The structural analysis of the films was carried out by Bruker D8 diffractometer with CuK_α radiation of wavelength 1.5481 Å. The diffraction angle (2θ) ranges from (10–80) with the speed of the detector at 2 per min. The tube current and voltage were 30 mA and 40 kV, respectively. For AC measurements disks from both materials were obtained. The dielectric constant and conductivity measurements were carried out by means of a high-resolution broadband impedance analyzer (Solartron 1260- Schlumberger Ltd. UK) interfaced with a PC. In addition to an electrometer, it was equipped with an amplifier and measuring cell. The frequency range of the applied AC electric field was between 0.1 Hz and 1 MHz. Good electromagnetic shielding was implemented to the whole sample holder to diminish the noise problems that are especially common at low frequencies. The measurements were automated by interfacing the impedance analyzer with a personal computer through a GPIB cable IEE488. A commercial interfacing and automation software Lab VIEW was used for the acquisition of data. Prior to the sample measurement, calibration was performed to eliminate the effect of stray capacitance. The percentage of error in permittivity ε and dielectric losses amount to 1% and 3%, respectively. The temperature of the samples was controlled by a temperature regulator with Pt 100 sensor. The percent of error in temperature measurements amounts to 0.5 °C. To avoid moisture, the samples were stored in desiccators in the presence of silica gel. Afterwards, the sample was transferred to the measuring cell and left with P₂O₅ until the measurements were carried out.

3. Results and discussion

3.1 Structural properties

3.1.1   X-ray diffraction

Fig. 1 shows the XRD patterns of the prepared PbS and PbTe nanopowders. Concerning PbS nanopowder, very sharp peaks are observed, indicating good polycrystalline structure. Our results are compared with the ICCD card number 65-0132 of PbS. They agree well with the corresponding values in this card. The maximum intensity XRD peak belongs to the (200) plane, followed by the planes (111), (220), (311), (222), (400), (420) and (422). The XRD pattern of PbTe, represented in Fig. 1(b), also shows very good polycrystalline structure when compared with the standard ICCD card number 65-0324. Very sharp peaks are present, with the highest maximum intensity corresponding to the (200) plane, followed by the (220), (222), (420), (422) and (400) planes. The present results agree well with those given in the card No. 65-0324 of the face-centered cubic PbTe. One phase is obtained, with the (200) plane of maximum relative X-ray intensity. The lattice constant “a” for PbS and PbTe with face-centered cubic is calculated according to relation^{[19, 20]}

$a = d\sqrt {{h^2} + {k^2} + {l^2}} ,$

where d is the interplanar spacing and (hkl) are the lattice planes. Table 1 shows that the lattice parameter of PbS = 5.91 Å, as obtained from the maximum intensity XRD peak which agrees with the reported value (5.914 Å) in the standard card, while the corresponding average value, which is estimated from all the peaks, is 5.92 Å, which agrees well with the standard given value. Concerning PbTe the estimated lattice parameter (a) is 6.47 Å as given from the highest intensity (200) peak which coincides with the corresponding value reported in the standard card. The average value is 6.57 Å, which is slightly higher than the reported value. The crystallite sizes “D” of both PbS and PbTe prepared nanomaterials were determined according to^[21]:

$D = \left( {0.9\lambda } \right)/\left( {\beta \cos \theta } \right),$

where λ is the X-ray wavelength (for Cu Kα, λ = 1.54 Å) and β is the full width at half maximum “FWHM” of the XRD peak and θ is the Bragg diffraction angle of the peak in radians. Table 1 shows the “D” values of PbS and PbTe nanomaterials. The crystallite size, determined from the XRD data of PbS is 5 nm as calculated from the highest XRD peak, while it is 6.01 nm as obtained from the average peaks. The corresponding values of PbTe are 2.69 and 1.95 nm, respectively. The dislocation density “dd” was also determined for PbS and PbTe nanomaterials from the following relation:

${\rm dd} = \frac{1}{{{D^2}}},$

where D is the crystallite size. The dislocation density is a measure of angular lattice misalignment of crystallites and consequently gives us information about the local density of the accumulated dislocations. The strain “ε” is calculated according to the following relation:

$\varepsilon = \frac{{\beta \cos \theta }}{4}.$

We use the previously mentioned data β and the Bragg diffraction angles for all the (hkl) planes present in the diffraction pattern. As the crystallinity of the polycrystalline material is high, the XRD peaks are very sharp, i.e. the XRD full peak width at half maximum is very small. As the crystallinity of the material decreases the peaks become less sharp. This indicates that the number of dislocations increases. This indicates the presence of defects in the material, which plays the role of creating some type of strain in the material. The strain and the dislocation density for both PbS and PbTe as determined from the highest peak intensities and from the average values are illustrated in Table 1. The dislocation densities of PbS are comparatively less than those of PbTe. Consequently, the strain of PbS is slightly less the corresponding strain of PbTe. This is logical since dislocation density is a function of the strain.

3.1.2   SEM and EDX

SEM micrographs of PbS are shown in Fig. 2. Star-shaped nanomaterial, micro-flowers, some cubes and semi-dendrites for our prepared PbS nanomaterial are observed. Fig. 2(b) shows an enlarged star-shaped particle. It is suggested that the growth of micro-flowers or star-shaped structures is accomplished in stages as follows. The cubes arrange themselves in the shape of a plus sign. Growth along the four branches continues until the star is formed. Incomplete branches form semi-dendrites. This process happens also in PbTe nanoparticles. Poudel et al.^[6] have studied the shape evolution of PbTe and PbSe nanostructures under different hydrothermal conditions. Face-open PbTe nano-boxes appeared when the reaction was at 100 and 125 °C, semi-micro flowers were formed at 160 °C and a flower-like morphology was observed at 185 °C. The micro flower size and shape did not change but secondary growth on the tips of the micro-flowers was obvious. Fig. 3(a) shows an SEM micrograph of PbTe nanomaterial. Clear microflowers of different sizes are spread all over the micrograph, as given previously by Poudel et al. (shape evolution) for PbTe and PbS. Furthermore some separate cubes are observed. They try to make branches of micro-flowers. Some of these cubes are open–face and the others are block shapes. Fig. 3(b) shows an enlarged part of PbTe nanomaterial. Micro-flowers are clearly seen and their formation from cubes is observed; furthermore, dendrites are observed. The EDX result shows the atomic ratio for Pb : S as 46.24 : 53.76. This confirms that the prepared PbS is nearly stoichiometric. The atomic ratio as obtained from the EDX result for Pb : Te is 48.81 : 51.19. Also, PbTe shows a nearly stoichiometric composition.

3.2 Electrical properties

3.2.1   AC conductivity

Fig. 4 illustrates the variation σ_ac with f for PbS and PbTe at temperatures from 298–423 K. Comparing both figures, it is clear that the total conductivity of PbTe is comparatively higher than that of PbS. The increase in conductivity of PbTe may be attributed to the small crystallite size for PbTe. Šalkus et al. and others reported that when average grain size decreases the total electric conductivity increases at room temperatures^[22–24]. This effect is due to the predominance of grain-boundary conduction in the nanostructured materials, coupled with an increase in the grain boundary ionic diffusivity with decreasing grain size^[22]. Moreover, the steady deviation from DC stage to AC dispersive area indicates the distribution of relaxation times. However, the total conductivity is the sum of DC-conductivity associated with free charges and conductivity associated with bound charges displacement. Furthermore, the conductivity increases with increasing temperature, which can be directly related to the mobility increase of the charge carriers. Jonscher’s power law correlates the conductivity with frequency according to the following equation^[25]

${\sigma _{\rm ac}} = {\sigma _{{\rm d}\rm c}} + A{\omega ^s},$

where A is a parameter which depends on the temperature and represents the polarizability strength in the material. The temperature dependent exponent, s, indicates the degree of interaction between mobile ions with the lattices. The origin of the frequency dependence of conductivity was explained by Jonscher^[25] as being due to the relaxation phenomenon of moving charge carriers. The frequency at which the conductivity increases with frequency increase is known as the hopping frequency. This process is due to polaron hopping of charged species. As the temperature increases, the hopping frequency tends to the higher frequency region. Due to the increase of temperature, charges accumulate at the grain boundaries with sufficient energy to overcome the barrier. According to Funke^[27], the physical parameter ‘s’ is given by s = back hop rate/site relaxation time, where back hop rate is the backward motion of an ion to its first site due to either bad site or coulomb repulsive interaction between mobile ions. When s < 1, the hopping motion is more probable, but when s > 1, the re-orientational motion happens and ions are well-localized. In the case of well-localized ions, the hopping of the mobile ion occurs only near to its initial site while the re-orientational motion means that the dipole is oriented with respect to the external electric field ^[28]. The frequency exponent “s” has been interpreted in terms of the models of conduction^[29–31]. For the small polaron tunneling (SPT) mechanism, the value of s increases as the temperature increases. According to the overlapping large polaron tunneling (OLPT) mechanism, the exponent s reaches a minimum value once temperature increases to a particular temperature, then increases with temperature. In the case of the correlated barrier hopping (CBH) mechanism, the exponent s decreases with temperature increase. The temperature dependence of the frequency exponent s for the PbS and PbTe nanomaterials is shown in Table 2. As shown, the values of the frequency exponent s are found to decrease with increasing temperature. This finding agrees well with the correlated barrier hopping CBH model, reported in Refs. [30–32]. The activation energy can be determined from the Arrhenius relationship^[28]:

${\sigma _{\rm ac}}\left( \omega \right) = {\sigma _0}{\rm{exp}}\frac{{\Delta {E_{\rm ac}}}}{{KT}},$

where ΔE_ac is the activation energy for AC conduction and σ₀ is the pre-exponential factor. The activation energies were calculated from the plot of conductivity and temperature at different frequencies. Fig. 5 shows the relationship between the activation energies and frequencies of PbS and PbTe. The activation energy ΔE_ac decreases with frequency increase due to frequency enhancement of the electronic jumps between localized states. Consequently, the AC activation energy decreases with increasing frequency as stated previously^[26]. Accordingly, this confirms that the CBH mechanism is a suitable mechanism for explaining the conduction mechanism of PbS and PbTe nanomaterials. Also, we note that the activation energy ΔE_ac for PbTe is larger that of PbS which agrees well with earlier investigations^[33].

3.2.2   Dielectric properties

Fig. 6 illustrates the real part of the complex dielectric constant ε’ as a function of frequency and temperature from 298 to 423 K for PbS and PbTe nanomaterials. It is obvious from the figure that the values of ε’ which correspond to PbTe are comparatively higher than the corresponding ones for PbS^[33]. Furthermore, ε’ increases with increasing temperature and decreases drastically with frequency increase up to a certain frequency f_t, after which it becomes constant. The frequency f_t shifts to a high value with increasing temperature^{[32, 34]}. The increase of the dielectric constant ε’ with a frequency decrease can be attributed to electrode polarization and space charge effects^[35–37]. At low frequencies, the charge carriers accumulate at the electrode-sample interface which, in turn, enhances electrode polarization. In addition, as the frequency of the applied field increases the orientational polarization decreases, since the dipoles are not capable of rotating rapidly, taking more time compared with the electronic and ionic polarizations. Therefore, their oscillations lag behind the field frequency. With further increase in frequency the permanent dipole, if present in the medium, cannot completely follow the field and orientational polarization ceases. As a result, the dielectric constant decreases, approaching a constant value at higher frequencies due to the interfacial or space charge polarization. It is also clear from these figures that the increase in the dielectric constant ε’ with temperature is much more pronounced in the low-frequency range compared with the higher frequency region. This behavior can be attributed to the increase of polarization and ionic conductivity as reported by earlier researchers^[38–40]. Fig. 7 shows the frequency dependence of the imaginary part of the complex dielectric constant, namely dielectric loss (ε”), at temperatures from 298 to 423 K for PbS and PbTe nanomaterials. The dielectric loss (ε”) decreases with frequency increase whereas it increases with temperature increase. The higher values of ε” at low frequencies are directly related to the conductivity of the material, which is due to the motion of the free charge carriers within the sample^{[41, 42]}. These conduction losses usually increase with the increase in DC conductivity towards high temperatures. That is why the recorded values for PbTe are higher than for PbS. The variation of dielectric loss ε” with frequency has been explained by some authors in a manner supporting our findings^[43–45].

3.2.3   Electric modulus spectrum

To eliminate the high loss effects, especially at low frequencies, electric modulus formalism has been studied. Electric modulus is a useful tool to extract the bulk response of the material because it inhibits the electrode polarization and eliminates the relaxation response due to conduction. The dynamic relaxation of charged carriers can be analyzed by the complex electric modulus^{[41, 45]}. According to Macedo et al.^[42], the complex electric modulus M* can be used for the analysis of electrical relaxation, which is given by the following relation:

${M^*}\left( \omega \right) = \frac{{{1}}}{{{\varepsilon ^*}\left( \omega \right)}} = {{ M' }}\left( \omega \right) + iM'' = \frac{{\varepsilon '}}{{{{\varepsilon '}^{{{2 }}}} + {{ }}{{\varepsilon ''}^2}}} + {{ i }}\frac{{\varepsilon ''}}{{{{\varepsilon '}^{{2}}} + {{\varepsilon ''}^{{2}}}}}.$ '/>

Here M* is the complex relative permittivity and M' and M'' are the real and imaginary parts of the complex electric modulus, respectively. The equation of the complex electric modulus is determined from the real and imaginary parts of dielectric measurements. Fig. 8 illustrates the dependence of the real part of the electric modulus M' on frequency. From Fig. 8, it can be seen that the M' values for PbTe are slightly higher those of the corresponding ones for PbS. Also, at the highest temperatures, the values of M' tend to be equal to zero at low frequencies and show a very slight increase with increasing frequency, reaching a maximum value at higher frequencies. Such behavior is an indication of a negligible electrode polarization phenomenon in the test materials^[46]. Fig. 9 shows the imaginary part of the electric modulus (M'') as a function of frequency. It is observed that M'' increases with frequency increase up to a maximum value at f_max and then decreases again. The peak position shifts to high frequencies with temperature increase. Furthermore, the peak amplitude increases with temperature increase for PbS while it decreases with temperature increase for PbTe. The shift towards the high-frequency region indicates the increase in the sample conductivity. As shown in the figure, two relaxation regions appear. At the f < f_max region, the ions are capable of moving large distances, giving rise to successful hopping from one site to another neighboring one. When f > f_max, the charge carriers are trapped in potential wells and they move short distances accompanied by the relaxation polarization process^[36]. Hence, f_max represents the transition from long range to short range mobility^{[37, 39]}. The graph of (M''/M''_max) versus (f/f_max) was plotted to check the dependence of the relaxation time on temperature as shown in Fig. 10. It demonstrates that in the temperature range from 298–423 K, the relaxation peaks collapse into one master curve, indicating the temperature independence of the dynamical relaxation processes^{[37, 38]}. The frequency dependence of loss modulus has been described by the Kohlraush–Williams–Watts function Φ (t)^[45] as follows:

$\varphi \left( t \right) = \exp \left\{ {{{\left( { - t/{\tau _M}} \right)}^\beta }} \right\},$

where the exponent β denotes the width of the distribution, whose value is 0 < β < 1 and τ_M is the time at which Φ (t) decays to (1/e). The parameters β and τ_M represent the Kohlraush exponential function and the conductivity relaxation time, respectively. The stretching parameter, β = 1.14/FWHM was determined from the full width at half maximum, FWHM, by fitting the Lorentzian function for the observed modulus curve in Fig. 9. The relaxation time was calculated from ω_max

${\tau _{\rm M}} = 1/{\omega _{\max }},$

Table 2 illustrates the relaxation time and β values as a function of temperature for PbS and PbTe. The value of τ_M of PbS is larger than that of PbTe while the β values of PbTe are larger than that of PbS. The β values indicate that non-Debye relaxation type is achieved for the whole temperature range.

4 Conclusions

PbS and PbTe nanomaterials were prepared by a hydrothermal method. From the XRD studies, both materials possess the polycrystalline structure of the face-centered cubic system. The lattice parameters, the crystallite sizes, the dislocation densities, and the resultant strains were determined. From the SEM analysis, star-shaped nanoparticles, micro-flowers and dendrites were observed. The dielectric properties were studied in a frequency range from 0.1 Hz to 1 MHz at temperatures from 298–423 K. The conductivity and dielectric constant of PbTe are comparatively higher than those of PbS. The electric modulus analysis demonstrates the non-Debye relaxation type for PbS and PbTe.